Average Error: 0.0 → 0.0
Time: 17.7s
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 r500382 = 0.5;
        double r500383 = re;
        double r500384 = sin(r500383);
        double r500385 = r500382 * r500384;
        double r500386 = 0.0;
        double r500387 = im;
        double r500388 = r500386 - r500387;
        double r500389 = exp(r500388);
        double r500390 = exp(r500387);
        double r500391 = r500389 + r500390;
        double r500392 = r500385 * r500391;
        return r500392;
}
↓
double f(double re, double im) {
        double r500393 = 0.5;
        double r500394 = re;
        double r500395 = sin(r500394);
        double r500396 = im;
        double r500397 = exp(r500396);
        double r500398 = r500395 / r500397;
        double r500399 = fma(r500395, r500397, r500398);
        double r500400 = r500393 * r500399;
        return r500400;
}

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 2019164 +o rules:numerics
(FPCore (re im)
  :name "math.sin on complex, real part"
  (* (* 0.5 (sin re)) (+ (exp (- 0 im)) (exp im))))