Average Error: 58.3 → 0.6
Time: 33.5s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]
↓
\[\left(\frac{-1}{3} \cdot \left(im \cdot \left(im \cdot im\right)\right) - \mathsf{fma}\left(\frac{1}{60}, {im}^{5}, im \cdot 2\right)\right) \cdot \left(0.5 \cdot \cos re\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)
↓
\left(\frac{-1}{3} \cdot \left(im \cdot \left(im \cdot im\right)\right) - \mathsf{fma}\left(\frac{1}{60}, {im}^{5}, im \cdot 2\right)\right) \cdot \left(0.5 \cdot \cos re\right)
double f(double re, double im) {
        double r6780781 = 0.5;
        double r6780782 = re;
        double r6780783 = cos(r6780782);
        double r6780784 = r6780781 * r6780783;
        double r6780785 = 0.0;
        double r6780786 = im;
        double r6780787 = r6780785 - r6780786;
        double r6780788 = exp(r6780787);
        double r6780789 = exp(r6780786);
        double r6780790 = r6780788 - r6780789;
        double r6780791 = r6780784 * r6780790;
        return r6780791;
}
↓
double f(double re, double im) {
        double r6780792 = -0.3333333333333333;
        double r6780793 = im;
        double r6780794 = r6780793 * r6780793;
        double r6780795 = r6780793 * r6780794;
        double r6780796 = r6780792 * r6780795;
        double r6780797 = 0.016666666666666666;
        double r6780798 = 5.0;
        double r6780799 = pow(r6780793, r6780798);
        double r6780800 = 2.0;
        double r6780801 = r6780793 * r6780800;
        double r6780802 = fma(r6780797, r6780799, r6780801);
        double r6780803 = r6780796 - r6780802;
        double r6780804 = 0.5;
        double r6780805 = re;
        double r6780806 = cos(r6780805);
        double r6780807 = r6780804 * r6780806;
        double r6780808 = r6780803 * r6780807;
        return r6780808;
}

Error

Bits error versus re

Bits error versus im

Target

Original58.3
Target0.3
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(0.1666666666666666574148081281236954964697 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333217685101601546193705872 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 58.3

    \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]

  2. Taylor expanded around 0 0.6

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

  3. Simplified0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (re im)
  :name "math.sin on complex, imaginary part"

  :herbie-target
  (if (< (fabs im) 1.0) (- (* (cos re) (+ (+ im (* (* (* 0.16666666666666666 im) im) im)) (* (* (* (* (* 0.008333333333333333 im) im) im) im) im)))) (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))

  (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))