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 r336477 = 0.5;
        double r336478 = re;
        double r336479 = sin(r336478);
        double r336480 = r336477 * r336479;
        double r336481 = 0.0;
        double r336482 = im;
        double r336483 = r336481 - r336482;
        double r336484 = exp(r336483);
        double r336485 = exp(r336482);
        double r336486 = r336484 + r336485;
        double r336487 = r336480 * r336486;
        return r336487;
}
↓
double f(double re, double im) {
        double r336488 = re;
        double r336489 = sin(r336488);
        double r336490 = im;
        double r336491 = exp(r336490);
        double r336492 = 0.5;
        double r336493 = r336492 / r336491;
        double r336494 = fma(r336491, r336492, r336493);
        double r336495 = r336489 * r336494;
        return r336495;
}

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 2019168 +o rules:numerics
(FPCore (re im)
  :name "math.sin on complex, real part"
  (* (* 0.5 (sin re)) (+ (exp (- 0 im)) (exp im))))