Average Error: 0.1 → 0.1
Time: 25.7s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0.0 - im} + e^{im}\right)\]
↓
\[0.5 \cdot \mathsf{fma}\left(\sin re, e^{im}, \frac{\sin re}{e^{im}}\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{0.0 - im} + e^{im}\right)
↓
0.5 \cdot \mathsf{fma}\left(\sin re, e^{im}, \frac{\sin re}{e^{im}}\right)
double f(double re, double im) {
        double r1140854 = 0.5;
        double r1140855 = re;
        double r1140856 = sin(r1140855);
        double r1140857 = r1140854 * r1140856;
        double r1140858 = 0.0;
        double r1140859 = im;
        double r1140860 = r1140858 - r1140859;
        double r1140861 = exp(r1140860);
        double r1140862 = exp(r1140859);
        double r1140863 = r1140861 + r1140862;
        double r1140864 = r1140857 * r1140863;
        return r1140864;
}
↓
double f(double re, double im) {
        double r1140865 = 0.5;
        double r1140866 = re;
        double r1140867 = sin(r1140866);
        double r1140868 = im;
        double r1140869 = exp(r1140868);
        double r1140870 = r1140867 / r1140869;
        double r1140871 = fma(r1140867, r1140869, r1140870);
        double r1140872 = r1140865 * r1140871;
        return r1140872;
}

Error

Bits error versus re

Bits error versus im

Derivation

  1. Initial program 0.1

    \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0.0 - im} + e^{im}\right)\]

  2. Taylor expanded around inf 0.1

    \[\leadsto \color{blue}{0.5 \cdot \left(\sin re \cdot \left(e^{-im} + e^{im}\right)\right)}\]

  3. Simplified0.1

    \[\leadsto \color{blue}{0.5 \cdot \mathsf{fma}\left(\sin re, e^{im}, \frac{\sin re}{e^{im}}\right)}\]

  4. Final simplification0.1

    \[\leadsto 0.5 \cdot \mathsf{fma}\left(\sin re, e^{im}, \frac{\sin re}{e^{im}}\right)\]

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (re im)
  :name "math.sin on complex, real part"
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))