Average Error: 43.8 → 0.7
Time: 55.5s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\mathsf{fma}\left(\left({im}^{5}\right), \frac{-1}{60}, \left(im \cdot \left(im \cdot \left(im \cdot \frac{-1}{3}\right) - 2\right)\right)\right) \cdot \left(0.5 \cdot \sin re\right)\]
\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 \left(im \cdot \left(im \cdot \frac{-1}{3}\right) - 2\right)\right)\right) \cdot \left(0.5 \cdot \sin re\right)
double f(double re, double im) {
        double r31096967 = 0.5;
        double r31096968 = re;
        double r31096969 = sin(r31096968);
        double r31096970 = r31096967 * r31096969;
        double r31096971 = im;
        double r31096972 = -r31096971;
        double r31096973 = exp(r31096972);
        double r31096974 = exp(r31096971);
        double r31096975 = r31096973 - r31096974;
        double r31096976 = r31096970 * r31096975;
        return r31096976;
}
↓
double f(double re, double im) {
        double r31096977 = im;
        double r31096978 = 5.0;
        double r31096979 = pow(r31096977, r31096978);
        double r31096980 = -0.016666666666666666;
        double r31096981 = -0.3333333333333333;
        double r31096982 = r31096977 * r31096981;
        double r31096983 = r31096977 * r31096982;
        double r31096984 = 2.0;
        double r31096985 = r31096983 - r31096984;
        double r31096986 = r31096977 * r31096985;
        double r31096987 = fma(r31096979, r31096980, r31096986);
        double r31096988 = 0.5;
        double r31096989 = re;
        double r31096990 = sin(r31096989);
        double r31096991 = r31096988 * r31096990;
        double r31096992 = r31096987 * r31096991;
        return r31096992;
}

Error

Bits error versus re

Bits error versus im

Target

Original43.8
Target0.2
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 43.8

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

  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

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019121 +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))))