Average Error: 0.0 → 0.0
Time: 19.6s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)\]
↓
\[\sin re \cdot \mathsf{fma}\left(e^{im}, 0.5, \frac{0.5}{e^{im}}\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)
↓
\sin re \cdot \mathsf{fma}\left(e^{im}, 0.5, \frac{0.5}{e^{im}}\right)
double f(double re, double im) {
        double r513682 = 0.5;
        double r513683 = re;
        double r513684 = sin(r513683);
        double r513685 = r513682 * r513684;
        double r513686 = 0.0;
        double r513687 = im;
        double r513688 = r513686 - r513687;
        double r513689 = exp(r513688);
        double r513690 = exp(r513687);
        double r513691 = r513689 + r513690;
        double r513692 = r513685 * r513691;
        return r513692;
}
↓
double f(double re, double im) {
        double r513693 = re;
        double r513694 = sin(r513693);
        double r513695 = im;
        double r513696 = exp(r513695);
        double r513697 = 0.5;
        double r513698 = r513697 / r513696;
        double r513699 = fma(r513696, r513697, r513698);
        double r513700 = r513694 * r513699;
        return r513700;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (re im)
  :name "math.sin on complex, real part"
  (* (* 0.5 (sin re)) (+ (exp (- 0 im)) (exp im))))