Average Error: 43.7 → 0.8
Time: 29.3s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(\left(\left(im \cdot im\right) \cdot im\right) \cdot \sin re\right) \cdot \left(-0.16666666666666666\right) - \left({im}^{5} \cdot \left(0.008333333333333333 \cdot \sin re\right) + \left(1.0 \cdot \sin re\right) \cdot im\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left(\left(\left(im \cdot im\right) \cdot im\right) \cdot \sin re\right) \cdot \left(-0.16666666666666666\right) - \left({im}^{5} \cdot \left(0.008333333333333333 \cdot \sin re\right) + \left(1.0 \cdot \sin re\right) \cdot im\right)
double f(double re, double im) {
        double r4885391 = 0.5;
        double r4885392 = re;
        double r4885393 = sin(r4885392);
        double r4885394 = r4885391 * r4885393;
        double r4885395 = im;
        double r4885396 = -r4885395;
        double r4885397 = exp(r4885396);
        double r4885398 = exp(r4885395);
        double r4885399 = r4885397 - r4885398;
        double r4885400 = r4885394 * r4885399;
        return r4885400;
}
↓
double f(double re, double im) {
        double r4885401 = im;
        double r4885402 = r4885401 * r4885401;
        double r4885403 = r4885402 * r4885401;
        double r4885404 = re;
        double r4885405 = sin(r4885404);
        double r4885406 = r4885403 * r4885405;
        double r4885407 = 0.16666666666666666;
        double r4885408 = -r4885407;
        double r4885409 = r4885406 * r4885408;
        double r4885410 = 5.0;
        double r4885411 = pow(r4885401, r4885410);
        double r4885412 = 0.008333333333333333;
        double r4885413 = r4885412 * r4885405;
        double r4885414 = r4885411 * r4885413;
        double r4885415 = 1.0;
        double r4885416 = r4885415 * r4885405;
        double r4885417 = r4885416 * r4885401;
        double r4885418 = r4885414 + r4885417;
        double r4885419 = r4885409 - r4885418;
        return r4885419;
}

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
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

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 0.8

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

  3. Simplified0.8

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

  4. Taylor expanded around inf 0.8

    \[\leadsto \color{blue}{-\left(0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) + \left(1.0 \cdot \left(\sin re \cdot im\right) + 0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right)\right)\right)}\]

  5. Simplified0.8

    \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot \left(\left(\left(im \cdot im\right) \cdot im\right) \cdot \sin re\right)\right) - \left(\left(\sin re \cdot 0.008333333333333333\right) \cdot {im}^{5} + \left(1.0 \cdot \sin re\right) \cdot im\right)}\]

  6. Final simplification0.8

    \[\leadsto \left(\left(\left(im \cdot im\right) \cdot im\right) \cdot \sin re\right) \cdot \left(-0.16666666666666666\right) - \left({im}^{5} \cdot \left(0.008333333333333333 \cdot \sin re\right) + \left(1.0 \cdot \sin re\right) \cdot im\right)\]

Reproduce

herbie shell --seed 2019153 
(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))))