Average Error: 0.0 → 0.0
Time: 27.4s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0.0 - im} + e^{im}\right)\]
↓
\[0.5 \cdot \mathsf{fma}\left(e^{im}, \sin re, \frac{\sin re}{e^{im - 0.0}}\right)\]
\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 - 0.0}}\right)
double f(double re, double im) {
        double r17590 = 0.5;
        double r17591 = re;
        double r17592 = sin(r17591);
        double r17593 = r17590 * r17592;
        double r17594 = 0.0;
        double r17595 = im;
        double r17596 = r17594 - r17595;
        double r17597 = exp(r17596);
        double r17598 = exp(r17595);
        double r17599 = r17597 + r17598;
        double r17600 = r17593 * r17599;
        return r17600;
}
↓
double f(double re, double im) {
        double r17601 = 0.5;
        double r17602 = im;
        double r17603 = exp(r17602);
        double r17604 = re;
        double r17605 = sin(r17604);
        double r17606 = 0.0;
        double r17607 = r17602 - r17606;
        double r17608 = exp(r17607);
        double r17609 = r17605 / r17608;
        double r17610 = fma(r17603, r17605, r17609);
        double r17611 = r17601 * r17610;
        return r17611;
}

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. Using strategy rm

  5. Applied exp-diff0.0

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

  6. Applied associate-*r/0.0

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

  7. Taylor expanded around inf 0.0

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

  8. Simplified0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2019323 +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))))