Average Error: 43.7 → 0.7
Time: 25.8s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(0.5 \cdot \sin re\right) \cdot \left({im}^{3} \cdot \frac{-1}{3} - \left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left(0.5 \cdot \sin re\right) \cdot \left({im}^{3} \cdot \frac{-1}{3} - \left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)\right)
double f(double re, double im) {
        double r182548 = 0.5;
        double r182549 = re;
        double r182550 = sin(r182549);
        double r182551 = r182548 * r182550;
        double r182552 = im;
        double r182553 = -r182552;
        double r182554 = exp(r182553);
        double r182555 = exp(r182552);
        double r182556 = r182554 - r182555;
        double r182557 = r182551 * r182556;
        return r182557;
}
↓
double f(double re, double im) {
        double r182558 = 0.5;
        double r182559 = re;
        double r182560 = sin(r182559);
        double r182561 = r182558 * r182560;
        double r182562 = im;
        double r182563 = 3.0;
        double r182564 = pow(r182562, r182563);
        double r182565 = -0.3333333333333333;
        double r182566 = r182564 * r182565;
        double r182567 = 0.016666666666666666;
        double r182568 = 5.0;
        double r182569 = pow(r182562, r182568);
        double r182570 = r182567 * r182569;
        double r182571 = 2.0;
        double r182572 = r182571 * r182562;
        double r182573 = r182570 + r182572;
        double r182574 = r182566 - r182573;
        double r182575 = r182561 * r182574;
        return r182575;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original43.7
Target0.3
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.1666666666666666574148081281236954964697 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333217685101601546193705872 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 43.7

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

  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. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019235 
(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))))