math.cos on complex, imaginary part

Average Error: 43.5 → 0.8

Time: 20.0s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

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

C

double f(double re, double im) {
        double r212287 = 0.5;
        double r212288 = re;
        double r212289 = sin(r212288);
        double r212290 = r212287 * r212289;
        double r212291 = im;
        double r212292 = -r212291;
        double r212293 = exp(r212292);
        double r212294 = exp(r212291);
        double r212295 = r212293 - r212294;
        double r212296 = r212290 * r212295;
        return r212296;
}

↓

double f(double re, double im) {
        double r212297 = 0.5;
        double r212298 = re;
        double r212299 = sin(r212298);
        double r212300 = r212297 * r212299;
        double r212301 = -0.3333333333333333;
        double r212302 = im;
        double r212303 = 3.0;
        double r212304 = pow(r212302, r212303);
        double r212305 = r212301 * r212304;
        double r212306 = r212300 * r212305;
        double r212307 = -2.0;
        double r212308 = 5.0;
        double r212309 = pow(r212302, r212308);
        double r212310 = -0.016666666666666666;
        double r212311 = r212309 * r212310;
        double r212312 = fma(r212302, r212307, r212311);
        double r212313 = r212300 * r212312;
        double r212314 = r212306 + r212313;
        return r212314;
}

Error

Bits error versus re

Bits error versus im

Target

Original	43.5
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 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

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

5. Applied fma-udef 0.8

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

6. Applied distribute-lft-in 0.8

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

7. Final simplification 0.8

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

Reproduce

herbie shell --seed 2019351 +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))))