cos2 (problem 3.4.1)

Average Error: 31.3 → 0.3

Time: 4.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.02680284925783882 \lor \neg \left(x \le 0.0326302713396224922\right):\\ \;\;\;\;\frac{\sqrt{1 - \cos x}}{x} \cdot \frac{\sqrt{{1}^{3} - {\left(\cos x\right)}^{3}}}{x \cdot \sqrt{1 \cdot 1 + \cos x \cdot \left(\cos x + 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)\\ \end{array}\]

C

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

↓

double code(double x) {
	double temp;
	if (((x <= -0.026802849257838823) || !(x <= 0.03263027133962249))) {
		temp = ((sqrt((1.0 - cos(x))) / x) * (sqrt((pow(1.0, 3.0) - pow(cos(x), 3.0))) / (x * sqrt(((1.0 * 1.0) + (cos(x) * (cos(x) + 1.0)))))));
	} else {
		temp = fma(pow(x, 4.0), 0.001388888888888889, (0.5 - (0.041666666666666664 * pow(x, 2.0))));
	}
	return temp;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -0.026802849257838823 or 0.03263027133962249 < x

   1. Initial program 1.1

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

   3. Applied add-sqr-sqrt 1.2

      \[\leadsto \frac{\color{blue}{\sqrt{1 - \cos x} \cdot \sqrt{1 - \cos x}}}{x \cdot x}\]

   4. Applied times-frac 0.6

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

   6. Applied flip3-- 0.6

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

   7. Applied sqrt-div 0.6

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

   8. Applied associate-/l/ 0.6

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

   9. Taylor expanded around inf 0.6

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

   10. Simplified 0.6

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

   if -0.026802849257838823 < x < 0.03263027133962249

   1. Initial program 62.3

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

   2. Taylor expanded around 0 0.0

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

   3. Simplified 0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.02680284925783882 \lor \neg \left(x \le 0.0326302713396224922\right):\\ \;\;\;\;\frac{\sqrt{1 - \cos x}}{x} \cdot \frac{\sqrt{{1}^{3} - {\left(\cos x\right)}^{3}}}{x \cdot \sqrt{1 \cdot 1 + \cos x \cdot \left(\cos x + 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))