Average Error: 43.2 → 0.8
Time: 37.8s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(\sin re \cdot \left(-0.5\right)\right) \cdot \mathsf{fma}\left({im}^{5}, \frac{1}{60}, im \cdot \left(2 + \left(im \cdot im\right) \cdot \frac{1}{3}\right)\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left(\sin re \cdot \left(-0.5\right)\right) \cdot \mathsf{fma}\left({im}^{5}, \frac{1}{60}, im \cdot \left(2 + \left(im \cdot im\right) \cdot \frac{1}{3}\right)\right)
double f(double re, double im) {
        double r7820481 = 0.5;
        double r7820482 = re;
        double r7820483 = sin(r7820482);
        double r7820484 = r7820481 * r7820483;
        double r7820485 = im;
        double r7820486 = -r7820485;
        double r7820487 = exp(r7820486);
        double r7820488 = exp(r7820485);
        double r7820489 = r7820487 - r7820488;
        double r7820490 = r7820484 * r7820489;
        return r7820490;
}
↓
double f(double re, double im) {
        double r7820491 = re;
        double r7820492 = sin(r7820491);
        double r7820493 = 0.5;
        double r7820494 = -r7820493;
        double r7820495 = r7820492 * r7820494;
        double r7820496 = im;
        double r7820497 = 5.0;
        double r7820498 = pow(r7820496, r7820497);
        double r7820499 = 0.016666666666666666;
        double r7820500 = 2.0;
        double r7820501 = r7820496 * r7820496;
        double r7820502 = 0.3333333333333333;
        double r7820503 = r7820501 * r7820502;
        double r7820504 = r7820500 + r7820503;
        double r7820505 = r7820496 * r7820504;
        double r7820506 = fma(r7820498, r7820499, r7820505);
        double r7820507 = r7820495 * r7820506;
        return r7820507;
}

Error

Bits error versus re

Bits error versus im

Target

Original43.2
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 43.2

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

  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

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019135 +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))))