Average Error: 58.0 → 0.8
Time: 35.8s
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 r6454173 = 0.5;
        double r6454174 = re;
        double r6454175 = cos(r6454174);
        double r6454176 = r6454173 * r6454175;
        double r6454177 = 0.0;
        double r6454178 = im;
        double r6454179 = r6454177 - r6454178;
        double r6454180 = exp(r6454179);
        double r6454181 = exp(r6454178);
        double r6454182 = r6454180 - r6454181;
        double r6454183 = r6454176 * r6454182;
        return r6454183;
}
↓
double f(double re, double im) {
        double r6454184 = -0.3333333333333333;
        double r6454185 = im;
        double r6454186 = r6454185 * r6454185;
        double r6454187 = r6454185 * r6454186;
        double r6454188 = r6454184 * r6454187;
        double r6454189 = 0.016666666666666666;
        double r6454190 = 5.0;
        double r6454191 = pow(r6454185, r6454190);
        double r6454192 = 2.0;
        double r6454193 = r6454185 * r6454192;
        double r6454194 = fma(r6454189, r6454191, r6454193);
        double r6454195 = r6454188 - r6454194;
        double r6454196 = 0.5;
        double r6454197 = re;
        double r6454198 = cos(r6454197);
        double r6454199 = r6454196 * r6454198;
        double r6454200 = r6454195 * r6454199;
        return r6454200;
}

Error

Bits error versus re

Bits error versus im

Target

Original58.0
Target0.3
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

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019179 +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))))