Average Error: 58.1 → 0.8
Time: 12.3s
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 r178313 = 0.5;
        double r178314 = re;
        double r178315 = cos(r178314);
        double r178316 = r178313 * r178315;
        double r178317 = 0.0;
        double r178318 = im;
        double r178319 = r178317 - r178318;
        double r178320 = exp(r178319);
        double r178321 = exp(r178318);
        double r178322 = r178320 - r178321;
        double r178323 = r178316 * r178322;
        return r178323;
}
↓
double f(double re, double im) {
        double r178324 = 0.5;
        double r178325 = re;
        double r178326 = cos(r178325);
        double r178327 = r178324 * r178326;
        double r178328 = 0.3333333333333333;
        double r178329 = im;
        double r178330 = 3.0;
        double r178331 = pow(r178329, r178330);
        double r178332 = r178328 * r178331;
        double r178333 = 0.016666666666666666;
        double r178334 = 5.0;
        double r178335 = pow(r178329, r178334);
        double r178336 = r178333 * r178335;
        double r178337 = 2.0;
        double r178338 = r178337 * r178329;
        double r178339 = r178336 + r178338;
        double r178340 = r178332 + r178339;
        double r178341 = -r178340;
        double r178342 = r178327 * r178341;
        return r178342;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.1
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 58.1

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

  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 2020089 
(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))))