cos2 (problem 3.4.1)

Average Error: 31.5 → 0.3

Time: 4.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.029824101963535701:\\ \;\;\;\;\frac{\frac{{e}^{\left(\log \left(1 - \cos x\right)\right)}}{x}}{x}\\ \mathbf{elif}\;x \le 0.0299197087399348839:\\ \;\;\;\;\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{e}^{\left(\log \left(\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\mathsf{fma}\left(\cos x, 1 + \cos x, 1 \cdot 1\right)}\right)\right)}}{x}}{x}\\ \end{array}\]

C

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

↓

double code(double x) {
	double VAR;
	if ((x <= -0.0298241019635357)) {
		VAR = ((pow(((double) M_E), log((1.0 - cos(x)))) / x) / x);
	} else {
		double VAR_1;
		if ((x <= 0.029919708739934884)) {
			VAR_1 = fma(pow(x, 4.0), 0.001388888888888889, (0.5 - (0.041666666666666664 * pow(x, 2.0))));
		} else {
			VAR_1 = ((pow(((double) M_E), log(((pow(1.0, 3.0) - pow(cos(x), 3.0)) / fma(cos(x), (1.0 + cos(x)), (1.0 * 1.0))))) / x) / x);
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if x < -0.0298241019635357

   1. Initial program 1.0

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

   3. Applied associate-/r* 0.5

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

   5. Applied add-exp-log 0.5

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

   7. Applied pow1 0.5

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

   8. Applied log-pow 0.5

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

   9. Applied exp-prod 0.5

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

   10. Simplified 0.5

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

   if -0.0298241019635357 < x < 0.029919708739934884

   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 0.029919708739934884 < x

   1. Initial program 0.8

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

   3. Applied associate-/r* 0.5

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

   5. Applied add-exp-log 0.5

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

   7. Applied pow1 0.5

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

   8. Applied log-pow 0.5

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

   9. Applied exp-prod 0.5

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

   10. Simplified 0.5

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

   12. Applied flip3-- 0.5

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

   13. Simplified 0.5

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.029824101963535701:\\ \;\;\;\;\frac{\frac{{e}^{\left(\log \left(1 - \cos x\right)\right)}}{x}}{x}\\ \mathbf{elif}\;x \le 0.0299197087399348839:\\ \;\;\;\;\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{e}^{\left(\log \left(\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\mathsf{fma}\left(\cos x, 1 + \cos x, 1 \cdot 1\right)}\right)\right)}}{x}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))