cos2 (problem 3.4.1)

Average Error: 30.8 → 0.4

Time: 4.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.02026115641790327:\\ \;\;\;\;\frac{\frac{\frac{\left(1 \cdot 1 + \frac{-1}{2}\right) - \frac{1}{2} \cdot \cos \left(x \cdot 2\right)}{1 + \cos x}}{x}}{x}\\ \mathbf{elif}\;x \le 0.033747369637332478:\\ \;\;\;\;{x}^{4} \cdot \frac{1}{720} + \left(\frac{1}{2} + x \cdot \left(x \cdot \frac{-1}{24}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{\cos x}{x}}{x}\\ \end{array}\]

C

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

↓

double code(double x) {
	double VAR;
	if ((x <= -0.02026115641790327)) {
		VAR = ((double) (((double) (((double) (((double) (((double) (((double) (1.0 * 1.0)) + -0.5)) - ((double) (0.5 * ((double) cos(((double) (x * 2.0)))))))) / ((double) (1.0 + ((double) cos(x)))))) / x)) / x));
	} else {
		double VAR_1;
		if ((x <= 0.03374736963733248)) {
			VAR_1 = ((double) (((double) (((double) pow(x, 4.0)) * 0.001388888888888889)) + ((double) (0.5 + ((double) (x * ((double) (x * -0.041666666666666664))))))));
		} else {
			VAR_1 = ((double) (((double) (((double) (1.0 / x)) - ((double) (((double) cos(x)) / x)))) / x));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if x < -0.02026115641790327

   1. Initial program 1.0

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

   3. Applied associate-/r* 0.4

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

   5. Applied flip-- 0.7

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

   7. Applied sqr-cos-a 0.9

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

   8. Applied associate--r+ 0.8

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

   9. Simplified 0.8

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

   if -0.02026115641790327 < x < 0.033747369637332478

   1. Initial program 61.5

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

   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

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

   if 0.033747369637332478 < x

   1. Initial program 1.1

      \[\frac{1 - \cos x}{x \cdot x}\]
   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 div-sub 0.6

      \[\leadsto \frac{\color{blue}{\frac{1}{x} - \frac{\cos x}{x}}}{x}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.02026115641790327:\\ \;\;\;\;\frac{\frac{\frac{\left(1 \cdot 1 + \frac{-1}{2}\right) - \frac{1}{2} \cdot \cos \left(x \cdot 2\right)}{1 + \cos x}}{x}}{x}\\ \mathbf{elif}\;x \le 0.033747369637332478:\\ \;\;\;\;{x}^{4} \cdot \frac{1}{720} + \left(\frac{1}{2} + x \cdot \left(x \cdot \frac{-1}{24}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{\cos x}{x}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))