| //===-- Single-precision cos 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 | |
| // | |
| //===----------------------------------------------------------------------===// | |
| namespace LIBC_NAMESPACE_DECL { | |
| // Exceptional cases for cosf. | |
| static constexpr size_t N_EXCEPTS = 6; | |
| static constexpr fputil::ExceptValues<float, N_EXCEPTS> COSF_EXCEPTS{{ | |
| // (inputs, RZ output, RU offset, RD offset, RN offset) | |
| // x = 0x1.64a032p43, cos(x) = 0x1.9d4ba4p-1 (RZ) | |
| {0x55325019, 0x3f4ea5d2, 1, 0, 0}, | |
| // x = 0x1.4555p51, cos(x) = 0x1.115d7cp-1 (RZ) | |
| {0x5922aa80, 0x3f08aebe, 1, 0, 1}, | |
| // x = 0x1.48a858p54, cos(x) = 0x1.f48148p-2 (RZ) | |
| {0x5aa4542c, 0x3efa40a4, 1, 0, 0}, | |
| // x = 0x1.3170fp63, cos(x) = 0x1.fe2976p-1 (RZ) | |
| {0x5f18b878, 0x3f7f14bb, 1, 0, 0}, | |
| // x = 0x1.2b9622p67, cos(x) = 0x1.f0285cp-1 (RZ) | |
| {0x6115cb11, 0x3f78142e, 1, 0, 1}, | |
| // x = 0x1.ddebdep120, cos(x) = 0x1.114438p-1 (RZ) | |
| {0x7beef5ef, 0x3f08a21c, 1, 0, 0}, | |
| }}; | |
| LLVM_LIBC_FUNCTION(float, cosf, (float x)) { | |
| using FPBits = typename fputil::FPBits<float>; | |
| FPBits xbits(x); | |
| xbits.set_sign(Sign::POS); | |
| uint32_t x_abs = xbits.uintval(); | |
| double xd = static_cast<double>(xbits.get_val()); | |
| // Range reduction: | |
| // For |x| > pi/16, we perform range reduction as follows: | |
| // Find k and y such that: | |
| // x = (k + y) * pi/32 | |
| // k is an integer | |
| // |y| < 0.5 | |
| // For small range (|x| < 2^45 when FMA instructions are available, 2^22 | |
| // otherwise), this is done by performing: | |
| // k = round(x * 32/pi) | |
| // y = x * 32/pi - k | |
| // For large range, we will omit all the higher parts of 16/pi such that the | |
| // least significant bits of their full products with x are larger than 63, | |
| // since cos((k + y + 64*i) * pi/32) = cos(x + i * 2pi) = cos(x). | |
| // | |
| // When FMA instructions are not available, we store the digits of 32/pi in | |
| // chunks of 28-bit precision. This will make sure that the products: | |
| // x * THIRTYTWO_OVER_PI_28[i] are all exact. | |
| // When FMA instructions are available, we simply store the digits of 32/pi in | |
| // chunks of doubles (53-bit of precision). | |
| // So when multiplying by the largest values of single precision, the | |
| // resulting output should be correct up to 2^(-208 + 128) ~ 2^-80. By the | |
| // worst-case analysis of range reduction, |y| >= 2^-38, so this should give | |
| // us more than 40 bits of accuracy. For the worst-case estimation of range | |
| // reduction, see for instances: | |
| // Elementary Functions by J-M. Muller, Chapter 11, | |
| // Handbook of Floating-Point Arithmetic by J-M. Muller et. al., | |
| // Chapter 10.2. | |
| // | |
| // Once k and y are computed, we then deduce the answer by the cosine of sum | |
| // formula: | |
| // cos(x) = cos((k + y)*pi/32) | |
| // = cos(y*pi/32) * cos(k*pi/32) - sin(y*pi/32) * sin(k*pi/32) | |
| // The values of sin(k*pi/32) and cos(k*pi/32) for k = 0..63 are precomputed | |
| // and stored using a vector of 32 doubles. Sin(y*pi/32) and cos(y*pi/32) are | |
| // computed using degree-7 and degree-6 minimax polynomials generated by | |
| // Sollya respectively. | |
| // |x| < 0x1.0p-12f | |
| if (LIBC_UNLIKELY(x_abs < 0x3980'0000U)) { | |
| // When |x| < 2^-12, the relative error of the approximation cos(x) ~ 1 | |
| // is: | |
| // |cos(x) - 1| < |x^2 / 2| = 2^-25 < epsilon(1)/2. | |
| // So the correctly rounded values of cos(x) are: | |
| // = 1 - eps(x) if rounding mode = FE_TOWARDZERO or FE_DOWWARD, | |
| // = 1 otherwise. | |
| // To simplify the rounding decision and make it more efficient and to | |
| // prevent compiler to perform constant folding, we use | |
| // fma(x, -2^-25, 1) instead. | |
| // Note: to use the formula 1 - 2^-25*x to decide the correct rounding, we | |
| // do need fma(x, -2^-25, 1) to prevent underflow caused by -2^-25*x when | |
| // |x| < 2^-125. For targets without FMA instructions, we simply use | |
| // double for intermediate results as it is more efficient than using an | |
| // emulated version of FMA. | |
| return fputil::multiply_add(xbits.get_val(), -0x1.0p-25f, 1.0f); | |
| return static_cast<float>(fputil::multiply_add(xd, -0x1.0p-25, 1.0)); | |
| } | |
| if (auto r = COSF_EXCEPTS.lookup(x_abs); LIBC_UNLIKELY(r.has_value())) | |
| return r.value(); | |
| // x is inf or nan. | |
| if (LIBC_UNLIKELY(x_abs >= 0x7f80'0000U)) { | |
| if (xbits.is_signaling_nan()) { | |
| fputil::raise_except_if_required(FE_INVALID); | |
| return FPBits::quiet_nan().get_val(); | |
| } | |
| 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(); | |
| } | |
| // Combine the results with the sine of sum formula: | |
| // cos(x) = cos((k + y)*pi/32) | |
| // = cos(y*pi/32) * cos(k*pi/32) - sin(y*pi/32) * sin(k*pi/32) | |
| // = cosm1_y * cos_k + sin_y * sin_k | |
| // = (cosm1_y * cos_k + cos_k) + sin_y * sin_k | |
| double sin_k, cos_k, sin_y, cosm1_y; | |
| sincosf_eval(xd, x_abs, sin_k, cos_k, sin_y, cosm1_y); | |
| return static_cast<float>(fputil::multiply_add( | |
| sin_y, -sin_k, fputil::multiply_add(cosm1_y, cos_k, cos_k))); | |
| } | |
| } // namespace LIBC_NAMESPACE_DECL | |