Average Error: 58.0 → 0.7
Time: 34.0s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]
↓
\[\left(\frac{-1}{3} \cdot \left(im \cdot \left(im \cdot im\right)\right) - \mathsf{fma}\left(\frac{1}{60}, {im}^{5}, im \cdot 2\right)\right) \cdot \left(0.5 \cdot \cos re\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)
↓
\left(\frac{-1}{3} \cdot \left(im \cdot \left(im \cdot im\right)\right) - \mathsf{fma}\left(\frac{1}{60}, {im}^{5}, im \cdot 2\right)\right) \cdot \left(0.5 \cdot \cos re\right)
double f(double re, double im) {
        double r7510231 = 0.5;
        double r7510232 = re;
        double r7510233 = cos(r7510232);
        double r7510234 = r7510231 * r7510233;
        double r7510235 = 0.0;
        double r7510236 = im;
        double r7510237 = r7510235 - r7510236;
        double r7510238 = exp(r7510237);
        double r7510239 = exp(r7510236);
        double r7510240 = r7510238 - r7510239;
        double r7510241 = r7510234 * r7510240;
        return r7510241;
}
↓
double f(double re, double im) {
        double r7510242 = -0.3333333333333333;
        double r7510243 = im;
        double r7510244 = r7510243 * r7510243;
        double r7510245 = r7510243 * r7510244;
        double r7510246 = r7510242 * r7510245;
        double r7510247 = 0.016666666666666666;
        double r7510248 = 5.0;
        double r7510249 = pow(r7510243, r7510248);
        double r7510250 = 2.0;
        double r7510251 = r7510243 * r7510250;
        double r7510252 = fma(r7510247, r7510249, r7510251);
        double r7510253 = r7510246 - r7510252;
        double r7510254 = 0.5;
        double r7510255 = re;
        double r7510256 = cos(r7510255);
        double r7510257 = r7510254 * r7510256;
        double r7510258 = r7510253 * r7510257;
        return r7510258;
}

Error

Bits error versus re

Bits error versus im

Target

Original58.0
Target0.2
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(0.1666666666666666574148081281236954964697 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333217685101601546193705872 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\\ \end{array}\]

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 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019192 +o rules:numerics
(FPCore (re im)
  :name "math.sin on complex, imaginary part"

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