2cos (problem 3.3.5)

Average Error: 39.5 → 15.9

Time: 7.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -4.95115446409742782 \cdot 10^{-9}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \sqrt[3]{{\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^{3}}\\ \mathbf{elif}\;\varepsilon \le 2.17240038846879428 \cdot 10^{-10}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \log \left(e^{\sin x \cdot \sin \varepsilon + \cos x}\right)\\ \end{array}\]

C

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

↓

double code(double x, double eps) {
	double temp;
	if ((eps <= -4.951154464097428e-09)) {
		temp = ((cos(x) * cos(eps)) - cbrt(pow(((sin(x) * sin(eps)) + cos(x)), 3.0)));
	} else {
		double temp_1;
		if ((eps <= 2.1724003884687943e-10)) {
			temp_1 = (eps * (((0.16666666666666666 * pow(x, 3.0)) - x) - (eps * 0.5)));
		} else {
			temp_1 = ((cos(x) * cos(eps)) - log(exp(((sin(x) * sin(eps)) + cos(x)))));
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus x

Bits error versus eps

Derivation

1. Split input into 3 regimes

2. if eps < -4.951154464097428e-09

   1. Initial program 30.8

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

   3. Applied cos-sum 1.4

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

   4. Applied associate--l- 1.4

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

   6. Applied add-cbrt-cube 1.5

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

   7. Simplified 1.5

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

   if -4.951154464097428e-09 < eps < 2.1724003884687943e-10

   1. Initial program 49.2

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

   2. Taylor expanded around 0 31.8

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \left({x}^{3} \cdot \varepsilon\right) - \left(x \cdot \varepsilon + \frac{1}{2} \cdot {\varepsilon}^{2}\right)}\]

   3. Simplified 31.8

      \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)}\]

   if 2.1724003884687943e-10 < eps

   1. Initial program 30.8

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

   3. Applied cos-sum 1.4

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

   4. Applied associate--l- 1.4

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

   6. Applied add-log-exp 1.5

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

   7. Applied add-log-exp 1.5

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

   8. Applied sum-log 1.6

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

   9. Simplified 1.5

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

4. Final simplification 15.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -4.95115446409742782 \cdot 10^{-9}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \sqrt[3]{{\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^{3}}\\ \mathbf{elif}\;\varepsilon \le 2.17240038846879428 \cdot 10^{-10}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \log \left(e^{\sin x \cdot \sin \varepsilon + \cos x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))