Average Error: 43.5 → 0.7
Time: 10.7s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(0.5 \cdot \sin re\right) \cdot \left(-\left(\frac{1}{3} \cdot {im}^{3} + \left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)\right)\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left(0.5 \cdot \sin 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 r209620 = 0.5;
        double r209621 = re;
        double r209622 = sin(r209621);
        double r209623 = r209620 * r209622;
        double r209624 = im;
        double r209625 = -r209624;
        double r209626 = exp(r209625);
        double r209627 = exp(r209624);
        double r209628 = r209626 - r209627;
        double r209629 = r209623 * r209628;
        return r209629;
}
↓
double f(double re, double im) {
        double r209630 = 0.5;
        double r209631 = re;
        double r209632 = sin(r209631);
        double r209633 = r209630 * r209632;
        double r209634 = 0.3333333333333333;
        double r209635 = im;
        double r209636 = 3.0;
        double r209637 = pow(r209635, r209636);
        double r209638 = r209634 * r209637;
        double r209639 = 0.016666666666666666;
        double r209640 = 5.0;
        double r209641 = pow(r209635, r209640);
        double r209642 = r209639 * r209641;
        double r209643 = 2.0;
        double r209644 = r209643 * r209635;
        double r209645 = r209642 + r209644;
        double r209646 = r209638 + r209645;
        double r209647 = -r209646;
        double r209648 = r209633 * r209647;
        return r209648;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original43.5
Target0.3
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.166666666666666657 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.00833333333333333322 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 43.5

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

  2. Taylor expanded around 0 0.7

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

  3. Final simplification0.7

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

Reproduce

herbie shell --seed 2020033 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64

  :herbie-target
  (if (< (fabs im) 1) (- (* (sin re) (+ (+ im (* (* (* 0.16666666666666666 im) im) im)) (* (* (* (* (* 0.008333333333333333 im) im) im) im) im)))) (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))

  (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))