Methodology and accuracy
How every calculator on DigitHelm is built, tested against published values, rounded and corrected.
Last updated October 28, 2026
Testing against reference values
Each calculator's math lives in a small engine module that is separate from the page. Every engine has a test file of golden cases: inputs paired with the answer a trusted source gives, and a note saying where that answer comes from. A test passes only if the engine reproduces the reference value to the stated precision.
Right now 512 of 512 calculators have engine tests, covering 3824 sourced reference values in total. These figures are counted from the test files when the site is built, so they are always current. Each tool page shows how many reference values its own engine was checked against.
Every tool is also tested with edge cases (zero, negative numbers, very large values, boundary values such as a discriminant of exactly zero) and with invalid input, which must produce a clear message rather than a blank or a wrong number.
Where reference values come from
- Math, algebra and calculus: Standard textbooks (for example Stewart, Precalculus and Calculus), the NIST Digital Library of Mathematical Functions, and closed-form results that can be checked by hand.
- Statistics and probability: Abramowitz and Stegun tables, the NIST/SEMATECH e-Handbook of Statistical Methods, and published critical-value tables.
- Finance: Closed-form time-value-of-money formulas as defined for spreadsheet PMT, FV, PV, RATE, NPV and IRR functions, and official publications (for example IRS publications, with the tax year stated) for limits and rates.
- Physics, chemistry and engineering: CODATA 2018 constants and SI 2019 definitions, IUPAC atomic weights, NIST Special Publication 811 for unit conversions, and standard engineering references cited on each page.
- Health and fitness: World Health Organization, CDC and NIH publications, and the original papers for named equations such as Mifflin-St Jeor or the US Navy body fat method.
- Unit conversions: Exact defined factors from NIST SP 811 and the 1959 international yard and pound agreement.
Where no published example exists, the expected value is computed independently by hand and the arithmetic is written into the test, so anyone can repeat it.
Rounding and precision
- Calculations use IEEE 754 double precision, which carries about 15 to 16 significant digits. Values are only rounded for display, never in the middle of a calculation.
- Most results show up to 6 decimal places with trailing zeros removed. Money is shown to the cent. Health measures are shown to one decimal place, because more digits would suggest precision the inputs do not have.
- Tiny floating-point artifacts, such as 0.1 + 0.2 showing as 0.30000000000000004, are cleaned to 12 significant digits before display.
- Unit conversions use exact defined factors (1 inch = 2.54 cm, 1 pound = 0.45359237 kg) rather than rounded approximations.
- Financial schedules compute interest on the unrounded balance; a lender that rounds every payment to the cent can differ by a few cents over a long loan.
Worked examples come from the calculator
The worked examples and reference tables on tool pages are generated at build time by running the same engine the calculator uses. They are not typed by hand, so the text and the calculator cannot drift apart.
Reporting an error
If a result looks wrong, email [email protected] with the tool name, the exact inputs, the result you got and the result you expected (and where your expected value comes from, if you know). Every report is checked against the reference sources. Confirmed errors are fixed, a new test case is added so the error cannot return, and the fix is listed below.
Corrections log
Fixes to calculations and to factual statements, newest first.
| Date | Where | What changed |
|---|---|---|
| October 27, 2026 | Alphabetize List Tool | Fixed: By number value read the wrong number from items with letters or thousands separators, so $1,200 sorted before $80 and item 10 before item 2. Each item is now sorted by the first number written in it (1,200 reads as 1200), and items with no number go after the rest. |
| October 27, 2026 | Bill Splitter | Fixed: tax and tip were split into cents separately, so leftover cents could pile up on one person (in an even split of $237.76 among six, one paid $39.65 while four paid $39.62); each share is now within a cent of its exact share, using the largest remainder method on the running totals. |
| October 27, 2026 | Bra Size Calculator | Fixed: European and French band sizes were one band too large (a 32 band showed as EU 75 and FR 90). A 32 band is EU 70 and FR 85, in the same row as the Australian 10 the tool already gave, so every converted size now sits on one band row. |
| October 27, 2026 | Reading Time Calculator | Fixed: in the pasted-text mode, numbers such as 3.5, 1,000 and 10:30 and abbreviations such as U.S. were counted as two or three words. Each now counts as one word, so reading time for text with many numbers is slightly shorter. The shared word counter behind the word count calculator was corrected. |
| October 27, 2026 | Word Count Calculator | Fixed: numbers such as 3.5, 1,000 and 10:30 were counted as two or three words, and abbreviations such as U.S. and e.g. as two; each now counts as one word, as in Microsoft Word. A hyphen or apostrophe left next to Chinese or Japanese characters no longer counts as a word. |
| October 27, 2026 | Word Count Calculator | Fixed: sentences that end inside quotation marks or brackets ("Stop." Then) and Chinese or Japanese sentences ending in 。 were joined to the next sentence, a heading above a blank line was joined to the paragraph below it, and Mr., Dr., e.g. and i.e. ended a sentence. The sentence count and the average sentence length now reflect real sentences. |
| October 26, 2026 | Logarithm Calculator | Fixed: finding the base for a very large logarithm value (for example log base b of 2 = 10¹⁷) returned a base of 1, which is not a valid base, because the true base rounds to exactly 1 in double precision. It now says the base is too close to 1 to represent. |
| October 26, 2026 | Random Number Generator | Fixed: with decimals, ranges of more than 4,294,967,296 possible values (for example 0 to 1,000,000 with 4 decimal places) ignored No repeats, so the same number could appear twice, and scaled a 53-bit fraction instead of rejecting the uneven tail, so some values were slightly more likely than others. These ranges now use an unbiased 53-bit rejection draw and keep every number different when asked. Ranges with more than 2^53 possible values (such as 0 to 1,000,000 with 10 decimal places), which cannot be stored exactly, are now refused with a message instead of returning numbers that could never land on most of the grid. |
| October 26, 2026 | Random Password Generator | Fixed: with Include every selected type on, the entropy ignored the passwords that the rule throws away, so it read slightly high, for example 52.4 bits instead of 51.3 for 8 characters of all four types and 131.1 instead of 130.9 for 20. It is now log2 of the number of passwords that can actually come out, counted by inclusion-exclusion; strength labels change only where a password sat just above a band edge. |
| October 23, 2026 | BMI Calculator | Fixed: a weight that gives a BMI of exactly 18.5 could be labelled Underweight because the division came out as 18.4999… (for example the lower end of the healthy range at 101.3 cm). The BMI is now rounded to 12 significant digits before it is put in a category. |
| October 23, 2026 | Credit Card Payoff Calculator | Fixed: on a minimum-payment plan the monthly minimum was not rounded to the cent (for example 2% of $4,991.67 was paid as $99.8334), while interest and the balance were. Minimums are now whole cents, which moves long schedules by a few cents in total. |
| October 23, 2026 | Date Calculator | Fixed: dates in the years 1 to 99 were read as 1901 to 1999 by the shared date helpers, so for example 15 June 0150 plus 100 years gave the wrong answer. Those years are now read as written; later years are unaffected. |
| October 23, 2026 | Date Difference Calculator | Fixed: dates in the years 1 to 99 were read as 1901 to 1999 by the shared date helpers, so for example 15 June 0150 plus 100 years gave the wrong answer. Those years are now read as written; later years are unaffected. |
| October 23, 2026 | Derivative Calculator | Fixed: for functions without a derivative rule (such as floor), the numerical fallback reported a slope even at a jump, for example 583.33 for floor(x) at x = 1. It now checks that the function is continuous and has matching one-sided slopes at the point, and says the derivative is undefined when it does not. |
| October 23, 2026 | Derivative Calculator | Fixed: the list of critical and inflection points showed false inflection points where the concavity does not change, for example about 40 rows for |x| between x = -5 and x = 5, a row at 0 for x^4, and rows next to the poles of gamma(x). An inflection point is now listed only when f'' clearly changes sign there, measured against the rounding error of the calculation, so genuine ones such as 0 for x^3 - 3x are still listed. |
| October 23, 2026 | Equation Graph Generator | Fixed: when a sample landed on a vertical asymptote (for example tan(x) on −π to π), the zero search listed about 2,000 false zeros, and turning points were reported at every asymptote. The searches now measure the function by its typical size and skip the jump across an asymptote, so tan(x) on −π to π lists only −π, 0 and π and no turning points. |
| October 23, 2026 | Equation Graph Generator | Fixed: a turning point at a removable hole was missed, so sin(x)/x from −10 to 10 did not list its maximum of 1 at x = 0, because the sample at 0 is 0/0. The turning-point search now reads such a point from the samples either side and refines it as usual, while poles such as 1/x² and ln|x| are still not reported. |
| October 23, 2026 | Exponent Calculator | Fixed: when solving for the base with a fraction exponent, the working printed b^2/3 = 4 and 4^(1/2/3), which read as (b squared) over 3 and an ambiguous 1/2/3. It now prints b^(2/3) = 4 and 4^(3/2); the answers themselves were always right. |
| October 23, 2026 | Grade Calculator | Fixed: the working showed the weighted sum as 88% × 20 + 82% × 15 + 79% × 25 = 4,965, which only adds up without the percent signs, and wrote the letter grade with a hyphen (B-). It now reads 88 × 20 + 82 × 15 + 79 × 25 = 4,965 and B−; the grades themselves were always right. |
| October 23, 2026 | Normal Distribution Calculator | Fixed: areas above a value far in the right tail were computed as 1 minus the area below, which loses all precision past about 8 standard deviations (z = 8 showed 6.66 × 10⁻¹⁶ instead of 6.22 × 10⁻¹⁶, and z ≥ 8.3 showed 0). Right tails are now computed directly, and the outside area is the sum of the two tails. |
| October 23, 2026 | Ovulation Calculator | Fixed: the due date shown if you conceive this cycle ignored the luteal phase you entered, so with an 11-day luteal phase on a 26-day cycle it came out 3 days before ovulation plus 266 days. It is now dated from the modelled ovulation (conception-based rule, ACOG Committee Opinion 700); with the default 14-day luteal phase nothing changes. |
| October 23, 2026 | Percentage Change Calculator | Fixed: the compound-growth path behind the rate chart stopped at period 200 when you asked for more periods, so a 300-period path ended at 7.32 times the start instead of 19.79. Long paths now use 200 evenly spaced points that end at the last period. |
| October 23, 2026 | Ratio Calculator | Fixed: the greatest-common-divisor step grouped thousands inside the list, as in GCD(1,920, 1,080), which is ambiguous. It now reads GCD(1920, 1080) = 120; the simplified ratios were always right. |
| October 23, 2026 | Recipe Scaling Calculator | Fixed: scaled amounts that land exactly on an eighth could show a nearby coarser measure instead, for example 3 3/8 cups (2 1/4 cups scaled by 1.5) appeared as 3 1/3 cups. The calculator now picks the closest standard measure. |
| October 21, 2026 | 3-Variable System of Equations Solver | Fixed: when a variable had a zero column (for example x missing from every equation), elimination skipped a row and back-substitution divided by zero, showing NaN instead of reporting infinitely many solutions. |
| October 21, 2026 | 3-Variable System of Equations Solver | Fixed: a system containing a row 0 = 0 before a row 0 = nonzero was reported as having infinitely many solutions instead of no solution. |
| October 21, 2026 | 401(k) & Roth IRA Calculator | Fixed: the breakeven tax rate was computed as 1 minus the Roth balance over the 401(k) balance, mixing different starting balances and contribution splits; it is now the retirement rate at which both paths leave equal after-tax money. |
| October 21, 2026 | 401(k) & Roth IRA Calculator | Fixed: the two accounts are now compared at the same take-home cost, the inflation input is now used for today's-dollar values, and contribution limit warnings use the 2026 IRS limits instead of 2024 figures. |
| October 21, 2026 | A/B Test Calculator | Fixed: the sample size used only the baseline variance p(1 - p) instead of p1(1 - p1) + p2(1 - p2), which understated the visitors needed by roughly half. |
| October 21, 2026 | A/B Test Calculator | Fixed: test duration ignored the share of traffic included in the test. |
| October 21, 2026 | A/B Test Calculator | Fixed: the expected loss of shipping B measured the gain when B wins instead of the loss when B is worse; it and the chance to beat control are now computed exactly instead of by random simulation. |
| October 21, 2026 | About page | Removed an inaccurate statement that the site carries no ads, an outdated tool count, and a claim of checking results against third-party tools that was not backed by any recorded test. |
| October 21, 2026 | Absolute Value Equation Solver | Fixed: equations true for every x (a = 0 with |b| = c, or two absolute values with equal or opposite insides) were summarized as having no valid solutions instead of all real numbers. |
| October 21, 2026 | Ad Reach & Frequency Calculator | Fixed: reach was derived from a frequency the user typed in, and 2+ and 3+ reach used ad hoc formulas; reach and the exposure distribution now come from Poisson or negative binomial models, or from measured reach. |
| October 21, 2026 | Ad Reach & Frequency Calculator | Fixed: CPM was computed as cost per GRP times 10, which ignores audience size; it is now cost divided by impressions in thousands. |
| October 21, 2026 | Ad Reach & Frequency Calculator | Removed the GRP / (GRP + K) reach shortcut and the Ostrow scoring grid, whose formulas could not be traced to their published sources. |
| October 21, 2026 | AdSense Earnings Calculator | Removed unsourced RPM tables by niche, country, device, traffic source, season and ad network; estimates now use the page RPM, CTR and CPC from your own AdSense reports. |
| October 21, 2026 | Advanced Combinatorics | Fixed: derangement, Catalan and multinomial results above 2^53 were computed in floating point and lost their last digits (D(20) showed 895014631192902100 instead of 895014631192902121); all counts now use exact integer arithmetic. |
| October 21, 2026 | Advanced Combinatorics | Fixed: stars and bars with zero bins returned a fractional count instead of reporting that at least one bin is needed. |
| October 21, 2026 | Advanced Scientific Calculator | Fixed: in degree mode tan 90° (and sec 90°, csc 180°, cot 180°) returned a huge finite number such as 1.633 × 10¹⁶ instead of being reported as undefined; angles at multiples of 90° now use exact values. |
| October 21, 2026 | All tools | Tool counts on the site are now computed from the tool list instead of being typed by hand, which had left them out of date. |
| October 21, 2026 | Amortization Calculator | Fixed: the milestone labeled 'more principal than interest' actually showed the month half the loan was repaid; both milestones are now computed and labeled separately. |
| October 21, 2026 | Amortization Calculator | Fixed: the payoff date was one month late because it counted from the month after the first payment. |
| October 21, 2026 | Annuity Calculator | Fixed: for an annuity due, the loan schedule charged interest before the start-of-period payment, so the balance did not reach zero; it now applies each payment before interest accrues. |
| October 21, 2026 | ANOVA Calculator | Fixed: when every group had zero internal spread but different means, F was infinite and the p-value was reported as 1 (not significant); it is now 0. |
| October 21, 2026 | ANOVA Calculator | Fixed: a group with a single value was silently dropped, which shifted the remaining group labels onto the wrong rows; such groups now produce a clear error. |
| October 21, 2026 | ANOVA Calculator | Fixed: the worked diet example listed a grand mean of 3.98 and SSB of 36.14; the correct values for that data are 4.03 and 25.56. |
| October 21, 2026 | APR Calculator | Fixed: the payment mode applied the entered rate to the loan amount minus fees, which understated the payment and could show negative interest; the payment now comes from the note rate on the full balance and the APR is solved from the amount financed. |
| October 21, 2026 | Arc Length Calculator | Fixed: the y = ln(x) example used log(x), which is the base-10 logarithm, so it measured the wrong curve; it now uses ln(x), and the π and e example bounds are exact instead of 3.14159 and 2.71828. |
| October 21, 2026 | Area Between Curves Calculator | Fixed: a crossing that fell exactly on a sampling point was skipped, so regions that cancel were not split (y = x and the x-axis on [-1, 1] gave an area of 0 instead of 1). |
| October 21, 2026 | Arithmetic Sequence Calculator | Fixed: with a common difference of 0, finding the number of terms to reach a target sum demanded an exact hit and returned an error, while other sequences correctly rounded up to the first term that reaches the target. |
| October 21, 2026 | Aspect Ratio Calculator | Fixed: decimal dimensions were rounded to whole numbers before simplifying, so 2.35 by 1 was reported as 2:1 instead of 47:20. |
| October 21, 2026 | Auto Loan Calculator | Fixed: when the trade-in was worth more than the price, sales tax came out negative and reduced the loan; taxable price is now never below zero, and money still owed on a trade-in is now included. |
| October 21, 2026 | Average Calculator | Fixed: a list made of one repeated value (such as 5, 5, 5) was reported as having no mode; its mode is now that value. |
| October 21, 2026 | Baby Percentile Calculator | Fixed: percentiles were estimated with a fixed standard deviation (1.5 kg for weight, 3.5 cm for length) at the nearest tabulated month, which misplaced babies, especially young infants. The calculator now uses the exact WHO LMS parameters for every month from birth to 24 months, interpolated between months, and adds head circumference. |
| October 21, 2026 | Basic Calculator | Fixed: the CE key cleared the whole calculation like AC; it now clears only the number being typed, as described. |
| October 21, 2026 | Basic Calculator | Fixed: results were rounded to 10 significant digits before being used in the next step, so 1 ÷ 3 × 3 gave 0.9999999999; full precision is now kept behind the display. |
| October 21, 2026 | Bayes Factor Calculator | Fixed: the t-test formula returned evidence for the null (BF01) labeled as BF10, so a t of 0 was reported as support for H1; it now uses the JZS default Bayes factor with the BIC approximation alongside. |
| October 21, 2026 | Bayes Factor Calculator | Fixed: the p-value route reported the Sellke bound -e p ln p as BF10 instead of its reciprocal, so p = 0.05 showed 0.41 instead of at most 2.46. |
| October 21, 2026 | Bayes Factor Calculator | Fixed: the chi-square route used exp((chi-square - df) / 2), which is not a recognized approximation; it now uses Raftery's BIC approximation, which needs the sample size. |
| October 21, 2026 | Beta Distribution Calculator | Fixed: the density at x = 0 (or x = 1) was reported as 0 when α (or β) equals 1; it now equals the other shape parameter, for example f(0) = 5 for Beta(1, 5). |
| October 21, 2026 | Beta Distribution Calculator | Fixed: an interval with the lower bound above the upper bound returned a negative probability; it now asks for a valid interval. |
| October 21, 2026 | Big Number Calculator | Fixed: the primality check used only the first 12 prime bases while claiming certainty up to 3.3 × 10^24, so numbers such as 318665857834031151167461 were wrongly reported prime; it now uses 13 bases and labels results above the proven bound as probable primes. |
| October 21, 2026 | Bill Splitter | Fixed: per-person amounts were rounded independently, so shares could add up to a cent less or more than the bill (for example 3 x $33.33 on $100.00); shares are now allocated in whole cents and always sum to the total. |
| October 21, 2026 | Binomial Distribution Calculator | Fixed: with p = 1 the mode was reported as n + 1, which is impossible; it is now n. |
| October 21, 2026 | BMI calculator | Pounds are now converted with the exact factor 0.45359237 kg (previously 1 ÷ 2.20462), and the healthy weight range uses the WHO upper limit of 24.9. |
| October 21, 2026 | Bond Price Calculator | Fixed: a 0% coupon was rejected as invalid, so the zero-coupon preset could not be calculated. |
| October 21, 2026 | Bond Price Calculator | Fixed: accrued interest was always assumed to be half a coupon and subtracted from a price computed on a coupon date; accrued interest, clean and dirty prices now follow from the time left to maturity. |
| October 21, 2026 | Boolean Algebra Calculator | Fixed: chained implications such as A → B → C were grouped as (A → B) → C; they now follow the standard right-associative reading A → (B → C). |
| October 21, 2026 | Box Plot Calculator | Fixed: the plot drew whiskers to the minimum and maximum even when those values were outliers; whiskers now stop at the most extreme values inside the Tukey fences. |
| October 21, 2026 | CAGR Calculator | Fixed: solving for years rejected negative growth rates, so the time for a declining value to reach a lower target could not be found; the year-by-year table also ran past the end value for fractional years and now stops exactly at it. |
| October 21, 2026 | Capital Gains Tax Calculator | Fixed: the long-term rate was chosen from other income alone and applied to the whole gain; the gain is now stacked on top of taxable income and split across the 0%, 15% and 20% bands. |
| October 21, 2026 | Capital Gains Tax Calculator | Fixed: short-term gains used the single-filer brackets for every filing status, and the 3.8% NIIT ignored the gain itself when testing income against the threshold. |
| October 21, 2026 | Capital Gains Tax Calculator | Fixed: a sale 366 days after a purchase in a leap year was treated as long-term although it is not held more than one year; brackets updated from 2024 to 2026 (Rev. Proc. 2025-32). |
| October 21, 2026 | Car Lease vs Buy Calculator | Fixed: the lease down payment was subtracted from the capitalized cost twice, and the acquisition fee was counted both in the payment and again in the lease total. |
| October 21, 2026 | CD Calculator | Fixed: the APY entered was compounded again as if it were a nominal rate, which overstated interest (5% APY on $10,000 for a year showed $512.67 instead of $500). |
| October 21, 2026 | CD Calculator | Fixed: the early withdrawal payout was never allowed to fall below the deposit, although penalties larger than the interest earned can reduce principal. |
| October 21, 2026 | CD Ladder Calculator | Fixed: each rung's APY was compounded daily again as if it were a nominal rate, overstating interest. |
| October 21, 2026 | CD Ladder Calculator | Fixed: the single-CD comparison set interest earned over different terms against one long CD; both are now valued at the same horizon, with early payouts reinvested at a stated APY. |
| October 21, 2026 | Chi-Square Calculator | Fixed: goodness of fit used expected values exactly as typed, so a ratio such as 9:3:3:1 or expected counts that did not add up to the observed total gave a wrong chi-square; expected values are now scaled to the observed total. |
| October 21, 2026 | Chi-Square Calculator | Fixed: the smoking and lung cancer example had its labels reversed (it showed 647 non-smokers without cancer); it now matches Doll and Hill (1950): 647 of 649 lung cancer patients and 622 of 649 controls were smokers. |
| October 21, 2026 | Chinese Remainder Theorem Calculator | Fixed: products of large moduli were computed in floating point, so systems whose combined modulus exceeded 2^53 returned wrong answers; all arithmetic is now exact. |
| October 21, 2026 | Chinese Remainder Theorem Calculator | Fixed: the preset x ≡ 0 (mod 2), x ≡ 0 (mod 3), x ≡ 1 (mod 5) was labeled with the answer 15 instead of 6, and the reference table repeated that error. |
| October 21, 2026 | Closing Cost Calculator | Fixed: the VA funding fee used the outdated 2.3% rate for every loan; it now follows the VA table (2.15% first use under 5% down, 3.3% subsequent use, 1.5% and 1.25% with larger down payments). |
| October 21, 2026 | Closing Cost Calculator | Fixed: prepaid interest assumed every month has 30 days; it now counts the actual days from the closing date to month end. |
| October 21, 2026 | Collatz Sequence Calculator | Fixed: sequences were computed in floating point, so starting values whose path rises above 2^53 (for example 319,804,831) produced wrong terms; all arithmetic is now exact. |
| October 21, 2026 | Collatz Sequence Calculator | Fixed: the reference table gave the wrong peak step for 27 (76 instead of 77) and 871 (170 instead of 31), and called 871 the highest peak below 1000 when that is 703. |
| October 21, 2026 | College Savings Calculator | Fixed: the four-year cost was four times the first-year price, ignoring cost increases during college and growth on money not yet spent; each year is now projected and discounted separately. |
| October 21, 2026 | College Savings Calculator | Fixed: a 0% return produced NaN for savings at the start of college. |
| October 21, 2026 | Completing the Square Calculator | Fixed: the roots panel said k ≥ 0 was needed for real roots when a is positive (it is k ≤ 0), and complex roots for a negative a were shown with a negative imaginary part such as 2 + −0.5i. |
| October 21, 2026 | Complex Number Calculator | Fixed: division, reciprocals and logarithms rejected any number with a squared modulus below 1e-15 (such as 0.00000001) as if it were zero, and whole-number powers now use exact repeated multiplication instead of trigonometry. |
| October 21, 2026 | Conic Section Analyzer | Fixed: rotated conics (B not zero) reported their center, vertices and foci in the rotated axes instead of the original x and y axes; points are now rotated back. |
| October 21, 2026 | Conic Section Analyzer | Fixed: vertical hyperbolas used the wrong semi-axis for the eccentricity, and parabolas with a negative squared coefficient were reported as opening the wrong way. |
| October 21, 2026 | Conic Section Analyzer | Fixed: equations with negative squared coefficients such as −x² − y² + 4 = 0 were classed as degenerate instead of a circle, and parabolas with no linear term returned NaN instead of a pair of lines. |
| October 21, 2026 | Content Marketing ROI Calculator | Replaced the built-in click-through rates by ranking position and the fixed decay rates per content type, which had no cited source, with inputs you set from your own data. |
| October 21, 2026 | Continued Fraction Calculator | Fixed: fractions were expanded through a floating-point decimal and cut off at 12 terms instead of exactly with the Euclidean algorithm, and negative fractions lost their sign; expansions and convergents are now exact. |
| October 21, 2026 | Continued Fraction Calculator | Fixed: evaluating a list of terms re-expanded the rounded decimal, so the convergents shown could differ from the terms entered; they now come directly from the entered terms. |
| October 21, 2026 | Cross Product Calculator | Fixed: the angle between the vectors came from arcsin, so obtuse angles were reported as acute (135° showed as 45°); it now uses atan2 of |a × b| and a · b. |
| October 21, 2026 | Crypto Profit Calculator | Fixed: buy fees are now added to the cost basis (they were ignored), a holding of exactly 12 months is no longer treated as long-term, and the fixed tax bracket presets were replaced with an optional tax rate you enter. |
| October 21, 2026 | Cubic Equation Solver | Fixed: the test for a repeated root used an absolute tolerance, so cubics with small distinct roots (such as 0.001, 0.002 and 0.003) were reported as having a double root; the substitution check for complex roots also evaluated only their real part. |
| October 21, 2026 | Customer Lifetime Value Calculator | Fixed: the predictive model ignored gross margin and started discounting after churn had already been applied to the first month; subscription LTV is now margin divided by churn, with the standard discounted formula as an option. |
| October 21, 2026 | Debt Payoff Calculator | Rebuilt as a planner for several debts that compares the avalanche and snowball methods, with minimum payments rolling over as each debt is paid off. Single-card questions are now handled by the credit card payoff calculator. |
| October 21, 2026 | Derivative Calculator | Fixed: the Taylor polynomial was never shown because of an always-true condition, and e(x + 1) was read as exp(x + 1) instead of e times (x + 1); derivatives are now computed from the same parser that evaluates the function. |
| October 21, 2026 | Differential Equation Solver | Fixed: the step-table ODE solver merged into this page stripped every letter except x, y and e before evaluating, so equations using sin, cos, exp, ln or sqrt were silently computed wrong; expressions now go through the safe math parser. |
| October 21, 2026 | Digital Root Calculator | Fixed: the multiplicative digital root multiplied digits in floating point, so numbers with more than about 16 nonzero digits produced rounded or invalid products; products are now exact. |
| October 21, 2026 | Discrete Fourier Transform (DFT) Calculator | Fixed: a trailing comma in the sample list added an extra 0 sample, which changed N and every bin, and non-numeric entries were silently dropped; both are now handled explicitly. |
| October 21, 2026 | Dollar-Cost Averaging Calculator | Fixed: the annualized return treated every contribution as invested on the first day; it is now the money-weighted return (IRR) of the actual purchase dates. |
| October 21, 2026 | DRIP Calculator | Fixed: total return added the reinvested dividends on top of a portfolio value that already contained them, double-counting every dividend. |
| October 21, 2026 | Effect Size Calculator | Fixed: the standard error of Cohen's d used 2(n1 + n2 − 2) instead of 2(n1 + n2) in its second term, and required sample size now uses the t distribution rather than the normal approximation, matching Cohen's power tables (64 per group for d = 0.5, not 63). |
| October 21, 2026 | Effect Size Calculator | Fixed: blank sample sizes silently defaulted to 30 per group; they are now required. |
| October 21, 2026 | Eigenvalue & Eigenvector Calculator | Fixed: the cubic solver used a fixed 1e-9 tolerance on the discriminant, so a 3×3 matrix with large repeated eigenvalues (for example diag(1000, 1000, 3000)) was reported as having complex eigenvalues, and close distinct eigenvalues such as 1 and 1.00001 were merged into one repeated value. |
| October 21, 2026 | Eigenvalue & Eigenvector Calculator | Fixed: a repeated eigenvalue with a two- or three-dimensional eigenspace now lists every independent eigenvector instead of only one. |
| October 21, 2026 | Email Revenue Calculator | Removed the list-quality tier discount, which cut revenue by up to 30% on top of the open rate already entered, so the same effect was being counted twice. Added conversion rate, margin and list growth inputs. |
| October 21, 2026 | Emergency Fund Calculator | Fixed: interest treated the savings APY as a nominal rate divided by 12, overstating growth; it now converts APY to a monthly rate. Also fixed: a monthly saving amount not in the preset list showed 0 months to goal. |
| October 21, 2026 | Equation Graph Generator | Fixed: the automatic y range was stretched to the most extreme sampled value, so a function with a pole such as tan(x) or 1/x was drawn as a flat line; the range now ignores the extreme tails, and curves are broken at jumps larger than half the visible range instead of 50 times it. |
| October 21, 2026 | Euler's Totient Calculator | Fixed: the Euler's theorem check reported 1 for n = 1 (every number is 0 mod 1), inputs were read with parseInt so values like 12.7 were silently truncated, and an FAQ wrongly called φ completely multiplicative. |
| October 21, 2026 | Exponent Calculator | Fixed: a negative base with an even-numerator fraction exponent was given a negative result, so (−8)^(2/3) showed −4 instead of 4. |
| October 21, 2026 | Exponent Calculator | Fixed: solving bⁿ = r for the base reported b³ = −8 as complex instead of b = −2, returned −16 for b^0.5 = −4 (which has no real solution), and omitted the negative base for even powers such as b² = 49. |
| October 21, 2026 | Exponent Rules Calculator | Fixed: for a negative base, b^(p/q) took its sign from the unreduced numerator, so (−8)^(2/6) showed +2 instead of −2; dividing powers of 0 now reports undefined instead of infinity. |
| October 21, 2026 | Exponential & Logarithmic Equation Solver | Fixed: the quadratic-in-exponential form divided by zero when A was 0 (returning no valid root) and listed a repeated root twice when the discriminant was 0. |
| October 21, 2026 | Exponential & Logarithmic Equation Solver | Fixed: blank or non-numeric coefficients were silently treated as 0; they are now flagged on the field. |
| October 21, 2026 | Exponential Growth & Decay Calculator | Fixed: per-period decay half-life used ln(1 + rate) instead of ln(1 − rate), overstating it (10% decay showed 7.27 periods instead of 6.58), and the bacteria example used a rate that doubled every 30 hours instead of every 30 minutes. |
| October 21, 2026 | Expression input (graphing and calculus tools) | Fixed operator precedence: −x² was evaluated as (−x)², so −2² gave 4 instead of −4. Expressions are now parsed with the standard rule that powers bind tighter than a leading minus. |
| October 21, 2026 | Fibonacci Calculator | Fixed: the Fibonacci membership test used an integer square root that returned a wrong value once 5n² exceeded 2^53, so Fibonacci numbers above about 42 million (for example F(40) = 102,334,155) were reported as not Fibonacci. |
| October 21, 2026 | Flooring Calculator | Fixed: the material cost used the area including waste instead of the whole boxes you have to buy, so it understated the cost whenever the boxes did not come out exact. |
| October 21, 2026 | Fraction Calculator | Fixed: decimal entries such as 2.5 were silently truncated to 2 before calculating; non-whole entries are now rejected with a message suggesting 5/2 or 2 1/2. |
| October 21, 2026 | Fraction to Decimal Converter | Fixed: fractions whose repeating block is longer than 500 digits (for example 1/503, period 502) were shown with an empty repeating block and a cycle length of 0; the period is now computed exactly from the multiplicative order of 10. |
| October 21, 2026 | Freelance Rate Calculator | Fixed: self-employment tax is now charged on 92.35% of net profit with the Social Security part capped at the 2026 wage base, instead of 15.3% of all revenue, and overhead is entered as a dollar amount rather than a percentage of gross pay. |
| October 21, 2026 | Frequency Distribution Calculator | Fixed: Sturges' rule was rounded to the nearest whole number (5 classes for 20 values) instead of rounded up (6 classes), class membership was decided by repeated floating-point addition so values on a boundary could land in the wrong class, and the grouped standard deviation divided by n instead of n − 1. |
| October 21, 2026 | Frustum Calculator | Fixed: when the top radius was larger than the bottom radius, the original cone height came out negative; it is now measured from the larger base. |
| October 21, 2026 | Game Theory Calculator | Fixed: mixed equilibria of 2 × 2 games used each player's own payoffs to set their probabilities, so asymmetric games such as Battle of the Sexes showed the row player at 1/3 instead of 2/3; each mix is now set by the opponent's payoffs, and mixed equilibria are also found for 3 × 3 games. |
| October 21, 2026 | GCD & LCM Calculator | Fixed: the LCM multiplied the numbers before dividing by the GCD in floating point, so results above 2^53 (for example lcm(10^12, 10^12 − 1)) were rounded; all arithmetic is now exact. |
| October 21, 2026 | Geometric Mean Calculator | Fixed: negative values were silently dropped and zeros were reported but then excluded, so the result was the geometric mean of a different data set; the mean is now computed from logarithms so long or large lists no longer overflow. |
| October 21, 2026 | Geometric Sequence Calculator | Fixed: finding the ratio or the position of a term failed for negative ratios (for example 3, −6, 12, −24), and solving for the ratio now lists both the positive and negative answers when they both fit. |
| October 21, 2026 | GPA Calculator | Fixed: the what-if planner picked the letter grade nearest to the points needed, which could be a grade that falls short (for example B+ when 3.4 points were needed); it now gives the lowest grade that reaches the target. |
| October 21, 2026 | GPA Calculator | Fixed: a term containing only pass, no-pass or withdrawn courses was shown as a 0.000 GPA; it now explains that no letter-graded credits were entered. |
| October 21, 2026 | Gram-Schmidt Calculator | Fixed: linear dependence was tested against a fixed cutoff of 1e-10, so independent vectors with very small components (such as (1e-11, 0) and (0, 1e-11)) were wrongly reported as dependent; the test is now relative to each vector's length. |
| October 21, 2026 | Guides | The five guides showed publication dates in January to March 2025, before the site existed. They now show their real publication date, May 13, 2026. |
| October 21, 2026 | Harmonic Mean Calculator | Fixed: weighted mode accepted negative values and returned a meaningless harmonic mean and an undefined geometric mean; values must now be positive, and the geometric mean is computed with logarithms so long lists no longer overflow. |
| October 21, 2026 | Heron's Formula Calculator, Triangle Area & Angles | Fixed: the triangle diagram labeled side a where side c was and side c where side a was. |
| October 21, 2026 | Heron's Formula Calculator, Triangle Area & Angles | Fixed: base-and-height mode invented an isosceles triangle and labeled its apex angle as the base angle; it now reports only the area, which is all a base and height determine. |
| October 21, 2026 | Heron's Formula Calculator, Triangle Area & Angles | Improved: area now uses Kahan's rearrangement of Heron's formula, which stays accurate for long, thin triangles. |
| October 21, 2026 | Hourly to Salary Calculator | Removed the take-home pay estimate, which used 2024 federal tax brackets; the page now converts gross pay only and links to the paycheck calculator for taxes. |
| October 21, 2026 | Inflation Calculator | Fixed: the "past value today" mode divided a past amount by (1 + rate)^years, which gives the present value of a future sum; a past amount converted to today's dollars must be multiplied. The modes are now labeled future cost and value today, and a CPI ratio method was added. |
| October 21, 2026 | Influencer Marketing ROI Calculator | Removed built-in earned media values, engagement benchmarks by follower tier and CPM ranges by platform, which had no cited source; the calculator now works from the reach, engagement, click and conversion figures you enter. |
| October 21, 2026 | Integral Calculator | Fixed: the 1/√(1 − x²) and 1/(1 + x²) options computed definite integrals of arcsin x and arctan x instead, so their numerical values did not match the displayed antiderivatives. |
| October 21, 2026 | Interpolation Calculator | Fixed: duplicate x-values were not detected, so the divided differences divided by zero and showed Infinity or NaN; they are now rejected with a message. |
| October 21, 2026 | Inverse Trig Calculator | Fixed: arcsinh used ln(x + √(x² + 1)), which returned no result for large negative inputs such as −100,000,000, and inverse hyperbolic results were wrongly converted to degrees as if they were angles. |
| October 21, 2026 | IRR Calculator | Fixed: sign changes separated by a zero cash flow were not counted, so some streams with more than one IRR showed a single rate without a warning. The calculator now finds and lists every IRR. |
| October 21, 2026 | Kruskal-Wallis H Test Calculator | Fixed: Dunn's pairwise test ignored tied values in its variance, which understated z when the data had ties; the p-values now use an accurate normal distribution instead of a three-term approximation. |
| October 21, 2026 | Laplace Transform Calculator | Fixed: the region of convergence for cosh(bt) was shown as s > b, which is wrong for negative b; it is now Re(s) > |b|. Non-integer n in tⁿ is now rejected instead of being silently rounded. |
| October 21, 2026 | Limit Calculator | Fixed: the limit was read off a single sample at x = c ± 0.00001, so infinite limits such as 1/x² at 0 were reported as a finite limit of about 10,000,000,000, and limits at infinity were only approximate; limits now use extrapolation and detect ±∞ and −∞ targets. |
| October 21, 2026 | Linear Equation Solver | Fixed: a two-equation system whose determinant was 0 was reported as having infinitely many solutions whenever both Cramer numerators were 0, even when one equation read 0 = 1 and had no solution. |
| October 21, 2026 | Linear Regression Calculator | Fixed: the prediction interval used a rough t value (2.5, 2.2 or 1.96 by sample size) instead of the exact Student t critical value, and non-numeric entries were silently dropped, which could pair x and y values from different rows. |
| October 21, 2026 | Logarithm Laws Calculator | Fixed: condensing logs with large arguments (for example 400 log 10) returned infinity because the combined argument overflowed; the value is now summed term by term and checked directly only when the argument fits. |
| October 21, 2026 | Logistic Regression Calculator | Fixed: coefficients came from 1,000 steps of slow gradient descent that usually stopped far from the maximum likelihood estimate; the model is now fitted by Newton-Raphson and matches R's glm. AUC now counts tied scores as one half, and data that perfectly separate the outcomes are reported instead of returning arbitrary coefficients (the old default example was perfectly separated). |
| October 21, 2026 | Mann-Whitney U Test Calculator | Fixed: p-values always came from the normal approximation with a three-term erf formula, even for small samples without ties; they now use the exact distribution of U in that case (matching R's wilcox.test) and an accurate normal distribution otherwise. |
| October 21, 2026 | Marketing Attribution Calculator | Fixed: touchpoints are now ordered by date before first and last touch are assigned (they were taken in the order typed), W-shaped credit with three touches is now split equally instead of 30/40/30, and time decay uses a half-life you can set instead of a fixed 0.1-per-day rate. |
| October 21, 2026 | Marketing Budget Mix Calculator | Fixed: a channel with revenue but no spend got an efficiency index of 0 and was flagged for a budget cut; it is now shown as having no spend and is left out of the reallocation. The projected lift no longer shows a plus sign when it is negative. |
| October 21, 2026 | Marketing Funnel Calculator | Fixed: the what-if table was labeled as a 5 percentage point improvement but multiplied the stage count by 1.05, which gives the same revenue gain for every stage; it now adds percentage points to one stage rate. The unused sales cycle input was removed and "annual ARR" is now labeled as revenue for twelve matching periods. |
| October 21, 2026 | Markov Chain Calculator | Fixed: the steady state was found by repeated multiplication from a uniform start, which never converges for periodic chains (for P = [[0, 1, 0], [1/2, 0, 1/2], [0, 1, 0]] it reported 1/3 each instead of 1/4, 1/2, 1/4); it is now solved exactly from π = πP, and chains with several closed classes are reported as having no unique steady state. |
| October 21, 2026 | Matrix Calculator | Fixed: the inverse treated any matrix with |det| below 1e-12 as singular, so invertible small-scale matrices such as diag(1e-7, 1e-7) were rejected; invertibility is now decided by Gauss-Jordan elimination with a tolerance relative to the matrix entries. |
| October 21, 2026 | Matrix Calculator | Fixed: the 2×2 inverse working rounded the determinant to a whole number when writing fractions, so a matrix with determinant 0.5 showed wrong rows (for example [1, 0] instead of [2, 0]). |
| October 21, 2026 | Matrix Inverse Calculator | Fixed: any matrix with |det| below 1e-10 was declared singular, so invertible small-scale matrices such as diag(1e-6, 1e-6) were rejected; invertibility is now decided by Gauss-Jordan elimination with a tolerance relative to the entries. |
| October 21, 2026 | Matrix LU Decomposition | Fixed: Doolittle elimination never exchanged rows, so invertible matrices with a zero pivot (such as [[0, 1], [1, 0]]) were reported as singular; the factorization now uses row exchanges and reports PA = LU, with the determinant sign adjusted for each swap. |
| October 21, 2026 | Matrix RREF Calculator | Fixed: entries were read with parseFloat, so a fraction such as 1/2 was silently treated as 1 and text such as 3x as 3; entries are now validated and fractions are parsed exactly. |
| October 21, 2026 | Mean, Median & Mode Calculator | Fixed: entries such as "12abc" were read as 12 without warning; any value that is not a number is now reported instead of being silently truncated. |
| October 21, 2026 | Mixed Number Calculator | Fixed: a denominator of 0 (or a blank denominator) was silently treated as 1, and decimal entries were truncated to whole numbers, so wrong answers were shown instead of an error; both are now rejected with a message. |
| October 21, 2026 | Modular Arithmetic Calculator | Fixed: power mod, multiplication and the two-congruence solver used floating-point products, so moduli above about 95 million gave wrong results, and exponents of 2^31 or more broke the bit loop; all arithmetic is now exact. |
| October 21, 2026 | Modulo Calculator | Fixed: with a negative divisor the floored (Python) result was forced to be non-negative while the quotient was floored, so 7 mod −3 showed 1 with quotient −3 instead of −2; floored remainders now take the sign of the divisor, and the Euclidean convention is offered separately. |
| October 21, 2026 | Modulo Calculator | Fixed: decimal inputs were divided in binary floating point, so 0.3 mod 0.1 gave 0.1 instead of 0; arithmetic is now exact. |
| October 21, 2026 | Multiple Regression Calculator | p-values came from a 1,000-step midpoint integration of the incomplete beta function over a singular integrand, which made them inaccurate, most of all with few degrees of freedom. They now use the tested Student t and F distribution functions, and the coefficients are solved with pivoted elimination. |
| October 21, 2026 | NPV Calculator | Fixed: MIRR ignored negative cash flows after year 0; they are now discounted to today at the discount rate, as the spreadsheet MIRR function does. |
| October 21, 2026 | NPV Calculator | Fixed: a blank cash flow row was skipped, which moved every later cash flow one year earlier; blank rows are now flagged instead. |
| October 21, 2026 | Nth Root Calculator | Fixed: for an even root of a negative number the complex answer was given as |x|^(1/n) × i, which is only right for square roots; ∜−16 showed 2i instead of the principal root 1.414 + 1.414i. |
| October 21, 2026 | Nth Root Calculator | Fixed: perfect powers could show floating-point error at high precision (∛1000 as 9.999999999999998); exact integer roots are now detected and shown exactly. |
| October 21, 2026 | Number Sequence Calculator | Fixed: Fibonacci and Lucas terms were computed in floating point, so terms beyond the 78th were rounded, and the Lucas sum formula was shown as L(n + 2) − 3 instead of the correct L(n + 1) − 1 for the listed terms; terms are now exact integers. |
| October 21, 2026 | Numerical Differentiation Calculator | Fixed: the reported truncation errors were differences from the Richardson estimate rather than true errors, and the Richardson error was never shown; errors are now measured against the exact derivative. Expressions no longer run through new Function(). |
| October 21, 2026 | Numerical Integration Calculator | Fixed: the outer 5-point Gauss-Legendre node was 0.9061798459366165 instead of 0.9061798459386640, and expressions are now parsed safely instead of being run as JavaScript code. |
| October 21, 2026 | Overtime Pay Calculator | Fixed: one week of pay was shown as the gross for a biweekly, semimonthly or monthly period and annualized from there, understating period and yearly pay; the week is now scaled by weeks per period and by 52. |
| October 21, 2026 | Overtime Pay Calculator | Fixed: California mode ignored the weekly 40-hour limit and the seventh consecutive day rule from Labor Code section 510. |
| October 21, 2026 | P-Value Calculator | Fixed: chi-square and F p-values ignored the selected tail, so a two-tailed choice returned the right-tail area. |
| October 21, 2026 | Paint Calculator | Fixed: in metric mode, trim area was computed in square feet and divided by a coverage in square meters per liter, overstating trim paint about tenfold. |
| October 21, 2026 | Paint Calculator | Fixed: ceiling and trim paint ignored the number of coats; they now use the same coat count as the walls. |
| October 21, 2026 | Paint Calculator | Changed: the economy, standard and premium coverage presets were replaced by the coverage figure from your can, and the default door area is now 20 sq ft (36 by 80 inches) instead of 21. |
| October 21, 2026 | Parametric Curve Grapher | Fixed: undefined points were dropped and their neighbors joined, so the curve and the arc length bridged gaps such as sqrt of a negative number; gaps are now left open and not counted, and arc length uses a Richardson-extrapolated 8,000-segment sum instead of a plain polyline that always ran short. |
| October 21, 2026 | Partial Derivative Calculator | Fixed: points where both first partials were merely below 0.01 were labeled as critical points and classified; classification now requires the gradient to be zero, and all derivatives are computed exactly instead of with a 0.00001 finite-difference step. |
| October 21, 2026 | Partial Fraction Decomposition Calculator | Fixed: choosing equal roots in a distinct-factor template, such as a = b in (x + a)(x + b), divided by a 1e-15 placeholder and showed huge meaningless coefficients; repeated factors are now detected and decomposed correctly. |
| October 21, 2026 | Pascal's Triangle Calculator | Fixed: expansions with a numeric or negative term, such as (x − 1)⁵, printed unsimplified terms like 10x^3·-1^2, which reads as −1; coefficients are now multiplied out exactly with the correct signs. |
| October 21, 2026 | Paycheck Calculator | Updated to 2026 federal brackets, standard deductions and the $184,500 Social Security wage base (the old version used 2024 figures). |
| October 21, 2026 | Paycheck Calculator | Fixed: Social Security and Medicare were charged on health insurance, HSA and FSA payroll deductions, which are exempt under a section 125 cafeteria plan. |
| October 21, 2026 | Paycheck Calculator | Fixed: Additional Medicare Tax was withheld from $250,000 for married filers; employers withhold it on wages over $200,000 regardless of filing status. |
| October 21, 2026 | Paycheck Calculator | Removed unsourced approximate state tax rates; you now enter your own state and local rate. |
| October 21, 2026 | Percentage Calculator | Fixed: the copied result and working could show floating-point noise (7% of 30 = 2.1000000000000005); results are now cleaned to 12 significant digits. |
| October 21, 2026 | Permutation Group Calculator | Fixed: cycles sharing a number, such as (1 2)(2 3), were overwritten instead of multiplied and could produce something that was not a permutation; overlapping cycles are now composed correctly. |
| October 21, 2026 | Permutation Group Calculator | Fixed: the product labeled σ∘τ (σ then τ) was actually computed as τ first, then σ; the label now matches the result, and both composition conventions are available. |
| October 21, 2026 | Piecewise Function Calculator & Evaluator | Fixed: a formula undefined at a breakpoint, such as (x² − 1)/(x − 1) at x = 1, was reported as a jump discontinuity because the limit was taken by substituting the endpoint; one-sided limits are now computed properly and the point is classified as removable. |
| October 21, 2026 | Piecewise Function Calculator & Evaluator | Formulas are now parsed by a restricted math parser instead of being run as JavaScript code. |
| October 21, 2026 | Platonic Solids Calculator | Fixed: the dodecahedron midradius used (1 + √5)/4 × a (0.809a) instead of (3 + √5)/4 × a (1.309a). |
| October 21, 2026 | Platonic Solids Calculator | Fixed: the dodecahedron dihedral angle was shown as arctan 2 = 63.43° instead of arccos(−1/√5) = 116.57°. |
| October 21, 2026 | Poisson Distribution Calculator | Fixed: probabilities for large rates (for example λ = 200 with k = 170) overflowed to infinity because λ^k and k! were computed separately; they are now computed in log space. |
| October 21, 2026 | Polynomial Long Division Calculator | Fixed: quotient terms with a zero coefficient were dropped, so (x⁴ + 1) ÷ (x² + 1) showed the quotient x − 1 instead of x² − 1. Invalid coefficients are now reported instead of being silently skipped. |
| October 21, 2026 | Portfolio Rebalancer | Fixed: drift was measured against holdings plus new cash, so a deposit made every holding look underweight; drift now uses current holdings only. |
| October 21, 2026 | Portfolio Rebalancer | Fixed: holdings inside the drift band were skipped individually, so the remaining trades no longer added up and the result did not reach target; the band now decides whether to rebalance the whole portfolio. |
| October 21, 2026 | Power Analysis Calculator | Fixed: t-test sample sizes used the normal approximation and came out one or two subjects short (63 instead of 64 per group for d = 0.5); they now use the exact noncentral t distribution. |
| October 21, 2026 | Power Analysis Calculator | Fixed: chi-square sample sizes ignored the degrees of freedom and were only correct for 1 df; they now use the noncentral chi-square distribution. |
| October 21, 2026 | PPC ROAS Calculator | Removed the Quality Score CPC discount and premium table, which was not from a published Google source. Break-even ROAS, CPA and CPC now come straight from your margin and conversion rate. |
| October 21, 2026 | Prime Number Checker | Fixed: 5 was not flagged as a safe prime, although (5 − 1) ÷ 2 = 2 is prime. |
| October 21, 2026 | Prime Number Checker | Fixed: for 1 and 2 the previous prime was shown as 2; there is no prime below 2, so it now says none. |
| October 21, 2026 | Prime Number Checker | Fixed: the prime gap shown for a prime spanned from the previous prime to the next one; the gaps before and after are now reported separately. |
| October 21, 2026 | Prime Number Checker | Fixed: listing primes in a range above 1,000,000 silently returned no primes; the range now works up to one trillion. |
| October 21, 2026 | Probability Calculator | Fixed: impossible inputs for P(A or B), such as P(A) = 0.8, P(B) = 0.7 and P(A and B) = 0.1, were silently clamped to a probability of 1 instead of being rejected. |
| October 21, 2026 | Process Capability Calculator | Fixed: the sigma level came from a coarse lookup table with wrong breakpoints (for example 10,700 DPMO was labeled 4σ instead of 6,210); it is now computed exactly from the defect rate. |
| October 21, 2026 | Process Capability Calculator | Fixed: Pp and Ppk were copies of Cp and Cpk; they now use the overall standard deviation, separate from the within-subgroup standard deviation. |
| October 21, 2026 | Proportion Solver | Fixed: when solving for a or b the solver checked the wrong value for zero, so a zero divisor returned infinity and some valid problems (such as a/5 = 0/10) were rejected. |
| October 21, 2026 | Quadrilateral Area Calculator | Fixed: convex quadrilaterals entered in clockwise order were reported as non-convex. |
| October 21, 2026 | Quadrilateral Area Calculator | Fixed: vertices entered out of order (a bow-tie with crossing sides) returned a meaningless shoelace area; they are now rejected with an explanation. |
| October 21, 2026 | Quadrilateral Area Calculator | Fixed: trapezoid leg lengths were added to the perimeter without checking that they fit the given height and parallel sides. |
| October 21, 2026 | Quartic Equation Solver | Fixed: repeated roots such as the quadruple root of x⁴ − 4x³ + 6x² − 4x + 1 = (x − 1)⁴ were shown as complex numbers with tiny imaginary parts; rational roots are now found exactly and repeated roots are reported with their multiplicity. The preset labeled "Triple + Simple" was actually a quadruple root and has been renamed. |
| October 21, 2026 | Rational Root Theorem Calculator | Fixed: when the constant term was 0 only the candidates 0 and ±1/q were tested, so roots such as 2 in x³ − 3x² + 2x were missed; the calculator now factors out x first. Roots are now tested with exact integer arithmetic and reported with their multiplicities. |
| October 21, 2026 | Regular Polygon Calculator | Fixed: a triangle was reported as having a longest diagonal equal to its side; triangles have no diagonals. |
| October 21, 2026 | Remainder Calculator | Fixed: decimal inputs were silently cut to whole numbers (12.7 was divided as 12, and 1e3 as 1); decimals are now divided exactly and other non-numbers are rejected with a message. |
| October 21, 2026 | Riemann Sum Calculator | Fixed: the sin(x) on [0, pi] and 1/x on [1, e] examples failed because the limits were read with parseFloat, so pi and e were rejected; limits now accept constants and expressions. |
| October 21, 2026 | Roman Numeral Converter | Fixed: invalid numerals such as IC, IIII and VV were converted as if they were correct; they are now flagged as non-standard and the standard form is shown. |
| October 21, 2026 | Salary Negotiation Calculator | Fixed: health coverage is now valued from the premiums you enter instead of an assumed $600 a month, commuting is counted as a cost of office days rather than as a saving, and paid time off now shows pay per hour worked instead of being added as extra cash. |
| October 21, 2026 | Salary to Hourly Calculator | Fixed: weekly pay was hourly x hours x 5 (five weeks of pay) and the daily rate showed a full week of pay; weekly is now salary / 52 and daily uses hours per day. |
| October 21, 2026 | Sample Size Calculator | The finite population correction was applied to n₀ after rounding it up, which could add one respondent too many (for example 279 instead of 278 for a population of 1,000). n₀ is now kept unrounded until the final step. |
| October 21, 2026 | Savings Goal Calculator | Fixed: the rate was labeled APY but divided by 12 as a nominal rate compounded monthly, which overstated growth; it is now converted with (1 + APY)^(1/12) - 1. |
| October 21, 2026 | Scientific Notation Calculator | Fixed: rounding that carried into a new power of ten was not renormalized, so 9.9999 at 4 significant figures showed 10.000 × 10^0 instead of 1.000 × 10¹. |
| October 21, 2026 | Scientific Notation Calculator | Fixed: engineering notation added decimal places instead of removing them, so 299,792,458 at 4 significant figures showed 299.79246 × 10^6 instead of 299.8 × 10⁶. |
| October 21, 2026 | Second Derivative Calculator | Fixed: the ln(x) example used log(x), which the parser reads as base 10, so f″(1) showed −0.434 instead of −1; derivatives are now exact instead of finite differences with h = 0.0001, which lost about half the digits. |
| October 21, 2026 | SEO ROI Calculator | Replaced the fixed click-through rates by ranking position, presented as Google benchmarks without a source, with click shares you enter per keyword, for example from Google Search Console. |
| October 21, 2026 | Sequence Analyzer | Cubic sequences predicted the same value for all five next terms because the difference table was never updated between steps. The analyzer now rebuilds the polynomial from the difference table and was checked against the cubes 1, 8, 27, 64, 125. |
| October 21, 2026 | Series Convergence Calculator | Fixed: terms written in n could not be evaluated because the expression parser only knew x, so every series failed; the ratio test also called the harmonic series convergent because it stopped at n = 25, where the ratio was 0.96. The tests now probe n up to 10⁷ and use a p-series comparison and the alternating series test. |
| October 21, 2026 | Simplex Method Calculator | Fixed: problems with ≥ constraints were converted to ≤ rows with negative right-hand sides and started from an infeasible basis, so the Diet preset (minimize 2x₁ + 3x₂ with x₁ + x₂ ≥ 4, x₁ + 3x₂ ≥ 6) returned x = 0, z = 0 instead of the true minimum 9 at (3, 1); the solver now uses the two-phase simplex method. |
| October 21, 2026 | Simplify Radicals Calculator | Fixed: sums of perfect powers such as √4 + √9 were left as 2 + 3 instead of 5, and radicals like ⁴√4 were not reduced to √2. Multiplying roots with different indices is now exact (∛2 × √2 = ⁶√32) and division with rationalized denominators was added. |
| October 21, 2026 | Slope Calculator | Fixed: two identical points were reported as a vertical line; they are now rejected because they do not define a line. |
| October 21, 2026 | Social Media Ads ROI Calculator | Removed built-in platform benchmark figures and the frequency-to-CTR adjustment, which had no cited source; the calculator now works from the CPM, click and conversion rates you enter. |
| October 21, 2026 | Social Security Calculator | Fixed: the PIA used the 2024 bend points for everyone and rounded to the nearest dime; it now uses the bend points for the year you turn 62 (2022 to 2026), applies cost-of-living adjustments since then, and rounds down to the dime as SSA does. |
| October 21, 2026 | Solar Panel Calculator | Fixed: the calculator assumed the 30% federal Residential Clean Energy Credit through 2032. That credit no longer applies to systems placed in service after 2025, so the incentive now defaults to 0% and can be set for any state or utility rebate. |
| October 21, 2026 | Solar Panel Calculator | Changed: production now uses an explicit performance ratio (default 86%, matching NREL PVWatts default losses) and includes panel degradation in the multi-year savings. |
| October 21, 2026 | Summation Calculator | Fixed: unsupported characters were silently deleted from the term (sqrt(n) was summed as n) and a leading minus before a power, as in -n^2, caused an error. Terms now go through the safe math parser, sums use compensated addition, and a step option was added. |
| October 21, 2026 | Surface Area Calculator | Fixed: triangular prism sides that could not form a triangle produced NaN instead of an error message. |
| October 21, 2026 | Systems of Equations Solver 2×2 | Fixed: a system containing an impossible equation such as 0x + 0y = 5 was reported as having infinitely many solutions; systems are now classified by comparing matrix ranks with a tolerance scaled to the coefficients. |
| October 21, 2026 | T-Test Calculator | The confidence interval used a lookup table of t critical values with gaps (for example degrees of freedom 11 to 14 all used the value for 10) and was always 95% regardless of the chosen significance level. It now uses the exact t quantile for the chosen level, and Welch degrees of freedom are no longer rounded down. |
| October 21, 2026 | Tangent Line Calculator | Fixed: f″(a) was computed with a step of 0.00001, whose rounding noise (about 10⁻⁶) exceeded the 10⁻⁸ threshold used to detect an inflection, so points such as sin(x) at 0 were labeled concave up or down at random; slopes and f″ are now exact. |
| October 21, 2026 | Taylor Series Calculator | Fixed: derivatives of order 4 and above were estimated by finite differences with h near 0.001, so dividing by h⁶ or more amplified rounding error and gave wrong coefficients from about order 5 on; coefficients are now exact by Taylor-mode automatic differentiation, up to order 20. |
| October 21, 2026 | Triangle Solver | Fixed: in the SAS case the second angle was found with arcsin, which always returns an acute angle, so triangles with an obtuse angle (for example a = 10, B = 20°, c = 3) got wrong angles and a wrong area. |
| October 21, 2026 | Trigonometric Equation Solver | Fixed: solutions were folded back into 0 to 2π with a 2π shift even when the period was not a divisor of 2π, which listed wrong angles for equations such as sin(1.5x) = 0.5, and lists were cut off after 8 solutions, so sin(5x) = 1/2 showed 8 of its 10 solutions. |
| October 21, 2026 | Trigonometry Calculator | Fixed: in radian mode the special-angle buttons entered rounded values such as 1.5708, so the 90° button showed tan as about −272,242 instead of undefined; angles can now be entered exactly as pi/2. |
| October 21, 2026 | Truth Table Generator | Fixed: expressions using NAND or NOR produced an all-false column because the operators were translated into invalid code; they now evaluate correctly. |
| October 21, 2026 | Truth Table Generator | Fixed: expressions are now read by a dedicated parser instead of being run as code, and unrecognized characters are reported instead of being silently dropped. |
| October 21, 2026 | Truth Table Generator | Fixed: XOR was evaluated with higher precedence than AND (A XOR B AND C was read as (A XOR B) AND C); AND now binds tighter, following the standard convention. |
| October 21, 2026 | Two-Way ANOVA Calculator | The tool accepted only one value per cell, so it could never test an interaction, and all three of its example data sets were perfectly additive, which made the error term zero and the F statistics undefined. It now supports replicate values with a full interaction model and refuses data with no residual variation instead of reporting infinite F values. |
| October 21, 2026 | Unit Circle Calculator | Fixed: exact values were looked up after rounding the angle to the nearest degree, so angles such as 30.4° were shown with the exact value sin = 1/2 that belongs to 30°. |
| October 21, 2026 | Vector Projection Calculator | Fixed: projecting onto the zero vector divided by zero and displayed NaN for every result; the calculator now explains that the zero vector has no direction to project onto. |
| October 21, 2026 | Volume Calculator | Fixed: cubic inches and cubic feet used rounded factors (1.6387e-5 and 0.0283168 m³) instead of the exact 1.6387064e-5 and 0.028316846592 m³, and dimensions had no unit, so every input was silently treated as meters. |
| October 21, 2026 | Volume of Revolution Calculator | Fixed: surface area used f(x) instead of |f(x)|, so curves below the axis gave negative areas, and the √x example failed because the derivative was evaluated at the endpoint x = 0; integrals now use adaptive quadrature that avoids endpoints, and washers and rotation about any line are supported. |
| October 21, 2026 | Wronskian Calculator | Fixed: expressions were run as JavaScript with Function(), so implicit products such as exp(2x) failed and log meant ln; second derivatives used a 0.00001 step whose rounding noise (about 10⁻⁶) matched the independence threshold. Expressions are now parsed safely and derivatives are exact. |