Average Error: 57.9 → 0.8
Time: 53.6s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]
↓
\[\left(0.5 \cdot \cos re\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot \frac{-1}{3}\right)\right) + \left(\frac{-1}{60} \cdot {im}^{5} - \left(im + im\right)\right) \cdot \left(0.5 \cdot \cos re\right)\]
\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 im\right) \cdot \left(im \cdot \frac{-1}{3}\right)\right) + \left(\frac{-1}{60} \cdot {im}^{5} - \left(im + im\right)\right) \cdot \left(0.5 \cdot \cos re\right)
double f(double re, double im) {
        double r9353390 = 0.5;
        double r9353391 = re;
        double r9353392 = cos(r9353391);
        double r9353393 = r9353390 * r9353392;
        double r9353394 = 0.0;
        double r9353395 = im;
        double r9353396 = r9353394 - r9353395;
        double r9353397 = exp(r9353396);
        double r9353398 = exp(r9353395);
        double r9353399 = r9353397 - r9353398;
        double r9353400 = r9353393 * r9353399;
        return r9353400;
}
↓
double f(double re, double im) {
        double r9353401 = 0.5;
        double r9353402 = re;
        double r9353403 = cos(r9353402);
        double r9353404 = r9353401 * r9353403;
        double r9353405 = im;
        double r9353406 = r9353405 * r9353405;
        double r9353407 = -0.3333333333333333;
        double r9353408 = r9353405 * r9353407;
        double r9353409 = r9353406 * r9353408;
        double r9353410 = r9353404 * r9353409;
        double r9353411 = -0.016666666666666666;
        double r9353412 = 5.0;
        double r9353413 = pow(r9353405, r9353412);
        double r9353414 = r9353411 * r9353413;
        double r9353415 = r9353405 + r9353405;
        double r9353416 = r9353414 - r9353415;
        double r9353417 = r9353416 * r9353404;
        double r9353418 = r9353410 + r9353417;
        return r9353418;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original57.9
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 57.9

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

  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

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

  5. Applied sub-neg0.8

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

  6. Applied distribute-lft-in0.8

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

  7. Final simplification0.8

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

Reproduce

herbie shell --seed 2019168 
(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))))