Average Error: 57.8 → 0.7
Time: 2.2m
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]
↓
\[\mathsf{fma}\left(\left({im}^{5}\right), \frac{-1}{60}, \left(im \cdot -2 + im \cdot \left(im \cdot \left(im \cdot \frac{-1}{3}\right)\right)\right)\right) \cdot \left(0.5 \cdot \cos re\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)
↓
\mathsf{fma}\left(\left({im}^{5}\right), \frac{-1}{60}, \left(im \cdot -2 + im \cdot \left(im \cdot \left(im \cdot \frac{-1}{3}\right)\right)\right)\right) \cdot \left(0.5 \cdot \cos re\right)
double f(double re, double im) {
        double r43752413 = 0.5;
        double r43752414 = re;
        double r43752415 = cos(r43752414);
        double r43752416 = r43752413 * r43752415;
        double r43752417 = 0.0;
        double r43752418 = im;
        double r43752419 = r43752417 - r43752418;
        double r43752420 = exp(r43752419);
        double r43752421 = exp(r43752418);
        double r43752422 = r43752420 - r43752421;
        double r43752423 = r43752416 * r43752422;
        return r43752423;
}
↓
double f(double re, double im) {
        double r43752424 = im;
        double r43752425 = 5.0;
        double r43752426 = pow(r43752424, r43752425);
        double r43752427 = -0.016666666666666666;
        double r43752428 = -2.0;
        double r43752429 = r43752424 * r43752428;
        double r43752430 = -0.3333333333333333;
        double r43752431 = r43752424 * r43752430;
        double r43752432 = r43752424 * r43752431;
        double r43752433 = r43752424 * r43752432;
        double r43752434 = r43752429 + r43752433;
        double r43752435 = fma(r43752426, r43752427, r43752434);
        double r43752436 = 0.5;
        double r43752437 = re;
        double r43752438 = cos(r43752437);
        double r43752439 = r43752436 * r43752438;
        double r43752440 = r43752435 * r43752439;
        return r43752440;
}

Error

Bits error versus re

Bits error versus im

Target

Original57.8
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 57.8

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

  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

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

  5. Applied sub-neg0.7

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

  6. Applied distribute-rgt-in0.7

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

  7. Simplified0.7

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

  8. Final simplification0.7

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

Reproduce

herbie shell --seed 2019125 +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))))