Average Error: 44.4 → 0.7
Time: 26.7s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(0.5 \cdot \sin re\right) \cdot \left(-\mathsf{fma}\left(\frac{1}{3}, {im}^{3}, \mathsf{fma}\left(\frac{1}{60}, {im}^{5}, 2 \cdot im\right)\right)\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left(0.5 \cdot \sin re\right) \cdot \left(-\mathsf{fma}\left(\frac{1}{3}, {im}^{3}, \mathsf{fma}\left(\frac{1}{60}, {im}^{5}, 2 \cdot im\right)\right)\right)
double f(double re, double im) {
        double r210987 = 0.5;
        double r210988 = re;
        double r210989 = sin(r210988);
        double r210990 = r210987 * r210989;
        double r210991 = im;
        double r210992 = -r210991;
        double r210993 = exp(r210992);
        double r210994 = exp(r210991);
        double r210995 = r210993 - r210994;
        double r210996 = r210990 * r210995;
        return r210996;
}
↓
double f(double re, double im) {
        double r210997 = 0.5;
        double r210998 = re;
        double r210999 = sin(r210998);
        double r211000 = r210997 * r210999;
        double r211001 = 0.3333333333333333;
        double r211002 = im;
        double r211003 = 3.0;
        double r211004 = pow(r211002, r211003);
        double r211005 = 0.016666666666666666;
        double r211006 = 5.0;
        double r211007 = pow(r211002, r211006);
        double r211008 = 2.0;
        double r211009 = r211008 * r211002;
        double r211010 = fma(r211005, r211007, r211009);
        double r211011 = fma(r211001, r211004, r211010);
        double r211012 = -r211011;
        double r211013 = r211000 * r211012;
        return r211013;
}

Error

Bits error versus re

Bits error versus im

Target

Original44.4
Target0.2
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 44.4

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

  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

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019208 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64

  :herbie-target
  (if (< (fabs im) 1) (- (* (sin re) (+ (+ im (* (* (* 0.166666666666666657 im) im) im)) (* (* (* (* (* 0.00833333333333333322 im) im) im) im) im)))) (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))

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