Average Error: 43.7 → 0.7
Time: 33.4s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\mathsf{fma}\left(\frac{-1}{3}, im \cdot \left(im \cdot im\right), \mathsf{fma}\left(\frac{-1}{60}, {im}^{5}, im \cdot -2\right)\right) \cdot \left(0.5 \cdot \sin re\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\mathsf{fma}\left(\frac{-1}{3}, im \cdot \left(im \cdot im\right), \mathsf{fma}\left(\frac{-1}{60}, {im}^{5}, im \cdot -2\right)\right) \cdot \left(0.5 \cdot \sin re\right)
double f(double re, double im) {
        double r8121819 = 0.5;
        double r8121820 = re;
        double r8121821 = sin(r8121820);
        double r8121822 = r8121819 * r8121821;
        double r8121823 = im;
        double r8121824 = -r8121823;
        double r8121825 = exp(r8121824);
        double r8121826 = exp(r8121823);
        double r8121827 = r8121825 - r8121826;
        double r8121828 = r8121822 * r8121827;
        return r8121828;
}
↓
double f(double re, double im) {
        double r8121829 = -0.3333333333333333;
        double r8121830 = im;
        double r8121831 = r8121830 * r8121830;
        double r8121832 = r8121830 * r8121831;
        double r8121833 = -0.016666666666666666;
        double r8121834 = 5.0;
        double r8121835 = pow(r8121830, r8121834);
        double r8121836 = -2.0;
        double r8121837 = r8121830 * r8121836;
        double r8121838 = fma(r8121833, r8121835, r8121837);
        double r8121839 = fma(r8121829, r8121832, r8121838);
        double r8121840 = 0.5;
        double r8121841 = re;
        double r8121842 = sin(r8121841);
        double r8121843 = r8121840 * r8121842;
        double r8121844 = r8121839 * r8121843;
        return r8121844;
}

Error

Bits error versus re

Bits error versus im

Target

Original43.7
Target0.3
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.1666666666666666574148081281236954964697 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333217685101601546193705872 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 43.7

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

  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

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, imaginary part"

  :herbie-target
  (if (< (fabs im) 1.0) (- (* (sin re) (+ (+ im (* (* (* 0.16666666666666666 im) im) im)) (* (* (* (* (* 0.008333333333333333 im) im) im) im) im)))) (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))

  (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))