Average Error: 43.4 → 0.7
Time: 10.9s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(0.5 \cdot \sin re\right) \cdot \left(\left(-\frac{1}{3} \cdot {im}^{3}\right) - \mathsf{fma}\left(\frac{1}{60}, {im}^{5}, 2 \cdot im\right)\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left(0.5 \cdot \sin re\right) \cdot \left(\left(-\frac{1}{3} \cdot {im}^{3}\right) - \mathsf{fma}\left(\frac{1}{60}, {im}^{5}, 2 \cdot im\right)\right)
double f(double re, double im) {
        double r229963 = 0.5;
        double r229964 = re;
        double r229965 = sin(r229964);
        double r229966 = r229963 * r229965;
        double r229967 = im;
        double r229968 = -r229967;
        double r229969 = exp(r229968);
        double r229970 = exp(r229967);
        double r229971 = r229969 - r229970;
        double r229972 = r229966 * r229971;
        return r229972;
}
↓
double f(double re, double im) {
        double r229973 = 0.5;
        double r229974 = re;
        double r229975 = sin(r229974);
        double r229976 = r229973 * r229975;
        double r229977 = 0.3333333333333333;
        double r229978 = im;
        double r229979 = 3.0;
        double r229980 = pow(r229978, r229979);
        double r229981 = r229977 * r229980;
        double r229982 = -r229981;
        double r229983 = 0.016666666666666666;
        double r229984 = 5.0;
        double r229985 = pow(r229978, r229984);
        double r229986 = 2.0;
        double r229987 = r229986 * r229978;
        double r229988 = fma(r229983, r229985, r229987);
        double r229989 = r229982 - r229988;
        double r229990 = r229976 * r229989;
        return r229990;
}

Error

Bits error versus re

Bits error versus im

Target

Original43.4
Target0.3
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.166666666666666657 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.00833333333333333322 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

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 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2020036 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64

  :herbie-target
  (if (< (fabs im) 1) (- (* (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))))