{ "category": "Math and trigonometry functions", "display_name": "ACOT", "entry_kind": "built_in_function", "eval_api": { "method": "POST", "url": "https://handbook.dnacalc.com/api/eval" }, "function_id": "FUNC.ACOT", "implementation_state": { "answer_bank": { "excel_builds": [ "20430" ], "larger_numeric": 0, "last_bit": 0, "matched": 16924, "other_mismatch": 0, "refused": 0, "rows": 16924, "unread": 0, "worst_ulp": 0 }, "detail": "Recorded: 16,924 in the answer bank and 3 workbook cells.", "dnacalc_commit": "9e1884369a678158dd82d3d74188daf986995e87", "in_catalogue": true, "state": "matches", "state_label": "Matches every recorded Excel answer", "summary": "Matches all 16,927 recorded answers", "workbook_corpus": { "cells": 3, "matched": 3 } }, "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/math.rs#L46-L46", "https://github.com/DnaCalc/DnaCalc/blob/9e1884369a678158dd82d3d74188daf986995e87/crates/calc-func/src/ported/acot.rs#L3-L22" ], "repository": "https://github.com/DnaCalc/DnaCalc", "shared_code": [ "catalogue/math::unary", "ported/excel_numeric::atan_with", "ported/excel_numeric::atan_reduced_with", "ported/excel_numeric::binary_with", "ported/excel_numeric::excel_x87_recip", "ported/excel_numeric::excel_atan_reduced", "ported/excel_numeric/soft87::pack_raw", "ported/excel_numeric/soft87::zero", "ported/excel_numeric/soft87::inf", "ported/excel_numeric/soft87::pack", "ported/excel_numeric/soft87::ext_pi", "ported/excel_numeric/soft87::ext_one", "ported/excel_numeric/x87::atan", "ported/excel_numeric/x87::atan_reduced", "ported/excel_numeric/x87::recip" ], "source": "https://github.com/DnaCalc/DnaCalc/blob/9e1884369a678158dd82d3d74188daf986995e87/crates/calc-func/src/catalogue/math.rs#L46-L46", "state": "matches", "suite": null, "summary": "Matches all 16,927 recorded answers" } ], "json_url": "https://handbook.dnacalc.com/api/functions/FUNC.ACOT.json", "name": "ACOT", "notes": [], "schema": "efh.function-reference/v1", "section_order": [ "what-it-does", "try-it", "quirks", "how-to-implement", "version-differences", "alternatives" ], "sections": { "alternatives": "\n### Relationships\n\n- **`COT`** — the forward function. `COT(ACOT(x)) = x` to rounding; `ACOT(COT(t))` recovers `t`\n only for `t` in `(0, pi)`.\n- **`ACOTH`** — the hyperbolic namesake, and a genuinely different shape: its domain excludes\n `[-1, 1]` entirely, where `ACOT` accepts everything.\n- **`ACOS`** — shares the stated range `0` to `pi`, and shares nothing else. The discriminator is\n the domain: `ACOS` takes `[-1, 1]`, `ACOT` takes all of `R`. Two surfaces with the same stated\n range are a live confusion, and this is the pair.\n- **`ATAN2`** — the four-quadrant route. `ACOT(x)` is `ATAN2(x, 1)` in Excel's `x`-first argument\n order, and that formulation is better behaved at large `|x|` than the subtraction is.\n- **Confused with**: `1/ATAN(x)`, which is not any inverse cotangent; and `ATAN(1/x)`, which is\n the *other* branch convention and differs by `pi` for negative arguments.\n\n\n", "how-to-implement": "\n### Numerical notes\n\n\n`ACOT` is analytically the tamest function on this batch of pages — bounded derivative, no\nsingularities, no domain edge — and it has one sharp numerical problem, which comes entirely from\nhow it is usually computed.\n\n**Catastrophic cancellation in `pi/2 - atan(x)`.** The identity is exact in mathematics. In binary64\nit is not: as `x` grows, `atan(x)` approaches `pi/2`, and the subtraction of two nearly equal\nquantities destroys the relative accuracy of a result that is itself near zero. The failure is\nprogressive and total:\n\n- The correctly rounded `atan(x)` becomes exactly the double nearest `pi/2` once `x` exceeds\n roughly `2^53` — beyond which the subtraction returns a fixed, tiny residue and then, past\n another threshold, exactly zero.\n- The true `acot(x)` at those arguments is approximately `1/x`, a perfectly representable positive\n number all the way down to the subnormal floor.\n- So there is a very wide band of arguments — everything above about `1e16`, which is more than\n 290 decades of the double range — on which `pi/2 - atan(x)` has no correct significant digits\n at all, and eventually returns a value outside the function's own open range.\n\nThis is the mirror image of the `ATAN` situation: `ATAN` is accurate for large `x` because its\nanswer is large; `ACOT`'s answer is small there, and small answers reached by subtraction are\nwhere relative accuracy goes to die.\n\n**The standard remedy** is the reciprocal reduction, and it is one line:\n\n acot(x) = atan(1/x) for x >= 1 (result in (0, pi/4])\n acot(x) = pi + atan(1/x) for x <= -1 (result in [3pi/4, pi))\n acot(x) = pi/2 - atan(x) for |x| < 1\n\nInside the unit interval the subtraction is harmless — both terms are of order one and the answer\nis of order one. Outside it, `1/x` is small and `atan` of a small argument is accurate to the last\nbit, so `acot(x)` inherits full relative accuracy right down to the subnormals. Cody & Waite give\nthis reduction for the `ATAN`/`ACOT` pair; it is the same trick, run the other way, that makes\n`ATAN` accurate for large arguments.\n\n**Cost of the remedy**: one division and one comparison. There is no accuracy trade-off, no extra\npolynomial, and no branch subtlety beyond getting the `x <= -1` offset right.\n\n**A self-test without an oracle.** `TAN(ACOT(x)) * x` should be `1`, and `ACOT(x) + ACOT(-x)`\nshould be `PI()`. Sweeping the first across many decades of `x` shows the exact point where the\nsubtraction stops carrying information, without needing any high-precision reference.\n\n\n", "quirks": "\n### Result and edge cases\n\n\nReturns `Number` — an angle in radians.\n\n- **Zero** returns `pi/2`. This is the case that separates the two branch conventions, and it is\n the one worth checking first in any implementation.\n- **Negative arguments** return values in `(pi/2, pi)` on the documented convention — never\n negative. A negative result from `ACOT` would be evidence of the other convention.\n- **Very large negative arguments** produce results approaching `pi` from below, and the\n corresponding question is whether they saturate at exactly `pi`.\n- **Subnormal arguments** return `pi/2`, indistinguishably from zero.\n- **Arrays** lift elementwise.\n\n### Version\n\n`ACOT` is one of the trigonometric surfaces added in Excel 2013, alongside `ACOTH`, `SEC`, `CSC`,\n`COT` and their hyperbolic partners. Workbooks that must open in earlier versions substitute\n`PI()/2 - ATAN(x)`, which is the same function on the documented convention.\n\n\n\n### Errors\n\n\n| Error | Condition | Source |\n|---|---|---|\n| `#VALUE!` | The argument does not convert to a number | Shared coercion rules |\n| propagated | An error value in the argument is returned unchanged | Shared coercion rules |\n\n", "sources": "- Microsoft Learn, \"WorksheetFunction.Acot method (Excel)\" —\n (retrieved: the\n description and the statement that the result is in radians in the range 0 to pi).\n- Abramowitz & Stegun, *Handbook of Mathematical Functions*, chapter 4 section 4.4 — inverse\n circular functions and the arccotangent branch conventions.\n- Kahan, *Branch Cuts for Complex Elementary Functions* — why the arccotangent branch is a decision\n with consequences rather than a notational preference.\n- Cody & Waite, *Software Manual for the Elementary Functions*, chapter on `ATAN`/`ACOT` — the\n reciprocal reduction.\n- Handbook projections `data/functions/FUNC.ACOT.json` (arity, `NumToNum` kernel signature,\n `real_result_policy` with `arg_domain_guard=none;non_finite=allow`) and\n `data/presence/FUNC.ACOT.json` (implementing module; no discrepancy, math-deviation or defect\n entries).\n- Handbook [Coercion and lifting](../model/02-coercion-and-lifting.md).\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| 19929 | Windows, bitness not recorded, CPU not recorded | 45 |\n| 20430 | Windows, 64-bit, CPU not recorded | 16,924 |\n\n- XC-ACOT-001: build 19929 versus build 20430; 2 same-input pairs, 0 differing answers. Compared axes: excel-build. Numeric answers were compared by their recorded binary64 bits; other answers by typed value.\n\nAgreement on these compared inputs does not establish agreement on unmeasured inputs.\n\n[Recorded environments and comparison details](https://github.com/DnaCalc/ExcelFunctionsHandbook/blob/5dc75391aa5125ccc5abdb095da85950980d6fd8/content/evidence/environments/FUNC.ACOT.json).", "what-it-does": "\n`ACOT(number)` is the inverse cotangent: the angle whose cotangent is *number*.\n\n acot: R -> (0, pi), cot(acot x) = x\n\n- **Domain**: all real numbers. Unlike `ACOS` and `ASIN`, there is no restriction — `cot` maps\n each period onto the whole real line.\n- **Range**: the open interval `(0, pi)`. Microsoft's Learn reference states the range as\n 0 to pi. Mathematically the endpoints are approached and never reached: `acot(x) -> 0+` as\n `x -> +infinity` and `acot(x) -> pi-` as `x -> -infinity`.\n- **Monotonicity**: strictly decreasing on the whole line. `acot(-1) = 3pi/4`, `acot(0) = pi/2`,\n `acot(1) = pi/4`.\n- **Reflection**: `acot(-x) = pi - acot(x)`. Antisymmetric about `(0, pi/2)`, like `ACOS`.\n- **Derivative**: `d/dx acot x = -1/(1 + x^2)` — bounded, never zero, and the negative of\n `ATAN`'s. `ACOT` is smooth and well conditioned everywhere; the difficulties on this page are\n about representation, not about conditioning.\n- **Asymptotics**: `acot(x) = 1/x - 1/(3x^3) + 1/(5x^5) - ...` for large `|x| > 1`; near zero,\n `acot(x) = pi/2 - x + x^3/3 - ...`.\n\n### The branch convention, which is the whole point\n\nThere are two incompatible conventions for arccotangent in the literature, and Excel picks one:\n\n| Convention | Range | Behaviour at `x -> 0-` |\n|---|---|---|\n| **Continuous** (Excel, per the Learn reference) | `(0, pi)` | continuous through zero: `acot(0) = pi/2` |\n| **Odd**, i.e. `atan(1/x)` | `[-pi/2, 0) ∪ (0, pi/2]` | jumps by `pi` across zero |\n\nOn the **continuous** convention the identity is\n\n acot(x) = pi/2 - atan(x) for every real x\n\nwhich is smooth everywhere. On the **odd** convention `acot(x) = atan(1/x)`, which is undefined at\nzero and discontinuous across it. The two agree for `x > 0` and differ by exactly `pi` for\n`x < 0`.\n\nThis is not a pedantic distinction. `ACOT(-1)` is `3pi/4` on Excel's stated convention and\n`-pi/4` on the other, and a formula ported from a language whose `acot` is `atan(1/x)` will\nsilently be wrong by `pi` on every negative input. Abramowitz & Stegun record both conventions;\nKahan's branch-cut paper is the standard argument for treating the choice as a real decision.\n\n\n\n### Arguments\n\n\n| Argument | Meaning | Default |\n|---|---|---|\n| `number` | The cotangent of the angle you want. Required. | — |\n\nThat description is Microsoft's, from the Learn reference: \"the cotangent of the angle that you\nwant\". No constraint on the argument is documented, and none is needed.\n\nThe misunderstood position is, again, not the slot: it is that `number` is a *cotangent*, not an\nangle, and that a reader who wants \"the angle whose tangent is x\" wants `ATAN`.\n\n\n" }, "signature": { "display": "ACOT(number)", "parameters": [ { "description": null, "name": "number", "optional": false, "repeats": false } ], "trailing_repeats": false }, "sources": { "metadata": { "admission": { "label": "implemented", "reason": null }, "arity": { "max": 1, "min": 1 }, "category": "Math and trigonometry functions", "classification": { "arg_preparation_profile": "ValuesOnlyPreAdapter", "coercion_lift_profile": "UnaryNumericScalarOrArrayElementwise", "determinism_class": "Deterministic", "error_collapse_profile": "None", "fec_dependency_profile": "None", "host_interaction_class": "None", "kernel_signature_class": "NumToNum", "rich_value_usage": "RichBlind", "surface_fec_dependency_profile": "RefOnly", "thread_safety_class": "SafePure", "volatility_class": "NonVolatile" }, "descriptions": { "long": null, "short": "Angle in radians, from 0 to pi, whose cotangent is the given number", "short_source": "dna-calc-own" }, "docs": { "english_description": "Returns the arccotangent of a number", "microsoft_url": "https://support.microsoft.com/en-us/office/acot-function-dc7e5008-fe6b-402e-bdd6-2eea8383d905" }, "entry_kind": "built_in_function", "localized_family_names": null, "localized_names": [ { "locale": "ar-SA", "name": "ACOT" }, { "locale": "bg-BG", "name": "ACOT" }, { "locale": "cs-CZ", "name": "ACOT" }, { "locale": "da-DK", "name": "ARCCOT" }, { "locale": "de-DE", "name": "ACOT" }, { "locale": "el-GR", "name": "ACOT" }, { "locale": "en-AU", "name": "ACOT" }, { "locale": "en-GB", "name": "ACOT" }, { "locale": "en-US", "name": "ACOT" }, { "locale": "es-ES", "name": "ACOT" }, { "locale": "et-EE", "name": "ACOT" }, { "locale": "fi-FI", "name": "ACOT" }, { "locale": "fr-FR", "name": "ACOT" }, { "locale": "he-IL", "name": "ACOT" }, { "locale": "hr-HR", "name": "ACOT" }, { "locale": "hu-HU", "name": "ARCCOT" }, { "locale": "id-ID", "name": "ACOT" }, { "locale": "it-IT", "name": "ACOT" }, { "locale": "ja-JP", "name": "ACOT" }, { "locale": "ko-KR", "name": "ACOT" }, { "locale": "lt-LT", "name": "ACOT" }, { "locale": "lv-LV", "name": "ACOT" }, { "locale": "nb-NO", "name": "ACOT" }, { "locale": "nl-NL", "name": "BOOGCOT" }, { "locale": "pl-PL", "name": "ACOT" }, { "locale": "pt-BR", "name": "ACOT" }, { "locale": "pt-PT", "name": "ACOT" }, { "locale": "ro-RO", "name": "ACOT" }, { "locale": "ru-RU", "name": "ACOT" }, { "locale": "sk-SK", "name": "ACOT" }, { "locale": "sl-SI", "name": "ACOT" }, { "locale": "sr-Latn-RS", "name": "ACOT" }, { "locale": "sv-SE", "name": "ARCCOT" }, { "locale": "th-TH", "name": "ACOT" }, { "locale": "tr-TR", "name": "ACOT" }, { "locale": "uk-UA", "name": "ACOT" }, { "locale": "vi-VN", "name": "ACOT" }, { "locale": "zh-CN", "name": "ACOT" }, { "locale": "zh-HK", "name": "ACOT" }, { "locale": "zh-TW", "name": "ACOT" } ], "signature": { "display": "ACOT(number)", "parameters": [ { "description": null, "name": "number", "optional": false, "repeats": false } ], "trailing_repeats": false }, "surface_name": "ACOT", "version_marker": null, "xlcall": { "code": "548", "symbol": "xlfAcot" } }, "microsoft": "https://support.microsoft.com/en-us/office/acot-function-dc7e5008-fe6b-402e-bdd6-2eea8383d905" }, "try_it": { "formula": "=ACOT(-1)" }, "url": "https://handbook.dnacalc.com/functions/ACOT.html", "variants": [], "versions": [ { "behaviour": "45 captured inputs are indexed for this environment. Capture dates are not recorded. This is a reading count, not a comparison with another build.", "excel_build": "16.0 build 19929", "platform": "Windows, bitness not recorded, CPU not recorded", "sources": [ "https://github.com/DnaCalc/ExcelFunctionsHandbook/blob/5dc75391aa5125ccc5abdb095da85950980d6fd8/content/evidence/environments/FUNC.ACOT.json" ] }, { "behaviour": "16,924 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.ACOT.json" ] } ] }