diff options
Diffstat (limited to 'libc/src/math/generic/atan.cpp')
| -rw-r--r-- | libc/src/math/generic/atan.cpp | 167 |
1 files changed, 2 insertions, 165 deletions
diff --git a/libc/src/math/generic/atan.cpp b/libc/src/math/generic/atan.cpp index cbca605..93bf2e1 100644 --- a/libc/src/math/generic/atan.cpp +++ b/libc/src/math/generic/atan.cpp @@ -7,173 +7,10 @@ //===----------------------------------------------------------------------===// #include "src/math/atan.h" -#include "atan_utils.h" -#include "src/__support/FPUtil/FEnvImpl.h" -#include "src/__support/FPUtil/FPBits.h" -#include "src/__support/FPUtil/double_double.h" -#include "src/__support/FPUtil/multiply_add.h" -#include "src/__support/FPUtil/nearest_integer.h" -#include "src/__support/macros/config.h" -#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY +#include "src/__support/math/atan.h" namespace LIBC_NAMESPACE_DECL { -// To compute atan(x), we divided it into the following cases: -// * |x| < 2^-26: -// Since |x| > atan(|x|) > |x| - |x|^3/3, and |x|^3/3 < ulp(x)/2, we simply -// return atan(x) = x - sign(x) * epsilon. -// * 2^-26 <= |x| < 1: -// We perform range reduction mod 2^-6 = 1/64 as follow: -// Let k = 2^(-6) * round(|x| * 2^6), then -// atan(x) = sign(x) * atan(|x|) -// = sign(x) * (atan(k) + atan((|x| - k) / (1 + |x|*k)). -// We store atan(k) in a look up table, and perform intermediate steps in -// double-double. -// * 1 < |x| < 2^53: -// First we perform the transformation y = 1/|x|: -// atan(x) = sign(x) * (pi/2 - atan(1/|x|)) -// = sign(x) * (pi/2 - atan(y)). -// Then we compute atan(y) using range reduction mod 2^-6 = 1/64 as the -// previous case: -// Let k = 2^(-6) * round(y * 2^6), then -// atan(y) = atan(k) + atan((y - k) / (1 + y*k)) -// = atan(k) + atan((1/|x| - k) / (1 + k/|x|) -// = atan(k) + atan((1 - k*|x|) / (|x| + k)). -// * |x| >= 2^53: -// Using the reciprocal transformation: -// atan(x) = sign(x) * (pi/2 - atan(1/|x|)). -// We have that: -// atan(1/|x|) <= 1/|x| <= 2^-53, -// which is smaller than ulp(pi/2) / 2. -// So we can return: -// atan(x) = sign(x) * (pi/2 - epsilon) - -LLVM_LIBC_FUNCTION(double, atan, (double x)) { - using FPBits = fputil::FPBits<double>; - - constexpr double IS_NEG[2] = {1.0, -1.0}; - constexpr DoubleDouble PI_OVER_2 = {0x1.1a62633145c07p-54, - 0x1.921fb54442d18p0}; - constexpr DoubleDouble MPI_OVER_2 = {-0x1.1a62633145c07p-54, - -0x1.921fb54442d18p0}; - - FPBits xbits(x); - bool x_sign = xbits.is_neg(); - xbits = xbits.abs(); - uint64_t x_abs = xbits.uintval(); - int x_exp = - static_cast<int>(x_abs >> FPBits::FRACTION_LEN) - FPBits::EXP_BIAS; - - // |x| < 1. - if (x_exp < 0) { - if (LIBC_UNLIKELY(x_exp < -26)) { -#ifdef LIBC_MATH_HAS_SKIP_ACCURATE_PASS - return x; -#else - if (x == 0.0) - return x; - // |x| < 2^-26 - return fputil::multiply_add(-0x1.0p-54, x, x); -#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS - } - - double x_d = xbits.get_val(); - // k = 2^-6 * round(2^6 * |x|) - double k = fputil::nearest_integer(0x1.0p6 * x_d); - unsigned idx = static_cast<unsigned>(k); - k *= 0x1.0p-6; - - // numerator = |x| - k - DoubleDouble num, den; - num.lo = 0.0; - num.hi = x_d - k; - - // denominator = 1 - k * |x| - den.hi = fputil::multiply_add(x_d, k, 1.0); - DoubleDouble prod = fputil::exact_mult(x_d, k); - // Using Dekker's 2SUM algorithm to compute the lower part. - den.lo = ((1.0 - den.hi) + prod.hi) + prod.lo; - - // x_r = (|x| - k) / (1 + k * |x|) - DoubleDouble x_r = fputil::div(num, den); - - // Approximating atan(x_r) using Taylor polynomial. - DoubleDouble p = atan_eval(x_r); - - // atan(x) = sign(x) * (atan(k) + atan(x_r)) - // = sign(x) * (atan(k) + atan( (|x| - k) / (1 + k * |x|) )) -#ifdef LIBC_MATH_HAS_SKIP_ACCURATE_PASS - return IS_NEG[x_sign] * (ATAN_I[idx].hi + (p.hi + (p.lo + ATAN_I[idx].lo))); -#else - - DoubleDouble c0 = fputil::exact_add(ATAN_I[idx].hi, p.hi); - double c1 = c0.lo + (ATAN_I[idx].lo + p.lo); - double r = IS_NEG[x_sign] * (c0.hi + c1); - - return r; -#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS - } - - // |x| >= 2^53 or x is NaN. - if (LIBC_UNLIKELY(x_exp >= 53)) { - // x is 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; - } - // |x| >= 2^53 - // atan(x) ~ sign(x) * pi/2. - if (x_exp >= 53) -#ifdef LIBC_MATH_HAS_SKIP_ACCURATE_PASS - return IS_NEG[x_sign] * PI_OVER_2.hi; -#else - return fputil::multiply_add(IS_NEG[x_sign], PI_OVER_2.hi, - IS_NEG[x_sign] * PI_OVER_2.lo); -#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS - } - - double x_d = xbits.get_val(); - double y = 1.0 / x_d; - - // k = 2^-6 * round(2^6 / |x|) - double k = fputil::nearest_integer(0x1.0p6 * y); - unsigned idx = static_cast<unsigned>(k); - k *= 0x1.0p-6; - - // denominator = |x| + k - DoubleDouble den = fputil::exact_add(x_d, k); - // numerator = 1 - k * |x| - DoubleDouble num; - num.hi = fputil::multiply_add(-x_d, k, 1.0); - DoubleDouble prod = fputil::exact_mult(x_d, k); - // Using Dekker's 2SUM algorithm to compute the lower part. - num.lo = ((1.0 - num.hi) - prod.hi) - prod.lo; - - // x_r = (1/|x| - k) / (1 - k/|x|) - // = (1 - k * |x|) / (|x| - k) - DoubleDouble x_r = fputil::div(num, den); - - // Approximating atan(x_r) using Taylor polynomial. - DoubleDouble p = atan_eval(x_r); - - // atan(x) = sign(x) * (pi/2 - atan(1/|x|)) - // = sign(x) * (pi/2 - atan(k) - atan(x_r)) - // = (-sign(x)) * (-pi/2 + atan(k) + atan((1 - k*|x|)/(|x| - k))) -#ifdef LIBC_MATH_HAS_SKIP_ACCURATE_PASS - double lo_part = p.lo + ATAN_I[idx].lo + MPI_OVER_2.lo; - return IS_NEG[!x_sign] * (MPI_OVER_2.hi + ATAN_I[idx].hi + (p.hi + lo_part)); -#else - DoubleDouble c0 = fputil::exact_add(MPI_OVER_2.hi, ATAN_I[idx].hi); - DoubleDouble c1 = fputil::exact_add(c0.hi, p.hi); - double c2 = c1.lo + (c0.lo + p.lo) + (ATAN_I[idx].lo + MPI_OVER_2.lo); - - double r = IS_NEG[!x_sign] * (c1.hi + c2); - - return r; -#endif -} +LLVM_LIBC_FUNCTION(double, atan, (double x)) { return math::atan(x); } } // namespace LIBC_NAMESPACE_DECL |