Average Error: 58.0 → 0.8
Time: 11.5s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]
↓
\[\left(0.5 \cdot \cos re\right) \cdot \left(-\left(\frac{1}{3} \cdot {im}^{3} + \left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)\right)\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)
↓
\left(0.5 \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 r210352 = 0.5;
        double r210353 = re;
        double r210354 = cos(r210353);
        double r210355 = r210352 * r210354;
        double r210356 = 0.0;
        double r210357 = im;
        double r210358 = r210356 - r210357;
        double r210359 = exp(r210358);
        double r210360 = exp(r210357);
        double r210361 = r210359 - r210360;
        double r210362 = r210355 * r210361;
        return r210362;
}
↓
double f(double re, double im) {
        double r210363 = 0.5;
        double r210364 = re;
        double r210365 = cos(r210364);
        double r210366 = r210363 * r210365;
        double r210367 = 0.3333333333333333;
        double r210368 = im;
        double r210369 = 3.0;
        double r210370 = pow(r210368, r210369);
        double r210371 = r210367 * r210370;
        double r210372 = 0.016666666666666666;
        double r210373 = 5.0;
        double r210374 = pow(r210368, r210373);
        double r210375 = r210372 * r210374;
        double r210376 = 2.0;
        double r210377 = r210376 * r210368;
        double r210378 = r210375 + r210377;
        double r210379 = r210371 + r210378;
        double r210380 = -r210379;
        double r210381 = r210366 * r210380;
        return r210381;
}

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
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(0.166666666666666657 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.00833333333333333322 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\\ \end{array}\]

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

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

Reproduce

herbie shell --seed 2020024 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"
  :precision binary64

  :herbie-target
  (if (< (fabs im) 1) (- (* (cos re) (+ (+ im (* (* (* 0.16666666666666666 im) im) im)) (* (* (* (* (* 0.008333333333333333 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))))