Average Error: 43.4 → 0.8
Time: 29.2s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(\frac{-1}{3} \cdot {im}^{3}\right) \cdot \left(0.5 \cdot \sin re\right) + \left(\left({im}^{5} \cdot \frac{-1}{60} + -2 \cdot im\right) \cdot 0.5\right) \cdot \sin re\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left(\frac{-1}{3} \cdot {im}^{3}\right) \cdot \left(0.5 \cdot \sin re\right) + \left(\left({im}^{5} \cdot \frac{-1}{60} + -2 \cdot im\right) \cdot 0.5\right) \cdot \sin re
double f(double re, double im) {
        double r212582 = 0.5;
        double r212583 = re;
        double r212584 = sin(r212583);
        double r212585 = r212582 * r212584;
        double r212586 = im;
        double r212587 = -r212586;
        double r212588 = exp(r212587);
        double r212589 = exp(r212586);
        double r212590 = r212588 - r212589;
        double r212591 = r212585 * r212590;
        return r212591;
}
↓
double f(double re, double im) {
        double r212592 = -0.3333333333333333;
        double r212593 = im;
        double r212594 = 3.0;
        double r212595 = pow(r212593, r212594);
        double r212596 = r212592 * r212595;
        double r212597 = 0.5;
        double r212598 = re;
        double r212599 = sin(r212598);
        double r212600 = r212597 * r212599;
        double r212601 = r212596 * r212600;
        double r212602 = 5.0;
        double r212603 = pow(r212593, r212602);
        double r212604 = -0.016666666666666666;
        double r212605 = r212603 * r212604;
        double r212606 = -2.0;
        double r212607 = r212606 * r212593;
        double r212608 = r212605 + r212607;
        double r212609 = r212608 * r212597;
        double r212610 = r212609 * r212599;
        double r212611 = r212601 + r212610;
        return r212611;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original43.4
Target0.2
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.1666666666666666574148081281236954964697 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333217685101601546193705872 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 43.4

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

  2. Taylor expanded around 0 0.8

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

  3. Simplified0.8

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

  5. Applied sub-neg0.8

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

  6. Applied distribute-lft-in0.8

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

  7. Simplified0.8

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

  8. Simplified0.8

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

  9. Final simplification0.8

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

Reproduce

herbie shell --seed 2019304 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64

  :herbie-target
  (if (< (fabs im) 1) (- (* (sin re) (+ (+ im (* (* (* 0.166666666666666657 im) im) im)) (* (* (* (* (* 0.00833333333333333322 im) im) im) im) im)))) (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))

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