1 /* 2 * Double-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_f64.h" 9 #include "math_config.h" 10 #include "pl_sig.h" 11 #include "pl_test.h" 12 13 #define AbsMask (0x7fffffffffffffff) 14 #define Half (0x3fe0000000000000) 15 #define One (0x3ff0000000000000) 16 #define PiOver2 (0x1.921fb54442d18p+0) 17 #define Small (0x3e50000000000000) /* 2^-26. */ 18 #define Small16 (0x3e50) 19 #define QNaN (0x7ff8) 20 21 /* Fast implementation of double-precision asin(x) based on polynomial 22 approximation. 23 24 For x < Small, approximate asin(x) by x. Small = 2^-26 for correct rounding. 25 26 For x in [Small, 0.5], use an order 11 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 1.01 ulps, 30 asin(0x1.da9735b5a9277p-2) got 0x1.ed78525a927efp-2 31 want 0x1.ed78525a927eep-2. 32 33 No cheap approximation can be obtained near x = 1, since the function is not 34 continuously differentiable on 1. 35 36 For x in [0.5, 1.0], we use a method based on a trigonometric identity 37 38 asin(x) = pi/2 - acos(x) 39 40 and a generalized power series expansion of acos(y) near y=1, that reads as 41 42 acos(y)/sqrt(2y) ~ 1 + 1/12 * y + 3/160 * y^2 + ... (1) 43 44 The Taylor series of asin(z) near z = 0, reads as 45 46 asin(z) ~ z + z^3 P(z^2) = z + z^3 * (1/6 + 3/40 z^2 + ...). 47 48 Therefore, (1) can be written in terms of P(y/2) or even asin(y/2) 49 50 acos(y) ~ sqrt(2y) (1 + y/2 * P(y/2)) = 2 * sqrt(y/2) (1 + y/2 * P(y/2) 51 52 Hence, if we write z = (1-x)/2, z is near 0 when x approaches 1 and 53 54 asin(x) ~ pi/2 - acos(x) ~ pi/2 - 2 * sqrt(z) (1 + z * P(z)). 55 56 The largest observed error in this region is 2.69 ulps, 57 asin(0x1.044ac9819f573p-1) got 0x1.110d7e85fdd5p-1 58 want 0x1.110d7e85fdd53p-1. */ 59 double 60 asin (double x) 61 { 62 uint64_t ix = asuint64 (x); 63 uint64_t ia = ix & AbsMask; 64 uint64_t ia16 = ia >> 48; 65 double ax = asdouble (ia); 66 uint64_t sign = ix & ~AbsMask; 67 68 /* Special values and invalid range. */ 69 if (unlikely (ia16 == QNaN)) 70 return x; 71 if (ia > One) 72 return __math_invalid (x); 73 if (ia16 < Small16) 74 return x; 75 76 /* Evaluate polynomial Q(x) = y + y * z * P(z) with 77 z2 = x ^ 2 and z = |x| , if |x| < 0.5 78 z2 = (1 - |x|) / 2 and z = sqrt(z2), if |x| >= 0.5. */ 79 double z2 = ax < 0.5 ? x * x : fma (-0.5, ax, 0.5); 80 double z = ax < 0.5 ? ax : sqrt (z2); 81 82 /* Use a single polynomial approximation P for both intervals. */ 83 double z4 = z2 * z2; 84 double z8 = z4 * z4; 85 double z16 = z8 * z8; 86 double p = estrin_11_f64 (z2, z4, z8, z16, __asin_poly); 87 88 /* Finalize polynomial: z + z * z2 * P(z2). */ 89 p = fma (z * z2, p, z); 90 91 /* asin(|x|) = Q(|x|) , for |x| < 0.5 92 = pi/2 - 2 Q(|x|), for |x| >= 0.5. */ 93 double y = ax < 0.5 ? p : fma (-2.0, p, PiOver2); 94 95 /* Copy sign. */ 96 return asdouble (asuint64 (y) | sign); 97 } 98 99 PL_SIG (S, D, 1, asin, -1.0, 1.0) 100 PL_TEST_ULP (asin, 2.19) 101 PL_TEST_INTERVAL (asin, 0, Small, 5000) 102 PL_TEST_INTERVAL (asin, Small, 0.5, 50000) 103 PL_TEST_INTERVAL (asin, 0.5, 1.0, 50000) 104 PL_TEST_INTERVAL (asin, 1.0, 0x1p11, 50000) 105 PL_TEST_INTERVAL (asin, 0x1p11, inf, 20000) 106 PL_TEST_INTERVAL (asin, -0, -inf, 20000) 107