Average Error: 43.7 → 0.7
Time: 34.0s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\mathsf{fma}\left(\frac{-1}{3}, im \cdot \left(im \cdot im\right), \mathsf{fma}\left(\frac{-1}{60}, {im}^{5}, im \cdot -2\right)\right) \cdot \left(0.5 \cdot \sin re\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\mathsf{fma}\left(\frac{-1}{3}, im \cdot \left(im \cdot im\right), \mathsf{fma}\left(\frac{-1}{60}, {im}^{5}, im \cdot -2\right)\right) \cdot \left(0.5 \cdot \sin re\right)
double f(double re, double im) {
        double r7910213 = 0.5;
        double r7910214 = re;
        double r7910215 = sin(r7910214);
        double r7910216 = r7910213 * r7910215;
        double r7910217 = im;
        double r7910218 = -r7910217;
        double r7910219 = exp(r7910218);
        double r7910220 = exp(r7910217);
        double r7910221 = r7910219 - r7910220;
        double r7910222 = r7910216 * r7910221;
        return r7910222;
}
↓
double f(double re, double im) {
        double r7910223 = -0.3333333333333333;
        double r7910224 = im;
        double r7910225 = r7910224 * r7910224;
        double r7910226 = r7910224 * r7910225;
        double r7910227 = -0.016666666666666666;
        double r7910228 = 5.0;
        double r7910229 = pow(r7910224, r7910228);
        double r7910230 = -2.0;
        double r7910231 = r7910224 * r7910230;
        double r7910232 = fma(r7910227, r7910229, r7910231);
        double r7910233 = fma(r7910223, r7910226, r7910232);
        double r7910234 = 0.5;
        double r7910235 = re;
        double r7910236 = sin(r7910235);
        double r7910237 = r7910234 * r7910236;
        double r7910238 = r7910233 * r7910237;
        return r7910238;
}

Error

Bits error versus re

Bits error versus im

Target

Original43.7
Target0.3
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 43.7

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

  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

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, imaginary part"

  :herbie-target
  (if (< (fabs im) 1.0) (- (* (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))))