Average Error: 0.1 → 0.1
Time: 26.6s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0.0 - im} + e^{im}\right)\]
↓
\[\mathsf{fma}\left(\sin re, e^{im}, \frac{\sin re}{e^{im}}\right) \cdot 0.5\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{0.0 - 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 r791346 = 0.5;
        double r791347 = re;
        double r791348 = sin(r791347);
        double r791349 = r791346 * r791348;
        double r791350 = 0.0;
        double r791351 = im;
        double r791352 = r791350 - r791351;
        double r791353 = exp(r791352);
        double r791354 = exp(r791351);
        double r791355 = r791353 + r791354;
        double r791356 = r791349 * r791355;
        return r791356;
}
↓
double f(double re, double im) {
        double r791357 = re;
        double r791358 = sin(r791357);
        double r791359 = im;
        double r791360 = exp(r791359);
        double r791361 = r791358 / r791360;
        double r791362 = fma(r791358, r791360, r791361);
        double r791363 = 0.5;
        double r791364 = r791362 * r791363;
        return r791364;
}

Error

Bits error versus re

Bits error versus im

Derivation

  1. Initial program 0.1

    \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0.0 - im} + e^{im}\right)\]

  2. Taylor expanded around inf 0.1

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

  3. Simplified0.1

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

  4. Final simplification0.1

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

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (re im)
  :name "math.sin on complex, real part"
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))