cos2 (problem 3.4.1)

Average Error: 31.7 → 0.3

Time: 4.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.0314231831300956152:\\ \;\;\;\;\frac{\frac{1}{x}}{x} \cdot \left(1 - \cos x\right)\\ \mathbf{elif}\;x \le 0.038605668716545485:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{x} - \frac{\cos x}{x}\right)\\ \end{array}\]

C

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

↓

double code(double x) {
	double VAR;
	if ((x <= -0.031423183130095615)) {
		VAR = ((double) (((double) (((double) (1.0 / x)) / x)) * ((double) (1.0 - ((double) cos(x))))));
	} else {
		double VAR_1;
		if ((x <= 0.038605668716545485)) {
			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 < -0.0314231831300956152

   1. Initial program 1.2

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

   3. Applied *-un-lft-identity 1.2

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

   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 0.7

      \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\frac{1}{x} - \frac{\cos x}{x}\right)}\]
   7. Using strategy rm

   8. Applied div-inv 0.6

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

   9. Applied div-inv 0.6

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

   10. Applied distribute-rgt-out-- 0.6

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

   11. Applied associate-*r* 0.6

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

   12. Simplified 0.5

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

   if -0.0314231831300956152 < x < 0.038605668716545485

   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 0.038605668716545485 < x

   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 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 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0314231831300956152:\\ \;\;\;\;\frac{\frac{1}{x}}{x} \cdot \left(1 - \cos x\right)\\ \mathbf{elif}\;x \le 0.038605668716545485:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{x} - \frac{\cos x}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020177 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))