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| author | Andrew Turner <andrew@FreeBSD.org> | 2024-02-20 09:02:15 +0000 |
|---|---|---|
| committer | Andrew Turner <andrew@FreeBSD.org> | 2024-02-20 09:02:15 +0000 |
| commit | edc5c0de794f521eb620d2b6cbaee2434442a8f3 (patch) | |
| tree | 64dfc547c0b6398e9cf94bd8175b21db8a74c814 /pl/math/asinf_2u5.c | |
| parent | 29866ecb89620f1c798b7f5ff6710255f13aa52e (diff) | |
Update the Arm Optimized Routinesvendor/arm-optimized-routines/v24.01
Import the v24.01 release of the Arm Optimized Routines [1].
[1] https://github.com/ARM-software/optimized-routines/tree/v24.01
Sponsored by: Arm Ltd
Diffstat (limited to 'pl/math/asinf_2u5.c')
| -rw-r--r-- | pl/math/asinf_2u5.c | 100 |
1 files changed, 100 insertions, 0 deletions
diff --git a/pl/math/asinf_2u5.c b/pl/math/asinf_2u5.c new file mode 100644 index 000000000000..ec608146ff66 --- /dev/null +++ b/pl/math/asinf_2u5.c @@ -0,0 +1,100 @@ +/* + * Single-precision asin(x) function. + * + * Copyright (c) 2023, Arm Limited. + * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception + */ + +#include "poly_scalar_f32.h" +#include "math_config.h" +#include "pl_sig.h" +#include "pl_test.h" + +#define AbsMask (0x7fffffff) +#define Half (0x3f000000) +#define One (0x3f800000) +#define PiOver2f (0x1.921fb6p+0f) +#define Small (0x39800000) /* 2^-12. */ +#define Small12 (0x398) +#define QNaN (0x7fc) + +/* Fast implementation of single-precision asin(x) based on polynomial + approximation. + + For x < Small, approximate asin(x) by x. Small = 2^-12 for correct rounding. + + For x in [Small, 0.5], use order 4 polynomial P such that the final + approximation is an odd polynomial: asin(x) ~ x + x^3 P(x^2). + + The largest observed error in this region is 0.83 ulps, + asinf(0x1.ea00f4p-2) got 0x1.fef15ep-2 want 0x1.fef15cp-2. + + No cheap approximation can be obtained near x = 1, since the function is not + continuously differentiable on 1. + + For x in [0.5, 1.0], we use a method based on a trigonometric identity + + asin(x) = pi/2 - acos(x) + + and a generalized power series expansion of acos(y) near y=1, that reads as + + acos(y)/sqrt(2y) ~ 1 + 1/12 * y + 3/160 * y^2 + ... (1) + + The Taylor series of asin(z) near z = 0, reads as + + asin(z) ~ z + z^3 P(z^2) = z + z^3 * (1/6 + 3/40 z^2 + ...). + + Therefore, (1) can be written in terms of P(y/2) or even asin(y/2) + + acos(y) ~ sqrt(2y) (1 + y/2 * P(y/2)) = 2 * sqrt(y/2) (1 + y/2 * P(y/2) + + Hence, if we write z = (1-x)/2, z is near 0 when x approaches 1 and + + asin(x) ~ pi/2 - acos(x) ~ pi/2 - 2 * sqrt(z) (1 + z * P(z)). + + The largest observed error in this region is 2.41 ulps, + asinf(0x1.00203ep-1) got 0x1.0c3a64p-1 want 0x1.0c3a6p-1. */ +float +asinf (float x) +{ + uint32_t ix = asuint (x); + uint32_t ia = ix & AbsMask; + uint32_t ia12 = ia >> 20; + float ax = asfloat (ia); + uint32_t sign = ix & ~AbsMask; + + /* Special values and invalid range. */ + if (unlikely (ia12 == QNaN)) + return x; + if (ia > One) + return __math_invalidf (x); + if (ia12 < Small12) + return x; + + /* Evaluate polynomial Q(x) = y + y * z * P(z) with + z2 = x ^ 2 and z = |x| , if |x| < 0.5 + z2 = (1 - |x|) / 2 and z = sqrt(z2), if |x| >= 0.5. */ + float z2 = ax < 0.5 ? x * x : fmaf (-0.5f, ax, 0.5f); + float z = ax < 0.5 ? ax : sqrtf (z2); + + /* Use a single polynomial approximation P for both intervals. */ + float p = horner_4_f32 (z2, __asinf_poly); + /* Finalize polynomial: z + z * z2 * P(z2). */ + p = fmaf (z * z2, p, z); + + /* asin(|x|) = Q(|x|) , for |x| < 0.5 + = pi/2 - 2 Q(|x|), for |x| >= 0.5. */ + float y = ax < 0.5 ? p : fmaf (-2.0f, p, PiOver2f); + + /* Copy sign. */ + return asfloat (asuint (y) | sign); +} + +PL_SIG (S, F, 1, asin, -1.0, 1.0) +PL_TEST_ULP (asinf, 1.91) +PL_TEST_INTERVAL (asinf, 0, Small, 5000) +PL_TEST_INTERVAL (asinf, Small, 0.5, 50000) +PL_TEST_INTERVAL (asinf, 0.5, 1.0, 50000) +PL_TEST_INTERVAL (asinf, 1.0, 0x1p11, 50000) +PL_TEST_INTERVAL (asinf, 0x1p11, inf, 20000) +PL_TEST_INTERVAL (asinf, -0, -inf, 20000) |