Average Error: 58.0 → 0.8
Time: 9.7s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]
↓
\[\left(0.5 \cdot \cos re\right) \cdot \left(-\left(\frac{1}{3} \cdot {im}^{3} + \left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)\right)\right)\]
\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(\frac{1}{3} \cdot {im}^{3} + \left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)\right)\right)
double f(double re, double im) {
        double r205584 = 0.5;
        double r205585 = re;
        double r205586 = cos(r205585);
        double r205587 = r205584 * r205586;
        double r205588 = 0.0;
        double r205589 = im;
        double r205590 = r205588 - r205589;
        double r205591 = exp(r205590);
        double r205592 = exp(r205589);
        double r205593 = r205591 - r205592;
        double r205594 = r205587 * r205593;
        return r205594;
}
↓
double f(double re, double im) {
        double r205595 = 0.5;
        double r205596 = re;
        double r205597 = cos(r205596);
        double r205598 = r205595 * r205597;
        double r205599 = 0.3333333333333333;
        double r205600 = im;
        double r205601 = 3.0;
        double r205602 = pow(r205600, r205601);
        double r205603 = r205599 * r205602;
        double r205604 = 0.016666666666666666;
        double r205605 = 5.0;
        double r205606 = pow(r205600, r205605);
        double r205607 = r205604 * r205606;
        double r205608 = 2.0;
        double r205609 = r205608 * r205600;
        double r205610 = r205607 + r205609;
        double r205611 = r205603 + r205610;
        double r205612 = -r205611;
        double r205613 = r205598 * r205612;
        return r205613;
}

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
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(0.166666666666666657 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.00833333333333333322 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\\ \end{array}\]

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 0.8

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

  3. Final simplification0.8

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

Reproduce

herbie shell --seed 2020027 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"
  :precision binary64

  :herbie-target
  (if (< (fabs im) 1) (- (* (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))))