1 //===-- Single-precision general inverse trigonometric functions ----------===// 2 // 3 // Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions. 4 // See https://llvm.org/LICENSE.txt for license information. 5 // SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception 6 // 7 //===----------------------------------------------------------------------===// 8 9 #ifndef LLVM_LIBC_SRC_MATH_GENERIC_INV_TRIGF_UTILS_H 10 #define LLVM_LIBC_SRC_MATH_GENERIC_INV_TRIGF_UTILS_H 11 12 #include "src/__support/FPUtil/PolyEval.h" 13 #include "src/__support/FPUtil/multiply_add.h" 14 #include "src/__support/common.h" 15 #include "src/__support/macros/config.h" 16 17 namespace LIBC_NAMESPACE_DECL { 18 19 // PI and PI / 2 20 static constexpr double M_MATH_PI = 0x1.921fb54442d18p+1; 21 static constexpr double M_MATH_PI_2 = 0x1.921fb54442d18p+0; 22 23 extern double ATAN_COEFFS[17][9]; 24 25 // Look-up table for atan(k/16) with k = 0..16. 26 static constexpr double ATAN_K_OVER_16[17] = { 27 0.0, 28 0x1.ff55bb72cfdeap-5, 29 0x1.fd5ba9aac2f6ep-4, 30 0x1.7b97b4bce5b02p-3, 31 0x1.f5b75f92c80ddp-3, 32 0x1.362773707ebccp-2, 33 0x1.6f61941e4def1p-2, 34 0x1.a64eec3cc23fdp-2, 35 0x1.dac670561bb4fp-2, 36 0x1.0657e94db30dp-1, 37 0x1.1e00babdefeb4p-1, 38 0x1.345f01cce37bbp-1, 39 0x1.4978fa3269ee1p-1, 40 0x1.5d58987169b18p-1, 41 0x1.700a7c5784634p-1, 42 0x1.819d0b7158a4dp-1, 43 0x1.921fb54442d18p-1, 44 }; 45 46 // For |x| <= 1/32 and 0 <= i <= 16, return Q(x) such that: 47 // Q(x) ~ (atan(x + i/16) - atan(i/16)) / x. 48 LIBC_INLINE static double atan_eval(double x, unsigned i) { 49 double x2 = x * x; 50 51 double c0 = fputil::multiply_add(x, ATAN_COEFFS[i][2], ATAN_COEFFS[i][1]); 52 double c1 = fputil::multiply_add(x, ATAN_COEFFS[i][4], ATAN_COEFFS[i][3]); 53 double c2 = fputil::multiply_add(x, ATAN_COEFFS[i][6], ATAN_COEFFS[i][5]); 54 double c3 = fputil::multiply_add(x, ATAN_COEFFS[i][8], ATAN_COEFFS[i][7]); 55 56 double x4 = x2 * x2; 57 double d1 = fputil::multiply_add(x2, c1, c0); 58 double d2 = fputil::multiply_add(x2, c3, c2); 59 double p = fputil::multiply_add(x4, d2, d1); 60 return p; 61 } 62 63 // Evaluate atan without big lookup table. 64 // atan(n/d) - atan(k/16) = atan((n/d - k/16) / (1 + (n/d) * (k/16))) 65 // = atan((n - d * k/16)) / (d + n * k/16)) 66 // So we let q = (n - d * k/16) / (d + n * k/16), 67 // and approximate with Taylor polynomial: 68 // atan(q) ~ q - q^3/3 + q^5/5 - q^7/7 + q^9/9 69 LIBC_INLINE static double atan_eval_no_table(double num, double den, 70 double k_over_16) { 71 double num_r = fputil::multiply_add(den, -k_over_16, num); 72 double den_r = fputil::multiply_add(num, k_over_16, den); 73 double q = num_r / den_r; 74 75 constexpr double ATAN_TAYLOR[] = { 76 -0x1.5555555555555p-2, 77 0x1.999999999999ap-3, 78 -0x1.2492492492492p-3, 79 0x1.c71c71c71c71cp-4, 80 }; 81 double q2 = q * q; 82 double q3 = q2 * q; 83 double q4 = q2 * q2; 84 double c0 = fputil::multiply_add(q2, ATAN_TAYLOR[1], ATAN_TAYLOR[0]); 85 double c1 = fputil::multiply_add(q2, ATAN_TAYLOR[3], ATAN_TAYLOR[2]); 86 double d = fputil::multiply_add(q4, c1, c0); 87 return fputil::multiply_add(q3, d, q); 88 } 89 90 // > Q = fpminimax(asin(x)/x, [|0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20|], 91 // [|1, D...|], [0, 0.5]); 92 static constexpr double ASIN_COEFFS[10] = { 93 0x1.5555555540fa1p-3, 0x1.333333512edc2p-4, 0x1.6db6cc1541b31p-5, 94 0x1.f1caff324770ep-6, 0x1.6e43899f5f4f4p-6, 0x1.1f847cf652577p-6, 95 0x1.9b60f47f87146p-7, 0x1.259e2634c494fp-6, -0x1.df946fa875ddp-8, 96 0x1.02311ecf99c28p-5}; 97 98 // Evaluate P(x^2) - 1, where P(x^2) ~ asin(x)/x 99 LIBC_INLINE static double asin_eval(double xsq) { 100 double x4 = xsq * xsq; 101 double r1 = fputil::polyeval(x4, ASIN_COEFFS[0], ASIN_COEFFS[2], 102 ASIN_COEFFS[4], ASIN_COEFFS[6], ASIN_COEFFS[8]); 103 double r2 = fputil::polyeval(x4, ASIN_COEFFS[1], ASIN_COEFFS[3], 104 ASIN_COEFFS[5], ASIN_COEFFS[7], ASIN_COEFFS[9]); 105 return fputil::multiply_add(xsq, r2, r1); 106 } 107 108 } // namespace LIBC_NAMESPACE_DECL 109 110 #endif // LLVM_LIBC_SRC_MATH_GENERIC_INV_TRIGF_UTILS_H 111