Average Error: 43.3 → 0.7
Time: 16.5s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(0.5 \cdot \sin re\right) \cdot \left(\frac{-1}{3} \cdot {im}^{3} - \left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left(0.5 \cdot \sin re\right) \cdot \left(\frac{-1}{3} \cdot {im}^{3} - \left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)\right)
double f(double re, double im) {
        double r362467 = 0.5;
        double r362468 = re;
        double r362469 = sin(r362468);
        double r362470 = r362467 * r362469;
        double r362471 = im;
        double r362472 = -r362471;
        double r362473 = exp(r362472);
        double r362474 = exp(r362471);
        double r362475 = r362473 - r362474;
        double r362476 = r362470 * r362475;
        return r362476;
}
↓
double f(double re, double im) {
        double r362477 = 0.5;
        double r362478 = re;
        double r362479 = sin(r362478);
        double r362480 = r362477 * r362479;
        double r362481 = -0.3333333333333333;
        double r362482 = im;
        double r362483 = 3.0;
        double r362484 = pow(r362482, r362483);
        double r362485 = r362481 * r362484;
        double r362486 = 0.016666666666666666;
        double r362487 = 5.0;
        double r362488 = pow(r362482, r362487);
        double r362489 = r362486 * r362488;
        double r362490 = 2.0;
        double r362491 = r362490 * r362482;
        double r362492 = r362489 + r362491;
        double r362493 = r362485 - r362492;
        double r362494 = r362480 * r362493;
        return r362494;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original43.3
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 43.3

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

  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

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019350 
(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))))