Average Error: 58.2 → 0.7
Time: 37.5s
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 r8716531 = 0.5;
        double r8716532 = re;
        double r8716533 = cos(r8716532);
        double r8716534 = r8716531 * r8716533;
        double r8716535 = 0.0;
        double r8716536 = im;
        double r8716537 = r8716535 - r8716536;
        double r8716538 = exp(r8716537);
        double r8716539 = exp(r8716536);
        double r8716540 = r8716538 - r8716539;
        double r8716541 = r8716534 * r8716540;
        return r8716541;
}
↓
double f(double re, double im) {
        double r8716542 = -0.016666666666666666;
        double r8716543 = im;
        double r8716544 = 5.0;
        double r8716545 = pow(r8716543, r8716544);
        double r8716546 = r8716542 * r8716545;
        double r8716547 = r8716543 + r8716543;
        double r8716548 = r8716546 - r8716547;
        double r8716549 = r8716543 * r8716543;
        double r8716550 = 0.3333333333333333;
        double r8716551 = r8716543 * r8716550;
        double r8716552 = r8716549 * r8716551;
        double r8716553 = r8716548 - r8716552;
        double r8716554 = 0.5;
        double r8716555 = re;
        double r8716556 = cos(r8716555);
        double r8716557 = r8716554 * r8716556;
        double r8716558 = r8716553 * r8716557;
        return r8716558;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.2
Target0.2
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 58.2

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019158 
(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))))