Average Error: 58.0 → 0.7
Time: 34.9s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]
↓
\[\left(\frac{-1}{3} \cdot \left(im \cdot \left(im \cdot im\right)\right) + \left({im}^{5} \cdot \frac{-1}{60} - \left(im + im\right)\right)\right) \cdot \left(0.5 \cdot \cos re\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)
↓
\left(\frac{-1}{3} \cdot \left(im \cdot \left(im \cdot im\right)\right) + \left({im}^{5} \cdot \frac{-1}{60} - \left(im + im\right)\right)\right) \cdot \left(0.5 \cdot \cos re\right)
double f(double re, double im) {
        double r9087386 = 0.5;
        double r9087387 = re;
        double r9087388 = cos(r9087387);
        double r9087389 = r9087386 * r9087388;
        double r9087390 = 0.0;
        double r9087391 = im;
        double r9087392 = r9087390 - r9087391;
        double r9087393 = exp(r9087392);
        double r9087394 = exp(r9087391);
        double r9087395 = r9087393 - r9087394;
        double r9087396 = r9087389 * r9087395;
        return r9087396;
}
↓
double f(double re, double im) {
        double r9087397 = -0.3333333333333333;
        double r9087398 = im;
        double r9087399 = r9087398 * r9087398;
        double r9087400 = r9087398 * r9087399;
        double r9087401 = r9087397 * r9087400;
        double r9087402 = 5.0;
        double r9087403 = pow(r9087398, r9087402);
        double r9087404 = -0.016666666666666666;
        double r9087405 = r9087403 * r9087404;
        double r9087406 = r9087398 + r9087398;
        double r9087407 = r9087405 - r9087406;
        double r9087408 = r9087401 + r9087407;
        double r9087409 = 0.5;
        double r9087410 = re;
        double r9087411 = cos(r9087410);
        double r9087412 = r9087409 * r9087411;
        double r9087413 = r9087408 * r9087412;
        return r9087413;
}

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
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(0.1666666666666666574148081281236954964697 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333217685101601546193705872 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\\ \end{array}\]

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 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019170 
(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))))