Average Error: 43.6 → 0.8
Time: 29.8s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(0.5 \cdot \sin re\right) \cdot \left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{3} - \left(\frac{1}{60} \cdot {im}^{5} + \left(im + im\right)\right)\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left(0.5 \cdot \sin re\right) \cdot \left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{3} - \left(\frac{1}{60} \cdot {im}^{5} + \left(im + im\right)\right)\right)
double f(double re, double im) {
        double r15290890 = 0.5;
        double r15290891 = re;
        double r15290892 = sin(r15290891);
        double r15290893 = r15290890 * r15290892;
        double r15290894 = im;
        double r15290895 = -r15290894;
        double r15290896 = exp(r15290895);
        double r15290897 = exp(r15290894);
        double r15290898 = r15290896 - r15290897;
        double r15290899 = r15290893 * r15290898;
        return r15290899;
}
↓
double f(double re, double im) {
        double r15290900 = 0.5;
        double r15290901 = re;
        double r15290902 = sin(r15290901);
        double r15290903 = r15290900 * r15290902;
        double r15290904 = im;
        double r15290905 = r15290904 * r15290904;
        double r15290906 = r15290904 * r15290905;
        double r15290907 = -0.3333333333333333;
        double r15290908 = r15290906 * r15290907;
        double r15290909 = 0.016666666666666666;
        double r15290910 = 5.0;
        double r15290911 = pow(r15290904, r15290910);
        double r15290912 = r15290909 * r15290911;
        double r15290913 = r15290904 + r15290904;
        double r15290914 = r15290912 + r15290913;
        double r15290915 = r15290908 - r15290914;
        double r15290916 = r15290903 * r15290915;
        return r15290916;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original43.6
Target0.3
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.1666666666666666574148081281236954964697 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333217685101601546193705872 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 43.6

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

  2. Taylor expanded around 0 0.8

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

  3. Simplified0.8

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019174 
(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))))