Average Error: 57.9 → 0.8
Time: 32.8s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]
↓
\[\left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{3} - \left({im}^{5} \cdot \frac{1}{60} + \left(im + im\right)\right)\right) \cdot \left(0.5 \cdot \cos re\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)
↓
\left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{3} - \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 r3541733 = 0.5;
        double r3541734 = re;
        double r3541735 = cos(r3541734);
        double r3541736 = r3541733 * r3541735;
        double r3541737 = 0.0;
        double r3541738 = im;
        double r3541739 = r3541737 - r3541738;
        double r3541740 = exp(r3541739);
        double r3541741 = exp(r3541738);
        double r3541742 = r3541740 - r3541741;
        double r3541743 = r3541736 * r3541742;
        return r3541743;
}
↓
double f(double re, double im) {
        double r3541744 = im;
        double r3541745 = r3541744 * r3541744;
        double r3541746 = r3541744 * r3541745;
        double r3541747 = -0.3333333333333333;
        double r3541748 = r3541746 * r3541747;
        double r3541749 = 5.0;
        double r3541750 = pow(r3541744, r3541749);
        double r3541751 = 0.016666666666666666;
        double r3541752 = r3541750 * r3541751;
        double r3541753 = r3541744 + r3541744;
        double r3541754 = r3541752 + r3541753;
        double r3541755 = r3541748 - r3541754;
        double r3541756 = 0.5;
        double r3541757 = re;
        double r3541758 = cos(r3541757);
        double r3541759 = r3541756 * r3541758;
        double r3541760 = r3541755 * r3541759;
        return r3541760;
}

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

Derivation

  1. Initial program 57.9

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

  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

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019152 
(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))))