diff options
Diffstat (limited to 'libc/src/math/generic/acospif16.cpp')
| -rw-r--r-- | libc/src/math/generic/acospif16.cpp | 122 |
1 files changed, 3 insertions, 119 deletions
diff --git a/libc/src/math/generic/acospif16.cpp b/libc/src/math/generic/acospif16.cpp index bfdf169..09cbd99 100644 --- a/libc/src/math/generic/acospif16.cpp +++ b/libc/src/math/generic/acospif16.cpp @@ -7,128 +7,12 @@ //===----------------------------------------------------------------------===// #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" +#include "src/__support/math/acospif16.h" namespace LIBC_NAMESPACE_DECL { LLVM_LIBC_FUNCTION(float16, acospif16, (float16 x)) { - using FPBits = fputil::FPBits<float16>; - 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<float16>(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<float16>(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<float16>(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<float>(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<float16>( - x_sign ? fputil::multiply_add(-2.0f, asin_sqrt_u, 1.0f) - : 2.0f * asin_sqrt_u); + return math::acospif16(x); } + } // namespace LIBC_NAMESPACE_DECL |