2cos (problem 3.3.5)

Average Error: 39.6 → 0.8

Time: 5.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.45763440746872374 \cdot 10^{-7}:\\ \;\;\;\;\frac{\left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\cos x \cdot \cos \varepsilon\right) - \left(\sin x \cdot \sin \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)}{\cos x \cdot \cos \varepsilon + \sin x \cdot \sin \varepsilon} - \cos x\\ \mathbf{elif}\;\varepsilon \le 5.23681108699865033 \cdot 10^{-6}:\\ \;\;\;\;\frac{-1}{2} \cdot {\varepsilon}^{2} - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon \cdot \cos x - \frac{{\left(\sin \varepsilon \cdot \sin x\right)}^{3} + {\left(\cos x\right)}^{3}}{\left(\sin \varepsilon \cdot \sin x\right) \cdot \left(\sin \varepsilon \cdot \sin x - \cos x\right) + \cos x \cdot \cos x}\\ \end{array}\]

C

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

↓

double code(double x, double eps) {
	double VAR;
	if ((eps <= -1.4576344074687237e-07)) {
		VAR = (((((cos(x) * cos(eps)) * (cos(x) * cos(eps))) - ((sin(x) * sin(eps)) * (sin(x) * sin(eps)))) / ((cos(x) * cos(eps)) + (sin(x) * sin(eps)))) - cos(x));
	} else {
		double VAR_1;
		if ((eps <= 5.23681108699865e-06)) {
			VAR_1 = ((-0.5 * pow(eps, 2.0)) - (sin(eps) * sin(x)));
		} else {
			VAR_1 = ((cos(eps) * cos(x)) - ((pow((sin(eps) * sin(x)), 3.0) + pow(cos(x), 3.0)) / (((sin(eps) * sin(x)) * ((sin(eps) * sin(x)) - 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 < -1.4576344074687237e-07

   1. Initial program 30.9

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

   3. Applied cos-sum 1.2

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

   5. Applied flip-- 1.3

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

   if -1.4576344074687237e-07 < eps < 5.23681108699865e-06

   1. Initial program 49.1

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

   3. Applied +-commutative 49.1

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

   4. Applied cos-sum 48.6

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

   5. Applied associate--l- 48.6

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

   7. Applied +-commutative 48.6

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

   8. Applied associate--r+ 11.7

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

   9. Taylor expanded around 0 0.4

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

   if 5.23681108699865e-06 < eps

   1. Initial program 30.4

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

   3. Applied +-commutative 30.4

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

   4. Applied cos-sum 0.9

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

   5. Applied associate--l- 1.0

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

   7. Applied flip3-+ 1.1

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

   8. Simplified 1.1

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.45763440746872374 \cdot 10^{-7}:\\ \;\;\;\;\frac{\left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\cos x \cdot \cos \varepsilon\right) - \left(\sin x \cdot \sin \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)}{\cos x \cdot \cos \varepsilon + \sin x \cdot \sin \varepsilon} - \cos x\\ \mathbf{elif}\;\varepsilon \le 5.23681108699865033 \cdot 10^{-6}:\\ \;\;\;\;\frac{-1}{2} \cdot {\varepsilon}^{2} - \sin \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon \cdot \cos x - \frac{{\left(\sin \varepsilon \cdot \sin x\right)}^{3} + {\left(\cos x\right)}^{3}}{\left(\sin \varepsilon \cdot \sin x\right) \cdot \left(\sin \varepsilon \cdot \sin x - \cos x\right) + \cos x \cdot \cos x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))