{ "category": "Math and trigonometry functions", "display_name": "CSCH", "entry_kind": "built_in_function", "eval_api": { "method": "POST", "url": "https://handbook.dnacalc.com/api/eval" }, "function_id": "FUNC.CSCH", "implementation_state": { "answer_bank": { "excel_builds": [ "20430" ], "larger_numeric": 0, "last_bit": 6, "matched": 33300, "other_mismatch": 0, "refused": 0, "rows": 33306, "unread": 0, "worst_ulp": 2 }, "detail": "Everything else matches (33,306 in the answer bank and 3 workbook cells). Last-bit means a finite number within 16 ULP of Excel's.", "dnacalc_commit": "9e1884369a678158dd82d3d74188daf986995e87", "in_catalogue": true, "state": "close", "state_label": "Close: last-bit deviations remain", "summary": "Close: 6 of 33,309 recorded answers differ in the last bits, worst 2 ULP", "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#L69-L69", "https://github.com/DnaCalc/DnaCalc/blob/9e1884369a678158dd82d3d74188daf986995e87/crates/calc-func/src/ported/csch.rs#L3-L11" ], "repository": "https://github.com/DnaCalc/DnaCalc", "shared_code": [ "catalogue/math::unary", "ported/excel_numeric::binary_with", "ported/excel_numeric::excel_expm1_internal", "ported/excel_numeric::expm1_internal_graph", "ported/excel_numeric::excel_x87_sub", "ported/excel_numeric::excel_x87_mul", "ported/excel_numeric::excel_x87_div", "ported/excel_numeric::excel_x87_recip", "ported/excel_numeric/x87::mul", "ported/excel_numeric/x87::sub", "ported/excel_numeric/x87::div", "ported/excel_numeric/x87::recip", "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/sinh::sinh_kernel" ], "source": "https://github.com/DnaCalc/DnaCalc/blob/9e1884369a678158dd82d3d74188daf986995e87/crates/calc-func/src/catalogue/math.rs#L69-L69", "state": "close", "suite": null, "summary": "Close: 6 of 33,309 recorded answers differ in the last bits, worst 2 ULP" } ], "json_url": "https://handbook.dnacalc.com/api/functions/FUNC.CSCH.json", "name": "CSCH", "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\n- **`SINH`** — the reciprocal, and the surface whose accuracy `CSCH` inherits. `SINH` has the\n classic cancellation problem near zero; see below.\n- **`SECH`** — the other hyperbolic reciprocal that decays to zero, but bounded above by 1 and\n with no pole. `SECH` is the well-behaved one; `CSCH` is not.\n- **[COTH](FUNC.COTH.md)** — the sibling from the same 2013 batch, sharing the pole at the\n origin, the documented `2^27` constraint, and the `arg_domain_guard=none` axis value. The two\n pages diverge from their documentation in the same way and should be read together.\n- **[COSH](FUNC.COSH.md)** — the identity `coth²x - csch²x = 1` ties this surface to the\n hyperbolic set; `csch x = 1/sinh x` and `sinh` is built from `exp`.\n- **`ASINH`** — there is no `ACSCH` on the worksheet surface; `ASINH(1/x)` is the substitute,\n and it is well conditioned, which makes it a good round-trip probe.\n- **[CSC](FUNC.CSC.md)** — the circular partner, `csch x = -i·csc(ix)`. Same `1/x` leading\n term; opposite behaviour at infinity.\n- **`IMCSCH`** — the complex version.\n\n\n", "how-to-implement": "\n### Numerical notes\n\n\n`CSCH` is `1/SINH`, and every hard question about it is really a question about `sinh`.\n\n**The cancellation, and the remedy.** Written as `(e^x - e^{-x})/2`, `sinh x` cancels\ncatastrophically for small `x`: the two exponentials are both close to 1, and their difference\nis close to `2x`, so the leading digits are lost. This is the textbook example of subtractive\ncancellation (A&S chapter 4.5; every numerical-analysis course). The standard remedy is\n`expm1`: with `t = expm1(x)`,\n\n sinh x = ( t + t/(t+1) ) / 2\n\nwhich never forms the difference of two near-equal quantities. fdlibm's `s_sinh.c` and Cephes'\n`sinh.c` both branch this way, using a polynomial in `x²` for the smallest arguments and the\n`expm1` form up to moderate size. Excel has no `EXPM1` on the worksheet surface, so a user who\nwrites `(EXP(x)-EXP(-x))/2` by hand will get the cancellation no matter how good `SINH` and\n`CSCH` are; the worksheet remedy is the series `x + x³/6 + x⁵/120`.\n\n**For `CSCH` specifically, the reciprocal reverses which end is dangerous.**\n\n- **Near zero**, `sinh` is small and its *relative* error is what propagates, so a stable\n `sinh` gives a stable `csch`. Better still, use the Laurent series `1/x - x/6 + 7x³/360`\n (A&S 4.5.66) directly below about `2^-13`: it avoids the division entirely and gives the\n deviation from `1/x` without cancellation.\n- **Underflow tail**: below the smallest subnormal the answer is `0`, gradually then abruptly.\n An implementation using `2·EXP(-x)` reaches that boundary smoothly; one using `1/SINH(x)`\n reaches it at whatever point `SINH` stopped being finite.\n\n**The one-line summary an implementer needs**: never compute `csch` as the reciprocal of a\nvalue that can overflow, and never compute `sinh` as a difference of exponentials.\n\n\n", "quirks": "\n### Result and edge cases\n\n\nReturns `Number`, never zero mathematically but zero by underflow for large arguments.\n\n- **`CSCH(0)`** — mathematically a pole. Microsoft's page states the magnitude constraint and\n the non-numeric condition and **says nothing about zero**. `#DIV/0!` follows by analogy with\n [COT](FUNC.COT.md), whose page does state it, but nothing establishes it here.\n- **Large `ABS(x)`** — the true value underflows to `0`. Whether Excel returns `0` or something\n else is unchecked; `0` is the mathematically correct rounded answer.\n- **Sign** — preserved: `CSCH` of a negative argument is negative, including in the underflow\n region, where the correct answer is negative zero if signed zeros are preserved and `0`\n otherwise. That distinction is invisible in most contexts and visible in `1/CSCH(x)`.\n- **Never zero for moderate arguments** — `csch` has no zeros, so a `0` returned for an\n argument below the underflow threshold would be a defect.\n- **Near zero** — `csch x ≈ 1/x`, so `CSCH` of a tiny argument is huge, overflowing only in the\n subnormal range.\n\n\n\n### Errors\n\n\nAs documented on Microsoft's `CSCH` page:\n\n| Error | Condition |\n|---|---|\n| `#NUM!` | `ABS(number)` is not less than `2^27` |\n| `#VALUE!` | `number` is non-numeric |\n\nThe zero case is not documented. Nothing else is documented.\n\n\n", "sources": "- Abramowitz & Stegun, chapter 4.5, in particular 4.5.66 — the `csch` expansion and the\n hyperbolic identity set.\n- fdlibm `s_sinh.c`, `s_expm1.c`, and Cephes `sinh.c` — the cancellation-free routes.\n- Handbook, [COTH](FUNC.COTH.md) — the sibling carrying the same documentation-versus-axis\n divergence; [Coercion and lifting](../model/02-coercion-and-lifting.md).\n- Handbook projections `data/functions/FUNC.CSCH.json` (the `arg_domain_guard=none` and\n `non_finite=allow` axis values) and `data/presence/FUNC.CSCH.json` (which records no\n discrepancy-catalogue, math-deviation or bug-stream mention for this surface).\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 | 36 |\n| 20430 | Windows, 64-bit, CPU not recorded | 33,306 |\n\n- XC-CSCH-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.CSCH.json).", "what-it-does": "\n`CSCH(number)` returns the hyperbolic cosecant.\n\n csch x = 1 / sinh x = 2 / (e^x - e^{-x})\n\nMicrosoft's page describes it as the hyperbolic cosecant of an angle specified in radians. That\nphrasing is worth a note: the argument of a hyperbolic function is not an angle in the circular\nsense and there are no radians involved — it is a hyperbolic angle, twice the area of the\ncorresponding hyperbolic sector. The wording appears to be inherited from the circular\nfunctions' documentation template. It changes nothing about the mathematics.\n\n| Property | Statement |\n|---|---|\n| Domain | all real `x` except `x = 0` |\n| Range | `(-∞, 0) ∪ (0, ∞)` — every nonzero real is attained exactly once |\n| Parity | odd: `csch(-x) = -csch(x)` |\n| Pole | one, simple, at `x = 0`, with residue 1 |\n| Laurent series at 0 | `csch x = 1/x - x/6 + 7x³/360 - …` (A&S 4.5.66) |\n| Asymptotics | `csch x ~ 2e^{-x}` as `x → +∞`: decays to zero, never reaching it |\n| Fundamental identity | `coth²x - csch²x = 1` |\n| Derivative | `d/dx csch x = -csch x · coth x` |\n| Relation to circular | `csch x = -i·csc(ix)` |\n| Monotone | strictly decreasing on `(-∞, 0)` and on `(0, ∞)` |\n\nThe shape is a single hyperbola-like curve on each side of the origin: unbounded at `0`,\ndecaying exponentially to `0` at `±∞`. Unlike its sibling [COTH](FUNC.COTH.md), which flattens\nonto `±1`, `CSCH` flattens onto `0` — so its large-argument behaviour is **underflow**, not\noverflow, and that is the whole difference in their numerical stories.\n\nThe practical thresholds: `csch x` falls below the smallest normal double once `x` is above\nabout 710, and below the smallest subnormal at about 745. The correct answer there is `0`, and\n`0` is representable, so nothing goes wrong.\n\n\n\n### Arguments\n\n\n| Argument | Meaning | Constraint (as documented) |\n|---|---|---|\n| `number` | The hyperbolic angle. Required. | `ABS(number) < 2^27` |\n\nAs on `COTH`, the constraint has no numerical motivation: `csch` needs no argument reduction,\nand its correct value at `2^27` is `0`. The bound reads like a documentation template applied\nacross the 2013 trigonometric batch. That reading is a hypothesis and is on the probe list.\n\nThe `non_finite=allow` value is, by contrast, entirely consistent with the mathematics here:\n`csch` of a large argument underflows to zero, which is finite, so no non-finite result arises\nfrom the value — only, potentially, from the pole.\n\n\n" }, "signature": { "display": "CSCH(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": "Custom", "rich_value_usage": "RichBlind", "surface_fec_dependency_profile": "RefOnly", "thread_safety_class": "SafePure", "volatility_class": "NonVolatile" }, "descriptions": { "long": null, "short": "Hyperbolic cosecant", "short_source": "dna-calc-own" }, "docs": { "english_description": "Returns the hyperbolic cosecant of an angle", "microsoft_url": "https://support.microsoft.com/en-us/office/csch-function-f58f2c22-eb75-4dd6-84f4-a503527f8eeb" }, "entry_kind": "built_in_function", "localized_family_names": null, "localized_names": [ { "locale": "ar-SA", "name": "CSCH" }, { "locale": "bg-BG", "name": "CSCH" }, { "locale": "cs-CZ", "name": "CSCH" }, { "locale": "da-DK", "name": "CSCH" }, { "locale": "de-DE", "name": "CSCH" }, { "locale": "el-GR", "name": "CSCH" }, { "locale": "en-AU", "name": "CSCH" }, { "locale": "en-GB", "name": "CSCH" }, { "locale": "en-US", "name": "CSCH" }, { "locale": "es-ES", "name": "CSCH" }, { "locale": "et-EE", "name": "CSCH" }, { "locale": "fi-FI", "name": "KOSEKH" }, { "locale": "fr-FR", "name": "CSCH*" }, { "locale": "he-IL", "name": "CSCH" }, { "locale": "hr-HR", "name": "CSCH" }, { "locale": "hu-HU", "name": "CSCH" }, { "locale": "id-ID", "name": "CSCH" }, { "locale": "it-IT", "name": "CSCH" }, { "locale": "ja-JP", "name": "CSCH" }, { "locale": "ko-KR", "name": "CSCH" }, { "locale": "lt-LT", "name": "CSCH" }, { "locale": "lv-LV", "name": "CSCH" }, { "locale": "nb-NO", "name": "CSCH" }, { "locale": "nl-NL", "name": "COSECH" }, { "locale": "pl-PL", "name": "CSCH" }, { "locale": "pt-BR", "name": "CSCH" }, { "locale": "pt-PT", "name": "CSCH" }, { "locale": "ro-RO", "name": "CSCH" }, { "locale": "ru-RU", "name": "CSCH" }, { "locale": "sk-SK", "name": "CSCH" }, { "locale": "sl-SI", "name": "CSCH" }, { "locale": "sr-Latn-RS", "name": "CSCH" }, { "locale": "sv-SE", "name": "CSCH" }, { "locale": "th-TH", "name": "CSCH" }, { "locale": "tr-TR", "name": "CSCH" }, { "locale": "uk-UA", "name": "CSCH" }, { "locale": "vi-VN", "name": "CSCH" }, { "locale": "zh-CN", "name": "CSCH" }, { "locale": "zh-HK", "name": "CSCH" }, { "locale": "zh-TW", "name": "CSCH" } ], "signature": { "display": "CSCH(number)", "parameters": [ { "description": null, "name": "number", "optional": false, "repeats": false } ], "trailing_repeats": false }, "surface_name": "CSCH", "version_marker": null, "xlcall": { "code": "553", "symbol": "xlfCsch" } }, "microsoft": "https://support.microsoft.com/en-us/office/csch-function-f58f2c22-eb75-4dd6-84f4-a503527f8eeb" }, "try_it": { "formula": "=CSCH(1)" }, "url": "https://handbook.dnacalc.com/functions/CSCH.html", "variants": [], "versions": [ { "behaviour": "36 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.CSCH.json" ] }, { "behaviour": "33,306 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.CSCH.json" ] } ] }