Average Error: 44.1 → 0.7
Time: 10.1s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(0.5 \cdot \sin re\right) \cdot \left(\left(-\frac{1}{3} \cdot {im}^{3}\right) - \mathsf{fma}\left(\frac{1}{60}, {im}^{5}, 2 \cdot im\right)\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left(0.5 \cdot \sin re\right) \cdot \left(\left(-\frac{1}{3} \cdot {im}^{3}\right) - \mathsf{fma}\left(\frac{1}{60}, {im}^{5}, 2 \cdot im\right)\right)
double f(double re, double im) {
        double r302152 = 0.5;
        double r302153 = re;
        double r302154 = sin(r302153);
        double r302155 = r302152 * r302154;
        double r302156 = im;
        double r302157 = -r302156;
        double r302158 = exp(r302157);
        double r302159 = exp(r302156);
        double r302160 = r302158 - r302159;
        double r302161 = r302155 * r302160;
        return r302161;
}
↓
double f(double re, double im) {
        double r302162 = 0.5;
        double r302163 = re;
        double r302164 = sin(r302163);
        double r302165 = r302162 * r302164;
        double r302166 = 0.3333333333333333;
        double r302167 = im;
        double r302168 = 3.0;
        double r302169 = pow(r302167, r302168);
        double r302170 = r302166 * r302169;
        double r302171 = -r302170;
        double r302172 = 0.016666666666666666;
        double r302173 = 5.0;
        double r302174 = pow(r302167, r302173);
        double r302175 = 2.0;
        double r302176 = r302175 * r302167;
        double r302177 = fma(r302172, r302174, r302176);
        double r302178 = r302171 - r302177;
        double r302179 = r302165 * r302178;
        return r302179;
}

Error

Bits error versus re

Bits error versus im

Target

Original44.1
Target0.2
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.166666666666666657 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.00833333333333333322 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 44.1

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

  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

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2020003 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64

  :herbie-target
  (if (< (fabs im) 1) (- (* (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))))