Average Error: 43.7 → 0.9
Time: 31.9s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(\frac{-1}{3} \cdot \left(im \cdot \left(im \cdot im\right)\right) - \mathsf{fma}\left(\frac{1}{60}, {im}^{5}, im + im\right)\right) \cdot \left(0.5 \cdot \sin re\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-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 + im\right)\right) \cdot \left(0.5 \cdot \sin re\right)
double f(double re, double im) {
        double r8087789 = 0.5;
        double r8087790 = re;
        double r8087791 = sin(r8087790);
        double r8087792 = r8087789 * r8087791;
        double r8087793 = im;
        double r8087794 = -r8087793;
        double r8087795 = exp(r8087794);
        double r8087796 = exp(r8087793);
        double r8087797 = r8087795 - r8087796;
        double r8087798 = r8087792 * r8087797;
        return r8087798;
}
↓
double f(double re, double im) {
        double r8087799 = -0.3333333333333333;
        double r8087800 = im;
        double r8087801 = r8087800 * r8087800;
        double r8087802 = r8087800 * r8087801;
        double r8087803 = r8087799 * r8087802;
        double r8087804 = 0.016666666666666666;
        double r8087805 = 5.0;
        double r8087806 = pow(r8087800, r8087805);
        double r8087807 = r8087800 + r8087800;
        double r8087808 = fma(r8087804, r8087806, r8087807);
        double r8087809 = r8087803 - r8087808;
        double r8087810 = 0.5;
        double r8087811 = re;
        double r8087812 = sin(r8087811);
        double r8087813 = r8087810 * r8087812;
        double r8087814 = r8087809 * r8087813;
        return r8087814;
}

Error

Bits error versus re

Bits error versus im

Target

Original43.7
Target0.3
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 43.7

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

  2. Taylor expanded around 0 0.9

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

  3. Simplified0.9

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

  4. Final simplification0.9

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

Reproduce

herbie shell --seed 2019165 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, imaginary part"

  :herbie-target
  (if (< (fabs im) 1) (- (* (sin re) (+ (+ im (* (* (* 1/6 im) im) im)) (* (* (* (* (* 1/120 im) im) im) im) im)))) (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))

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