Average Error: 0.0 → 0.0
Time: 26.7s
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 r511480 = 0.5;
        double r511481 = re;
        double r511482 = sin(r511481);
        double r511483 = r511480 * r511482;
        double r511484 = 0.0;
        double r511485 = im;
        double r511486 = r511484 - r511485;
        double r511487 = exp(r511486);
        double r511488 = exp(r511485);
        double r511489 = r511487 + r511488;
        double r511490 = r511483 * r511489;
        return r511490;
}
↓
double f(double re, double im) {
        double r511491 = re;
        double r511492 = sin(r511491);
        double r511493 = 0.5;
        double r511494 = im;
        double r511495 = exp(r511494);
        double r511496 = r511493 / r511495;
        double r511497 = fma(r511493, r511495, r511496);
        double r511498 = r511492 * r511497;
        return r511498;
}

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 2019152 +o rules:numerics
(FPCore (re im)
  :name "math.sin on complex, real part"
  (* (* 0.5 (sin re)) (+ (exp (- 0 im)) (exp im))))