Average Error: 0.0 → 0.0
Time: 24.9s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)\]
↓
\[\sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)
↓
\sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)
double f(double re, double im) {
        double r524058 = 0.5;
        double r524059 = re;
        double r524060 = sin(r524059);
        double r524061 = r524058 * r524060;
        double r524062 = 0.0;
        double r524063 = im;
        double r524064 = r524062 - r524063;
        double r524065 = exp(r524064);
        double r524066 = exp(r524063);
        double r524067 = r524065 + r524066;
        double r524068 = r524061 * r524067;
        return r524068;
}
↓
double f(double re, double im) {
        double r524069 = re;
        double r524070 = sin(r524069);
        double r524071 = 0.5;
        double r524072 = im;
        double r524073 = exp(r524072);
        double r524074 = r524071 / r524073;
        double r524075 = fma(r524071, r524073, r524074);
        double r524076 = r524070 * r524075;
        return r524076;
}

Error

Bits error versus re

Bits error versus im

Derivation

  1. Initial program 0.0

    \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)\]

  2. Simplified0.0

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019144 +o rules:numerics
(FPCore (re im)
  :name "math.sin on complex, real part"
  (* (* 0.5 (sin re)) (+ (exp (- 0 im)) (exp im))))