math.cos on complex, imaginary part

Average Error: 43.2 → 0.7

Time: 31.2s

Precision: 64

Math

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

↓

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

C

double f(double re, double im) {
        double r169294 = 0.5;
        double r169295 = re;
        double r169296 = sin(r169295);
        double r169297 = r169294 * r169296;
        double r169298 = im;
        double r169299 = -r169298;
        double r169300 = exp(r169299);
        double r169301 = exp(r169298);
        double r169302 = r169300 - r169301;
        double r169303 = r169297 * r169302;
        return r169303;
}

↓

double f(double re, double im) {
        double r169304 = 0.5;
        double r169305 = re;
        double r169306 = sin(r169305);
        double r169307 = r169304 * r169306;
        double r169308 = -0.3333333333333333;
        double r169309 = im;
        double r169310 = 3.0;
        double r169311 = pow(r169309, r169310);
        double r169312 = r169308 * r169311;
        double r169313 = r169307 * r169312;
        double r169314 = -2.0;
        double r169315 = 5.0;
        double r169316 = pow(r169309, r169315);
        double r169317 = -0.016666666666666666;
        double r169318 = r169316 * r169317;
        double r169319 = fma(r169309, r169314, r169318);
        double r169320 = r169307 * r169319;
        double r169321 = r169313 + r169320;
        return r169321;
}

Error

Bits error versus re

Bits error versus im

Target

Original	43.2
Target	0.3
Herbie	0.7

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

Derivation

1. Initial program 43.2

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

2. Taylor expanded around 0 0.7

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

3. Simplified 0.7

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

5. Applied fma-udef 0.7

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

6. Applied distribute-lft-in 0.7

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

7. Final simplification 0.7

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

Reproduce

herbie shell --seed 2019323 +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))))