math.cos on complex, imaginary part

Average Error: 43.5 → 0.9

Time: 35.4s

Precision: 64

Math

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

↓

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

C

double f(double re, double im) {
        double r6845389 = 0.5;
        double r6845390 = re;
        double r6845391 = sin(r6845390);
        double r6845392 = r6845389 * r6845391;
        double r6845393 = im;
        double r6845394 = -r6845393;
        double r6845395 = exp(r6845394);
        double r6845396 = exp(r6845393);
        double r6845397 = r6845395 - r6845396;
        double r6845398 = r6845392 * r6845397;
        return r6845398;
}

↓

double f(double re, double im) {
        double r6845399 = 1.0;
        double r6845400 = re;
        double r6845401 = sin(r6845400);
        double r6845402 = r6845399 * r6845401;
        double r6845403 = im;
        double r6845404 = r6845402 * r6845403;
        double r6845405 = 0.16666666666666666;
        double r6845406 = r6845403 * r6845403;
        double r6845407 = r6845406 * r6845403;
        double r6845408 = r6845405 * r6845407;
        double r6845409 = 0.008333333333333333;
        double r6845410 = r6845407 * r6845406;
        double r6845411 = r6845409 * r6845410;
        double r6845412 = r6845408 + r6845411;
        double r6845413 = r6845401 * r6845412;
        double r6845414 = r6845404 + r6845413;
        double r6845415 = -r6845414;
        return r6845415;
}

Error

Bits error versus re

Bits error versus im

Target

Original	43.5
Target	0.3
Herbie	0.9

\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

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 0.9

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

3. Simplified 0.9

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

4. Taylor expanded around -inf 0.9

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

5. Simplified 0.9

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

6. Final simplification 0.9

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

Reproduce

herbie shell --seed 2019143 
(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))))