cos2 (problem 3.4.1)

Average Error: 31.5 → 0.4

Time: 3.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.0358400916521245741:\\ \;\;\;\;\left(\sqrt[3]{\frac{1}{x}} \cdot \sqrt[3]{\frac{1}{x}}\right) \cdot \left(\sqrt[3]{\frac{1}{x}} \cdot \frac{1 - \cos x}{x}\right)\\ \mathbf{elif}\;x \le 0.032312263032406779:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{x} - \frac{\cos x}{x}\right)\\ \end{array}\]

C

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

↓

double code(double x) {
	double VAR;
	if ((x <= -0.035840091652124574)) {
		VAR = ((double) (((double) (((double) cbrt(((double) (1.0 / x)))) * ((double) cbrt(((double) (1.0 / x)))))) * ((double) (((double) cbrt(((double) (1.0 / x)))) * ((double) (((double) (1.0 - ((double) cos(x)))) / x))))));
	} else {
		double VAR_1;
		if ((x <= 0.03231226303240678)) {
			VAR_1 = ((double) (((double) (((double) (0.001388888888888889 * ((double) pow(x, 4.0)))) + 0.5)) - ((double) (0.041666666666666664 * ((double) pow(x, 2.0))))));
		} else {
			VAR_1 = ((double) (((double) (1.0 / x)) * ((double) (((double) (1.0 / x)) - ((double) (((double) cos(x)) / x))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if x < -0.035840091652124574

   1. Initial program 1.1

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

   3. Applied *-un-lft-identity 1.1

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

   4. Applied times-frac 0.5

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

   6. Applied add-cube-cbrt 0.9

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

   7. Applied associate-*l* 0.9

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

   if -0.035840091652124574 < x < 0.03231226303240678

   1. Initial program 62.2

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

   2. Taylor expanded around 0 0.0

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

   if 0.03231226303240678 < x

   1. Initial program 0.9

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

   3. Applied *-un-lft-identity 0.9

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

   4. Applied times-frac 0.5

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

   6. Applied div-sub 0.6

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0358400916521245741:\\ \;\;\;\;\left(\sqrt[3]{\frac{1}{x}} \cdot \sqrt[3]{\frac{1}{x}}\right) \cdot \left(\sqrt[3]{\frac{1}{x}} \cdot \frac{1 - \cos x}{x}\right)\\ \mathbf{elif}\;x \le 0.032312263032406779:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{x} - \frac{\cos x}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020140 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))