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

↓

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

double f(double re, double im) {
        double r40247380 = 0.5;
        double r40247381 = re;
        double r40247382 = cos(r40247381);
        double r40247383 = r40247380 * r40247382;
        double r40247384 = 0.0;
        double r40247385 = im;
        double r40247386 = r40247384 - r40247385;
        double r40247387 = exp(r40247386);
        double r40247388 = exp(r40247385);
        double r40247389 = r40247387 - r40247388;
        double r40247390 = r40247383 * r40247389;
        return r40247390;
}

↓

double f(double re, double im) {
        double r40247391 = 0.5;
        double r40247392 = re;
        double r40247393 = cos(r40247392);
        double r40247394 = r40247391 * r40247393;
        double r40247395 = im;
        double r40247396 = 5.0;
        double r40247397 = pow(r40247395, r40247396);
        double r40247398 = -0.016666666666666666;
        double r40247399 = r40247397 * r40247398;
        double r40247400 = exp(r40247399);
        double r40247401 = log(r40247400);
        double r40247402 = 0.3333333333333333;
        double r40247403 = r40247395 * r40247402;
        double r40247404 = r40247403 * r40247395;
        double r40247405 = 2.0;
        double r40247406 = r40247404 + r40247405;
        double r40247407 = r40247406 * r40247395;
        double r40247408 = r40247401 - r40247407;
        double r40247409 = r40247394 * r40247408;
        return r40247409;
}

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

↓

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

Target

Original	57.8
Target	0.3
Herbie	0.9

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

Derivation

Initial program 57.8
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]
Taylor expanded around 0 0.8
\[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-\left(\frac{1}{3} \cdot {im}^{3} + \left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)\right)\right)}\]
Simplified0.8
\[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\frac{-1}{60} \cdot {im}^{5} - \left(\left(\frac{1}{3} \cdot im\right) \cdot im + 2\right) \cdot im\right)}\]
Using strategy rm
Applied add-log-exp0.9
\[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\log \left(e^{\frac{-1}{60} \cdot {im}^{5}}\right)} - \left(\left(\frac{1}{3} \cdot im\right) \cdot im + 2\right) \cdot im\right)\]
Final simplification0.9
\[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\log \left(e^{{im}^{5} \cdot \frac{-1}{60}}\right) - \left(\left(im \cdot \frac{1}{3}\right) \cdot im + 2\right) \cdot im\right)\]

Reproduce

herbie shell --seed 2019101 
(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))))

Error

Target

Derivation

Reproduce