Average Error: 43.5 → 0.7
Time: 47.4s
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 r5934920 = 0.5;
        double r5934921 = re;
        double r5934922 = sin(r5934921);
        double r5934923 = r5934920 * r5934922;
        double r5934924 = im;
        double r5934925 = -r5934924;
        double r5934926 = exp(r5934925);
        double r5934927 = exp(r5934924);
        double r5934928 = r5934926 - r5934927;
        double r5934929 = r5934923 * r5934928;
        return r5934929;
}
↓
double f(double re, double im) {
        double r5934930 = -0.3333333333333333;
        double r5934931 = im;
        double r5934932 = r5934931 * r5934931;
        double r5934933 = r5934931 * r5934932;
        double r5934934 = r5934930 * r5934933;
        double r5934935 = 0.016666666666666666;
        double r5934936 = 5.0;
        double r5934937 = pow(r5934931, r5934936);
        double r5934938 = r5934931 + r5934931;
        double r5934939 = fma(r5934935, r5934937, r5934938);
        double r5934940 = r5934934 - r5934939;
        double r5934941 = 0.5;
        double r5934942 = re;
        double r5934943 = sin(r5934942);
        double r5934944 = r5934941 * r5934943;
        double r5934945 = r5934940 * r5934944;
        return r5934945;
}

Error

Bits error versus re

Bits error versus im

Target

Original43.5
Target0.3
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 43.5

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

  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

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019149 +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))))