Average Error: 43.4 → 0.8
Time: 53.2s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\mathsf{fma}\left(\left({im}^{5}\right), \frac{-1}{60}, \left(im \cdot -2 + im \cdot \left(im \cdot \left(im \cdot \frac{-1}{3}\right)\right)\right)\right) \cdot \left(0.5 \cdot \sin re\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-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 \sin re\right)
double f(double re, double im) {
        double r34685829 = 0.5;
        double r34685830 = re;
        double r34685831 = sin(r34685830);
        double r34685832 = r34685829 * r34685831;
        double r34685833 = im;
        double r34685834 = -r34685833;
        double r34685835 = exp(r34685834);
        double r34685836 = exp(r34685833);
        double r34685837 = r34685835 - r34685836;
        double r34685838 = r34685832 * r34685837;
        return r34685838;
}
↓
double f(double re, double im) {
        double r34685839 = im;
        double r34685840 = 5.0;
        double r34685841 = pow(r34685839, r34685840);
        double r34685842 = -0.016666666666666666;
        double r34685843 = -2.0;
        double r34685844 = r34685839 * r34685843;
        double r34685845 = -0.3333333333333333;
        double r34685846 = r34685839 * r34685845;
        double r34685847 = r34685839 * r34685846;
        double r34685848 = r34685839 * r34685847;
        double r34685849 = r34685844 + r34685848;
        double r34685850 = fma(r34685841, r34685842, r34685849);
        double r34685851 = 0.5;
        double r34685852 = re;
        double r34685853 = sin(r34685852);
        double r34685854 = r34685851 * r34685853;
        double r34685855 = r34685850 * r34685854;
        return r34685855;
}

Error

Bits error versus re

Bits error versus im

Target

Original43.4
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 43.4

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

  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 sub-neg0.8

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

  6. Applied distribute-lft-in0.8

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

  7. Simplified0.8

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

  8. Final simplification0.8

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

Reproduce

herbie shell --seed 2019107 +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))))