math.sin on complex, imaginary part

Average Error: 58.3 → 0.6

Time: 35.7s

Precision: 64

Math

\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]

↓

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

C

double f(double re, double im) {
        double r6710746 = 0.5;
        double r6710747 = re;
        double r6710748 = cos(r6710747);
        double r6710749 = r6710746 * r6710748;
        double r6710750 = 0.0;
        double r6710751 = im;
        double r6710752 = r6710750 - r6710751;
        double r6710753 = exp(r6710752);
        double r6710754 = exp(r6710751);
        double r6710755 = r6710753 - r6710754;
        double r6710756 = r6710749 * r6710755;
        return r6710756;
}

↓

double f(double re, double im) {
        double r6710757 = im;
        double r6710758 = r6710757 * r6710757;
        double r6710759 = r6710757 * r6710758;
        double r6710760 = -0.3333333333333333;
        double r6710761 = r6710759 * r6710760;
        double r6710762 = 0.016666666666666666;
        double r6710763 = 5.0;
        double r6710764 = pow(r6710757, r6710763);
        double r6710765 = r6710757 + r6710757;
        double r6710766 = fma(r6710762, r6710764, r6710765);
        double r6710767 = r6710761 - r6710766;
        double r6710768 = 0.5;
        double r6710769 = re;
        double r6710770 = cos(r6710769);
        double r6710771 = r6710768 * r6710770;
        double r6710772 = r6710767 * r6710771;
        return r6710772;
}

Error

Bits error versus re

Bits error versus im

Target

Original	58.3
Target	0.3
Herbie	0.6

\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(0.1666666666666666574148081281236954964697 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333217685101601546193705872 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\\ \end{array}\]

Derivation

1. Initial program 58.3

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]

2. Taylor expanded around 0 0.6

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

3. Simplified 0.6

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

4. Final simplification 0.6

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (re im)
  :name "math.sin on complex, imaginary part"

  :herbie-target
  (if (< (fabs im) 1.0) (- (* (cos re) (+ (+ im (* (* (* 0.16666666666666666 im) im) im)) (* (* (* (* (* 0.008333333333333333 im) im) im) im) im)))) (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))

  (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))