Average Error: 58.0 → 0.8
Time: 55.4s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]
↓
\[\left(0.5 \cdot \cos re\right) \cdot \left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{3} + \left(\frac{-1}{60} \cdot {im}^{5} - \left(im + im\right)\right)\right)\]
\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(im \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{3} + \left(\frac{-1}{60} \cdot {im}^{5} - \left(im + im\right)\right)\right)
double f(double re, double im) {
        double r9618303 = 0.5;
        double r9618304 = re;
        double r9618305 = cos(r9618304);
        double r9618306 = r9618303 * r9618305;
        double r9618307 = 0.0;
        double r9618308 = im;
        double r9618309 = r9618307 - r9618308;
        double r9618310 = exp(r9618309);
        double r9618311 = exp(r9618308);
        double r9618312 = r9618310 - r9618311;
        double r9618313 = r9618306 * r9618312;
        return r9618313;
}
↓
double f(double re, double im) {
        double r9618314 = 0.5;
        double r9618315 = re;
        double r9618316 = cos(r9618315);
        double r9618317 = r9618314 * r9618316;
        double r9618318 = im;
        double r9618319 = r9618318 * r9618318;
        double r9618320 = r9618318 * r9618319;
        double r9618321 = -0.3333333333333333;
        double r9618322 = r9618320 * r9618321;
        double r9618323 = -0.016666666666666666;
        double r9618324 = 5.0;
        double r9618325 = pow(r9618318, r9618324);
        double r9618326 = r9618323 * r9618325;
        double r9618327 = r9618318 + r9618318;
        double r9618328 = r9618326 - r9618327;
        double r9618329 = r9618322 + r9618328;
        double r9618330 = r9618317 * r9618329;
        return r9618330;
}

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

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. Simplified0.8

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019200 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"

  :herbie-target
  (if (< (fabs im) 1.0) (- (* (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))))