2cos (problem 3.3.5)

Average Error: 39.1 → 15.5

Time: 7.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -2.14002637841950426 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt[3]{{\left({\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^{3}\right)}^{3}} - {\left(\cos x\right)}^{3}}{\left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon\right) \cdot \left(\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x\right) + \cos x \cdot \cos x}\\ \mathbf{elif}\;\varepsilon \le 2.673419722608438 \cdot 10^{-8}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(e^{\cos x \cdot \cos \varepsilon}\right) - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

C

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

↓

double code(double x, double eps) {
	double temp;
	if ((eps <= -2.1400263784195043e-07)) {
		temp = ((cbrt(pow(pow(((cos(x) * cos(eps)) - (sin(x) * sin(eps))), 3.0), 3.0)) - pow(cos(x), 3.0)) / ((((cos(eps) * cos(x)) - (sin(x) * sin(eps))) * (((cos(x) * cos(eps)) - (sin(x) * sin(eps))) + cos(x))) + (cos(x) * cos(x))));
	} else {
		double temp_1;
		if ((eps <= 2.6734197226084376e-08)) {
			temp_1 = (eps * (((0.16666666666666666 * pow(x, 3.0)) - x) - (eps * 0.5)));
		} else {
			temp_1 = ((log(exp((cos(x) * cos(eps)))) - (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 < -2.1400263784195043e-07

   1. Initial program 31.0

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

   3. Applied cos-sum 1.1

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

   5. Applied flip3-- 1.3

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

   6. Simplified 1.3

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

   8. Applied add-cbrt-cube 1.3

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

   9. Simplified 1.3

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

   if -2.1400263784195043e-07 < eps < 2.6734197226084376e-08

   1. Initial program 48.7

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

   2. Taylor expanded around 0 30.9

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

   3. Simplified 30.9

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

   if 2.6734197226084376e-08 < eps

   1. Initial program 29.7

      \[\cos \left(x + \varepsilon\right) - \cos x\]
   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 add-log-exp 1.4

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

4. Final simplification 15.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -2.14002637841950426 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt[3]{{\left({\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}^{3}\right)}^{3}} - {\left(\cos x\right)}^{3}}{\left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon\right) \cdot \left(\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x\right) + \cos x \cdot \cos x}\\ \mathbf{elif}\;\varepsilon \le 2.673419722608438 \cdot 10^{-8}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(e^{\cos x \cdot \cos \varepsilon}\right) - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))