Average Error: 43.5 → 0.8
Time: 31.6s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(\frac{-1}{3}, {im}^{3}, \mathsf{fma}\left(im, -2, {im}^{5} \cdot \frac{-1}{60}\right)\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(\frac{-1}{3}, {im}^{3}, \mathsf{fma}\left(im, -2, {im}^{5} \cdot \frac{-1}{60}\right)\right)
double f(double re, double im) {
        double r128681 = 0.5;
        double r128682 = re;
        double r128683 = sin(r128682);
        double r128684 = r128681 * r128683;
        double r128685 = im;
        double r128686 = -r128685;
        double r128687 = exp(r128686);
        double r128688 = exp(r128685);
        double r128689 = r128687 - r128688;
        double r128690 = r128684 * r128689;
        return r128690;
}
↓
double f(double re, double im) {
        double r128691 = 0.5;
        double r128692 = re;
        double r128693 = sin(r128692);
        double r128694 = r128691 * r128693;
        double r128695 = -0.3333333333333333;
        double r128696 = im;
        double r128697 = 3.0;
        double r128698 = pow(r128696, r128697);
        double r128699 = -2.0;
        double r128700 = 5.0;
        double r128701 = pow(r128696, r128700);
        double r128702 = -0.016666666666666666;
        double r128703 = r128701 * r128702;
        double r128704 = fma(r128696, r128699, r128703);
        double r128705 = fma(r128695, r128698, r128704);
        double r128706 = r128694 * r128705;
        return r128706;
}

Error

Bits error versus re

Bits error versus im

Target

Original43.5
Target0.3
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.1666666666666666574148081281236954964697 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333217685101601546193705872 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 43.5

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

  2. Taylor expanded around 0 0.8

    \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(-\left(\frac{1}{3} \cdot {im}^{3} + \left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)\right)\right)}\]

  3. Simplified0.8

    \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{3}, {im}^{3}, \mathsf{fma}\left(im, -2, {im}^{5} \cdot \frac{-1}{60}\right)\right)}\]

  4. Final simplification0.8

    \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(\frac{-1}{3}, {im}^{3}, \mathsf{fma}\left(im, -2, {im}^{5} \cdot \frac{-1}{60}\right)\right)\]

Reproduce

herbie shell --seed 2019306 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64

  :herbie-target
  (if (< (fabs im) 1) (- (* (sin re) (+ (+ im (* (* (* 0.166666666666666657 im) im) im)) (* (* (* (* (* 0.00833333333333333322 im) im) im) im) im)))) (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))

  (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))