Hyperbolic functions and their inverses: precision from the condition number, as in WMA by tombouctou · Pull Request #2056 · Mathics3/mathics-core

tombouctou · 2026-10-19T20:27:53Z

Part of step 6 of the outline in #2047. On top of the inverse trigonometric PR, #2055: this diff is against its branch.

The hyperbolic functions and their inverses get their precision from the condition number Abs[x f'[x] / f[x]], as WMA does.

One WMA detail: for Tanh and Coth with Abs[x] < 1, WMA uses the condition number of Sinh[x] / Cosh[x] with both errors added, x (Tanh[x] + Coth[x]). From Abs[x] >= 1 on it uses the true one. I checked at 0.05, 0.3, 0.5, 0.9, 0.99, 1., 1.01, 1.1, 1.2, 1.4, 2. and 5.; the switch is exactly at 1. This PR follows it, so Precision[Tanh[0.3`20]] is 19.9519, as in WMA, not 20.0258. For large x, x (1 - Tanh[x]^2) / Tanh[x] would cancel, so it is computed as 2 x / Sinh[2 x].

Input	before	this PR	WMA 15
Precision[Sinh[25.`20]]	`20.`	`18.6021`	`18.6021`
Precision[Tanh[0.3`20]]	`20.`	`19.9519`	`19.9519`
Precision[Tanh[25.`20]]	`20.`	`39.7147`	`39.7147`
Precision[ArcTanh[0.99`20]]	`20.`	`18.7259`	`18.7259`
Precision[ArcSech[0.99`20]]	`20.`	`18.3018`	`18.3018`

Zeros

With the rule from #2054 for an inexact zero argument, as in WMA 15: Sinh, Tanh, ArcSinh and ArcTanh of 020are020, and Cosh[020]andSech[020] are 1 with precision 40.301. Csch of 0. or 020`` is now ComplexInfinity, as `Coth[0.]` already was; it stayed unevaluated. Tested in `test_hyperbolic_real_valued_precision`.

Tests

test_hyperbolic_real_valued_precision in test/builtin/atomic/test_numbers.py, with WMA 15 values. Full run on kimmy: doctests all pass. The pytest failures are the ones that depend on the order the tests run in; the failing files that touch numbers pass on their own.

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rocky · 2026-10-19T21:46:02Z

LGTM - merge when you feel is ready.

Decide if you want to add edge-case tests around exact zero / near-zero / pole-adjacent values, where formulas divide by value or by sqrt(1-x^2).

… number Part of step 6 of #2047 (MATH-227), on top of the inverse trig PR. As in WMA, the relative error of x is multiplied by Abs[x f'[x] / f[x]]. For Tanh and Coth, WMA uses the condition number of Sinh[x] / Cosh[x], with both errors added, when Abs[x] < 1: Precision[Tanh[0.3`20]] is 19.9519, not 20.0258. This follows it. Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>

As asked in review: mpmath numbers in and out, as for Power. Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>

As for Cot and Csc: Csch of an inexact zero is ComplexInfinity, as in WMA (Csch[0.] stayed unevaluated; Coth[0.] already had the rule). The hyperbolic functions of 0``20 get their error from how much they change near 0: Sinh[0``20] is 0``20, Cosh[0``20] is 1 with precision 40.301. Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>

As suggested in review. Coth and Csch set has_poles, and Tanh and Sech are bounded by 1 (see #2054). ArcCosh[1.`20] and ArcSech[1.`20] are 0``9.849, as in WMA, and the values next to 1 or to a pole are tested too, as are 0`20 (an exact 0) and 0.0`20 (a machine 0.). Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>

As suggested in review, edge cases next to poles and where the formulas divide by the value or by Sqrt[1 - x^2]. As in WMA: - Where the condition factor does not exist (a division by zero, an infinite factor, or a value of 0), the error is how much f changes within the error of its argument, as was already done for an inexact zero argument; a side where f has no real value is left out. This is what the inverse functions need at the ends of their domain (#2055, #2056), where ArcSin[1.`20] raised ZeroDivisionError. - A function with poles gives ComplexInfinity when no digit is left: Tan[N[Pi/2, 20]] and Tan[1.5707963`5] (was 0.`1.). Tan, Sec, Cot and Csc set has_poles. - A bounded function has an error no larger than its bound: Sin[1.`20*^30] is 0``0. Sin and Cos set value_bound. Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>

tombouctou · 2026-10-19T22:05:54Z

Thanks, added (03d2771, with the shared part in #2054), checked against WMA 15:

Exact and machine zeros: Sinh[020]is0, and Cosh[0.020] is a machine 1..
Near zero: Sinh[020]is020, and Cosh[020]`` is 1 with precision 40.301.
Poles: Coth and Csch of 0. or 020`` are ComplexInfinity; `Csch[0.]` stayed unevaluated before. `Coth` and `Csch` are also marked as having poles: like `Tan` in Sin, Cos, Tan, Cot, Sec, Csc: precision from the condition number, as in WMA #2054, they give `ComplexInfinity` when no digit is left.
Ends of the domain, where the formula divides by Sqrt[x^2 - 1]: ArcCosh[1.`20] and ArcSech[1.`20] are 09.849, from the fallback in #2054. Next to the end, ArcCosh[1.0000000120]`` has precision 12.301, and next to a pole ``ArcTanh[0.9999999920]`` has 13.2813.
Bounds: Tanh and Sech are bounded by 1, so when no digit is left their error is at most 1, as for Sin (Sin[1.`20*^30] is 00). Sinh[1000.`2]is unbounded and gives0``-434.993``.

Left as it was: ArcTanh[1.`20] is Infinity here and Indeterminate in WMA. That comes from SymPy, not from this code.

…n WMA (#2063) A follow-up to #2056 (part of #2047). `ArcTanh` and `ArcCoth` have poles at 1 and -1. For an inexact argument there, SymPy gave `Infinity`. Near a pole, the old path gave `0.` or a complex number even when the argument's error covered the pole. In WMA, if the error of an inexact argument covers a pole, the result is `Indeterminate`: within that error, the function goes to `Infinity` on one side and takes an imaginary part on the other. A machine number has no error, so only `1.` and `-1.` themselves are covered. Exact `ArcTanh[1]` is still `Infinity`. This is a new `MPMathFunction` attribute, `indeterminate_at`, checked by `covers_point` in `mathics/eval/numeric.py`. | Input | master | this PR | WMA 15 | |---|---|---|---| | ``ArcTanh[1.`20]`` | `Infinity` | `Indeterminate` | `Indeterminate` | | `ArcTanh[-1.]` | `-Infinity` | `Indeterminate` | `Indeterminate` | | `ArcCoth[1.]` | `Infinity` | `Indeterminate` | `Indeterminate` | | ``ArcTanh[1.0000001`5]`` | `Infinity` | `Indeterminate` | `Indeterminate` | | ``ArcTanh[0.9999999`5]`` | `0.` | `Indeterminate` | `Indeterminate` | | ``ArcTanh[1.000001`6]`` | `7.27805 - 1.5708 I` | `Indeterminate` | `Indeterminate` | | ``Precision[ArcTanh[0.999999`6.1]]`` | `1.26163` | `1.26163` | `1.26163` | | `ArcTanh[1]` | `Infinity` | `Infinity` | `Infinity` | Right at the boundary, WMA gives ``ArcTanh[0.999999`6]`` = ``0``-6.``, which looks like an artifact. Here it is a number with precision about 1.16. ## Tests In `test_hyperbolic_real_valued_precision` (`test/builtin/atomic/test_numbers.py`), with WMA 15 values, plus a doctest in `ArcTanh`. Full run on kimmy next to master: all 4620 doctests pass. The pytest failures are the ones that depend on test order: each failing file passes when run on its own. 🤖 Generated with [Claude Code](https://claude.com/claude-code) Co-authored-by: Claude Opus 5.5 <noreply@anthropic.com> Co-authored-by: R. Bernstein <rocky@users.noreply.github.com>

The rest of #2047 step 6, for complex numbers: complex arithmetic, and functions with a complex value at a real argument, now get their precision the way WMA does. In WMA each part of a `Complex` keeps its own error. `Accuracy` combines the two errors in quadrature, and `Precision` is that accuracy plus `Log10` of the modulus; an exact part also counts toward the modulus. Before this PR, Mathics3 gave a complex result the precision of its least precise input, or machine precision, which is why ``1.5`20.3 + 2.5`20.3 I`` had precision 15. - `Plus` and `Times` on complex numbers with exact or `PrecisionReal` parts work out each part with the real rules. `Times` multiplies as a balanced tree and squares a pair of equal factors. `z^n` squares starting from the top bit of `n`, and `1/z` is `(a - b I)/(a^2 + b^2)`. This is the order WMA uses, and it changes the last digits of the precision. ``2.`20 I`` keeps an exact zero real part. - A function with a complex value at a real argument (``ArcSin[2.`20]``, ``Log[-1.`20]``, ``ArcCosh[0``20]``, ``ArcSec[0.5`20]``, …) gives each part the error from that part of `x f'[x]`, as #2052–#2056 do for real values. A part that does not change gets `2^-54` of the error, so the `Pi` part of ``Log[-1.`20]`` is 16 digits more precise than the real part, as in WMA. At a zero or a branch point, the change of `f` within the error goes to the nonzero parts. - A negative `PrecisionReal` to a `Rational` power: `x^(1/q)` shares the error of the whole value between its parts, and `x^(p/q)` is `x^n (x^(1/q))^r`. Checked against WMA for `p/q` from −5/2 to 7/3. - `Log[x]` is no longer rewritten as `Log[E, x]`: the numerified `E` base added an error of its own. - An inexact zero real part is now shown, as in WMA (``0``19.4 + 2.`19.7 I``). `N[I, 30]` is ``1.`30 I``. | Input | master | this PR | WMA 15 | |---|---|---|---| | ``Precision[ArcSec[0.5`20]]`` | `15.1196` | `20.0571` | `20.0571` | | ``Precision[1.5`20.3 + 2.5`20.3 I]`` | `15.` | `20.3` | `20.3` | | ``ArcCosh[0``20]`` | ``1.5707963…`20.*I`` | ``1.5707963…`20.1961*I`` | ``1.5707963…`20.1961*I`` | | ``Sqrt[-2.`20]`` | ``1.4142135…`20.*I`` | ``1.4142135…`20.301*I`` | ``1.4142135…`20.301*I`` | | ``Precision[Log[-1.`20]]`` | `15.4971` | `20.4971` | `20.4971` | | ``Accuracy[Im[Log[-1.`20]]] - Accuracy[Re[Log[-1.`20]]]`` | `4.50285` | `16.2556` | `16.2556` | | ``Precision[(1.`20 + 2.`25 I) (3.`22 - 1.`18 I)]`` | `15.` | `18.4939` | `18.4939` | | ``Precision[(1.`20 + 2.`25 I)^5]`` | `15.` | `19.3049` | `19.3049` | | ``Precision[(-8.`20)^(1/3)]`` | `20.` | `20.4771` | `20.4771` | Not in this PR: functions of a complex argument (``Sqrt[1.`20 + I]``, ``Exp[1.`20 I]``), a negative base with an inexact exponent, and `N` of an exact complex number with two nonzero parts, where WMA splits the precision between the parts. ## Tests `test_complex_precision` and `test_complex_valued_precision` in `test/builtin/atomic/test_numbers.py`, next to `test_real_valued_precision`, with WMA 15 values. Full run on kimmy: all doctests pass. pytest under xdist (`-n 4`) fails a different handful of tests on each run, on master as well, because those tests depend on symbols that earlier tests in the same worker define or clear (`f`, `F`, `x`). With `--dist loadfile`, master fails 15 (`test_forms.py`, `test_options.py`, `test_testing_expressions.py`, `test_latex.py`) and this branch fails 7 (`test_list.py`, `test_diffeqns.py`, `test_constructing.py`). None of them is about numbers, and each of these files passes when run on its own. Benchmark from #2050 on kimmy, best of 5, ms: | Case | master | this PR | |---|---|---| | ``Total[Table[N[1/k, 50], {k, 2000}]]`` | 2645 | 2738 | | ``Times @@ Table[N[1 + 1/k, 50], {k, 2000}]`` | 3255 | 3324 | | ``Do[x = N[k, 50] + Pi, {k, 2000}]`` | 5388 | 5238 | | ``Total[Table[1./k, {k, 2000}]]`` | 1988 | 2056 | | ``Total[Table[N[1/k + I/(k+1), 50], {k, 2000}]]`` | 4884 | 5127 | | ``Times @@ Table[N[1 + I/k, 50], {k, 500}]`` | 883 | 1070 | | ``Table[N[1 + I/k, 30]^5, {k, 500}]`` | 1435 | 1820 | | ``Table[1 / N[1 + I/k, 30], {k, 500}]`` | 1592 | 1670 | | ``Table[ArcSin[N[1 + k, 30]], {k, 500}]`` | 1112 | 1504 | | ``Table[Log[N[-k, 30]], {k, 500}]`` | 1421 | 1322 | The real cases are unchanged within noise. Complex `Times`, `z^5` and `ArcSin` outside its domain are 20–35% slower: their parts are now computed from real operations, and `ArcSin` takes more derivatives. 🤖 Generated with [Claude Code](https://claude.com/claude-code) Co-authored-by: Claude Opus 5.5 <noreply@anthropic.com>

tombouctou mentioned this pull request Oct 19, 2026

Add SetPrecision and SetAccuracy, as in WMA #2057

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This was referenced Oct 19, 2026

Sin, Cos, Tan, Cot, Sec, Csc: precision from the condition number, as in WMA #2054

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Power, Sqrt: precision from the condition number, as in WMA #2052

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tombouctou and others added 4 commits October 20, 2026 02:02

condition_numbers of the hyperbolic functions: type annotations

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As asked in review: mpmath numbers in and out, as for Power. Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>

tombouctou force-pushed the arctrig-real-valued-precision branch from 48907e4 to 58715b3 Compare October 19, 2026 22:04

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Base automatically changed from arctrig-real-valued-precision to master October 19, 2026 22:33

tombouctou merged commit ad3a9b5 into master Oct 19, 2026
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tombouctou mentioned this pull request Oct 20, 2026

ArcTanh, ArcCoth: Indeterminate where the error covers the pole, as in WMA #2063

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Hyperbolic functions and their inverses: precision from the condition number, as in WMA - #2056

Hyperbolic functions and their inverses: precision from the condition number, as in WMA#2056
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