math.cos on complex, imaginary part

Average Error: 44.1 → 0.8

Time: 10.2s

Precision: 64

Math

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

↓

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

C

double f(double re, double im) {
        double r309514 = 0.5;
        double r309515 = re;
        double r309516 = sin(r309515);
        double r309517 = r309514 * r309516;
        double r309518 = im;
        double r309519 = -r309518;
        double r309520 = exp(r309519);
        double r309521 = exp(r309518);
        double r309522 = r309520 - r309521;
        double r309523 = r309517 * r309522;
        return r309523;
}

↓

double f(double re, double im) {
        double r309524 = 0.5;
        double r309525 = re;
        double r309526 = sin(r309525);
        double r309527 = r309524 * r309526;
        double r309528 = 0.3333333333333333;
        double r309529 = im;
        double r309530 = 3.0;
        double r309531 = pow(r309529, r309530);
        double r309532 = r309528 * r309531;
        double r309533 = 0.016666666666666666;
        double r309534 = 5.0;
        double r309535 = pow(r309529, r309534);
        double r309536 = cbrt(r309535);
        double r309537 = r309536 * r309536;
        double r309538 = r309533 * r309537;
        double r309539 = r309538 * r309536;
        double r309540 = 2.0;
        double r309541 = r309540 * r309529;
        double r309542 = r309539 + r309541;
        double r309543 = r309532 + r309542;
        double r309544 = -r309543;
        double r309545 = r309527 * r309544;
        return r309545;
}

Error

Bits error versus re

Bits error versus im

Target

Original	44.1
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 44.1

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

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 add-cube-cbrt 0.8

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

5. Applied associate-*r* 0.8

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

6. Final simplification 0.8

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

Reproduce

herbie shell --seed 2019353 
(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))))