fix(libc): compute tanh with FreeBSD msun's algorithm, as titles' libc.prx does

tanh returned the host's tanh, whose last bit differs from the titles' copy
on Windows. The titles' copy is FreeBSD msun's s_tanh.c over msun's expm1,
and it sets no errno. tanh now computes that algorithm over this file's
Expm1.

Co-Authored-By: Claude Opus 5.5 <[email protected]>

docs/dev/debt/libc.md

@@ -114,6 +114,7 @@
 - [expm1](../../../core/libs/prx/libc/src/Math.cpp) (libc) does not compare floating-point exceptions, and its double inputs are checked by sampling, not exhaustively. It is FreeBSD msun's [`s_expm1.c`](https://github.com/freebsd/freebsd-src/blob/release/11.4.0/lib/msun/src/s_expm1.c) with `huge` read at run time, and sets `ERANGE` only when a finite argument overflows to infinity, as the titles' `libc.prx` does. It gives the same bits and `errno` as the `expm1` bundled with PPSA07923 and PPSA05916 on 320 million random and 47,222 structured arguments on a Windows host; the FMA-contracted build bundled with PPSA13934 differs on 43,433 of the random ones.
 - [log1p](../../../core/libs/prx/libc/src/Math.cpp) (libc) does not compare floating-point exceptions, and its double inputs are checked by sampling, not exhaustively. It is FreeBSD msun's [`s_log1p.c`](https://github.com/freebsd/freebsd-src/blob/release/11.4.0/lib/msun/src/s_log1p.c). As the titles' `libc.prx` does, it sets `EDOM` at msun's NaN return (arguments below -1, `-inf` and negative NaNs) and leaves `errno` alone for `log1p(-1)`, where it returns `-inf`. It gives the same bits, NaN payloads included, and `errno` as the `log1p` bundled with PPSA07923 and PPSA05916 on 320 million random and 68,216 structured arguments on a Windows host. The FMA-contracted build bundled with PPSA13934 differs on 73,682 of the random ones.
 - [nextafter and nextafterf](../../../core/libs/prx/libc/src/Math.cpp) (libc) do not compare floating-point exceptions, and their inputs are checked by sampling, not exhaustively. They are FreeBSD msun's [`s_nextafter.c`](https://github.com/freebsd/freebsd-src/blob/release/11.4.0/lib/msun/src/s_nextafter.c) and [`s_nextafterf.c`](https://github.com/freebsd/freebsd-src/blob/release/11.4.0/lib/msun/src/s_nextafterf.c), with `ERANGE` at msun's overflow and underflow returns as the titles' `libc.prx` sets it; a step away from zero returns early and sets none. The copies bundled with eleven titles give the same bits and `errno` on 4,000,784 sampled pairs per function on a Windows host.
+- [tanh](../../../core/libs/prx/libc/src/Math.cpp) (libc) does not compare floating-point exceptions, and its double inputs are checked by sampling, not exhaustively. It is FreeBSD msun's [`s_tanh.c`](https://github.com/freebsd/freebsd-src/blob/release/11.4.0/lib/msun/src/s_tanh.c) over this file's msun `Expm1`, with `tiny` read at run time; it sets no `errno`, as the titles' `libc.prx` does not. It gives the same bits as the `tanh` bundled with PPSA07923 and PPSA05916 on 16 million random and 68,211 structured arguments on a Windows host; the FMA-contracted build bundled with PPSA13934 differs on 1,054 of them.
 - [sinhf](../../../core/libs/prx/libc/src/Math.cpp) (libc) follows the `sinhf` of the `libc.prx` bundled with PPSA28319 (disassembly): FreeBSD msun's [`e_sinhf.c`](https://github.com/freebsd/freebsd-src/blob/5aa554c7e5b6fd9042986df07f12ac5687e41392/lib/msun/src/e_sinhf.c) from before `__ldexp_expf`, which squares `expf(|x| / 2)` between `logf(FLT_MAX)` and the overflow threshold, calling msun's [`expm1f`](https://github.com/freebsd/freebsd-src/blob/59bc2f095289a10b8cfa55cbb2b45c3bd6c2d4ae/lib/msun/src/s_expm1f.c) and [`expf`](https://github.com/freebsd/freebsd-src/blob/59bc2f095289a10b8cfa55cbb2b45c3bd6c2d4ae/lib/msun/src/e_expf.c), whose paths for the arguments it passes are inlined here. Past the threshold Sony's error helper sets `errno` to `ERANGE` before returning `x * 1e37`; msun itself sets no `errno`. That code, run on a Windows host, gives the same bits as this implementation for every finite argument up to the threshold, of both signs; it was not run on Linux. That the system `libSceLibcInternal`, which Ruffle (PPSA68091) and RetroArch (PPSA99169) import while bundling a zero-return `libc.prx`, has the same code is assumed
 - [acoshf](../../../core/libs/prx/libc/src/Math.cpp) (libc) does not reproduce bundled invalid-domain errno or x87 exception flags (#10610); exact system `libSceLibcInternal` behavior and FMA-contracted copies remain unverified. Positive finite inputs use [msun](https://github.com/freebsd/freebsd-src/blob/5aa554c7e5b6fd9042986df07f12ac5687e41392/lib/msun/src/e_acoshf.c) range handling and this file's `logf`; the middle range divides -1 before addition under directed rounding. Bits, errno and SSE/x87 status and controls match the supplied PPSA03766/PPSA07923 copies on 236,592 positive finite calls (4,929 inputs, four rounding modes, DAZ/FTZ combinations and sticky flags); these samples are not exhaustive.
 - [acoshl, asinhl, atanhl, atan2l, cbrtl, erfl, erfcl, expm1l, hypotl, log1pl, log2l, lgammal, tgammal, j0f, j1f, jnf, y0f, y1f and ynf](../../../core/libs/prx/libc/src/Math.cpp) (libc) return the host C library's results (the float Bessel functions through its double ones), not those of the console's own code, so the last bits can differ (`y0f`, `y1f`, `ynf` follow the console at `+-0`). As in the console's `libSceLibcInternal.sprx` (SDK 4.5x build of `J03950001`, 0x56410), `lgammal` does not store the sign in `signgam`; `hypotl` returns infinity for an infinite argument even when the other is a NaN, as C11 F.10.4.3 requires. [hypot3, hypot3f and hypot3l](../../../core/libs/prx/libc/src/Math.cpp) follow that module (0x51D30, 0x520D0, 0x52420): the first NaN of x, y and z is returned unchanged, any infinity gives positive infinity, a zero argument gives the two-argument hypot of the other two, and otherwise the largest magnitude is combined with the hypot of the other two