math.cos on complex, imaginary part

Average Error: 42.8 → 0.8

Time: 51.5s

Precision: 64

Math

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

↓

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

C

double f(double re, double im) {
        double r31952281 = 0.5;
        double r31952282 = re;
        double r31952283 = sin(r31952282);
        double r31952284 = r31952281 * r31952283;
        double r31952285 = im;
        double r31952286 = -r31952285;
        double r31952287 = exp(r31952286);
        double r31952288 = exp(r31952285);
        double r31952289 = r31952287 - r31952288;
        double r31952290 = r31952284 * r31952289;
        return r31952290;
}

↓

double f(double re, double im) {
        double r31952291 = im;
        double r31952292 = 5.0;
        double r31952293 = pow(r31952291, r31952292);
        double r31952294 = -0.016666666666666666;
        double r31952295 = -2.0;
        double r31952296 = r31952291 * r31952295;
        double r31952297 = -0.3333333333333333;
        double r31952298 = r31952291 * r31952297;
        double r31952299 = r31952291 * r31952298;
        double r31952300 = r31952291 * r31952299;
        double r31952301 = r31952296 + r31952300;
        double r31952302 = fma(r31952293, r31952294, r31952301);
        double r31952303 = 0.5;
        double r31952304 = re;
        double r31952305 = sin(r31952304);
        double r31952306 = r31952303 * r31952305;
        double r31952307 = r31952302 * r31952306;
        return r31952307;
}

Error

Bits error versus re

Bits error versus im

Target

Original	42.8
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 42.8

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

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

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

5. Applied sub-neg 0.8

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

6. Applied distribute-lft-in 0.8

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

7. Simplified 0.8

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

8. Final simplification 0.8

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

Reproduce

herbie shell --seed 2019112 +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))))