Average Error: 58.3 → 0.6
Time: 34.7s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]
↓
\[\left(\left(\frac{-1}{3} \cdot \left(im \cdot \left(im \cdot im\right)\right) - \left(im + im\right)\right) - \frac{1}{60} \cdot {im}^{5}\right) \cdot \left(0.5 \cdot \cos re\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)
↓
\left(\left(\frac{-1}{3} \cdot \left(im \cdot \left(im \cdot im\right)\right) - \left(im + im\right)\right) - \frac{1}{60} \cdot {im}^{5}\right) \cdot \left(0.5 \cdot \cos re\right)
double f(double re, double im) {
        double r8849556 = 0.5;
        double r8849557 = re;
        double r8849558 = cos(r8849557);
        double r8849559 = r8849556 * r8849558;
        double r8849560 = 0.0;
        double r8849561 = im;
        double r8849562 = r8849560 - r8849561;
        double r8849563 = exp(r8849562);
        double r8849564 = exp(r8849561);
        double r8849565 = r8849563 - r8849564;
        double r8849566 = r8849559 * r8849565;
        return r8849566;
}
↓
double f(double re, double im) {
        double r8849567 = -0.3333333333333333;
        double r8849568 = im;
        double r8849569 = r8849568 * r8849568;
        double r8849570 = r8849568 * r8849569;
        double r8849571 = r8849567 * r8849570;
        double r8849572 = r8849568 + r8849568;
        double r8849573 = r8849571 - r8849572;
        double r8849574 = 0.016666666666666666;
        double r8849575 = 5.0;
        double r8849576 = pow(r8849568, r8849575);
        double r8849577 = r8849574 * r8849576;
        double r8849578 = r8849573 - r8849577;
        double r8849579 = 0.5;
        double r8849580 = re;
        double r8849581 = cos(r8849580);
        double r8849582 = r8849579 * r8849581;
        double r8849583 = r8849578 * r8849582;
        return r8849583;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.3
Target0.3
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(0.1666666666666666574148081281236954964697 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333217685101601546193705872 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 58.3

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

  2. Taylor expanded around 0 0.6

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

  3. Simplified0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019172 
(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))))