2cos (problem 3.3.5)

Average Error: 40.0 → 0.7

Time: 6.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.0550978972917479 \cdot 10^{-4}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 4.0014569438137001 \cdot 10^{-7}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \left(x + \left(\varepsilon + x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \end{array}\]

C

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

↓

double code(double x, double eps) {
	double VAR;
	if ((eps <= -0.00010550978972917479)) {
		VAR = ((double) (((double) (((double) (((double) cos(x)) * ((double) cos(eps)))) - ((double) (((double) sin(x)) * ((double) sin(eps)))))) - ((double) cos(x))));
	} else {
		double VAR_1;
		if ((eps <= 4.0014569438137e-07)) {
			VAR_1 = ((double) (-2.0 * ((double) (((double) sin(((double) (eps * 0.5)))) * ((double) sin(((double) (0.5 * ((double) (x + ((double) (eps + x))))))))))));
		} else {
			VAR_1 = ((double) (((double) (((double) cos(x)) * ((double) cos(eps)))) - ((double) (((double) cos(x)) + ((double) (((double) sin(x)) * ((double) sin(eps))))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus eps

Derivation

1. Split input into 3 regimes

2. if eps < -1.0550978972917479e-4

   1. Initial program 30.8

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

   3. Applied cos-sum 0.9

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

   if -1.0550978972917479e-4 < eps < 4.0014569438137001e-7

   1. Initial program 49.3

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

   3. Applied diff-cos 37.3

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

   4. Simplified 0.4

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

   if 4.0014569438137001e-7 < eps

   1. Initial program 31.0

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

   3. Applied cos-sum 1.0

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

   4. Applied associate--l- 1.0

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

   5. Simplified 1.0

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.0550978972917479 \cdot 10^{-4}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 4.0014569438137001 \cdot 10^{-7}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \left(x + \left(\varepsilon + x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))