Average Error: 43.4 → 0.7
Time: 1.5m
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(\frac{-1}{60} \cdot {im}^{5}\right) \cdot \left(0.5 \cdot \sin re\right) + \left(im \cdot \sin re\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{3}, \left(im \cdot im\right), -2\right) \cdot 0.5\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left(\frac{-1}{60} \cdot {im}^{5}\right) \cdot \left(0.5 \cdot \sin re\right) + \left(im \cdot \sin re\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{3}, \left(im \cdot im\right), -2\right) \cdot 0.5\right)
double f(double re, double im) {
        double r46957518 = 0.5;
        double r46957519 = re;
        double r46957520 = sin(r46957519);
        double r46957521 = r46957518 * r46957520;
        double r46957522 = im;
        double r46957523 = -r46957522;
        double r46957524 = exp(r46957523);
        double r46957525 = exp(r46957522);
        double r46957526 = r46957524 - r46957525;
        double r46957527 = r46957521 * r46957526;
        return r46957527;
}
↓
double f(double re, double im) {
        double r46957528 = -0.016666666666666666;
        double r46957529 = im;
        double r46957530 = 5.0;
        double r46957531 = pow(r46957529, r46957530);
        double r46957532 = r46957528 * r46957531;
        double r46957533 = 0.5;
        double r46957534 = re;
        double r46957535 = sin(r46957534);
        double r46957536 = r46957533 * r46957535;
        double r46957537 = r46957532 * r46957536;
        double r46957538 = r46957529 * r46957535;
        double r46957539 = -0.3333333333333333;
        double r46957540 = r46957529 * r46957529;
        double r46957541 = -2.0;
        double r46957542 = fma(r46957539, r46957540, r46957541);
        double r46957543 = r46957542 * r46957533;
        double r46957544 = r46957538 * r46957543;
        double r46957545 = r46957537 + r46957544;
        return r46957545;
}

Error

Bits error versus re

Bits error versus im

Target

Original43.4
Target0.3
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 43.4

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

  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. Simplified0.7

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

  5. Applied sub-neg0.7

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

  6. Applied distribute-rgt-in0.7

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

  7. Simplified0.7

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

  9. Applied fma-udef0.7

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

  10. Applied distribute-lft-in0.7

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

  11. Simplified0.7

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

  12. Final simplification0.7

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

Reproduce

herbie shell --seed 2019128 +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))))