Average Error: 43.7 → 0.7
Time: 32.7s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left({im}^{5} \cdot \frac{-1}{60} - \left(2 - \frac{-1}{3} \cdot \left(im \cdot im\right)\right) \cdot im\right) \cdot \left(0.5 \cdot \sin re\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left({im}^{5} \cdot \frac{-1}{60} - \left(2 - \frac{-1}{3} \cdot \left(im \cdot im\right)\right) \cdot im\right) \cdot \left(0.5 \cdot \sin re\right)
double f(double re, double im) {
        double r9799927 = 0.5;
        double r9799928 = re;
        double r9799929 = sin(r9799928);
        double r9799930 = r9799927 * r9799929;
        double r9799931 = im;
        double r9799932 = -r9799931;
        double r9799933 = exp(r9799932);
        double r9799934 = exp(r9799931);
        double r9799935 = r9799933 - r9799934;
        double r9799936 = r9799930 * r9799935;
        return r9799936;
}
↓
double f(double re, double im) {
        double r9799937 = im;
        double r9799938 = 5.0;
        double r9799939 = pow(r9799937, r9799938);
        double r9799940 = -0.016666666666666666;
        double r9799941 = r9799939 * r9799940;
        double r9799942 = 2.0;
        double r9799943 = -0.3333333333333333;
        double r9799944 = r9799937 * r9799937;
        double r9799945 = r9799943 * r9799944;
        double r9799946 = r9799942 - r9799945;
        double r9799947 = r9799946 * r9799937;
        double r9799948 = r9799941 - r9799947;
        double r9799949 = 0.5;
        double r9799950 = re;
        double r9799951 = sin(r9799950);
        double r9799952 = r9799949 * r9799951;
        double r9799953 = r9799948 * r9799952;
        return r9799953;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original43.7
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 43.7

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

  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

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

  5. Applied pow10.7

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

  6. Applied pow10.7

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

  7. Applied pow10.7

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

  8. Applied pow-prod-down0.7

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

  9. Applied pow-prod-down0.7

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

  10. Simplified0.7

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

  11. Final simplification0.7

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

Reproduce

herbie shell --seed 2019171 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"

  :herbie-target
  (if (< (fabs im) 1.0) (- (* (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))))