Average Error: 43.4 → 0.7
Time: 32.8s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(\frac{-1}{3} \cdot \left(im \cdot \left(im \cdot im\right)\right) - \mathsf{fma}\left(\frac{1}{60}, \left({im}^{5}\right), \left(im \cdot 2\right)\right)\right) \cdot \left(0.5 \cdot \sin re\right)\]
\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(\frac{1}{60}, \left({im}^{5}\right), \left(im \cdot 2\right)\right)\right) \cdot \left(0.5 \cdot \sin re\right)
double f(double re, double im) {
        double r5964159 = 0.5;
        double r5964160 = re;
        double r5964161 = sin(r5964160);
        double r5964162 = r5964159 * r5964161;
        double r5964163 = im;
        double r5964164 = -r5964163;
        double r5964165 = exp(r5964164);
        double r5964166 = exp(r5964163);
        double r5964167 = r5964165 - r5964166;
        double r5964168 = r5964162 * r5964167;
        return r5964168;
}
↓
double f(double re, double im) {
        double r5964169 = -0.3333333333333333;
        double r5964170 = im;
        double r5964171 = r5964170 * r5964170;
        double r5964172 = r5964170 * r5964171;
        double r5964173 = r5964169 * r5964172;
        double r5964174 = 0.016666666666666666;
        double r5964175 = 5.0;
        double r5964176 = pow(r5964170, r5964175);
        double r5964177 = 2.0;
        double r5964178 = r5964170 * r5964177;
        double r5964179 = fma(r5964174, r5964176, r5964178);
        double r5964180 = r5964173 - r5964179;
        double r5964181 = 0.5;
        double r5964182 = re;
        double r5964183 = sin(r5964182);
        double r5964184 = r5964181 * r5964183;
        double r5964185 = r5964180 * r5964184;
        return r5964185;
}

Error

Bits error versus re

Bits error versus im

Target

Original43.4
Target0.3
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 43.4

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, imaginary part"

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

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