Average Error: 0.0 → 0.0
Time: 21.7s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)\]
↓
\[\mathsf{fma}\left(\sin re, e^{im}, \frac{\sin re}{e^{im}}\right) \cdot 0.5\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)
↓
\mathsf{fma}\left(\sin re, e^{im}, \frac{\sin re}{e^{im}}\right) \cdot 0.5
double f(double re, double im) {
        double r399753 = 0.5;
        double r399754 = re;
        double r399755 = sin(r399754);
        double r399756 = r399753 * r399755;
        double r399757 = 0.0;
        double r399758 = im;
        double r399759 = r399757 - r399758;
        double r399760 = exp(r399759);
        double r399761 = exp(r399758);
        double r399762 = r399760 + r399761;
        double r399763 = r399756 * r399762;
        return r399763;
}
↓
double f(double re, double im) {
        double r399764 = re;
        double r399765 = sin(r399764);
        double r399766 = im;
        double r399767 = exp(r399766);
        double r399768 = r399765 / r399767;
        double r399769 = fma(r399765, r399767, r399768);
        double r399770 = 0.5;
        double r399771 = r399769 * r399770;
        return r399771;
}

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. 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

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019146 +o rules:numerics
(FPCore (re im)
  :name "math.sin on complex, real part"
  (* (* 0.5 (sin re)) (+ (exp (- 0 im)) (exp im))))