Average Error: 0.0 → 0.0
Time: 29.7s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0.0 - im} + e^{im}\right)\]
↓
\[\mathsf{fma}\left(e^{im}, \sin re, \frac{\sin re}{e^{im}}\right) \cdot 0.5\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{0.0 - im} + e^{im}\right)
↓
\mathsf{fma}\left(e^{im}, \sin re, \frac{\sin re}{e^{im}}\right) \cdot 0.5
double f(double re, double im) {
        double r605891 = 0.5;
        double r605892 = re;
        double r605893 = sin(r605892);
        double r605894 = r605891 * r605893;
        double r605895 = 0.0;
        double r605896 = im;
        double r605897 = r605895 - r605896;
        double r605898 = exp(r605897);
        double r605899 = exp(r605896);
        double r605900 = r605898 + r605899;
        double r605901 = r605894 * r605900;
        return r605901;
}
↓
double f(double re, double im) {
        double r605902 = im;
        double r605903 = exp(r605902);
        double r605904 = re;
        double r605905 = sin(r605904);
        double r605906 = r605905 / r605903;
        double r605907 = fma(r605903, r605905, r605906);
        double r605908 = 0.5;
        double r605909 = r605907 * r605908;
        return r605909;
}

Error

Bits error versus re

Bits error versus im

Derivation

  1. Initial program 0.0

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

  2. Taylor expanded around inf 0.0

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

  3. Simplified0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (re im)
  :name "math.sin on complex, real part"
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))