2cos (problem 3.3.5)

Average Error: 39.7 → 16.5

Time: 7.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.7648464678937232 \cdot 10^{-14}:\\ \;\;\;\;\left(\log \left(e^{\cos x \cdot \cos \varepsilon}\right) - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 1.08087522352913595 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{24} \cdot {\varepsilon}^{4} - \left(x \cdot \varepsilon + \frac{1}{2} \cdot {\varepsilon}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon \cdot \cos x - \sqrt[3]{{\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^{3}}\\ \end{array}\]

C

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

↓

double code(double x, double eps) {
	double temp;
	if ((eps <= -1.7648464678937232e-14)) {
		temp = ((log(exp((cos(x) * cos(eps)))) - (sin(x) * sin(eps))) - cos(x));
	} else {
		double temp_1;
		if ((eps <= 1.080875223529136e-08)) {
			temp_1 = ((0.041666666666666664 * pow(eps, 4.0)) - ((x * eps) + (0.5 * pow(eps, 2.0))));
		} else {
			temp_1 = ((cos(eps) * cos(x)) - cbrt(pow(((sin(x) * sin(eps)) + cos(x)), 3.0)));
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus x

Bits error versus eps

Derivation

1. Split input into 3 regimes

2. if eps < -1.7648464678937232e-14

   1. Initial program 31.5

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

   3. Applied cos-sum 2.5

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

   5. Applied add-log-exp 2.7

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

   if -1.7648464678937232e-14 < eps < 1.080875223529136e-08

   1. Initial program 48.9

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

   3. Applied cos-sum 48.8

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

   4. Taylor expanded around inf 48.8

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

   5. Taylor expanded around 0 31.1

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {\varepsilon}^{4} - \left(x \cdot \varepsilon + \frac{1}{2} \cdot {\varepsilon}^{2}\right)}\]

   if 1.080875223529136e-08 < eps

   1. Initial program 29.7

      \[\cos \left(x + \varepsilon\right) - \cos x\]
   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. Taylor expanded around inf 1.1

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

   6. Applied add-cbrt-cube 1.2

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

   7. Simplified 1.2

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

4. Final simplification 16.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.7648464678937232 \cdot 10^{-14}:\\ \;\;\;\;\left(\log \left(e^{\cos x \cdot \cos \varepsilon}\right) - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 1.08087522352913595 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{24} \cdot {\varepsilon}^{4} - \left(x \cdot \varepsilon + \frac{1}{2} \cdot {\varepsilon}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon \cdot \cos x - \sqrt[3]{{\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))