//===-- Single-precision acos function ------------------------------------===// // // Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions. // See https://llvm.org/LICENSE.txt for license information. // SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception // //===----------------------------------------------------------------------===// #include "src/math/acosf.h" #include "src/__support/FPUtil/FEnvImpl.h" #include "src/__support/FPUtil/FPBits.h" #include "src/__support/FPUtil/PolyEval.h" #include "src/__support/FPUtil/except_value_utils.h" #include "src/__support/FPUtil/multiply_add.h" #include "src/__support/FPUtil/sqrt.h" #include "src/__support/macros/config.h" #include "src/__support/macros/optimization.h" // LIBC_UNLIKELY #include "inv_trigf_utils.h" namespace LIBC_NAMESPACE_DECL { static constexpr size_t N_EXCEPTS = 4; // Exceptional values when |x| <= 0.5 static constexpr fputil::ExceptValues ACOSF_EXCEPTS = {{ // (inputs, RZ output, RU offset, RD offset, RN offset) // x = 0x1.110b46p-26, acosf(x) = 0x1.921fb4p0 (RZ) {0x328885a3, 0x3fc90fda, 1, 0, 1}, // x = -0x1.110b46p-26, acosf(x) = 0x1.921fb4p0 (RZ) {0xb28885a3, 0x3fc90fda, 1, 0, 1}, // x = 0x1.04c444p-12, acosf(x) = 0x1.920f68p0 (RZ) {0x39826222, 0x3fc907b4, 1, 0, 1}, // x = -0x1.04c444p-12, acosf(x) = 0x1.923p0 (RZ) {0xb9826222, 0x3fc91800, 1, 0, 1}, }}; LLVM_LIBC_FUNCTION(float, acosf, (float x)) { using FPBits = typename fputil::FPBits; FPBits xbits(x); uint32_t x_uint = xbits.uintval(); uint32_t x_abs = xbits.uintval() & 0x7fff'ffffU; uint32_t x_sign = x_uint >> 31; // |x| <= 0.5 if (LIBC_UNLIKELY(x_abs <= 0x3f00'0000U)) { // |x| < 0x1p-10 if (LIBC_UNLIKELY(x_abs < 0x3a80'0000U)) { // When |x| < 2^-10, we use the following approximation: // acos(x) = pi/2 - asin(x) // ~ pi/2 - x - x^3 / 6 // Check for exceptional values if (auto r = ACOSF_EXCEPTS.lookup(x_uint); LIBC_UNLIKELY(r.has_value())) return r.value(); double xd = static_cast(x); return static_cast(fputil::multiply_add( -0x1.5555555555555p-3 * xd, xd * xd, M_MATH_PI_2 - xd)); } // For |x| <= 0.5, we approximate acosf(x) by: // acos(x) = pi/2 - asin(x) = pi/2 - x * P(x^2) // Where P(X^2) = Q(X) is a degree-20 minimax even polynomial approximating // asin(x)/x on [0, 0.5] generated by Sollya with: // > Q = fpminimax(asin(x)/x, [|0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20|], // [|1, D...|], [0, 0.5]); double xd = static_cast(x); double xsq = xd * xd; double x3 = xd * xsq; double r = asin_eval(xsq); return static_cast(fputil::multiply_add(-x3, r, M_MATH_PI_2 - xd)); } // |x| >= 1, return 0, 2pi, or NaNs. if (LIBC_UNLIKELY(x_abs >= 0x3f80'0000U)) { if (x_abs == 0x3f80'0000U) return x_sign ? /* x == -1.0f */ fputil::round_result_slightly_down( 0x1.921fb6p+1f) : /* x == 1.0f */ 0.0f; if (x_abs <= 0x7f80'0000U) { fputil::set_errno_if_required(EDOM); fputil::raise_except_if_required(FE_INVALID); } return x + FPBits::quiet_nan().get_val(); } // When 0.5 < |x| < 1, we perform range reduction as follow: // // Assume further that 0.5 < x <= 1, and let: // y = acos(x) // We use the double angle formula: // x = cos(y) = 1 - 2 sin^2(y/2) // So: // sin(y/2) = sqrt( (1 - x)/2 ) // And hence: // y = 2 * asin( sqrt( (1 - x)/2 ) ) // Let u = (1 - x)/2, then // acos(x) = 2 * asin( sqrt(u) ) // Moreover, since 0.5 < x <= 1, // 0 <= u < 1/4, and 0 <= sqrt(u) < 0.5, // And hence we can reuse the same polynomial approximation of asin(x) when // |x| <= 0.5: // acos(x) ~ 2 * sqrt(u) * P(u). // // When -1 < x <= -0.5, we use the identity: // acos(x) = pi - acos(-x) // which is reduced to the postive case. xbits.set_sign(Sign::POS); double xd = static_cast(xbits.get_val()); double u = fputil::multiply_add(-0.5, xd, 0.5); double cv = 2 * fputil::sqrt(u); double r3 = asin_eval(u); double r = fputil::multiply_add(cv * u, r3, cv); return static_cast(x_sign ? M_MATH_PI - r : r); } } // namespace LIBC_NAMESPACE_DECL