cos2 (problem 3.4.1)

Average Error: 31.9 → 0.4

Time: 9.4s

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

Math

\[\frac{1 - \cos x}{x \cdot x}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.034533322447672377:\\ \;\;\;\;\frac{e^{\frac{2}{3} \cdot \log \left(1 - \cos x\right)}}{x} \cdot \frac{\sqrt[3]{\log \left(e^{1 - \cos x}\right)}}{x}\\ \mathbf{elif}\;x \le 0.0342323272118313024:\\ \;\;\;\;\left(\frac{1}{2} + \frac{1}{720} \cdot {x}^{4}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\left(x \cdot x\right) \cdot \left(\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1\right)}\\ \end{array}\]

C

double code(double x) {
	return (((double) (1.0 - ((double) cos(x)))) / ((double) (x * x)));
}

↓

double code(double x) {
	double VAR;
	if ((x <= -0.03453332244767238)) {
		VAR = ((double) ((((double) exp(((double) (0.6666666666666666 * ((double) log(((double) (1.0 - ((double) cos(x)))))))))) / x) * (((double) cbrt(((double) log(((double) exp(((double) (1.0 - ((double) cos(x)))))))))) / x)));
	} else {
		double VAR_1;
		if ((x <= 0.0342323272118313)) {
			VAR_1 = ((double) (((double) (0.5 + ((double) (0.001388888888888889 * ((double) pow(x, 4.0)))))) - ((double) (0.041666666666666664 * ((double) pow(x, 2.0))))));
		} else {
			VAR_1 = (((double) (((double) pow(1.0, 3.0)) - ((double) pow(((double) cos(x)), 3.0)))) / ((double) (((double) (x * x)) * ((double) (((double) (((double) cos(x)) * ((double) (1.0 + ((double) cos(x)))))) + ((double) (1.0 * 1.0)))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if x < -0.034533322447672377

   1. Initial program 1.0

      \[\frac{1 - \cos x}{x \cdot x}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 1.4

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1 - \cos x} \cdot \sqrt[3]{1 - \cos x}\right) \cdot \sqrt[3]{1 - \cos x}}}{x \cdot x}\]

   4. Applied times-frac 0.8

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1 - \cos x} \cdot \sqrt[3]{1 - \cos x}}{x} \cdot \frac{\sqrt[3]{1 - \cos x}}{x}}\]
   5. Using strategy rm

   6. Applied add-log-exp 0.8

      \[\leadsto \frac{\color{blue}{\log \left(e^{\sqrt[3]{1 - \cos x} \cdot \sqrt[3]{1 - \cos x}}\right)}}{x} \cdot \frac{\sqrt[3]{1 - \cos x}}{x}\]

   7. Simplified 0.7

      \[\leadsto \frac{\log \color{blue}{\left(e^{\sqrt[3]{{\left(1 - \cos x\right)}^{2}}}\right)}}{x} \cdot \frac{\sqrt[3]{1 - \cos x}}{x}\]
   8. Using strategy rm

   9. Applied add-log-exp 0.7

      \[\leadsto \frac{\log \left(e^{\sqrt[3]{{\left(1 - \cos x\right)}^{2}}}\right)}{x} \cdot \frac{\sqrt[3]{1 - \color{blue}{\log \left(e^{\cos x}\right)}}}{x}\]

   10. Applied add-log-exp 0.7

       \[\leadsto \frac{\log \left(e^{\sqrt[3]{{\left(1 - \cos x\right)}^{2}}}\right)}{x} \cdot \frac{\sqrt[3]{\color{blue}{\log \left(e^{1}\right)} - \log \left(e^{\cos x}\right)}}{x}\]

   11. Applied diff-log 0.7

       \[\leadsto \frac{\log \left(e^{\sqrt[3]{{\left(1 - \cos x\right)}^{2}}}\right)}{x} \cdot \frac{\sqrt[3]{\color{blue}{\log \left(\frac{e^{1}}{e^{\cos x}}\right)}}}{x}\]

   12. Simplified 0.7

       \[\leadsto \frac{\log \left(e^{\sqrt[3]{{\left(1 - \cos x\right)}^{2}}}\right)}{x} \cdot \frac{\sqrt[3]{\log \color{blue}{\left(e^{1 - \cos x}\right)}}}{x}\]
   13. Using strategy rm

   14. Applied add-exp-log 0.7

       \[\leadsto \frac{\color{blue}{e^{\log \left(\log \left(e^{\sqrt[3]{{\left(1 - \cos x\right)}^{2}}}\right)\right)}}}{x} \cdot \frac{\sqrt[3]{\log \left(e^{1 - \cos x}\right)}}{x}\]

   15. Simplified 0.6

       \[\leadsto \frac{e^{\color{blue}{\frac{2}{3} \cdot \log \left(1 - \cos x\right)}}}{x} \cdot \frac{\sqrt[3]{\log \left(e^{1 - \cos x}\right)}}{x}\]

   if -0.034533322447672377 < x < 0.0342323272118313024

   1. Initial program 62.1

      \[\frac{1 - \cos x}{x \cdot x}\]

   2. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{1}{720} \cdot {x}^{4}\right) - \frac{1}{24} \cdot {x}^{2}}\]

   if 0.0342323272118313024 < x

   1. Initial program 0.9

      \[\frac{1 - \cos x}{x \cdot x}\]
   2. Using strategy rm

   3. Applied flip3-- 1.0

      \[\leadsto \frac{\color{blue}{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}}}{x \cdot x}\]

   4. Applied associate-/l/ 1.0

      \[\leadsto \color{blue}{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\left(x \cdot x\right) \cdot \left(1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)\right)}}\]

   5. Simplified 1.0

      \[\leadsto \frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\color{blue}{\left(x \cdot x\right) \cdot \left(\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1\right)}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.034533322447672377:\\ \;\;\;\;\frac{e^{\frac{2}{3} \cdot \log \left(1 - \cos x\right)}}{x} \cdot \frac{\sqrt[3]{\log \left(e^{1 - \cos x}\right)}}{x}\\ \mathbf{elif}\;x \le 0.0342323272118313024:\\ \;\;\;\;\left(\frac{1}{2} + \frac{1}{720} \cdot {x}^{4}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\left(x \cdot x\right) \cdot \left(\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))