Average Error: 58.1 → 0.7
Time: 34.6s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]
↓
\[\left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{3} - \mathsf{fma}\left(\frac{1}{60}, {im}^{5}, im + im\right)\right) \cdot \left(0.5 \cdot \cos re\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)
↓
\left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{3} - \mathsf{fma}\left(\frac{1}{60}, {im}^{5}, im + im\right)\right) \cdot \left(0.5 \cdot \cos re\right)
double f(double re, double im) {
        double r7338278 = 0.5;
        double r7338279 = re;
        double r7338280 = cos(r7338279);
        double r7338281 = r7338278 * r7338280;
        double r7338282 = 0.0;
        double r7338283 = im;
        double r7338284 = r7338282 - r7338283;
        double r7338285 = exp(r7338284);
        double r7338286 = exp(r7338283);
        double r7338287 = r7338285 - r7338286;
        double r7338288 = r7338281 * r7338287;
        return r7338288;
}
↓
double f(double re, double im) {
        double r7338289 = im;
        double r7338290 = r7338289 * r7338289;
        double r7338291 = r7338289 * r7338290;
        double r7338292 = -0.3333333333333333;
        double r7338293 = r7338291 * r7338292;
        double r7338294 = 0.016666666666666666;
        double r7338295 = 5.0;
        double r7338296 = pow(r7338289, r7338295);
        double r7338297 = r7338289 + r7338289;
        double r7338298 = fma(r7338294, r7338296, r7338297);
        double r7338299 = r7338293 - r7338298;
        double r7338300 = 0.5;
        double r7338301 = re;
        double r7338302 = cos(r7338301);
        double r7338303 = r7338300 * r7338302;
        double r7338304 = r7338299 * r7338303;
        return r7338304;
}

Error

Bits error versus re

Bits error versus im

Target

Original58.1
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 58.1

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

  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

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019169 +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))))