cos2 (problem 3.4.1)

Average Error: 31.0 → 0.4

Time: 4.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.033320062920463786:\\ \;\;\;\;\frac{\frac{{1}^{3} - \mathsf{expm1}\left(\mathsf{log1p}\left({\left(\cos x\right)}^{3}\right)\right)}{x}}{\mathsf{fma}\left(1, x \cdot 1, \left(x \cdot \cos x\right) \cdot \left(1 + \cos x\right)\right)}\\ \mathbf{elif}\;x \le 0.0329960873997033124:\\ \;\;\;\;\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(1 - \cos x\right)}}{x \cdot x}\\ \end{array}\]

C

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

↓

double code(double x) {
	double temp;
	if ((x <= -0.033320062920463786)) {
		temp = (((pow(1.0, 3.0) - expm1(log1p(pow(cos(x), 3.0)))) / x) / fma(1.0, (x * 1.0), ((x * cos(x)) * (1.0 + cos(x)))));
	} else {
		double temp_1;
		if ((x <= 0.03299608739970331)) {
			temp_1 = fma(pow(x, 4.0), 0.001388888888888889, (0.5 - (0.041666666666666664 * pow(x, 2.0))));
		} else {
			temp_1 = (exp(log((1.0 - cos(x)))) / (x * x));
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if x < -0.033320062920463786

   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-inv 0.6

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

   8. Applied flip3-- 0.6

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

   9. Applied associate-*l/ 0.6

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

   10. Applied frac-times 0.6

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

   11. Simplified 0.5

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

   12. Simplified 0.5

       \[\leadsto \frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{x}}{\color{blue}{\mathsf{fma}\left(1, x \cdot 1, \left(x \cdot \cos x\right) \cdot \left(1 + \cos x\right)\right)}}\]
   13. Using strategy rm

   14. Applied expm1-log1p-u 0.5

       \[\leadsto \frac{\frac{{1}^{3} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\cos x\right)}^{3}\right)\right)}}{x}}{\mathsf{fma}\left(1, x \cdot 1, \left(x \cdot \cos x\right) \cdot \left(1 + \cos x\right)\right)}\]

   if -0.033320062920463786 < x < 0.03299608739970331

   1. Initial program 62.2

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

   2. Taylor expanded around 0 0.0

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

   3. Simplified 0.0

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

   if 0.03299608739970331 < x

   1. Initial program 1.0

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

   3. Applied add-exp-log 1.0

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.033320062920463786:\\ \;\;\;\;\frac{\frac{{1}^{3} - \mathsf{expm1}\left(\mathsf{log1p}\left({\left(\cos x\right)}^{3}\right)\right)}{x}}{\mathsf{fma}\left(1, x \cdot 1, \left(x \cdot \cos x\right) \cdot \left(1 + \cos x\right)\right)}\\ \mathbf{elif}\;x \le 0.0329960873997033124:\\ \;\;\;\;\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(1 - \cos x\right)}}{x \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))