cos2 (problem 3.4.1)

Average Error: 31.5 → 0.4

Time: 4.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.029536016732455837:\\ \;\;\;\;\frac{\log \left(e^{1 - \cos x}\right)}{x \cdot x}\\ \mathbf{elif}\;x \le 0.03411142251183559:\\ \;\;\;\;\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{{e}^{\left(\log \left(1 - \cos x\right)\right)}}{x}\\ \end{array}\]

C

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

↓

double code(double x) {
	double VAR;
	if ((x <= -0.029536016732455837)) {
		VAR = ((double) (((double) log(((double) exp(((double) (1.0 - ((double) cos(x)))))))) / ((double) (x * x))));
	} else {
		double VAR_1;
		if ((x <= 0.03411142251183559)) {
			VAR_1 = ((double) fma(((double) pow(x, 4.0)), 0.001388888888888889, ((double) (0.5 - ((double) (0.041666666666666664 * ((double) pow(x, 2.0))))))));
		} else {
			VAR_1 = ((double) (((double) (1.0 / x)) * ((double) (((double) pow(((double) M_E), ((double) log(((double) (1.0 - ((double) cos(x)))))))) / x))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if x < -0.029536016732455837

   1. Initial program 1.0

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

   3. Applied add-log-exp 1.0

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

   4. Applied add-log-exp 1.0

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

   5. Applied diff-log 1.1

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

   6. Simplified 1.1

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

   if -0.029536016732455837 < x < 0.03411142251183559

   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 0.03411142251183559 < x

   1. Initial program 1.0

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

   3. Applied *-un-lft-identity 1.0

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

   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-exp-log 0.5

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

   8. Applied pow1 0.5

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

   9. Applied log-pow 0.5

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

   10. Applied exp-prod 0.5

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

   11. Simplified 0.5

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.029536016732455837:\\ \;\;\;\;\frac{\log \left(e^{1 - \cos x}\right)}{x \cdot x}\\ \mathbf{elif}\;x \le 0.03411142251183559:\\ \;\;\;\;\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{{e}^{\left(\log \left(1 - \cos x\right)\right)}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020114 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))