Average Error: 58.0 → 0.7
Time: 38.2s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]
↓
\[\left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{3}\right) \cdot \left(0.5 \cdot \cos re\right) + \mathsf{fma}\left(-2, im, \frac{-1}{60} \cdot {im}^{5}\right) \cdot \left(0.5 \cdot \cos re\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)
↓
\left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{3}\right) \cdot \left(0.5 \cdot \cos re\right) + \mathsf{fma}\left(-2, im, \frac{-1}{60} \cdot {im}^{5}\right) \cdot \left(0.5 \cdot \cos re\right)
double f(double re, double im) {
        double r7128492 = 0.5;
        double r7128493 = re;
        double r7128494 = cos(r7128493);
        double r7128495 = r7128492 * r7128494;
        double r7128496 = 0.0;
        double r7128497 = im;
        double r7128498 = r7128496 - r7128497;
        double r7128499 = exp(r7128498);
        double r7128500 = exp(r7128497);
        double r7128501 = r7128499 - r7128500;
        double r7128502 = r7128495 * r7128501;
        return r7128502;
}
↓
double f(double re, double im) {
        double r7128503 = im;
        double r7128504 = r7128503 * r7128503;
        double r7128505 = r7128503 * r7128504;
        double r7128506 = -0.3333333333333333;
        double r7128507 = r7128505 * r7128506;
        double r7128508 = 0.5;
        double r7128509 = re;
        double r7128510 = cos(r7128509);
        double r7128511 = r7128508 * r7128510;
        double r7128512 = r7128507 * r7128511;
        double r7128513 = -2.0;
        double r7128514 = -0.016666666666666666;
        double r7128515 = 5.0;
        double r7128516 = pow(r7128503, r7128515);
        double r7128517 = r7128514 * r7128516;
        double r7128518 = fma(r7128513, r7128503, r7128517);
        double r7128519 = r7128518 * r7128511;
        double r7128520 = r7128512 + r7128519;
        return r7128520;
}

Error

Bits error versus re

Bits error versus im

Target

Original58.0
Target0.3
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 58.0

    \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  5. Applied fma-udef0.7

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

  6. Applied distribute-lft-in0.7

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

  7. Simplified0.7

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

  8. Final simplification0.7

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

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (re im)
  :name "math.sin on complex, imaginary part"

  :herbie-target
  (if (< (fabs im) 1) (- (* (cos re) (+ (+ im (* (* (* 1/6 im) im) im)) (* (* (* (* (* 1/120 im) im) im) im) im)))) (* (* 0.5 (cos re)) (- (exp (- 0 im)) (exp im))))

  (* (* 0.5 (cos re)) (- (exp (- 0 im)) (exp im))))