Average Error: 0.0 → 0.0
Time: 5.3s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0.0 - im} + e^{im}\right)\]
↓
\[\mathsf{fma}\left(0.5 \cdot \sin re, e^{0.0 - im}, \left(0.5 \cdot \sin re\right) \cdot e^{im}\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{0.0 - im} + e^{im}\right)
↓
\mathsf{fma}\left(0.5 \cdot \sin re, e^{0.0 - im}, \left(0.5 \cdot \sin re\right) \cdot e^{im}\right)
double f(double re, double im) {
        double r30767 = 0.5;
        double r30768 = re;
        double r30769 = sin(r30768);
        double r30770 = r30767 * r30769;
        double r30771 = 0.0;
        double r30772 = im;
        double r30773 = r30771 - r30772;
        double r30774 = exp(r30773);
        double r30775 = exp(r30772);
        double r30776 = r30774 + r30775;
        double r30777 = r30770 * r30776;
        return r30777;
}
↓
double f(double re, double im) {
        double r30778 = 0.5;
        double r30779 = re;
        double r30780 = sin(r30779);
        double r30781 = r30778 * r30780;
        double r30782 = 0.0;
        double r30783 = im;
        double r30784 = r30782 - r30783;
        double r30785 = exp(r30784);
        double r30786 = exp(r30783);
        double r30787 = r30781 * r30786;
        double r30788 = fma(r30781, r30785, r30787);
        return r30788;
}

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 fma-def0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2020021 +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))))