math.cos on complex, imaginary part

Average Error: 43.5 → 0.8

Time: 42.2s

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 r7610286 = 0.5;
        double r7610287 = re;
        double r7610288 = sin(r7610287);
        double r7610289 = r7610286 * r7610288;
        double r7610290 = im;
        double r7610291 = -r7610290;
        double r7610292 = exp(r7610291);
        double r7610293 = exp(r7610290);
        double r7610294 = r7610292 - r7610293;
        double r7610295 = r7610289 * r7610294;
        return r7610295;
}

↓

double f(double re, double im) {
        double r7610296 = im;
        double r7610297 = r7610296 * r7610296;
        double r7610298 = r7610296 * r7610297;
        double r7610299 = r7610298 * r7610297;
        double r7610300 = 0.008333333333333333;
        double r7610301 = r7610299 * r7610300;
        double r7610302 = 1.0;
        double r7610303 = r7610296 * r7610302;
        double r7610304 = r7610301 + r7610303;
        double r7610305 = re;
        double r7610306 = sin(r7610305);
        double r7610307 = r7610304 * r7610306;
        double r7610308 = 0.16666666666666666;
        double r7610309 = r7610298 * r7610308;
        double r7610310 = r7610306 * r7610309;
        double r7610311 = r7610307 + r7610310;
        double r7610312 = -r7610311;
        return r7610312;
}

Error

Bits error versus re

Bits error versus im

Target

Original	43.5
Target	0.2
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 43.5

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

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. Simplified 0.8

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

4. Taylor expanded around -inf 0.8

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

5. Simplified 0.8

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

6. Final simplification 0.8

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

Reproduce

herbie shell --seed 2019142 
(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))))