Average Error: 57.9 → 0.8
Time: 34.7s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]
↓
\[\mathsf{fma}\left(\frac{-1}{3}, im \cdot \left(im \cdot im\right), {im}^{5} \cdot \frac{-1}{60} - \left(im + im\right)\right) \cdot \left(0.5 \cdot \cos re\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)
↓
\mathsf{fma}\left(\frac{-1}{3}, im \cdot \left(im \cdot im\right), {im}^{5} \cdot \frac{-1}{60} - \left(im + im\right)\right) \cdot \left(0.5 \cdot \cos re\right)
double f(double re, double im) {
        double r7418174 = 0.5;
        double r7418175 = re;
        double r7418176 = cos(r7418175);
        double r7418177 = r7418174 * r7418176;
        double r7418178 = 0.0;
        double r7418179 = im;
        double r7418180 = r7418178 - r7418179;
        double r7418181 = exp(r7418180);
        double r7418182 = exp(r7418179);
        double r7418183 = r7418181 - r7418182;
        double r7418184 = r7418177 * r7418183;
        return r7418184;
}
↓
double f(double re, double im) {
        double r7418185 = -0.3333333333333333;
        double r7418186 = im;
        double r7418187 = r7418186 * r7418186;
        double r7418188 = r7418186 * r7418187;
        double r7418189 = 5.0;
        double r7418190 = pow(r7418186, r7418189);
        double r7418191 = -0.016666666666666666;
        double r7418192 = r7418190 * r7418191;
        double r7418193 = r7418186 + r7418186;
        double r7418194 = r7418192 - r7418193;
        double r7418195 = fma(r7418185, r7418188, r7418194);
        double r7418196 = 0.5;
        double r7418197 = re;
        double r7418198 = cos(r7418197);
        double r7418199 = r7418196 * r7418198;
        double r7418200 = r7418195 * r7418199;
        return r7418200;
}

Error

Bits error versus re

Bits error versus im

Target

Original57.9
Target0.2
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 57.9

    \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]

  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

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (re im)
  :name "math.sin on complex, imaginary part"

  :herbie-target
  (if (< (fabs im) 1) (- (* (cos re) (+ (+ im (* (* (* 1/6 im) im) im)) (* (* (* (* (* 1/120 im) im) im) im) im)))) (* (* 0.5 (cos re)) (- (exp (- 0 im)) (exp im))))

  (* (* 0.5 (cos re)) (- (exp (- 0 im)) (exp im))))