Average Error: 58.0 → 0.8
Time: 23.3s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]
↓
\[\left(\frac{1}{2} \cdot \cos re\right) \cdot \left(-\left(\frac{1}{3} \cdot {im}^{3} + \left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)\right)\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)
↓
\left(\frac{1}{2} \cdot \cos re\right) \cdot \left(-\left(\frac{1}{3} \cdot {im}^{3} + \left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)\right)\right)
double f(double re, double im) {
        double r211437 = 0.5;
        double r211438 = re;
        double r211439 = cos(r211438);
        double r211440 = r211437 * r211439;
        double r211441 = 0.0;
        double r211442 = im;
        double r211443 = r211441 - r211442;
        double r211444 = exp(r211443);
        double r211445 = exp(r211442);
        double r211446 = r211444 - r211445;
        double r211447 = r211440 * r211446;
        return r211447;
}
↓
double f(double re, double im) {
        double r211448 = 1.0;
        double r211449 = 2.0;
        double r211450 = r211448 / r211449;
        double r211451 = re;
        double r211452 = cos(r211451);
        double r211453 = r211450 * r211452;
        double r211454 = 0.3333333333333333;
        double r211455 = im;
        double r211456 = 3.0;
        double r211457 = pow(r211455, r211456);
        double r211458 = r211454 * r211457;
        double r211459 = 0.016666666666666666;
        double r211460 = 5.0;
        double r211461 = pow(r211455, r211460);
        double r211462 = r211459 * r211461;
        double r211463 = 2.0;
        double r211464 = r211463 * r211455;
        double r211465 = r211462 + r211464;
        double r211466 = r211458 + r211465;
        double r211467 = -r211466;
        double r211468 = r211453 * r211467;
        return r211468;
}

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
Target0.2
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(0.1666666666666666574148081281236954964697 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333217685101601546193705872 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 58.0

    \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]

  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. Final simplification0.8

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

Reproduce

herbie shell --seed 2019304 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"
  :precision binary64

  :herbie-target
  (if (< (fabs im) 1) (- (* (cos re) (+ (+ im (* (* (* 0.166666666666666657 im) im) im)) (* (* (* (* (* 0.00833333333333333322 im) im) im) im) im)))) (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))

  (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))