Average Error: 43.6 → 0.8
Time: 30.7s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(\frac{-1}{3} \cdot \left(im \cdot \left(im \cdot im\right)\right) - \mathsf{fma}\left({im}^{5}, \frac{1}{60}, im + im\right)\right) \cdot \left(0.5 \cdot \sin re\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left(\frac{-1}{3} \cdot \left(im \cdot \left(im \cdot im\right)\right) - \mathsf{fma}\left({im}^{5}, \frac{1}{60}, im + im\right)\right) \cdot \left(0.5 \cdot \sin re\right)
double f(double re, double im) {
        double r8839129 = 0.5;
        double r8839130 = re;
        double r8839131 = sin(r8839130);
        double r8839132 = r8839129 * r8839131;
        double r8839133 = im;
        double r8839134 = -r8839133;
        double r8839135 = exp(r8839134);
        double r8839136 = exp(r8839133);
        double r8839137 = r8839135 - r8839136;
        double r8839138 = r8839132 * r8839137;
        return r8839138;
}
↓
double f(double re, double im) {
        double r8839139 = -0.3333333333333333;
        double r8839140 = im;
        double r8839141 = r8839140 * r8839140;
        double r8839142 = r8839140 * r8839141;
        double r8839143 = r8839139 * r8839142;
        double r8839144 = 5.0;
        double r8839145 = pow(r8839140, r8839144);
        double r8839146 = 0.016666666666666666;
        double r8839147 = r8839140 + r8839140;
        double r8839148 = fma(r8839145, r8839146, r8839147);
        double r8839149 = r8839143 - r8839148;
        double r8839150 = 0.5;
        double r8839151 = re;
        double r8839152 = sin(r8839151);
        double r8839153 = r8839150 * r8839152;
        double r8839154 = r8839149 * r8839153;
        return r8839154;
}

Error

Bits error versus re

Bits error versus im

Target

Original43.6
Target0.3
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.1666666666666666574148081281236954964697 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333217685101601546193705872 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 43.6

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

  2. Taylor expanded around 0 0.8

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

  3. Simplified0.8

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, imaginary part"

  :herbie-target
  (if (< (fabs im) 1.0) (- (* (sin re) (+ (+ im (* (* (* 0.16666666666666666 im) im) im)) (* (* (* (* (* 0.008333333333333333 im) im) im) im) im)))) (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))

  (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))