2cos (problem 3.3.5)

Average Error: 40.0 → 16.1

Time: 7.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.5330496627903165 \cdot 10^{-7}:\\ \;\;\;\;\log \left(e^{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x}\right)\\ \mathbf{elif}\;\varepsilon \le 9.38913363863520236 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \frac{\left(\sin x \cdot \sin \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right) - \sqrt[3]{{\left(\cos x\right)}^{6}}}{\sin x \cdot \sin \varepsilon - \cos x}\\ \end{array}\]

C

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

↓

double code(double x, double eps) {
	double temp;
	if ((eps <= -1.5330496627903165e-07)) {
		temp = log(exp((((cos(x) * cos(eps)) - (sin(x) * sin(eps))) - cos(x))));
	} else {
		double temp_1;
		if ((eps <= 9.389133638635202e-07)) {
			temp_1 = (eps * (((0.16666666666666666 * pow(x, 3.0)) - x) - (eps * 0.5)));
		} else {
			temp_1 = ((cos(x) * cos(eps)) - ((((sin(x) * sin(eps)) * (sin(x) * sin(eps))) - cbrt(pow(cos(x), 6.0))) / ((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 < -1.5330496627903165e-07

   1. Initial program 30.6

      \[\cos \left(x + \varepsilon\right) - \cos x\]
   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. Applied associate--l- 1.2

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

   6. Applied add-log-exp 1.3

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

   7. Applied add-log-exp 1.3

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

   8. Applied sum-log 1.4

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

   9. Applied add-log-exp 1.5

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

   10. Applied diff-log 1.6

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

   11. Simplified 1.4

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

   13. Applied associate--r+ 1.3

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

   if -1.5330496627903165e-07 < eps < 9.389133638635202e-07

   1. Initial program 49.8

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

   2. Taylor expanded around 0 31.4

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

   3. Simplified 31.4

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

   if 9.389133638635202e-07 < eps

   1. Initial program 30.5

      \[\cos \left(x + \varepsilon\right) - \cos x\]
   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. Applied associate--l- 1.2

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

   6. Applied flip-+ 1.3

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

   8. Applied add-cbrt-cube 1.4

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

   9. Applied add-cbrt-cube 1.5

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

   10. Applied cbrt-unprod 1.4

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

   11. Simplified 1.5

       \[\leadsto \cos x \cdot \cos \varepsilon - \frac{\left(\sin x \cdot \sin \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right) - \sqrt[3]{\color{blue}{{\left(\cos x\right)}^{6}}}}{\sin x \cdot \sin \varepsilon - \cos x}\]
3. Recombined 3 regimes into one program.

4. Final simplification 16.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.5330496627903165 \cdot 10^{-7}:\\ \;\;\;\;\log \left(e^{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x}\right)\\ \mathbf{elif}\;\varepsilon \le 9.38913363863520236 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \frac{\left(\sin x \cdot \sin \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right) - \sqrt[3]{{\left(\cos x\right)}^{6}}}{\sin x \cdot \sin \varepsilon - \cos x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020058 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))