{
  "category": "Statistical functions",
  "display_name": "FISHER",
  "entry_kind": "built_in_function",
  "eval_api": {
    "method": "POST",
    "url": "https://handbook.dnacalc.com/api/eval"
  },
  "function_id": "FUNC.FISHER",
  "implementation_state": {
    "answer_bank": {
      "excel_builds": [
        "20430"
      ],
      "larger_numeric": 0,
      "last_bit": 0,
      "matched": 34166,
      "other_mismatch": 0,
      "refused": 0,
      "rows": 34166,
      "unread": 0,
      "worst_ulp": 0
    },
    "detail": "Recorded: 34,166 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 34,169 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/statistics.rs#L406-L408",
        "https://github.com/DnaCalc/DnaCalc/blob/9e1884369a678158dd82d3d74188daf986995e87/crates/calc-func/src/ported/fisher_fn.rs#L3-L16"
      ],
      "repository": "https://github.com/DnaCalc/DnaCalc",
      "shared_code": [
        "ported/excel_numeric::binary_with",
        "ported/excel_numeric::excel_x87_add",
        "ported/excel_numeric::excel_x87_sub",
        "ported/excel_numeric::excel_x87_div",
        "ported/excel_numeric/x87::add",
        "ported/excel_numeric/x87::sub",
        "ported/excel_numeric/x87::div"
      ],
      "source": "https://github.com/DnaCalc/DnaCalc/blob/9e1884369a678158dd82d3d74188daf986995e87/crates/calc-func/src/catalogue/statistics.rs#L406-L408",
      "state": "matches",
      "suite": null,
      "summary": "Matches all 34,169 recorded answers"
    }
  ],
  "json_url": "https://handbook.dnacalc.com/api/functions/FUNC.FISHER.json",
  "name": "FISHER",
  "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- **[FISHERINV](FUNC.FISHERINV.md)** — the inverse, tanh. Microsoft states the round trip\n  directly: if y = FISHER(x) then FISHERINV(y) = x. The pair is asymmetric in a way that\n  matters to implementers: `FISHER` has a bounded domain and a documented `#NUM!` guard,\n  while `FISHERINV` has an unbounded domain and no documented `#NUM!` condition at all.\n- **`ATANH`** — the Math-category spelling of the same mathematical function. Excel publishes\n  both. They are documented to compute the same thing; whether they compute it with the same\n  code, and therefore return the same bits, is not established here and would need evidence.\n  Treating `FISHER` and `ATANH` as interchangeable is an assumption, not a fact.\n- **`CORREL` / `PEARSON`** — the usual producers of the r that gets fed to `FISHER`.\n- **`NORM.S.INV`, `NORM.S.DIST`** — the usual next step: the transformed value is compared\n  against a standard normal quantile, and `FISHERINV` maps the interval endpoints back.\n- **`TANH` / `ATANH`** — the hyperbolic pair `FISHER`/`FISHERINV` shadows.\n\n\n",
    "how-to-implement": "\n### Numerical notes\n\n\nThe interesting part of `FISHER` is that the textbook formula is the wrong way to evaluate it.\n\n**Cancellation near zero.** Written as ½·ln((1 + x)/(1 − x)), the quotient is a number close to\n1 for small x, and taking a logarithm of something close to 1 discards the leading significant\ndigits: the information about x lives in the difference between the quotient and 1, and that\ndifference has already been rounded away by the division. The accurate rearrangement is\n\n    artanh(x) = ½ · log1p( 2x / (1 − x) )\n\nwhich keeps the small quantity small and hands it to a routine designed for arguments near 1.\nThis is what fdlibm's `atanh` does, and it is the standard treatment in the literature — the\nsame cancellation lesson that governs `LN(1+x)` throughout the Handbook. Cephes and Boost use\nthe equivalent form; Numerical Recipes discusses the general pattern under the heading of\navoiding subtractive cancellation.\n\n**Overflow is not the failure mode; loss of the domain is.** As |x| → 1 the quotient grows, but\nonly to about 2/(1 − |x|), and the logarithm tames it, so a double-precision evaluation stays\nfinite until x is within a rounding of ±1. The practical hazard is the other side: 1 − x is\ncomputed exactly only when x is not extremely close to 1 (Sterbenz), and once x is within half\nan ulp of 1 the subtraction returns the rounded gap and the result is dominated by that single\nrounding.\n\n**What a careful implementation does.** Use the `log1p` form; return x unchanged (or x + x³/3)\nbelow the threshold where the series is indistinguishable from the function at working\nprecision; preserve the sign of zero; and decide deliberately whether the endpoints raise or\nsaturate, since Excel's documented answer there is an error rather than an infinity.\n\n\n",
    "quirks": "\n### Result and edge cases\n\n\nReturns `Number`.\n\n- **At the origin**, artanh(0) = 0, and 0 is exactly representable, so an implementation that\n  gets anything other than a signed zero here is doing something wrong.\n- **Approaching the poles**, the true value diverges. The documented behaviour is not a\n  saturating value and not an infinity: it is `#NUM!` for the whole closed complement\n  |x| ≥ 1, which means the endpoints themselves error rather than returning ±∞.\n- **Very small x**, including subnormals, should return x itself to within a rounding, because\n  artanh(x) = x + O(x³).\n\n\n\n### Errors\n\n\nAs documented by Microsoft:\n\n| Error | Condition |\n|---|---|\n| `#VALUE!` | `x` is nonnumeric |\n| `#NUM!` | `x` ≤ −1, or `x` ≥ 1 |\n\nThe `#NUM!` condition is stated with non-strict inequalities on both sides, so both endpoints\nare errors. Error values arriving in the argument propagate under the shared coercion\ndiscipline; the Handbook has not observed any of this in Excel itself.\n\n\n",
    "sources": "- Abramowitz & Stegun, *Handbook of Mathematical Functions*, chapter 4 — the inverse\n  hyperbolic tangent, its series, and its relation to the logarithm.\n- fdlibm `s_atanh.c`; Cephes; Boost.Math — the `log1p` rearrangement named under\n  **Numerical notes**.\n- Handbook, [The value universe](../model/01-value-universe.md) and\n  [Coercion and lifting](../model/02-coercion-and-lifting.md).\n- Handbook projections `data/functions/FUNC.FISHER.json` (arity, unary numeric scalar-or-array\n  lift profile, deterministic and non-volatile classification) and\n  `data/presence/FUNC.FISHER.json` (implementing module, no sibling surfaces).\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| 19929 | Windows, bitness not recorded, CPU not recorded | 21 |\n| 20430 | Windows, 64-bit, CPU not recorded | 34,166 |\n\n- XC-FISHER-001: build 19929 versus build 20430; 3 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.FISHER.json).",
    "what-it-does": "\n`FISHER(x)` is the inverse hyperbolic tangent:\n\n    FISHER(x)  =  artanh(x)  =  ½ · ln( (1 + x) / (1 − x) )\n\n- **Domain:** the open interval −1 < x < 1.\n- **Range:** the whole real line.\n- **Poles:** logarithmic singularities at x = −1 and x = +1, where the function tends to −∞\n  and +∞ respectively. There is no branch cut inside the domain.\n- **Symmetry:** odd. artanh(−x) = −artanh(x), and artanh(0) = 0 exactly.\n- **Derivative:** 1 / (1 − x²), which is 1 at the origin and grows without bound at the ends.\n- **Limiting form near zero:** artanh(x) = x + x³/3 + x⁵/5 + … , so for small x the function\n  is x to first order.\n\nMicrosoft's documentation states the purpose rather than the formula in text: the\ntransformation \"produces a function that is normally distributed rather than skewed\", and it\nis for hypothesis testing on the correlation coefficient. That is the statistical reason the\nfunction exists. Pearson's sample correlation r is confined to [−1, 1] and its sampling\ndistribution is badly skewed near the ends; artanh(r) is approximately normal with variance\n1/(n − 3) more or less independently of the true correlation, which is what makes confidence\nintervals for a correlation tractable. This is R. A. Fisher's variance-stabilizing\nz-transformation, and the function is a spreadsheet spelling of it.\n\nThe mathematics and the statistics are separable. `FISHER` is artanh; the normal-approximation\nstory is why Excel ships it under a statistician's name rather than as `ATANH` — which Excel\nalso ships, in the Math category. See **Relationships**.\n\n\n\n### Arguments\n\n\n| Argument | Meaning | Default |\n|---|---|---|\n| `x` | The value to transform. Required. | — |\n\nThe trap in that sentence is worth naming: **`TRUE` converts to 1, and 1 is outside the\ndomain.** `FISHER(TRUE)` is therefore a domain failure rather than a value, and so is\n`FISHER(\"1\")`. Nothing in the documentation warns about this, because the documentation\ndiscusses the domain in terms of x and the coercion in terms of \"nonnumeric\".\n\n\n"
  },
  "signature": {
    "display": "FISHER(x)",
    "parameters": [
      {
        "description": null,
        "name": "x",
        "optional": false,
        "repeats": false
      }
    ],
    "trailing_repeats": false
  },
  "sources": {
    "metadata": {
      "admission": {
        "label": "implemented",
        "reason": null
      },
      "arity": {
        "max": 1,
        "min": 1
      },
      "category": "Statistical 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": "Fisher transformation of a correlation, artanh(x)",
        "short_source": "dna-calc-own"
      },
      "docs": {
        "english_description": "Returns the Fisher transformation",
        "microsoft_url": "https://support.microsoft.com/en-us/office/fisher-function-d656523c-5076-4f95-b87b-7741bf236c69"
      },
      "entry_kind": "built_in_function",
      "localized_family_names": null,
      "localized_names": [
        {
          "locale": "ar-SA",
          "name": "FISHER"
        },
        {
          "locale": "bg-BG",
          "name": "FISHER"
        },
        {
          "locale": "cs-CZ",
          "name": "FISHER"
        },
        {
          "locale": "da-DK",
          "name": "FISHER"
        },
        {
          "locale": "de-DE",
          "name": "FISHER"
        },
        {
          "locale": "el-GR",
          "name": "FISHER"
        },
        {
          "locale": "en-AU",
          "name": "FISHER"
        },
        {
          "locale": "en-GB",
          "name": "FISHER"
        },
        {
          "locale": "en-US",
          "name": "FISHER"
        },
        {
          "locale": "es-ES",
          "name": "FISHER"
        },
        {
          "locale": "et-EE",
          "name": "FISHER"
        },
        {
          "locale": "fi-FI",
          "name": "FISHER"
        },
        {
          "locale": "fr-FR",
          "name": "FISHER*"
        },
        {
          "locale": "he-IL",
          "name": "FISHER"
        },
        {
          "locale": "hr-HR",
          "name": "FISHER"
        },
        {
          "locale": "hu-HU",
          "name": "FISHER"
        },
        {
          "locale": "id-ID",
          "name": "FISHER"
        },
        {
          "locale": "it-IT",
          "name": "FISHER"
        },
        {
          "locale": "ja-JP",
          "name": "FISHER"
        },
        {
          "locale": "ko-KR",
          "name": "FISHER"
        },
        {
          "locale": "lt-LT",
          "name": "FISHER"
        },
        {
          "locale": "lv-LV",
          "name": "FISHER"
        },
        {
          "locale": "nb-NO",
          "name": "FISHER"
        },
        {
          "locale": "nl-NL",
          "name": "FISHER"
        },
        {
          "locale": "pl-PL",
          "name": "ROZKŁAD.FISHER"
        },
        {
          "locale": "pt-BR",
          "name": "FISHER"
        },
        {
          "locale": "pt-PT",
          "name": "FISHER"
        },
        {
          "locale": "ro-RO",
          "name": "FISHER"
        },
        {
          "locale": "ru-RU",
          "name": "ФИШЕР"
        },
        {
          "locale": "sk-SK",
          "name": "FISHER"
        },
        {
          "locale": "sl-SI",
          "name": "FISHER"
        },
        {
          "locale": "sr-Latn-RS",
          "name": "FISHER"
        },
        {
          "locale": "sv-SE",
          "name": "FISHER"
        },
        {
          "locale": "th-TH",
          "name": "FISHER"
        },
        {
          "locale": "tr-TR",
          "name": "FISHER işlevi"
        },
        {
          "locale": "uk-UA",
          "name": "FISHER"
        },
        {
          "locale": "vi-VN",
          "name": "FISHER"
        },
        {
          "locale": "zh-CN",
          "name": "FISHER"
        },
        {
          "locale": "zh-HK",
          "name": "FISHER"
        },
        {
          "locale": "zh-TW",
          "name": "FISHER"
        }
      ],
      "signature": {
        "display": "FISHER(x)",
        "parameters": [
          {
            "description": null,
            "name": "x",
            "optional": false,
            "repeats": false
          }
        ],
        "trailing_repeats": false
      },
      "surface_name": "FISHER",
      "version_marker": null,
      "xlcall": {
        "code": "283",
        "symbol": "xlfFisher"
      }
    },
    "microsoft": "https://support.microsoft.com/en-us/office/fisher-function-d656523c-5076-4f95-b87b-7741bf236c69"
  },
  "try_it": {
    "formula": "=FISHER(0.5)"
  },
  "url": "https://handbook.dnacalc.com/functions/FISHER.html",
  "variants": [],
  "versions": [
    {
      "behaviour": "21 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.FISHER.json"
      ]
    },
    {
      "behaviour": "34,166 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.FISHER.json"
      ]
    }
  ]
}