Average Error: 58.1 → 0.7
Time: 28.1s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]
↓
\[\left(0.5 \cdot \cos re\right) \cdot \left(im \cdot -2 + \frac{-1}{60} \cdot {im}^{5}\right) + \left(\left(\left(im \cdot im\right) \cdot \frac{-1}{3}\right) \cdot im\right) \cdot \left(0.5 \cdot \cos re\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)
↓
\left(0.5 \cdot \cos re\right) \cdot \left(im \cdot -2 + \frac{-1}{60} \cdot {im}^{5}\right) + \left(\left(\left(im \cdot im\right) \cdot \frac{-1}{3}\right) \cdot im\right) \cdot \left(0.5 \cdot \cos re\right)
double f(double re, double im) {
        double r7102427 = 0.5;
        double r7102428 = re;
        double r7102429 = cos(r7102428);
        double r7102430 = r7102427 * r7102429;
        double r7102431 = 0.0;
        double r7102432 = im;
        double r7102433 = r7102431 - r7102432;
        double r7102434 = exp(r7102433);
        double r7102435 = exp(r7102432);
        double r7102436 = r7102434 - r7102435;
        double r7102437 = r7102430 * r7102436;
        return r7102437;
}
↓
double f(double re, double im) {
        double r7102438 = 0.5;
        double r7102439 = re;
        double r7102440 = cos(r7102439);
        double r7102441 = r7102438 * r7102440;
        double r7102442 = im;
        double r7102443 = -2.0;
        double r7102444 = r7102442 * r7102443;
        double r7102445 = -0.016666666666666666;
        double r7102446 = 5.0;
        double r7102447 = pow(r7102442, r7102446);
        double r7102448 = r7102445 * r7102447;
        double r7102449 = r7102444 + r7102448;
        double r7102450 = r7102441 * r7102449;
        double r7102451 = r7102442 * r7102442;
        double r7102452 = -0.3333333333333333;
        double r7102453 = r7102451 * r7102452;
        double r7102454 = r7102453 * r7102442;
        double r7102455 = r7102454 * r7102441;
        double r7102456 = r7102450 + r7102455;
        return r7102456;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.1
Target0.3
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 58.1

    \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  5. Applied sub-neg0.7

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

  6. Applied associate--l+0.7

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

  7. Applied distribute-lft-in0.7

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

  8. Simplified0.7

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

  9. Final simplification0.7

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

Reproduce

herbie shell --seed 2019132 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"

  :herbie-target
  (if (< (fabs im) 1) (- (* (cos re) (+ (+ im (* (* (* 1/6 im) im) im)) (* (* (* (* (* 1/120 im) im) im) im) im)))) (* (* 0.5 (cos re)) (- (exp (- 0 im)) (exp im))))

  (* (* 0.5 (cos re)) (- (exp (- 0 im)) (exp im))))