Average Error: 57.9 → 0.8
Time: 38.7s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]
↓
\[\left(\left(\frac{-1}{60} \cdot {im}^{5} - \left(im + im\right)\right) - \left(im \cdot im\right) \cdot \left(im \cdot \frac{1}{3}\right)\right) \cdot \left(0.5 \cdot \cos re\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)
↓
\left(\left(\frac{-1}{60} \cdot {im}^{5} - \left(im + im\right)\right) - \left(im \cdot im\right) \cdot \left(im \cdot \frac{1}{3}\right)\right) \cdot \left(0.5 \cdot \cos re\right)
double f(double re, double im) {
        double r7368087 = 0.5;
        double r7368088 = re;
        double r7368089 = cos(r7368088);
        double r7368090 = r7368087 * r7368089;
        double r7368091 = 0.0;
        double r7368092 = im;
        double r7368093 = r7368091 - r7368092;
        double r7368094 = exp(r7368093);
        double r7368095 = exp(r7368092);
        double r7368096 = r7368094 - r7368095;
        double r7368097 = r7368090 * r7368096;
        return r7368097;
}
↓
double f(double re, double im) {
        double r7368098 = -0.016666666666666666;
        double r7368099 = im;
        double r7368100 = 5.0;
        double r7368101 = pow(r7368099, r7368100);
        double r7368102 = r7368098 * r7368101;
        double r7368103 = r7368099 + r7368099;
        double r7368104 = r7368102 - r7368103;
        double r7368105 = r7368099 * r7368099;
        double r7368106 = 0.3333333333333333;
        double r7368107 = r7368099 * r7368106;
        double r7368108 = r7368105 * r7368107;
        double r7368109 = r7368104 - r7368108;
        double r7368110 = 0.5;
        double r7368111 = re;
        double r7368112 = cos(r7368111);
        double r7368113 = r7368110 * r7368112;
        double r7368114 = r7368109 * r7368113;
        return r7368114;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original57.9
Target0.2
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 57.9

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

  2. Taylor expanded around 0 0.8

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

  3. Simplified0.8

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019162 
(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))))