Average Error: 58.0 → 0.7
Time: 35.2s
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 r7707366 = 0.5;
        double r7707367 = re;
        double r7707368 = cos(r7707367);
        double r7707369 = r7707366 * r7707368;
        double r7707370 = 0.0;
        double r7707371 = im;
        double r7707372 = r7707370 - r7707371;
        double r7707373 = exp(r7707372);
        double r7707374 = exp(r7707371);
        double r7707375 = r7707373 - r7707374;
        double r7707376 = r7707369 * r7707375;
        return r7707376;
}
↓
double f(double re, double im) {
        double r7707377 = -0.3333333333333333;
        double r7707378 = im;
        double r7707379 = r7707378 * r7707378;
        double r7707380 = r7707378 * r7707379;
        double r7707381 = r7707377 * r7707380;
        double r7707382 = 0.016666666666666666;
        double r7707383 = 5.0;
        double r7707384 = pow(r7707378, r7707383);
        double r7707385 = 2.0;
        double r7707386 = r7707378 * r7707385;
        double r7707387 = fma(r7707382, r7707384, r7707386);
        double r7707388 = r7707381 - r7707387;
        double r7707389 = 0.5;
        double r7707390 = re;
        double r7707391 = cos(r7707390);
        double r7707392 = r7707389 * r7707391;
        double r7707393 = r7707388 * r7707392;
        return r7707393;
}

Error

Bits error versus re

Bits error versus im

Target

Original58.0
Target0.3
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 2019174 +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))))