Average Error: 43.4 → 0.7
Time: 35.4s
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 r193799 = 0.5;
        double r193800 = re;
        double r193801 = sin(r193800);
        double r193802 = r193799 * r193801;
        double r193803 = im;
        double r193804 = -r193803;
        double r193805 = exp(r193804);
        double r193806 = exp(r193803);
        double r193807 = r193805 - r193806;
        double r193808 = r193802 * r193807;
        return r193808;
}
↓
double f(double re, double im) {
        double r193809 = 0.5;
        double r193810 = re;
        double r193811 = sin(r193810);
        double r193812 = r193809 * r193811;
        double r193813 = -0.3333333333333333;
        double r193814 = im;
        double r193815 = 3.0;
        double r193816 = pow(r193814, r193815);
        double r193817 = -2.0;
        double r193818 = 5.0;
        double r193819 = pow(r193814, r193818);
        double r193820 = -0.016666666666666666;
        double r193821 = r193819 * r193820;
        double r193822 = fma(r193814, r193817, r193821);
        double r193823 = fma(r193813, r193816, r193822);
        double r193824 = r193812 * r193823;
        return r193824;
}

Error

Bits error versus re

Bits error versus im

Target

Original43.4
Target0.3
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.1666666666666666574148081281236954964697 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333217685101601546193705872 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 43.4

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64

  :herbie-target
  (if (< (fabs im) 1) (- (* (sin re) (+ (+ im (* (* (* 0.16666666666666666 im) im) im)) (* (* (* (* (* 0.008333333333333333 im) im) im) im) im)))) (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))

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