2cos (problem 3.3.5)

Average Error: 39.4 → 18.4

Time: 8.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -8.56774410847329512 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{{\left({\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^{3}\right)}^{3} - {\left({\left(\cos x\right)}^{3}\right)}^{3}}{\left({\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^{6} + {\left(\cos x\right)}^{6}\right) + {\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^{3} \cdot {\left(\cos x\right)}^{3}}}{\left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon\right) \cdot \sqrt[3]{{\left(\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x\right)}^{3}} + \cos x \cdot \cos x}\\ \mathbf{elif}\;x \le 1.44278688709749027 \cdot 10^{-22}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^{2} \cdot \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - {\left(\cos x\right)}^{3}}{\left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon\right) \cdot \left(\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x\right) + \cos x \cdot \cos x}\\ \end{array}\]

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 ((x <= -8.567744108473295e-16)) {
		VAR = ((double) (((double) (((double) (((double) pow(((double) pow(((double) (((double) (((double) cos(x)) * ((double) cos(eps)))) - ((double) (((double) sin(x)) * ((double) sin(eps)))))), 3.0)), 3.0)) - ((double) pow(((double) pow(((double) cos(x)), 3.0)), 3.0)))) / ((double) (((double) (((double) pow(((double) (((double) (((double) cos(x)) * ((double) cos(eps)))) - ((double) (((double) sin(x)) * ((double) sin(eps)))))), 6.0)) + ((double) pow(((double) cos(x)), 6.0)))) + ((double) (((double) pow(((double) (((double) (((double) cos(x)) * ((double) cos(eps)))) - ((double) (((double) sin(x)) * ((double) sin(eps)))))), 3.0)) * ((double) pow(((double) cos(x)), 3.0)))))))) / ((double) (((double) (((double) (((double) (((double) cos(eps)) * ((double) cos(x)))) - ((double) (((double) sin(x)) * ((double) sin(eps)))))) * ((double) cbrt(((double) pow(((double) (((double) (((double) (((double) cos(x)) * ((double) cos(eps)))) - ((double) (((double) sin(x)) * ((double) sin(eps)))))) + ((double) cos(x)))), 3.0)))))) + ((double) (((double) cos(x)) * ((double) cos(x))))))));
	} else {
		double VAR_1;
		if ((x <= 1.4427868870974903e-22)) {
			VAR_1 = ((double) (-2.0 * ((double) (((double) sin(((double) (((double) (((double) (x + eps)) - x)) / 2.0)))) * ((double) sin(((double) (((double) (((double) (x + eps)) + x)) / 2.0))))))));
		} else {
			VAR_1 = ((double) (((double) (((double) (((double) pow(((double) (((double) (((double) cos(x)) * ((double) cos(eps)))) - ((double) (((double) sin(x)) * ((double) sin(eps)))))), 2.0)) * ((double) (((double) (((double) cos(x)) * ((double) cos(eps)))) - ((double) (((double) sin(x)) * ((double) sin(eps)))))))) - ((double) pow(((double) cos(x)), 3.0)))) / ((double) (((double) (((double) (((double) (((double) cos(eps)) * ((double) cos(x)))) - ((double) (((double) sin(x)) * ((double) sin(eps)))))) * ((double) (((double) (((double) (((double) cos(x)) * ((double) cos(eps)))) - ((double) (((double) sin(x)) * ((double) sin(eps)))))) + ((double) cos(x)))))) + ((double) (((double) cos(x)) * ((double) 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 x < -8.567744108473295e-16

   1. Initial program 58.4

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

   3. Applied cos-sum 29.2

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

   5. Applied flip3-- 29.3

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

   6. Simplified 29.3

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

   8. Applied add-cbrt-cube 29.3

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

   9. Simplified 29.3

      \[\leadsto \frac{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^{3} - {\left(\cos x\right)}^{3}}{\left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon\right) \cdot \sqrt[3]{\color{blue}{{\left(\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x\right)}^{3}}} + \cos x \cdot \cos x}\]
   10. Using strategy rm

   11. Applied flip3-- 29.4

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

   12. Simplified 29.4

       \[\leadsto \frac{\frac{{\left({\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^{3}\right)}^{3} - {\left({\left(\cos x\right)}^{3}\right)}^{3}}{\color{blue}{\left({\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^{6} + {\left(\cos x\right)}^{6}\right) + {\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^{3} \cdot {\left(\cos x\right)}^{3}}}}{\left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon\right) \cdot \sqrt[3]{{\left(\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x\right)}^{3}} + \cos x \cdot \cos x}\]

   if -8.567744108473295e-16 < x < 1.4427868870974903e-22

   1. Initial program 18.6

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

   3. Applied diff-cos 6.4

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

   if 1.4427868870974903e-22 < x

   1. Initial program 57.6

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

   3. Applied cos-sum 28.8

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

   5. Applied flip3-- 28.9

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

   6. Simplified 28.9

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

   8. Applied add-cube-cbrt 29.9

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

   9. Applied unpow-prod-down 30.0

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

   10. Simplified 29.5

       \[\leadsto \frac{\color{blue}{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^{2}} \cdot {\left(\sqrt[3]{\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon}\right)}^{3} - {\left(\cos x\right)}^{3}}{\left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon\right) \cdot \left(\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x\right) + \cos x \cdot \cos x}\]

   11. Simplified 29.0

       \[\leadsto \frac{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^{2} \cdot \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - {\left(\cos x\right)}^{3}}{\left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon\right) \cdot \left(\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x\right) + \cos x \cdot \cos x}\]
3. Recombined 3 regimes into one program.

4. Final simplification 18.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -8.56774410847329512 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{{\left({\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^{3}\right)}^{3} - {\left({\left(\cos x\right)}^{3}\right)}^{3}}{\left({\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^{6} + {\left(\cos x\right)}^{6}\right) + {\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^{3} \cdot {\left(\cos x\right)}^{3}}}{\left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon\right) \cdot \sqrt[3]{{\left(\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x\right)}^{3}} + \cos x \cdot \cos x}\\ \mathbf{elif}\;x \le 1.44278688709749027 \cdot 10^{-22}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^{2} \cdot \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - {\left(\cos x\right)}^{3}}{\left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon\right) \cdot \left(\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x\right) + \cos x \cdot \cos x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020126 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))