Average Error: 58.0 → 0.8
Time: 38.1s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]
↓
\[\left(0.5 \cdot \cos re\right) \cdot \left(-\mathsf{fma}\left(\frac{1}{3}, {im}^{3}, \mathsf{fma}\left(\frac{1}{60}, {im}^{5}, 2 \cdot im\right)\right)\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)
↓
\left(0.5 \cdot \cos 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 r115209 = 0.5;
        double r115210 = re;
        double r115211 = cos(r115210);
        double r115212 = r115209 * r115211;
        double r115213 = 0.0;
        double r115214 = im;
        double r115215 = r115213 - r115214;
        double r115216 = exp(r115215);
        double r115217 = exp(r115214);
        double r115218 = r115216 - r115217;
        double r115219 = r115212 * r115218;
        return r115219;
}
↓
double f(double re, double im) {
        double r115220 = 0.5;
        double r115221 = re;
        double r115222 = cos(r115221);
        double r115223 = r115220 * r115222;
        double r115224 = 0.3333333333333333;
        double r115225 = im;
        double r115226 = 3.0;
        double r115227 = pow(r115225, r115226);
        double r115228 = 0.016666666666666666;
        double r115229 = 5.0;
        double r115230 = pow(r115225, r115229);
        double r115231 = 2.0;
        double r115232 = r115231 * r115225;
        double r115233 = fma(r115228, r115230, r115232);
        double r115234 = fma(r115224, r115227, r115233);
        double r115235 = -r115234;
        double r115236 = r115223 * r115235;
        return r115236;
}

Error

Bits error versus re

Bits error versus im

Target

Original58.0
Target0.2
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(0.1666666666666666574148081281236954964697 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333217685101601546193705872 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 58.0

    \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]

  2. Taylor expanded around 0 0.8

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

  3. Simplified0.8

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (re im)
  :name "math.sin on complex, imaginary part"
  :precision binary64

  :herbie-target
  (if (< (fabs im) 1) (- (* (cos re) (+ (+ im (* (* (* 0.16666666666666666 im) im) im)) (* (* (* (* (* 0.008333333333333333 im) im) im) im) im)))) (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))

  (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))