Average Error: 58.1 → 0.8
Time: 31.8s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]
↓
\[\left(0.5 \cdot \cos re\right) \cdot \left({im}^{3} \cdot \frac{-1}{3} - \left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)\right)\]
\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({im}^{3} \cdot \frac{-1}{3} - \left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)\right)
double f(double re, double im) {
        double r170380 = 0.5;
        double r170381 = re;
        double r170382 = cos(r170381);
        double r170383 = r170380 * r170382;
        double r170384 = 0.0;
        double r170385 = im;
        double r170386 = r170384 - r170385;
        double r170387 = exp(r170386);
        double r170388 = exp(r170385);
        double r170389 = r170387 - r170388;
        double r170390 = r170383 * r170389;
        return r170390;
}
↓
double f(double re, double im) {
        double r170391 = 0.5;
        double r170392 = re;
        double r170393 = cos(r170392);
        double r170394 = r170391 * r170393;
        double r170395 = im;
        double r170396 = 3.0;
        double r170397 = pow(r170395, r170396);
        double r170398 = -0.3333333333333333;
        double r170399 = r170397 * r170398;
        double r170400 = 0.016666666666666666;
        double r170401 = 5.0;
        double r170402 = pow(r170395, r170401);
        double r170403 = r170400 * r170402;
        double r170404 = 2.0;
        double r170405 = r170404 * r170395;
        double r170406 = r170403 + r170405;
        double r170407 = r170399 - r170406;
        double r170408 = r170394 * r170407;
        return r170408;
}

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

Derivation

  1. Initial program 58.1

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

  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. Simplified0.8

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019303 
(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))))