Average Error: 43.1 → 0.8
Time: 9.3s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(0.5 \cdot \sin re\right) \cdot \left(-\left(\left(\frac{1}{3} \cdot {im}^{3} + \frac{1}{60} \cdot {im}^{5}\right) + 2 \cdot im\right)\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left(0.5 \cdot \sin re\right) \cdot \left(-\left(\left(\frac{1}{3} \cdot {im}^{3} + \frac{1}{60} \cdot {im}^{5}\right) + 2 \cdot im\right)\right)
double f(double re, double im) {
        double r658416 = 0.5;
        double r658417 = re;
        double r658418 = sin(r658417);
        double r658419 = r658416 * r658418;
        double r658420 = im;
        double r658421 = -r658420;
        double r658422 = exp(r658421);
        double r658423 = exp(r658420);
        double r658424 = r658422 - r658423;
        double r658425 = r658419 * r658424;
        return r658425;
}
↓
double f(double re, double im) {
        double r658426 = 0.5;
        double r658427 = re;
        double r658428 = sin(r658427);
        double r658429 = r658426 * r658428;
        double r658430 = 0.3333333333333333;
        double r658431 = im;
        double r658432 = 3.0;
        double r658433 = pow(r658431, r658432);
        double r658434 = r658430 * r658433;
        double r658435 = 0.016666666666666666;
        double r658436 = 5.0;
        double r658437 = pow(r658431, r658436);
        double r658438 = r658435 * r658437;
        double r658439 = r658434 + r658438;
        double r658440 = 2.0;
        double r658441 = r658440 * r658431;
        double r658442 = r658439 + r658441;
        double r658443 = -r658442;
        double r658444 = r658429 * r658443;
        return r658444;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original43.1
Target0.3
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.166666666666666657 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.00833333333333333322 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 43.1

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

  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. Using strategy rm

  4. Applied associate-+r+0.8

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

  5. Final simplification0.8

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

Reproduce

herbie shell --seed 2020039 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64

  :herbie-target
  (if (< (fabs im) 1) (- (* (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))))