Average Error: 58.2 → 0.6
Time: 44.1s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]
↓
\[\left(\frac{-1}{3} \cdot \left(im \cdot \left(im \cdot im\right)\right) + \left({im}^{5} \cdot \frac{-1}{60} - \left(im + im\right)\right)\right) \cdot \left(0.5 \cdot \cos re\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - 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 r8426589 = 0.5;
        double r8426590 = re;
        double r8426591 = cos(r8426590);
        double r8426592 = r8426589 * r8426591;
        double r8426593 = 0.0;
        double r8426594 = im;
        double r8426595 = r8426593 - r8426594;
        double r8426596 = exp(r8426595);
        double r8426597 = exp(r8426594);
        double r8426598 = r8426596 - r8426597;
        double r8426599 = r8426592 * r8426598;
        return r8426599;
}
↓
double f(double re, double im) {
        double r8426600 = -0.3333333333333333;
        double r8426601 = im;
        double r8426602 = r8426601 * r8426601;
        double r8426603 = r8426601 * r8426602;
        double r8426604 = r8426600 * r8426603;
        double r8426605 = 5.0;
        double r8426606 = pow(r8426601, r8426605);
        double r8426607 = -0.016666666666666666;
        double r8426608 = r8426606 * r8426607;
        double r8426609 = r8426601 + r8426601;
        double r8426610 = r8426608 - r8426609;
        double r8426611 = r8426604 + r8426610;
        double r8426612 = 0.5;
        double r8426613 = re;
        double r8426614 = cos(r8426613);
        double r8426615 = r8426612 * r8426614;
        double r8426616 = r8426611 * r8426615;
        return r8426616;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.2
Target0.2
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 58.2

    \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]

  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

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019149 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"

  :herbie-target
  (if (< (fabs im) 1) (- (* (cos re) (+ (+ im (* (* (* 1/6 im) im) im)) (* (* (* (* (* 1/120 im) im) im) im) im)))) (* (* 0.5 (cos re)) (- (exp (- 0 im)) (exp im))))

  (* (* 0.5 (cos re)) (- (exp (- 0 im)) (exp im))))