Average Error: 44.0 → 0.8
Time: 38.4s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(\frac{-1}{3} \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) \cdot \left(0.5 \cdot \sin re\right) + \left(0.5 \cdot \sin re\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) \cdot \frac{-1}{60} - \left(im + im\right)\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left(\frac{-1}{3} \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) \cdot \left(0.5 \cdot \sin re\right) + \left(0.5 \cdot \sin re\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) \cdot \frac{-1}{60} - \left(im + im\right)\right)
double f(double re, double im) {
        double r8290547 = 0.5;
        double r8290548 = re;
        double r8290549 = sin(r8290548);
        double r8290550 = r8290547 * r8290549;
        double r8290551 = im;
        double r8290552 = -r8290551;
        double r8290553 = exp(r8290552);
        double r8290554 = exp(r8290551);
        double r8290555 = r8290553 - r8290554;
        double r8290556 = r8290550 * r8290555;
        return r8290556;
}
↓
double f(double re, double im) {
        double r8290557 = -0.3333333333333333;
        double r8290558 = im;
        double r8290559 = r8290558 * r8290558;
        double r8290560 = r8290558 * r8290559;
        double r8290561 = r8290557 * r8290560;
        double r8290562 = 0.5;
        double r8290563 = re;
        double r8290564 = sin(r8290563);
        double r8290565 = r8290562 * r8290564;
        double r8290566 = r8290561 * r8290565;
        double r8290567 = r8290559 * r8290560;
        double r8290568 = -0.016666666666666666;
        double r8290569 = r8290567 * r8290568;
        double r8290570 = r8290558 + r8290558;
        double r8290571 = r8290569 - r8290570;
        double r8290572 = r8290565 * r8290571;
        double r8290573 = r8290566 + r8290572;
        return r8290573;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original44.0
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 44.0

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

  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

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

  5. Applied sub-neg0.8

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

  6. Applied distribute-lft-in0.8

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

  7. Simplified0.8

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

  8. Final simplification0.8

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

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(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))))