2cos (problem 3.3.5)

Average Error: 39.1 → 0.7

Time: 5.3s

Precision: 64

Math

\[\cos \left(x + \varepsilon\right) - \cos x\]

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -3.893780865416545 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(\cos x \cdot \cos \varepsilon - \cos x\right) \cdot \left(\cos x \cdot \cos \varepsilon - \cos x\right) - \left(\sin x \cdot \sin \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)}{\left(\cos x \cdot \cos \varepsilon - \cos x\right) + \sin x \cdot \sin \varepsilon}\\ \mathbf{elif}\;\varepsilon \le 1.14867786703334021 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1}{2} \cdot {\varepsilon}^{2} - \sin x \cdot \sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3} + {\left(\cos x\right)}^{3}}{\left(\sin x \cdot \sin \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon - \cos x\right) + \cos x \cdot \cos x}\\ \end{array}\]

C

double code(double x, double eps) {
	return (cos((x + eps)) - cos(x));
}

↓

double code(double x, double eps) {
	double VAR;
	if ((eps <= -3.893780865416545e-05)) {
		VAR = (((((cos(x) * cos(eps)) - cos(x)) * ((cos(x) * cos(eps)) - cos(x))) - ((sin(x) * sin(eps)) * (sin(x) * sin(eps)))) / (((cos(x) * cos(eps)) - cos(x)) + (sin(x) * sin(eps))));
	} else {
		double VAR_1;
		if ((eps <= 1.1486778670333402e-05)) {
			VAR_1 = ((-0.5 * pow(eps, 2.0)) - (sin(x) * sin(eps)));
		} else {
			VAR_1 = ((cos(x) * cos(eps)) - ((pow((sin(x) * sin(eps)), 3.0) + pow(cos(x), 3.0)) / (((sin(x) * sin(eps)) * ((sin(x) * sin(eps)) - cos(x))) + (cos(x) * cos(x)))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus eps

Derivation

1. Split input into 3 regimes

2. if eps < -3.893780865416545e-05

   1. Initial program 29.9

      \[\cos \left(x + \varepsilon\right) - \cos x\]
   2. Using strategy rm

   3. Applied cos-sum 0.9

      \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   4. Using strategy rm

   5. Applied associate--l- 0.9

      \[\leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
   6. Using strategy rm

   7. Applied +-commutative 0.9

      \[\leadsto \cos x \cdot \cos \varepsilon - \color{blue}{\left(\cos x + \sin x \cdot \sin \varepsilon\right)}\]

   8. Applied associate--r+ 0.9

      \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \cos x\right) - \sin x \cdot \sin \varepsilon}\]
   9. Using strategy rm

   10. Applied flip-- 1.0

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

   if -3.893780865416545e-05 < eps < 1.1486778670333402e-05

   1. Initial program 48.8

      \[\cos \left(x + \varepsilon\right) - \cos x\]
   2. Using strategy rm

   3. Applied cos-sum 48.4

      \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   4. Using strategy rm

   5. Applied associate--l- 48.4

      \[\leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
   6. Using strategy rm

   7. Applied +-commutative 48.4

      \[\leadsto \cos x \cdot \cos \varepsilon - \color{blue}{\left(\cos x + \sin x \cdot \sin \varepsilon\right)}\]

   8. Applied associate--r+ 11.8

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

   9. Taylor expanded around 0 0.4

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot {\varepsilon}^{2}} - \sin x \cdot \sin \varepsilon\]

   if 1.1486778670333402e-05 < eps

   1. Initial program 29.6

      \[\cos \left(x + \varepsilon\right) - \cos x\]
   2. Using strategy rm

   3. Applied cos-sum 0.9

      \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   4. Using strategy rm

   5. Applied associate--l- 0.9

      \[\leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
   6. Using strategy rm

   7. Applied flip3-+ 1.1

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

   8. Simplified 1.1

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -3.893780865416545 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(\cos x \cdot \cos \varepsilon - \cos x\right) \cdot \left(\cos x \cdot \cos \varepsilon - \cos x\right) - \left(\sin x \cdot \sin \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)}{\left(\cos x \cdot \cos \varepsilon - \cos x\right) + \sin x \cdot \sin \varepsilon}\\ \mathbf{elif}\;\varepsilon \le 1.14867786703334021 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1}{2} \cdot {\varepsilon}^{2} - \sin x \cdot \sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3} + {\left(\cos x\right)}^{3}}{\left(\sin x \cdot \sin \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon - \cos x\right) + \cos x \cdot \cos x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020071 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))