Average Error: 43.4 → 0.7
Time: 9.2s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(0.5 \cdot \sin re\right) \cdot \left(-\left(\frac{1}{3} \cdot {im}^{3} + \left({im}^{5} \cdot \frac{1}{60} + 2 \cdot im\right)\right)\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left(0.5 \cdot \sin re\right) \cdot \left(-\left(\frac{1}{3} \cdot {im}^{3} + \left({im}^{5} \cdot \frac{1}{60} + 2 \cdot im\right)\right)\right)
double f(double re, double im) {
        double r387567 = 0.5;
        double r387568 = re;
        double r387569 = sin(r387568);
        double r387570 = r387567 * r387569;
        double r387571 = im;
        double r387572 = -r387571;
        double r387573 = exp(r387572);
        double r387574 = exp(r387571);
        double r387575 = r387573 - r387574;
        double r387576 = r387570 * r387575;
        return r387576;
}
↓
double f(double re, double im) {
        double r387577 = 0.5;
        double r387578 = re;
        double r387579 = sin(r387578);
        double r387580 = r387577 * r387579;
        double r387581 = 0.3333333333333333;
        double r387582 = im;
        double r387583 = 3.0;
        double r387584 = pow(r387582, r387583);
        double r387585 = r387581 * r387584;
        double r387586 = 5.0;
        double r387587 = pow(r387582, r387586);
        double r387588 = 0.016666666666666666;
        double r387589 = r387587 * r387588;
        double r387590 = 2.0;
        double r387591 = r387590 * r387582;
        double r387592 = r387589 + r387591;
        double r387593 = r387585 + r387592;
        double r387594 = -r387593;
        double r387595 = r387580 * r387594;
        return r387595;
}

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
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.1666666666666666574148081281236954964697 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333217685101601546193705872 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

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. Using strategy rm

  4. Applied *-commutative0.7

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

  5. Final simplification0.7

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

Reproduce

herbie shell --seed 2019344 
(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))))