Average Error: 57.8 → 0.9
Time: 1.5m
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]
↓
\[\mathsf{fma}\left(\left({im}^{5}\right), \frac{-1}{60}, \left(im \cdot \left(im \cdot \left(im \cdot \frac{-1}{3}\right) - 2\right)\right)\right) \cdot \left(0.5 \cdot \cos re\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)
↓
\mathsf{fma}\left(\left({im}^{5}\right), \frac{-1}{60}, \left(im \cdot \left(im \cdot \left(im \cdot \frac{-1}{3}\right) - 2\right)\right)\right) \cdot \left(0.5 \cdot \cos re\right)
double f(double re, double im) {
        double r30217149 = 0.5;
        double r30217150 = re;
        double r30217151 = cos(r30217150);
        double r30217152 = r30217149 * r30217151;
        double r30217153 = 0.0;
        double r30217154 = im;
        double r30217155 = r30217153 - r30217154;
        double r30217156 = exp(r30217155);
        double r30217157 = exp(r30217154);
        double r30217158 = r30217156 - r30217157;
        double r30217159 = r30217152 * r30217158;
        return r30217159;
}
↓
double f(double re, double im) {
        double r30217160 = im;
        double r30217161 = 5.0;
        double r30217162 = pow(r30217160, r30217161);
        double r30217163 = -0.016666666666666666;
        double r30217164 = -0.3333333333333333;
        double r30217165 = r30217160 * r30217164;
        double r30217166 = r30217160 * r30217165;
        double r30217167 = 2.0;
        double r30217168 = r30217166 - r30217167;
        double r30217169 = r30217160 * r30217168;
        double r30217170 = fma(r30217162, r30217163, r30217169);
        double r30217171 = 0.5;
        double r30217172 = re;
        double r30217173 = cos(r30217172);
        double r30217174 = r30217171 * r30217173;
        double r30217175 = r30217170 * r30217174;
        return r30217175;
}

Error

Bits error versus re

Bits error versus im

Target

Original57.8
Target0.3
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 57.8

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

  2. Taylor expanded around 0 0.9

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

  3. Simplified0.9

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

  4. Final simplification0.9

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

Reproduce

herbie shell --seed 2019124 +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))))