math.cos on complex, imaginary part

Average Error: 44.1 → 0.8

Time: 9.7s

Precision: 64

Math

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

↓

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

C

double f(double re, double im) {
        double r287519 = 0.5;
        double r287520 = re;
        double r287521 = sin(r287520);
        double r287522 = r287519 * r287521;
        double r287523 = im;
        double r287524 = -r287523;
        double r287525 = exp(r287524);
        double r287526 = exp(r287523);
        double r287527 = r287525 - r287526;
        double r287528 = r287522 * r287527;
        return r287528;
}

↓

double f(double re, double im) {
        double r287529 = 0.5;
        double r287530 = re;
        double r287531 = sin(r287530);
        double r287532 = r287529 * r287531;
        double r287533 = 0.3333333333333333;
        double r287534 = im;
        double r287535 = r287533 * r287534;
        double r287536 = r287535 * r287534;
        double r287537 = r287536 * r287534;
        double r287538 = -r287537;
        double r287539 = 0.016666666666666666;
        double r287540 = 5.0;
        double r287541 = pow(r287534, r287540);
        double r287542 = 2.0;
        double r287543 = r287542 * r287534;
        double r287544 = fma(r287539, r287541, r287543);
        double r287545 = r287538 - r287544;
        double r287546 = r287532 * r287545;
        return r287546;
}

Error

Bits error versus re

Bits error versus im

Target

Original	44.1
Target	0.3
Herbie	0.8

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

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 0.8

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

3. Simplified 0.8

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

5. Applied unpow3 0.8

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

6. Applied associate-*r* 0.8

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

7. Simplified 0.8

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

8. Final simplification 0.8

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

Reproduce

herbie shell --seed 2019353 +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))))