Average Error: 57.9 → 0.8
Time: 10.3s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]
↓
\[\left(0.5 \cdot \cos re\right) \cdot \left(\left(-\frac{1}{3} \cdot {im}^{3}\right) - \mathsf{fma}\left(\frac{1}{60}, {im}^{5}, 2 \cdot im\right)\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)
↓
\left(0.5 \cdot \cos 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 r251226 = 0.5;
        double r251227 = re;
        double r251228 = cos(r251227);
        double r251229 = r251226 * r251228;
        double r251230 = 0.0;
        double r251231 = im;
        double r251232 = r251230 - r251231;
        double r251233 = exp(r251232);
        double r251234 = exp(r251231);
        double r251235 = r251233 - r251234;
        double r251236 = r251229 * r251235;
        return r251236;
}
↓
double f(double re, double im) {
        double r251237 = 0.5;
        double r251238 = re;
        double r251239 = cos(r251238);
        double r251240 = r251237 * r251239;
        double r251241 = 0.3333333333333333;
        double r251242 = im;
        double r251243 = 3.0;
        double r251244 = pow(r251242, r251243);
        double r251245 = r251241 * r251244;
        double r251246 = -r251245;
        double r251247 = 0.016666666666666666;
        double r251248 = 5.0;
        double r251249 = pow(r251242, r251248);
        double r251250 = 2.0;
        double r251251 = r251250 * r251242;
        double r251252 = fma(r251247, r251249, r251251);
        double r251253 = r251246 - r251252;
        double r251254 = r251240 * r251253;
        return r251254;
}

Error

Bits error versus re

Bits error versus im

Target

Original57.9
Target0.3
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(0.166666666666666657 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.00833333333333333322 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 57.9

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

  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

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2020018 +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))))