Average Error: 0.0 → 0.0
Time: 5.1s
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 r22701 = 0.5;
        double r22702 = re;
        double r22703 = sin(r22702);
        double r22704 = r22701 * r22703;
        double r22705 = 0.0;
        double r22706 = im;
        double r22707 = r22705 - r22706;
        double r22708 = exp(r22707);
        double r22709 = exp(r22706);
        double r22710 = r22708 + r22709;
        double r22711 = r22704 * r22710;
        return r22711;
}
↓
double f(double re, double im) {
        double r22712 = 0.5;
        double r22713 = re;
        double r22714 = sin(r22713);
        double r22715 = r22712 * r22714;
        double r22716 = 0.0;
        double r22717 = im;
        double r22718 = r22716 - r22717;
        double r22719 = exp(r22718);
        double r22720 = exp(r22717);
        double r22721 = r22715 * r22720;
        double r22722 = fma(r22715, r22719, r22721);
        return r22722;
}

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 2020039 +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))))