math.cos on complex, imaginary part

Average Error: 43.7 → 0.8

Time: 26.3s

Precision: 64

Math

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

↓

\[-\mathsf{fma}\left({im}^{5}, \sin re \cdot 0.008333333333333333, \mathsf{fma}\left(\sin re \cdot im, 1.0, \sin re \cdot \left(\left(\left(im \cdot im\right) \cdot im\right) \cdot 0.16666666666666666\right)\right)\right)\]

C

double f(double re, double im) {
        double r3087260 = 0.5;
        double r3087261 = re;
        double r3087262 = sin(r3087261);
        double r3087263 = r3087260 * r3087262;
        double r3087264 = im;
        double r3087265 = -r3087264;
        double r3087266 = exp(r3087265);
        double r3087267 = exp(r3087264);
        double r3087268 = r3087266 - r3087267;
        double r3087269 = r3087263 * r3087268;
        return r3087269;
}

↓

double f(double re, double im) {
        double r3087270 = im;
        double r3087271 = 5.0;
        double r3087272 = pow(r3087270, r3087271);
        double r3087273 = re;
        double r3087274 = sin(r3087273);
        double r3087275 = 0.008333333333333333;
        double r3087276 = r3087274 * r3087275;
        double r3087277 = r3087274 * r3087270;
        double r3087278 = 1.0;
        double r3087279 = r3087270 * r3087270;
        double r3087280 = r3087279 * r3087270;
        double r3087281 = 0.16666666666666666;
        double r3087282 = r3087280 * r3087281;
        double r3087283 = r3087274 * r3087282;
        double r3087284 = fma(r3087277, r3087278, r3087283);
        double r3087285 = fma(r3087272, r3087276, r3087284);
        double r3087286 = -r3087285;
        return r3087286;
}

Error

Bits error versus re

Bits error versus im

Target

Original	43.7
Target	0.3
Herbie	0.8

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

Derivation

1. Initial program 43.7

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

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

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

4. Taylor expanded around inf 0.8

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

5. Simplified 0.8

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

6. Final simplification 0.8

   \[\leadsto -\mathsf{fma}\left({im}^{5}, \sin re \cdot 0.008333333333333333, \mathsf{fma}\left(\sin re \cdot im, 1.0, \sin re \cdot \left(\left(\left(im \cdot im\right) \cdot im\right) \cdot 0.16666666666666666\right)\right)\right)\]

Reproduce

herbie shell --seed 2019153 +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))))