//===-- Half-precision acospi 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/acospif16.h" #include "hdr/errno_macros.h" #include "hdr/fenv_macros.h" #include "src/__support/FPUtil/FEnvImpl.h" #include "src/__support/FPUtil/FPBits.h" #include "src/__support/FPUtil/PolyEval.h" #include "src/__support/FPUtil/cast.h" #include "src/__support/FPUtil/multiply_add.h" #include "src/__support/FPUtil/sqrt.h" #include "src/__support/macros/optimization.h" namespace LIBC_NAMESPACE_DECL { LLVM_LIBC_FUNCTION(float16, acospif16, (float16 x)) { using FPBits = fputil::FPBits; FPBits xbits(x); uint16_t x_u = xbits.uintval(); uint16_t x_abs = x_u & 0x7fff; uint16_t x_sign = x_u >> 15; // |x| > 0x1p0, |x| > 1, or x is NaN. if (LIBC_UNLIKELY(x_abs > 0x3c00)) { // acospif16(NaN) = NaN if (xbits.is_nan()) { if (xbits.is_signaling_nan()) { fputil::raise_except_if_required(FE_INVALID); return FPBits::quiet_nan().get_val(); } return x; } // 1 < |x| <= +inf fputil::raise_except_if_required(FE_INVALID); fputil::set_errno_if_required(EDOM); return FPBits::quiet_nan().get_val(); } // |x| == 0x1p0, x is 1 or -1 // if x is (-)1, return 1 // if x is (+)1, return 0 if (LIBC_UNLIKELY(x_abs == 0x3c00)) return fputil::cast(x_sign ? 1.0f : 0.0f); float xf = x; float xsq = xf * xf; // Degree-6 minimax polynomial coefficients of asin(x) generated by Sollya // with: > P = fpminimax(asin(x)/(pi * x), [|0, 2, 4, 6, 8|], [|SG...|], [0, // 0.5]); constexpr float POLY_COEFFS[5] = {0x1.45f308p-2f, 0x1.b2900cp-5f, 0x1.897e36p-6f, 0x1.9efafcp-7f, 0x1.06d884p-6f}; // |x| <= 0x1p-1, |x| <= 0.5 if (x_abs <= 0x3800) { // if x is 0, return 0.5 if (LIBC_UNLIKELY(x_abs == 0)) return fputil::cast(0.5f); // Note that: acos(x) = pi/2 + asin(-x) = pi/2 - asin(x), then // acospi(x) = 0.5 - asin(x)/pi float interm = fputil::polyeval(xsq, POLY_COEFFS[0], POLY_COEFFS[1], POLY_COEFFS[2], POLY_COEFFS[3], POLY_COEFFS[4]); return fputil::cast(fputil::multiply_add(-xf, interm, 0.5f)); } // When |x| > 0.5, assume that 0.5 < |x| <= 1 // // Step-by-step range-reduction proof: // 1: Let y = asin(x), such that, x = sin(y) // 2: From complimentary angle identity: // x = sin(y) = cos(pi/2 - y) // 3: Let z = pi/2 - y, such that x = cos(z) // 4: From double angle formula; cos(2A) = 1 - 2 * sin^2(A): // z = 2A, z/2 = A // cos(z) = 1 - 2 * sin^2(z/2) // 5: Make sin(z/2) subject of the formula: // sin(z/2) = sqrt((1 - cos(z))/2) // 6: Recall [3]; x = cos(z). Therefore: // sin(z/2) = sqrt((1 - x)/2) // 7: Let u = (1 - x)/2 // 8: Therefore: // asin(sqrt(u)) = z/2 // 2 * asin(sqrt(u)) = z // 9: Recall [3]; z = pi/2 - y. Therefore: // y = pi/2 - z // y = pi/2 - 2 * asin(sqrt(u)) // 10: Recall [1], y = asin(x). Therefore: // asin(x) = pi/2 - 2 * asin(sqrt(u)) // 11: Recall that: acos(x) = pi/2 + asin(-x) = pi/2 - asin(x) // Therefore: // acos(x) = pi/2 - (pi/2 - 2 * asin(sqrt(u))) // acos(x) = 2 * asin(sqrt(u)) // acospi(x) = 2 * (asin(sqrt(u)) / pi) // // THE RANGE REDUCTION, HOW? // 12: Recall [7], u = (1 - x)/2 // 13: Since 0.5 < x <= 1, therefore: // 0 <= u <= 0.25 and 0 <= sqrt(u) <= 0.5 // // Hence, we can reuse the same [0, 0.5] domain polynomial approximation for // Step [11] as `sqrt(u)` is in range. // When -1 < x <= -0.5, the identity: // acos(x) = pi - acos(-x) // acospi(x) = 1 - acos(-x)/pi // allows us to compute for the negative x value (lhs) // with a positive x value instead (rhs). float xf_abs = (xf < 0 ? -xf : xf); float u = fputil::multiply_add(-0.5f, xf_abs, 0.5f); float sqrt_u = fputil::sqrt(u); float asin_sqrt_u = sqrt_u * fputil::polyeval(u, POLY_COEFFS[0], POLY_COEFFS[1], POLY_COEFFS[2], POLY_COEFFS[3], POLY_COEFFS[4]); // Same as acos(x), but devided the expression with pi return fputil::cast( x_sign ? fputil::multiply_add(-2.0f, asin_sqrt_u, 1.0f) : 2.0f * asin_sqrt_u); } } // namespace LIBC_NAMESPACE_DECL