Average Error: 43.2 → 0.7
Time: 34.9s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(\frac{-1}{3} \cdot \left(im \cdot \left(im \cdot im\right)\right) - \mathsf{fma}\left(\frac{1}{60}, {im}^{5}, im + im\right)\right) \cdot \left(0.5 \cdot \sin re\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left(\frac{-1}{3} \cdot \left(im \cdot \left(im \cdot im\right)\right) - \mathsf{fma}\left(\frac{1}{60}, {im}^{5}, im + im\right)\right) \cdot \left(0.5 \cdot \sin re\right)
double f(double re, double im) {
        double r3960780 = 0.5;
        double r3960781 = re;
        double r3960782 = sin(r3960781);
        double r3960783 = r3960780 * r3960782;
        double r3960784 = im;
        double r3960785 = -r3960784;
        double r3960786 = exp(r3960785);
        double r3960787 = exp(r3960784);
        double r3960788 = r3960786 - r3960787;
        double r3960789 = r3960783 * r3960788;
        return r3960789;
}
↓
double f(double re, double im) {
        double r3960790 = -0.3333333333333333;
        double r3960791 = im;
        double r3960792 = r3960791 * r3960791;
        double r3960793 = r3960791 * r3960792;
        double r3960794 = r3960790 * r3960793;
        double r3960795 = 0.016666666666666666;
        double r3960796 = 5.0;
        double r3960797 = pow(r3960791, r3960796);
        double r3960798 = r3960791 + r3960791;
        double r3960799 = fma(r3960795, r3960797, r3960798);
        double r3960800 = r3960794 - r3960799;
        double r3960801 = 0.5;
        double r3960802 = re;
        double r3960803 = sin(r3960802);
        double r3960804 = r3960801 * r3960803;
        double r3960805 = r3960800 * r3960804;
        return r3960805;
}

Error

Bits error versus re

Bits error versus im

Target

Original43.2
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 43.2

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

  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

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019151 +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))))