1 /* 2 * Single-precision asin(x) function. 3 * 4 * Copyright (c) 2023, Arm Limited. 5 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception 6 */ 7 8 #include "poly_scalar_f32.h" 9 #include "math_config.h" 10 #include "pl_sig.h" 11 #include "pl_test.h" 12 13 #define AbsMask (0x7fffffff) 14 #define Half (0x3f000000) 15 #define One (0x3f800000) 16 #define PiOver2f (0x1.921fb6p+0f) 17 #define Small (0x39800000) /* 2^-12. */ 18 #define Small12 (0x398) 19 #define QNaN (0x7fc) 20 21 /* Fast implementation of single-precision asin(x) based on polynomial 22 approximation. 23 24 For x < Small, approximate asin(x) by x. Small = 2^-12 for correct rounding. 25 26 For x in [Small, 0.5], use order 4 polynomial P such that the final 27 approximation is an odd polynomial: asin(x) ~ x + x^3 P(x^2). 28 29 The largest observed error in this region is 0.83 ulps, 30 asinf(0x1.ea00f4p-2) got 0x1.fef15ep-2 want 0x1.fef15cp-2. 31 32 No cheap approximation can be obtained near x = 1, since the function is not 33 continuously differentiable on 1. 34 35 For x in [0.5, 1.0], we use a method based on a trigonometric identity 36 37 asin(x) = pi/2 - acos(x) 38 39 and a generalized power series expansion of acos(y) near y=1, that reads as 40 41 acos(y)/sqrt(2y) ~ 1 + 1/12 * y + 3/160 * y^2 + ... (1) 42 43 The Taylor series of asin(z) near z = 0, reads as 44 45 asin(z) ~ z + z^3 P(z^2) = z + z^3 * (1/6 + 3/40 z^2 + ...). 46 47 Therefore, (1) can be written in terms of P(y/2) or even asin(y/2) 48 49 acos(y) ~ sqrt(2y) (1 + y/2 * P(y/2)) = 2 * sqrt(y/2) (1 + y/2 * P(y/2) 50 51 Hence, if we write z = (1-x)/2, z is near 0 when x approaches 1 and 52 53 asin(x) ~ pi/2 - acos(x) ~ pi/2 - 2 * sqrt(z) (1 + z * P(z)). 54 55 The largest observed error in this region is 2.41 ulps, 56 asinf(0x1.00203ep-1) got 0x1.0c3a64p-1 want 0x1.0c3a6p-1. */ 57 float 58 asinf (float x) 59 { 60 uint32_t ix = asuint (x); 61 uint32_t ia = ix & AbsMask; 62 uint32_t ia12 = ia >> 20; 63 float ax = asfloat (ia); 64 uint32_t sign = ix & ~AbsMask; 65 66 /* Special values and invalid range. */ 67 if (unlikely (ia12 == QNaN)) 68 return x; 69 if (ia > One) 70 return __math_invalidf (x); 71 if (ia12 < Small12) 72 return x; 73 74 /* Evaluate polynomial Q(x) = y + y * z * P(z) with 75 z2 = x ^ 2 and z = |x| , if |x| < 0.5 76 z2 = (1 - |x|) / 2 and z = sqrt(z2), if |x| >= 0.5. */ 77 float z2 = ax < 0.5 ? x * x : fmaf (-0.5f, ax, 0.5f); 78 float z = ax < 0.5 ? ax : sqrtf (z2); 79 80 /* Use a single polynomial approximation P for both intervals. */ 81 float p = horner_4_f32 (z2, __asinf_poly); 82 /* Finalize polynomial: z + z * z2 * P(z2). */ 83 p = fmaf (z * z2, p, z); 84 85 /* asin(|x|) = Q(|x|) , for |x| < 0.5 86 = pi/2 - 2 Q(|x|), for |x| >= 0.5. */ 87 float y = ax < 0.5 ? p : fmaf (-2.0f, p, PiOver2f); 88 89 /* Copy sign. */ 90 return asfloat (asuint (y) | sign); 91 } 92 93 PL_SIG (S, F, 1, asin, -1.0, 1.0) 94 PL_TEST_ULP (asinf, 1.91) 95 PL_TEST_INTERVAL (asinf, 0, Small, 5000) 96 PL_TEST_INTERVAL (asinf, Small, 0.5, 50000) 97 PL_TEST_INTERVAL (asinf, 0.5, 1.0, 50000) 98 PL_TEST_INTERVAL (asinf, 1.0, 0x1p11, 50000) 99 PL_TEST_INTERVAL (asinf, 0x1p11, inf, 20000) 100 PL_TEST_INTERVAL (asinf, -0, -inf, 20000) 101