{ "category": "Statistical functions", "display_name": "GAMMA.DIST", "entry_kind": "built_in_function", "eval_api": { "method": "POST", "url": "https://handbook.dnacalc.com/api/eval" }, "function_id": "FUNC.GAMMA.DIST", "implementation_state": { "answer_bank": { "excel_builds": [ "20131", "20430", "20527", "?" ], "larger_numeric": 164, "last_bit": 29708, "matched": 48103, "other_mismatch": 0, "refused": 0, "rows": 77975, "unread": 0, "worst_ulp": 58 }, "detail": "Of 78,071 recorded answers (77,975 in the answer bank and 96 workbook cells), 29,887 do not match: 164 differ by more than 16 ULP, 29,723 differ by at most 16 ULP.", "dnacalc_commit": "9e1884369a678158dd82d3d74188daf986995e87", "in_catalogue": true, "state": "differs", "state_label": "Known larger differences", "summary": "Known larger differences on 29,887 of 78,071 recorded answers", "workbook_corpus": { "cells": 96, "last_bit": 15, "matched": 81, "worst_ulp": 14 } }, "implementations": [ { "commit": "9e1884369a678158dd82d3d74188daf986995e87", "flavour": "excel-compatibility-candidate", "language": "Rust", "name": "CalcFunc", "own_code": [ "https://github.com/DnaCalc/DnaCalc/blob/9e1884369a678158dd82d3d74188daf986995e87/crates/calc-func/src/catalogue/distributions.rs#L223-L226" ], "repository": "https://github.com/DnaCalc/DnaCalc", "shared_code": [ "catalogue/distributions::nums", "ported/beta_gamma_stats_family::validate_gamma_shape", "ported/beta_gamma_stats_family::gamma_dist_kernel", "ported/discrete_dist::finite", "ported/discrete_dist::expon_dist_kernel", "ported/excel_numeric::binary_with", "ported/excel_numeric::excel_expm1_internal", "ported/excel_numeric::excel_expm1_spilled", "ported/excel_numeric::expm1_internal_graph", "ported/excel_numeric::excel_x87_add", "ported/excel_numeric::excel_x87_sub", "ported/excel_numeric::excel_x87_mul", "ported/excel_numeric::excel_x87_temme_finish", "ported/excel_numeric::excel_x87_div", "ported/excel_numeric/x87::mul", "ported/excel_numeric/x87::add", "ported/excel_numeric/x87::sub", "ported/excel_numeric/x87::div", "ported/excel_numeric/x87_chains::pow2_positive", "ported/excel_numeric/x87_chains::pow2", "ported/excel_numeric/x87_chains::exp_ext", "ported/excel_numeric/x87_chains::exp_with", "ported/excel_numeric/x87_chains::excel_exp", "ported/excel_numeric/x87_chains::log_with", "ported/excel_numeric/x87_chains::excel_log", "ported/normal_dist_common::stored_horner", "ported/normal_dist_common::erf_approx", "ported/normal_dist_common::erfc_upper", "ported/special_dist_family::erf_precise_kernel", "ported/special_math_common::ln_gamma", "ported/special_math_common::gratio_norm_gamma", "ported/special_math_common::gratio_rexp", "ported/special_math_common::gratio_rlog", "ported/special_math_common::gratio_gam1", "ported/special_math_common::gratio", "ported/special_math_common::series_prefactor", "ported/special_math_common::gratio_after40", "ported/special_math_common::gratio_cf", "ported/special_math_common::gratio_temme_270", "ported/special_math_common::gratio_temme_340", "ported/special_math_common::gratio_temme_330", "ported/special_math_common::gratio_temme_finish", "ported/special_math_common::regularized_gamma_p" ], "source": "https://github.com/DnaCalc/DnaCalc/blob/9e1884369a678158dd82d3d74188daf986995e87/crates/calc-func/src/catalogue/distributions.rs#L223-L226", "state": "differs", "suite": null, "summary": "Known larger differences on 29,887 of 78,071 recorded answers" } ], "json_url": "https://handbook.dnacalc.com/api/functions/FUNC.GAMMA.DIST.json", "name": "GAMMA.DIST", "notes": [ { "basis": "Noise lines on windows of consecutive doubles (Excel build 20527) put ln x stored and multiplied by a, not a power of x. Storing t - x adds a slow line of amplitude 5, and leaving it unstored takes the noise at small x from 13 units of 2^-53 to 1.5. A scan of one constant factor per shape peaks at -0.875, +0.25, -1.25, +1.625 and -2 of 2^-53 for a = 2 to 6, and the first is the error of GAMMALN(3). This form matches 15,364 of 26,776 rows at a = 2, 15,588 of 26,770 at a = 3, 1,088 of 2,051 at a = 4, 654 of 1,142 at a = 5, 691 of 1,151 at a = 6, and over all shapes 47,647 of 77,975 recorded rows.", "documentation": null, "evidence": [ { "commit": "b5d84a80a5a4640eca9ce04f232c0bc115503f34", "locator": null, "note": "research write-up: how the stored stages were read", "path": "content/notes/evidence/2026-11-04/gamma-EVIDENCE.md", "repository": "ExcelFunctionsHandbook" }, { "commit": "78de0f0a49cbb65d43c57f7cd4f4276848ed7eda", "locator": null, "note": "the prefactor in the product; the commit message carries the row counts", "path": "crates/calc-func/src/ported/special_math_common.rs", "repository": "DnaCalc" }, { "commit": "af74247e1ed54f0b535143f1305261b98d80d18e", "locator": null, "note": "rows matched per function", "path": "crates/calc-func/witness-status.tsv", "repository": "DnaCalc" } ], "functions": [ "FUNC.GAMMA.DIST" ], "inferred": "The form is read from noise and scored in part: one stored stage of the prefactor or the sum is still missing, with a root mean square of about 0.8 of 2^-53 at small x. Non-integer shapes depend on GAMMALN below 3, which is not identified. An earlier reading of a chopped exponential in this function was this constant.", "kind": "excel-differs-from-math", "note_id": "FN-GAMMA.DIST-001", "open": [ "The missing stored stage of the prefactor or the sum.", "Shapes outside 2 to 6, and the upper arm of the distribution." ], "recorded_on": "2026-11-04", "repros": [ { "call": "GAMMA.DIST(0.010000000000003386, 2, 1, TRUE)", "correct": { "bits": "0x3f0a0a5081f5d23f", "display": "4.966791334029942e-05", "how": "mpmath regularized lower incomplete gamma at 800 bits, correctly rounded.", "kind": "number" }, "excel": { "bits": "0x3f0a0a5081f5d243", "display": "4.9667913340299445e-05", "kind": "number" }, "input_route": "cell-value2", "inputs": [ { "name": "x", "value": { "bits": "0x3f847ae147ae1c1b", "display": "0.010000000000003386", "kind": "number" } }, { "name": "alpha", "value": { "bits": "0x4000000000000000", "display": "2", "kind": "number" } }, { "name": "beta", "value": { "bits": "0x3ff0000000000000", "display": "1", "kind": "number" } }, { "name": "cumulative", "value": { "bits": null, "display": "TRUE", "kind": "logical" } } ], "recorded_in": "Excel 16.0 build 20527, Windows; bitness and CPU not recorded in this reading", "source": { "commit": "b5d84a80a5a4640eca9ce04f232c0bc115503f34", "locator": "GAMMA.DIST 3f847ae147ae1c1b 4000000000000000", "note": null, "path": "content/notes/evidence/2026-11-04/gamma-readings.tsv", "repository": "ExcelFunctionsHandbook" } }, { "call": "GAMMA.DIST(0.5000000000001851, 3, 1, TRUE)", "correct": { "bits": "0x3f8d774972ab2b89", "display": "0.014387677966984718", "how": "mpmath regularized lower incomplete gamma at 800 bits, correctly rounded.", "kind": "number" }, "excel": { "bits": "0x3f8d774972ab2b85", "display": "0.01438767796698471", "kind": "number" }, "input_route": "cell-value2", "inputs": [ { "name": "x", "value": { "bits": "0x3fe0000000000683", "display": "0.5000000000001851", "kind": "number" } }, { "name": "alpha", "value": { "bits": "0x4008000000000000", "display": "3", "kind": "number" } }, { "name": "beta", "value": { "bits": "0x3ff0000000000000", "display": "1", "kind": "number" } }, { "name": "cumulative", "value": { "bits": null, "display": "TRUE", "kind": "logical" } } ], "recorded_in": "Excel 16.0 build 20527, Windows; bitness and CPU not recorded in this reading", "source": { "commit": "b5d84a80a5a4640eca9ce04f232c0bc115503f34", "locator": "GAMMA.DIST 3fe0000000000683 4008000000000000", "note": null, "path": "content/notes/evidence/2026-11-04/gamma-readings.tsv", "repository": "ExcelFunctionsHandbook" } } ], "schema": "efh.function-note/v1", "statement": "With RN(v) meaning v rounded to 64 bits on the x87 unit and then stored to binary64, the series arm and the density of GAMMA.DIST at integer shape a from 2 to 6 share r = RN(exp(t - x - GAMMALN(a+1))), with ln x stored, t = RN(a*ln x), and the difference held in the 64-bit registers. The cumulative answer is RN(r*S) with S the stored forward sum, and the density RN(r*a)/x, the division last; with a scale, y = RN(x/beta) comes first. So GAMMALN's own error is in every answer: at shape 2 the constant is GAMMALN(3), one unit above stored ln 2 (FN-GAMMALN-001).", "status": "inferred", "title": "GAMMA.DIST's prefactor carries GAMMALN(a+1) in the exponent, with its error" } ], "schema": "efh.function-reference/v1", "section_order": [ "what-it-does", "try-it", "quirks", "how-to-implement", "version-differences", "alternatives" ], "sections": { "alternatives": "\n[GAMMADIST](FUNC.GAMMADIST.md) is the compatibility spelling. [GAMMA.INV](FUNC.GAMMA.INV.md) finds a quantile; [EXPON.DIST](FUNC.EXPON.DIST.md) covers shape one; [CHISQ.DIST](FUNC.CHISQ.DIST.md) selects shape n/2 and scale two. [GAMMA](FUNC.GAMMA.md) and [GAMMALN](FUNC.GAMMALN.md) evaluate related special functions rather than a CDF.\n\n\n\n\n", "how-to-implement": "\nUse D for ordinary binary64 storage and [R](../model/09-staged-arithmetic.md) for the stored 64-bit-significand operation.\n\nFor the current cumulative route, validate x and the two positive parameters, then store `y=D(x/beta)`:\n\n1. At alpha=0.5, compute the ordinary binary64 square root of y and pass it to ERF.PRECISE. Keep divide before square root. Do not silently replace this code’s root by a separately staged public SQRT call at unmeasured boundaries.\n2. At alpha=1, call EXPON.DIST with `(y,1,TRUE)`, retaining its corrected exponential-difference stages. Calling it with `(x,1/beta,TRUE)` changes a ratio into a stored reciprocal/product.\n3. For other shapes, evaluate the generic regularized lower gamma ratio `P(alpha,y)`. Integer alpha does not select a finite-sum shortcut in the current code.\n\nThe source retains 68 half-shape composition controls on scales one, two and four, and shape-one controls including non-dyadic scales three and five. Those observations constrain the named compositions, rather than every inner approximation or scale.\n\nFor the current density candidate at positive x and alpha other than one:\n\n```text\ny = R(x/beta)\nt = R(alpha*LN(y))\nG = stored GAMMALN(alpha+1)\nprefix = current exponential-of-difference candidate(t,y,G)\nratio = R(alpha/x)\ndensity = R(prefix*ratio)\n```\n\nThe prefix corresponds mathematically to `exp(t-y-G)`. Its producer and return stage are not fully identified; the current source retains a particular extended difference/exponential candidate. The independently stored alpha/x ratio and final multiplication are a separately measured consumer. Shapes one and x=0 retain a log-density fallback, rather than this branch.\n\nFor the cumulative example, y is exactly one; the CDF is about `1-exp(-1)`. Its density is a separate calculation. Do not transfer a supported CDF composition into the density merely because both come from the same distribution.\n\nThe notes retain earlier integer-shape prefactor findings and their scope. General gamma ratios, GAMMALN, the density prefix, endpoint publication and remaining report mismatches remain open. No published Handbook suite establishes whole-function agreement.\n\n\n", "quirks": "\nMicrosoft documents `#VALUE!` for nonnumeric x or shape/scale, and `#NUM!` for negative x or a nonpositive shape or scale. Beta is a scale, so replacing it by an exponential rate without taking a reciprocal changes the distribution.\n\nAt x=0 the mathematical density’s right-hand limit is zero for alpha>1, `1/beta` for alpha=1 and unbounded for smaller positive alpha. The current fallback produces zero in the first case but `#NUM!` for the other two. Those are code facts; a mathematical limit alone is not evidence of Excel’s actual boundary return.\n\nThe cumulative half-shape branch currently uses an ordinary binary64 square root after dividing by beta. Its retained public-composition controls do not settle every root precision boundary. Large finite parameters can also overflow an intermediate ratio; finite inputs do not by themselves prove a successful candidate result.\n\n\n", "sources": "\n- [Microsoft: GAMMA.DIST function](https://support.microsoft.com/en-us/excel/functions/gamma-dist-function), opened 7 November 2026; syntax, documented argument rules and errors.\n- [DnaCalc implementation](https://github.com/DnaCalc/DnaCalc/blob/da3ecc4fb19a3a5529708c311d0e8f05688ac165/crates/calc-func/src/ported/beta_gamma_stats_family.rs); implementation steps and the limits identified in the text.\n- [Current exponential difference candidate](https://github.com/DnaCalc/DnaCalc/blob/da3ecc4fb19a3a5529708c311d0e8f05688ac165/crates/calc-func/src/ported/excel_numeric/x87_chains.rs); `excel_exp_difference` and its resident difference.\n- [Remaining numerical failures](https://github.com/DnaCalc/DnaCalc/blob/da3ecc4fb19a3a5529708c311d0e8f05688ac165/docs/generated/problem-functions.md); counts keep their source scopes.\n\n", "try-it": "\n`=GAMMA.DIST(2,1,2,TRUE)` is about `0.6321205588`. The scale ratio is one, giving the exponential CDF `1-exp(-1)`.\n\n`=GAMMA.DIST(2,1,2,FALSE)` is about `0.1839397206`. That is a density, mathematically `exp(-1)/2`; it is not the probability of observing exactly the real-valued point two.\n\n\n", "version-differences": "The recorded environments below distinguish a reading from a same-input comparison. Counts describe captured inputs in each environment; overlapping captures and comparisons are not totals of unique worksheet calls.\n\n| Excel 16.0 build | Platform and CPU | Captured inputs |\n|---|---|---:|\n| 20131 | Windows, bitness not recorded, AMD Ryzen 7 4800H with Radeon Graphics | 77,653 |\n| 20430 | Windows, 64-bit, CPU not recorded | 194 |\n\nThe record contains no same-input comparison between these environments. Reading different inputs on different builds does not establish either a version change or cross-build agreement.\n\n[Recorded environments and comparison details](https://github.com/DnaCalc/ExcelFunctionsHandbook/blob/5dc75391aa5125ccc5abdb095da85950980d6fd8/content/evidence/environments/FUNC.GAMMA.DIST.json).", "what-it-does": "\n`GAMMA.DIST(x,alpha,beta,cumulative)` evaluates a gamma distribution with positive shape alpha and positive scale beta at nonnegative x. TRUE returns a cumulative probability; FALSE returns a density. Scale beta has the same units as x.\n\nShape one has the exponential distribution with rate `1/beta`. A shape `n/2` for positive integer n, with scale two, has the chi-square distribution with n degrees of freedom. These are mathematical relations; particular worksheet compositions still have their own stores.\n\n\n" }, "signature": { "display": "GAMMA.DIST(x, alpha, beta, cumulative)", "parameters": [ { "description": null, "name": "x", "optional": false, "repeats": false }, { "description": null, "name": "alpha", "optional": false, "repeats": false }, { "description": null, "name": "beta", "optional": false, "repeats": false }, { "description": null, "name": "cumulative", "optional": false, "repeats": false } ], "trailing_repeats": false }, "sources": { "metadata": { "admission": { "label": "implemented", "reason": null }, "arity": { "max": 4, "min": 4 }, "category": "Statistical functions", "classification": { "arg_preparation_profile": "ValuesOnlyPreAdapter", "coercion_lift_profile": "Custom", "determinism_class": "Deterministic", "error_collapse_profile": "None", "fec_dependency_profile": "None", "host_interaction_class": "None", "kernel_signature_class": "Custom", "rich_value_usage": "RichBlind", "surface_fec_dependency_profile": "None", "thread_safety_class": "SafePure", "volatility_class": "NonVolatile" }, "descriptions": { "long": null, "short": "Gamma distribution: the density or the cumulative probability", "short_source": "dna-calc-own" }, "docs": { "english_description": "Returns the gamma distribution", "microsoft_url": "https://support.microsoft.com/en-us/office/gamma-dist-function-9b6f1538-d11c-4d5f-8966-21f6a2201def" }, "entry_kind": "built_in_function", "localized_family_names": null, "localized_names": [ { "locale": "ar-SA", "name": "GAMMA.DIST" }, { "locale": "bg-BG", "name": "GAMMA.DIST" }, { "locale": "cs-CZ", "name": "GAMMA.DIST" }, { "locale": "da-DK", "name": "GaMMA.FORDELING" }, { "locale": "de-DE", "name": "GAMMA.DIST" }, { "locale": "el-GR", "name": "GAMMA.DIST" }, { "locale": "en-AU", "name": "GAMMA.DIST" }, { "locale": "en-GB", "name": "GAMMA.DIST" }, { "locale": "en-US", "name": "GAMMA.DIST" }, { "locale": "es-ES", "name": "DISTR.GAMMA.N" }, { "locale": "et-EE", "name": "GAMMA.DIST" }, { "locale": "fi-FI", "name": "GAMMA.JAKAUMA" }, { "locale": "fr-FR", "name": "LOI.GAMMA.N*" }, { "locale": "he-IL", "name": "GAMMA.DIST" }, { "locale": "hr-HR", "name": "GAMMA.DIST" }, { "locale": "hu-HU", "name": "GAMMA.ELOSZL" }, { "locale": "id-ID", "name": "GAMMA.DIST" }, { "locale": "it-IT", "name": "GAMMA.DIST" }, { "locale": "ja-JP", "name": "GAMMA.DIST" }, { "locale": "ko-KR", "name": "GAMMA.DIST" }, { "locale": "lt-LT", "name": "GAMMA.DIST" }, { "locale": "lv-LV", "name": "GAMMA.DIST" }, { "locale": "nb-NO", "name": "GAMMA.FORDELING" }, { "locale": "nl-NL", "name": "GAMMA.VERD.N" }, { "locale": "pl-PL", "name": "ROZKŁ.GAMMA" }, { "locale": "pt-BR", "name": "GAMMA.DIST" }, { "locale": "pt-PT", "name": "DIST.GAMA" }, { "locale": "ro-RO", "name": "GAMMA.DIST" }, { "locale": "ru-RU", "name": "ГАММА.РАСП" }, { "locale": "sk-SK", "name": "GAMMA.DIST" }, { "locale": "sl-SI", "name": "GAMMA.DIST" }, { "locale": "sr-Latn-RS", "name": "GAMMA.DIST" }, { "locale": "sv-SE", "name": "GAMMA.FÖRD" }, { "locale": "th-TH", "name": "GAMMA.DIST" }, { "locale": "tr-TR", "name": "GAMA.DAĞ" }, { "locale": "uk-UA", "name": "GAMMA.DIST" }, { "locale": "vi-VN", "name": "GAMMA.DIST" }, { "locale": "zh-CN", "name": "GAMMA.DIST" }, { "locale": "zh-HK", "name": "GAMMA.DIST" }, { "locale": "zh-TW", "name": "GAMMA.DIST" } ], "signature": { "display": "GAMMA.DIST(x, alpha, beta, cumulative)", "parameters": [ { "description": null, "name": "x", "optional": false, "repeats": false }, { "description": null, "name": "alpha", "optional": false, "repeats": false }, { "description": null, "name": "beta", "optional": false, "repeats": false }, { "description": null, "name": "cumulative", "optional": false, "repeats": false } ], "trailing_repeats": false }, "surface_name": "GAMMA.DIST", "version_marker": null, "xlcall": { "code": "495", "symbol": "xlfGamma_dist" } }, "microsoft": "https://support.microsoft.com/en-us/office/gamma-dist-function-9b6f1538-d11c-4d5f-8966-21f6a2201def" }, "try_it": { "formula": "=GAMMA.DIST(2,1,2,TRUE)" }, "url": "https://handbook.dnacalc.com/functions/GAMMA.DIST.html", "variants": [], "versions": [ { "behaviour": "GAMMA.DIST(0.010000000000003386, 2, 1, TRUE) returned 4.9667913340299445e-05 (0x3f0a0a5081f5d243). GAMMA.DIST(0.5000000000001851, 3, 1, TRUE) returned 0.01438767796698471 (0x3f8d774972ab2b85). Inputs were supplied through cells using Value2. These are controls from this environment, not a same-input comparison with another build.", "excel_build": "16.0 build 20527", "observations": [ { "call": "GAMMA.DIST(0.010000000000003386, 2, 1, TRUE)", "excel": { "bits": "0x3f0a0a5081f5d243", "display": "4.9667913340299445e-05", "kind": "number" }, "input_route": "cell-value2", "inputs": [ { "name": "x", "value": { "bits": "0x3f847ae147ae1c1b", "display": "0.010000000000003386", "kind": "number" } }, { "name": "alpha", "value": { "bits": "0x4000000000000000", "display": "2", "kind": "number" } }, { "name": "beta", "value": { "bits": "0x3ff0000000000000", "display": "1", "kind": "number" } }, { "name": "cumulative", "value": { "bits": null, "display": "TRUE", "kind": "logical" } } ], "recorded_in": "Excel 16.0 build 20527, Windows; bitness and CPU not recorded in this reading" }, { "call": "GAMMA.DIST(0.5000000000001851, 3, 1, TRUE)", "excel": { "bits": "0x3f8d774972ab2b85", "display": "0.01438767796698471", "kind": "number" }, "input_route": "cell-value2", "inputs": [ { "name": "x", "value": { "bits": "0x3fe0000000000683", "display": "0.5000000000001851", "kind": "number" } }, { "name": "alpha", "value": { "bits": "0x4008000000000000", "display": "3", "kind": "number" } }, { "name": "beta", "value": { "bits": "0x3ff0000000000000", "display": "1", "kind": "number" } }, { "name": "cumulative", "value": { "bits": null, "display": "TRUE", "kind": "logical" } } ], "recorded_in": "Excel 16.0 build 20527, Windows; bitness and CPU not recorded in this reading" } ], "platform": "Windows; bitness and CPU not recorded in this reading", "sources": [ "https://github.com/DnaCalc/ExcelFunctionsHandbook/blob/3345a87/content/notes/FN-GAMMA.DIST-001.json" ] }, { "behaviour": "77,653 captured inputs are indexed for this environment. Capture dates: 2026-07-16 to 2026-07-18. This is a reading count, not a comparison with another build.", "excel_build": "16.0 build 20131", "platform": "Windows, bitness not recorded, AMD Ryzen 7 4800H with Radeon Graphics", "sources": [ "https://github.com/DnaCalc/ExcelFunctionsHandbook/blob/5dc75391aa5125ccc5abdb095da85950980d6fd8/content/evidence/environments/FUNC.GAMMA.DIST.json" ] }, { "behaviour": "194 captured inputs are indexed for this environment. Capture dates: 2026-09-29 to 2026-09-29. This is a reading count, not a comparison with another build.", "excel_build": "16.0 build 20430", "platform": "Windows, 64-bit, CPU not recorded", "sources": [ "https://github.com/DnaCalc/ExcelFunctionsHandbook/blob/5dc75391aa5125ccc5abdb095da85950980d6fd8/content/evidence/environments/FUNC.GAMMA.DIST.json" ] } ] }