math.cos on complex, imaginary part

Average Error: 43.8 → 0.8

Time: 9.7s

Precision: 64

Math

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

↓

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

C

double f(double re, double im) {
        double r387359 = 0.5;
        double r387360 = re;
        double r387361 = sin(r387360);
        double r387362 = r387359 * r387361;
        double r387363 = im;
        double r387364 = -r387363;
        double r387365 = exp(r387364);
        double r387366 = exp(r387363);
        double r387367 = r387365 - r387366;
        double r387368 = r387362 * r387367;
        return r387368;
}

↓

double f(double re, double im) {
        double r387369 = 0.5;
        double r387370 = re;
        double r387371 = sin(r387370);
        double r387372 = r387369 * r387371;
        double r387373 = im;
        double r387374 = 3.0;
        double r387375 = pow(r387373, r387374);
        double r387376 = r387372 * r387375;
        double r387377 = -0.3333333333333333;
        double r387378 = r387376 * r387377;
        double r387379 = 0.016666666666666666;
        double r387380 = 5.0;
        double r387381 = pow(r387373, r387380);
        double r387382 = r387379 * r387381;
        double r387383 = 2.0;
        double r387384 = r387383 * r387373;
        double r387385 = r387382 + r387384;
        double r387386 = -r387385;
        double r387387 = r387372 * r387386;
        double r387388 = r387378 + r387387;
        return r387388;
}

Error

Bits error versus re

Bits error versus im

Target

Original	43.8
Target	0.3
Herbie	0.8

\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.166666666666666657 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.00833333333333333322 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

1. Initial program 43.8

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

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. Using strategy rm

4. Applied distribute-neg-in 0.8

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

5. Applied distribute-lft-in 0.8

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

6. Simplified 0.8

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

7. Final simplification 0.8

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

Reproduce

herbie shell --seed 2020035 
(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))))