Average Error: 43.7 → 0.8
Time: 45.9s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(\left({im}^{5}\right), \frac{-1}{60}, \left(\frac{im \cdot \left({\left(im \cdot \left(im \cdot \frac{-1}{3}\right)\right)}^{3} - 8\right)}{\mathsf{fma}\left(2, \left(im \cdot \left(im \cdot \frac{-1}{3}\right)\right), \left(\mathsf{fma}\left(\left(im \cdot \left(im \cdot \frac{-1}{3}\right)\right), \left(im \cdot \left(im \cdot \frac{-1}{3}\right)\right), 4\right)\right)\right)}\right)\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(\left({im}^{5}\right), \frac{-1}{60}, \left(\frac{im \cdot \left({\left(im \cdot \left(im \cdot \frac{-1}{3}\right)\right)}^{3} - 8\right)}{\mathsf{fma}\left(2, \left(im \cdot \left(im \cdot \frac{-1}{3}\right)\right), \left(\mathsf{fma}\left(\left(im \cdot \left(im \cdot \frac{-1}{3}\right)\right), \left(im \cdot \left(im \cdot \frac{-1}{3}\right)\right), 4\right)\right)\right)}\right)\right)
double f(double re, double im) {
        double r24411568 = 0.5;
        double r24411569 = re;
        double r24411570 = sin(r24411569);
        double r24411571 = r24411568 * r24411570;
        double r24411572 = im;
        double r24411573 = -r24411572;
        double r24411574 = exp(r24411573);
        double r24411575 = exp(r24411572);
        double r24411576 = r24411574 - r24411575;
        double r24411577 = r24411571 * r24411576;
        return r24411577;
}
↓
double f(double re, double im) {
        double r24411578 = 0.5;
        double r24411579 = re;
        double r24411580 = sin(r24411579);
        double r24411581 = r24411578 * r24411580;
        double r24411582 = im;
        double r24411583 = 5.0;
        double r24411584 = pow(r24411582, r24411583);
        double r24411585 = -0.016666666666666666;
        double r24411586 = -0.3333333333333333;
        double r24411587 = r24411582 * r24411586;
        double r24411588 = r24411582 * r24411587;
        double r24411589 = 3.0;
        double r24411590 = pow(r24411588, r24411589);
        double r24411591 = 8.0;
        double r24411592 = r24411590 - r24411591;
        double r24411593 = r24411582 * r24411592;
        double r24411594 = 2.0;
        double r24411595 = 4.0;
        double r24411596 = fma(r24411588, r24411588, r24411595);
        double r24411597 = fma(r24411594, r24411588, r24411596);
        double r24411598 = r24411593 / r24411597;
        double r24411599 = fma(r24411584, r24411585, r24411598);
        double r24411600 = r24411581 * r24411599;
        return r24411600;
}

Error

Bits error versus re

Bits error versus im

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

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

  5. Applied flip3--0.8

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

  6. Applied associate-*r/0.8

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

  7. Simplified0.8

    \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(\left({im}^{5}\right), \frac{-1}{60}, \left(\frac{im \cdot \left({\left(\left(im \cdot \frac{-1}{3}\right) \cdot im\right)}^{3} - {2}^{3}\right)}{\color{blue}{\mathsf{fma}\left(2, \left(im \cdot \left(im \cdot \frac{-1}{3}\right)\right), \left(\mathsf{fma}\left(\left(im \cdot \left(im \cdot \frac{-1}{3}\right)\right), \left(im \cdot \left(im \cdot \frac{-1}{3}\right)\right), 4\right)\right)\right)}}\right)\right)\]

  8. Final simplification0.8

    \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(\left({im}^{5}\right), \frac{-1}{60}, \left(\frac{im \cdot \left({\left(im \cdot \left(im \cdot \frac{-1}{3}\right)\right)}^{3} - 8\right)}{\mathsf{fma}\left(2, \left(im \cdot \left(im \cdot \frac{-1}{3}\right)\right), \left(\mathsf{fma}\left(\left(im \cdot \left(im \cdot \frac{-1}{3}\right)\right), \left(im \cdot \left(im \cdot \frac{-1}{3}\right)\right), 4\right)\right)\right)}\right)\right)\]

Reproduce

herbie shell --seed 2019120 +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))))