Average Error: 0.0 → 0.0
Time: 26.9s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0.0 - im} + e^{im}\right)\]
↓
\[0.5 \cdot \mathsf{fma}\left(e^{im}, \sin re, \frac{\sin re}{e^{im}}\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{0.0 - im} + e^{im}\right)
↓
0.5 \cdot \mathsf{fma}\left(e^{im}, \sin re, \frac{\sin re}{e^{im}}\right)
double f(double re, double im) {
        double r1255634 = 0.5;
        double r1255635 = re;
        double r1255636 = sin(r1255635);
        double r1255637 = r1255634 * r1255636;
        double r1255638 = 0.0;
        double r1255639 = im;
        double r1255640 = r1255638 - r1255639;
        double r1255641 = exp(r1255640);
        double r1255642 = exp(r1255639);
        double r1255643 = r1255641 + r1255642;
        double r1255644 = r1255637 * r1255643;
        return r1255644;
}
↓
double f(double re, double im) {
        double r1255645 = 0.5;
        double r1255646 = im;
        double r1255647 = exp(r1255646);
        double r1255648 = re;
        double r1255649 = sin(r1255648);
        double r1255650 = r1255649 / r1255647;
        double r1255651 = fma(r1255647, r1255649, r1255650);
        double r1255652 = r1255645 * r1255651;
        return r1255652;
}

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

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (re im)
  :name "math.sin on complex, real part"
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))