Average Error: 58.0 → 0.9
Time: 34.6s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]
↓
\[\left(\left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{3} - \left(im + im\right)\right) - \frac{1}{60} \cdot {im}^{5}\right) \cdot \left(0.5 \cdot \cos re\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)
↓
\left(\left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{3} - \left(im + im\right)\right) - \frac{1}{60} \cdot {im}^{5}\right) \cdot \left(0.5 \cdot \cos re\right)
double f(double re, double im) {
        double r7917959 = 0.5;
        double r7917960 = re;
        double r7917961 = cos(r7917960);
        double r7917962 = r7917959 * r7917961;
        double r7917963 = 0.0;
        double r7917964 = im;
        double r7917965 = r7917963 - r7917964;
        double r7917966 = exp(r7917965);
        double r7917967 = exp(r7917964);
        double r7917968 = r7917966 - r7917967;
        double r7917969 = r7917962 * r7917968;
        return r7917969;
}
↓
double f(double re, double im) {
        double r7917970 = im;
        double r7917971 = r7917970 * r7917970;
        double r7917972 = r7917970 * r7917971;
        double r7917973 = -0.3333333333333333;
        double r7917974 = r7917972 * r7917973;
        double r7917975 = r7917970 + r7917970;
        double r7917976 = r7917974 - r7917975;
        double r7917977 = 0.016666666666666666;
        double r7917978 = 5.0;
        double r7917979 = pow(r7917970, r7917978);
        double r7917980 = r7917977 * r7917979;
        double r7917981 = r7917976 - r7917980;
        double r7917982 = 0.5;
        double r7917983 = re;
        double r7917984 = cos(r7917983);
        double r7917985 = r7917982 * r7917984;
        double r7917986 = r7917981 * r7917985;
        return r7917986;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.0
Target0.3
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 58.0

    \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]

  2. Taylor expanded around 0 0.9

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

  3. Simplified0.9

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

  4. Final simplification0.9

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

Reproduce

herbie shell --seed 2019165 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"

  :herbie-target
  (if (< (fabs im) 1) (- (* (cos re) (+ (+ im (* (* (* 1/6 im) im) im)) (* (* (* (* (* 1/120 im) im) im) im) im)))) (* (* 0.5 (cos re)) (- (exp (- 0 im)) (exp im))))

  (* (* 0.5 (cos re)) (- (exp (- 0 im)) (exp im))))