math.cos on complex, imaginary part

Average Error: 43.3 → 0.7

Time: 35.6s

Precision: 64

Math

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

↓

\[-\left(\left(\left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \left(im \cdot im\right)\right) \cdot 0.008333333333333333 + im \cdot 1.0\right) \cdot \sin re + \sin re \cdot \left(\left(im \cdot \left(im \cdot im\right)\right) \cdot 0.16666666666666666\right)\right)\]

C

double f(double re, double im) {
        double r4933875 = 0.5;
        double r4933876 = re;
        double r4933877 = sin(r4933876);
        double r4933878 = r4933875 * r4933877;
        double r4933879 = im;
        double r4933880 = -r4933879;
        double r4933881 = exp(r4933880);
        double r4933882 = exp(r4933879);
        double r4933883 = r4933881 - r4933882;
        double r4933884 = r4933878 * r4933883;
        return r4933884;
}

↓

double f(double re, double im) {
        double r4933885 = im;
        double r4933886 = r4933885 * r4933885;
        double r4933887 = r4933885 * r4933886;
        double r4933888 = r4933887 * r4933886;
        double r4933889 = 0.008333333333333333;
        double r4933890 = r4933888 * r4933889;
        double r4933891 = 1.0;
        double r4933892 = r4933885 * r4933891;
        double r4933893 = r4933890 + r4933892;
        double r4933894 = re;
        double r4933895 = sin(r4933894);
        double r4933896 = r4933893 * r4933895;
        double r4933897 = 0.16666666666666666;
        double r4933898 = r4933887 * r4933897;
        double r4933899 = r4933895 * r4933898;
        double r4933900 = r4933896 + r4933899;
        double r4933901 = -r4933900;
        return r4933901;
}

Error

Bits error versus re

Bits error versus im

Target

Original	43.3
Target	0.3
Herbie	0.7

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

Derivation

1. Initial program 43.3

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

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. Simplified 0.7

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

4. Taylor expanded around inf 0.7

   \[\leadsto \color{blue}{-\left(0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) + \left(1.0 \cdot \left(\sin re \cdot im\right) + 0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right)\right)\right)}\]

5. Simplified 0.7

   \[\leadsto \color{blue}{-\left(\left(\left(\left(im \cdot im\right) \cdot im\right) \cdot 0.16666666666666666\right) \cdot \sin re + \sin re \cdot \left(im \cdot 1.0 + \left(\left(\left(im \cdot im\right) \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot 0.008333333333333333\right)\right)}\]

6. Final simplification 0.7

   \[\leadsto -\left(\left(\left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \left(im \cdot im\right)\right) \cdot 0.008333333333333333 + im \cdot 1.0\right) \cdot \sin re + \sin re \cdot \left(\left(im \cdot \left(im \cdot im\right)\right) \cdot 0.16666666666666666\right)\right)\]

Reproduce

herbie shell --seed 2019146 
(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))))