Average Error: 43.4 → 0.7
Time: 24.6s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left({im}^{3} \cdot \frac{-1}{3}\right) \cdot \left(0.5 \cdot \sin re\right) + \left(im \cdot -2 + {im}^{5} \cdot \frac{-1}{60}\right) \cdot \left(0.5 \cdot \sin re\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left({im}^{3} \cdot \frac{-1}{3}\right) \cdot \left(0.5 \cdot \sin re\right) + \left(im \cdot -2 + {im}^{5} \cdot \frac{-1}{60}\right) \cdot \left(0.5 \cdot \sin re\right)
double f(double re, double im) {
        double r221416 = 0.5;
        double r221417 = re;
        double r221418 = sin(r221417);
        double r221419 = r221416 * r221418;
        double r221420 = im;
        double r221421 = -r221420;
        double r221422 = exp(r221421);
        double r221423 = exp(r221420);
        double r221424 = r221422 - r221423;
        double r221425 = r221419 * r221424;
        return r221425;
}
↓
double f(double re, double im) {
        double r221426 = im;
        double r221427 = 3.0;
        double r221428 = pow(r221426, r221427);
        double r221429 = -0.3333333333333333;
        double r221430 = r221428 * r221429;
        double r221431 = 0.5;
        double r221432 = re;
        double r221433 = sin(r221432);
        double r221434 = r221431 * r221433;
        double r221435 = r221430 * r221434;
        double r221436 = -2.0;
        double r221437 = r221426 * r221436;
        double r221438 = 5.0;
        double r221439 = pow(r221426, r221438);
        double r221440 = -0.016666666666666666;
        double r221441 = r221439 * r221440;
        double r221442 = r221437 + r221441;
        double r221443 = r221442 * r221434;
        double r221444 = r221435 + r221443;
        return r221444;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original43.4
Target0.3
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.166666666666666657 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.00833333333333333322 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 43.4

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

  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

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

  5. Applied sub-neg0.7

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

  6. Applied distribute-lft-in0.7

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

  7. Simplified0.7

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

  8. Simplified0.7

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

  9. Final simplification0.7

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

Reproduce

herbie shell --seed 2019198 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"

  :herbie-target
  (if (< (fabs im) 1.0) (- (* (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))))