exp_reduced_() [1/2]

double fl::detail::exp_reduced_ ( double x )

inline

Definition at line 156 of file math.cpp.hpp.

                                                  {
    // AVR builds double as 32 bits: the float path is the double path there,
    // and the 64-bit bit patterns below would read past the value.
    if (sizeof(double) == sizeof(float)) {
        return static_cast<double>(exp_reduced_(static_cast<float>(x)));
    }
    DoubleBits xb;
    xb.f = x;
    const u64 ax = xb.u & 0x7fffffffffffffffull;
    if (ax >= 0x40862e42fefa39efull) {          // |x| >= 709.7827, or NaN
        if (ax > 0x7ff0000000000000ull) return x;  // NaN
        if ((xb.u >> 63) == 0u) {                  // overflow -> +inf
            DoubleBits inf;
            inf.u = 0x7ff0000000000000ull;
            return inf.f;
        }
        if (ax >= 0x40874910d52d3052ull) return 0.0;  // x <= -745.133: underflow
    }
    const double kMagic = 6755399441055744.0;  // 1.5 * 2^52
    DoubleBits t;
    t.f = x * 1.44269504088896338700e+00 + kMagic;
    const int k = static_cast<int>(t.i - 0x4338000000000000ll);
    const double kd = t.f - kMagic;
    // fdlibm e_exp: e^r = 1 + 2r/(2 - c(r)) with c a degree-5 polynomial in
    // r^2. Five multiply-adds and one division reach < 1 ulp, half the work of
    // a Taylor series of the same accuracy; that matters on cores that do
    // double in software.
    const double hi = x - kd * 6.93147180369123816490e-01;
    const double lo = kd * 1.90821492927058770002e-10;
    const double r = hi - lo;
    const double r2 = r * r;
    const double c = r - r2 * (1.66666666666666019037e-01 +
                              r2 * (-2.77777777770155933842e-03 +
                                   r2 * (6.61375632143793436117e-05 +
                                        r2 * (-1.65339022054652515390e-06 +
                                             r2 * 4.13813679705723846039e-08))));
    const double p = 1.0 - ((lo - (r * c) / (2.0 - c)) - hi);
    return scale_pow2_(p, k);
}

References exp_reduced_(), fl::detail::DoubleBits::f, fl::FL_NO_EXCEPT, scale_pow2_(), fl::t, fl::detail::DoubleBits::u, and fl::x.

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