Average Error: 0.0 → 0.0
Time: 29.9s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)\]
↓
\[0.5 \cdot \mathsf{fma}\left(\sin re, e^{im}, \frac{\sin re}{e^{im}}\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - 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 r785866 = 0.5;
        double r785867 = re;
        double r785868 = sin(r785867);
        double r785869 = r785866 * r785868;
        double r785870 = 0.0;
        double r785871 = im;
        double r785872 = r785870 - r785871;
        double r785873 = exp(r785872);
        double r785874 = exp(r785871);
        double r785875 = r785873 + r785874;
        double r785876 = r785869 * r785875;
        return r785876;
}
↓
double f(double re, double im) {
        double r785877 = 0.5;
        double r785878 = re;
        double r785879 = sin(r785878);
        double r785880 = im;
        double r785881 = exp(r785880);
        double r785882 = r785879 / r785881;
        double r785883 = fma(r785879, r785881, r785882);
        double r785884 = r785877 * r785883;
        return r785884;
}

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. Using strategy rm

  3. Applied associate-*l*0.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019138 +o rules:numerics
(FPCore (re im)
  :name "math.sin on complex, real part"
  (* (* 0.5 (sin re)) (+ (exp (- 0 im)) (exp im))))