Average Error: 43.5 → 0.8
Time: 44.6s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(\frac{-1}{3} \cdot \left(im \cdot \left(im \cdot im\right)\right) - \mathsf{fma}\left(\frac{1}{60}, {im}^{5}, im + im\right)\right) \cdot \left(0.5 \cdot \sin re\right)\]
\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(\frac{1}{60}, {im}^{5}, im + im\right)\right) \cdot \left(0.5 \cdot \sin re\right)
double f(double re, double im) {
        double r6035112 = 0.5;
        double r6035113 = re;
        double r6035114 = sin(r6035113);
        double r6035115 = r6035112 * r6035114;
        double r6035116 = im;
        double r6035117 = -r6035116;
        double r6035118 = exp(r6035117);
        double r6035119 = exp(r6035116);
        double r6035120 = r6035118 - r6035119;
        double r6035121 = r6035115 * r6035120;
        return r6035121;
}
↓
double f(double re, double im) {
        double r6035122 = -0.3333333333333333;
        double r6035123 = im;
        double r6035124 = r6035123 * r6035123;
        double r6035125 = r6035123 * r6035124;
        double r6035126 = r6035122 * r6035125;
        double r6035127 = 0.016666666666666666;
        double r6035128 = 5.0;
        double r6035129 = pow(r6035123, r6035128);
        double r6035130 = r6035123 + r6035123;
        double r6035131 = fma(r6035127, r6035129, r6035130);
        double r6035132 = r6035126 - r6035131;
        double r6035133 = 0.5;
        double r6035134 = re;
        double r6035135 = sin(r6035134);
        double r6035136 = r6035133 * r6035135;
        double r6035137 = r6035132 * r6035136;
        return r6035137;
}

Error

Bits error versus re

Bits error versus im

Target

Original43.5
Target0.3
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 43.5

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

  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

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019144 +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))))