Average Error: 43.5 → 0.8
Time: 40.7s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\mathsf{fma}\left(im \cdot im, \frac{-1}{3} \cdot im, \mathsf{fma}\left(\frac{-1}{60}, im \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right), -2 \cdot im\right)\right) \cdot \left(0.5 \cdot \sin re\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\mathsf{fma}\left(im \cdot im, \frac{-1}{3} \cdot im, \mathsf{fma}\left(\frac{-1}{60}, im \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right), -2 \cdot im\right)\right) \cdot \left(0.5 \cdot \sin re\right)
double f(double re, double im) {
        double r4252078 = 0.5;
        double r4252079 = re;
        double r4252080 = sin(r4252079);
        double r4252081 = r4252078 * r4252080;
        double r4252082 = im;
        double r4252083 = -r4252082;
        double r4252084 = exp(r4252083);
        double r4252085 = exp(r4252082);
        double r4252086 = r4252084 - r4252085;
        double r4252087 = r4252081 * r4252086;
        return r4252087;
}
↓
double f(double re, double im) {
        double r4252088 = im;
        double r4252089 = r4252088 * r4252088;
        double r4252090 = -0.3333333333333333;
        double r4252091 = r4252090 * r4252088;
        double r4252092 = -0.016666666666666666;
        double r4252093 = r4252089 * r4252089;
        double r4252094 = r4252088 * r4252093;
        double r4252095 = -2.0;
        double r4252096 = r4252095 * r4252088;
        double r4252097 = fma(r4252092, r4252094, r4252096);
        double r4252098 = fma(r4252089, r4252091, r4252097);
        double r4252099 = 0.5;
        double r4252100 = re;
        double r4252101 = sin(r4252100);
        double r4252102 = r4252099 * r4252101;
        double r4252103 = r4252098 * r4252102;
        return r4252103;
}

Error

Bits error versus re

Bits error versus im

Target

Original43.5
Target0.2
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 43.5

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

  2. Taylor expanded around 0 0.8

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

  3. Simplified0.8

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

  4. Taylor expanded around -inf 0.8

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

  5. Simplified0.8

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

  6. Final simplification0.8

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

Reproduce

herbie shell --seed 2019142 +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))))