Average Error: 0.0 → 0.0
Time: 19.5s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0.0 - im} + e^{im}\right)\]
↓
\[\mathsf{fma}\left(e^{0.0 - im}, 0.5 \cdot \sin re, e^{im} \cdot \left(0.5 \cdot \sin re\right)\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{0.0 - im} + e^{im}\right)
↓
\mathsf{fma}\left(e^{0.0 - im}, 0.5 \cdot \sin re, e^{im} \cdot \left(0.5 \cdot \sin re\right)\right)
double f(double re, double im) {
        double r70356 = 0.5;
        double r70357 = re;
        double r70358 = sin(r70357);
        double r70359 = r70356 * r70358;
        double r70360 = 0.0;
        double r70361 = im;
        double r70362 = r70360 - r70361;
        double r70363 = exp(r70362);
        double r70364 = exp(r70361);
        double r70365 = r70363 + r70364;
        double r70366 = r70359 * r70365;
        return r70366;
}
↓
double f(double re, double im) {
        double r70367 = 0.0;
        double r70368 = im;
        double r70369 = r70367 - r70368;
        double r70370 = exp(r70369);
        double r70371 = 0.5;
        double r70372 = re;
        double r70373 = sin(r70372);
        double r70374 = r70371 * r70373;
        double r70375 = exp(r70368);
        double r70376 = r70375 * r70374;
        double r70377 = fma(r70370, r70374, r70376);
        return r70377;
}

Error

Bits error versus re

Bits error versus im

Derivation

  1. Initial program 0.0

    \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0.0 - im} + e^{im}\right)\]
  2. Using strategy rm

  3. Applied distribute-lft-in0.0

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

  4. Simplified0.0

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

  5. Simplified0.0

    \[\leadsto e^{0.0 - im} \cdot \left(0.5 \cdot \sin re\right) + \color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right)}\]
  6. Using strategy rm

  7. Applied fma-def0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(e^{0.0 - im}, 0.5 \cdot \sin re, e^{im} \cdot \left(0.5 \cdot \sin re\right)\right)}\]

  8. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(e^{0.0 - im}, 0.5 \cdot \sin re, e^{im} \cdot \left(0.5 \cdot \sin re\right)\right)\]

Reproduce

herbie shell --seed 2020043 +o rules:numerics
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))