Average Error: 43.7 → 0.9
Time: 30.8s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{3} - \left(\left(im + {im}^{5} \cdot \frac{1}{60}\right) + im\right)\right) \cdot \left(0.5 \cdot \sin re\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{3} - \left(\left(im + {im}^{5} \cdot \frac{1}{60}\right) + im\right)\right) \cdot \left(0.5 \cdot \sin re\right)
double f(double re, double im) {
        double r10102319 = 0.5;
        double r10102320 = re;
        double r10102321 = sin(r10102320);
        double r10102322 = r10102319 * r10102321;
        double r10102323 = im;
        double r10102324 = -r10102323;
        double r10102325 = exp(r10102324);
        double r10102326 = exp(r10102323);
        double r10102327 = r10102325 - r10102326;
        double r10102328 = r10102322 * r10102327;
        return r10102328;
}
↓
double f(double re, double im) {
        double r10102329 = im;
        double r10102330 = r10102329 * r10102329;
        double r10102331 = r10102329 * r10102330;
        double r10102332 = -0.3333333333333333;
        double r10102333 = r10102331 * r10102332;
        double r10102334 = 5.0;
        double r10102335 = pow(r10102329, r10102334);
        double r10102336 = 0.016666666666666666;
        double r10102337 = r10102335 * r10102336;
        double r10102338 = r10102329 + r10102337;
        double r10102339 = r10102338 + r10102329;
        double r10102340 = r10102333 - r10102339;
        double r10102341 = 0.5;
        double r10102342 = re;
        double r10102343 = sin(r10102342);
        double r10102344 = r10102341 * r10102343;
        double r10102345 = r10102340 * r10102344;
        return r10102345;
}

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
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

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 0.9

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

  3. Simplified0.9

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

  4. Final simplification0.9

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

Reproduce

herbie shell --seed 2019165 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"

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

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