Average Error: 0.0 → 0.0
Time: 12.7s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0.0 - im} + e^{im}\right)\]
↓
\[\mathsf{fma}\left(e^{im}, \sin re, \frac{\sin re}{e^{im}}\right) \cdot 0.5\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{0.0 - im} + e^{im}\right)
↓
\mathsf{fma}\left(e^{im}, \sin re, \frac{\sin re}{e^{im}}\right) \cdot 0.5
double f(double re, double im) {
        double r631539 = 0.5;
        double r631540 = re;
        double r631541 = sin(r631540);
        double r631542 = r631539 * r631541;
        double r631543 = 0.0;
        double r631544 = im;
        double r631545 = r631543 - r631544;
        double r631546 = exp(r631545);
        double r631547 = exp(r631544);
        double r631548 = r631546 + r631547;
        double r631549 = r631542 * r631548;
        return r631549;
}
↓
double f(double re, double im) {
        double r631550 = im;
        double r631551 = exp(r631550);
        double r631552 = re;
        double r631553 = sin(r631552);
        double r631554 = r631553 / r631551;
        double r631555 = fma(r631551, r631553, r631554);
        double r631556 = 0.5;
        double r631557 = r631555 * r631556;
        return r631557;
}

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. Taylor expanded around inf 0.0

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

  3. Simplified0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (re im)
  :name "math.sin on complex, real part"
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))