2cos (problem 3.3.5)

Average Error: 39.5 → 15.6

Time: 7.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -8.5411563429199795 \cdot 10^{-10}:\\ \;\;\;\;\log \left(e^{\mathsf{fma}\left(\cos \varepsilon, \cos x, -\sqrt[3]{{\left(\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right)}^{3}}\right)}\right)\\ \mathbf{elif}\;\varepsilon \le 1.1618945885559161 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{24}, {\varepsilon}^{4}, -\mathsf{fma}\left(x, \varepsilon, \frac{1}{2} \cdot {\varepsilon}^{2}\right)\right) \cdot \left(\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x\right)}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} - {\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{\mathsf{fma}\left(\sin x \cdot \sin \varepsilon, \mathsf{fma}\left(\cos x, \cos \varepsilon, \sin x \cdot \sin \varepsilon\right), \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\cos x \cdot \cos \varepsilon\right)\right)} - \cos x\\ \end{array}\]

C

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

↓

double code(double x, double eps) {
	double temp;
	if ((eps <= -8.54115634291998e-10)) {
		temp = log(exp(fma(cos(eps), cos(x), -cbrt(pow(fma(sin(x), sin(eps), cos(x)), 3.0)))));
	} else {
		double temp_1;
		if ((eps <= 1.1618945885559161e-07)) {
			temp_1 = ((fma(0.041666666666666664, pow(eps, 4.0), -fma(x, eps, (0.5 * pow(eps, 2.0)))) * (((cos(x) * cos(eps)) - (sin(x) * sin(eps))) + cos(x))) / (((cos(x) * cos(eps)) - (sin(x) * sin(eps))) + cos(x)));
		} else {
			temp_1 = (((pow((cos(x) * cos(eps)), 3.0) - pow((sin(x) * sin(eps)), 3.0)) / fma((sin(x) * sin(eps)), fma(cos(x), cos(eps), (sin(x) * sin(eps))), ((cos(x) * cos(eps)) * (cos(x) * cos(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 < -8.54115634291998e-10

   1. Initial program 29.8

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

   5. Applied add-log-exp 1.5

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

   6. Applied add-log-exp 1.6

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

   7. Applied add-log-exp 1.7

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

   8. Applied diff-log 1.8

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

   9. Applied diff-log 1.8

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

   10. Simplified 1.5

       \[\leadsto \log \color{blue}{\left(e^{\mathsf{fma}\left(\cos \varepsilon, \cos x, -\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right)}\right)}\]
   11. Using strategy rm

   12. Applied add-cbrt-cube 1.6

       \[\leadsto \log \left(e^{\mathsf{fma}\left(\cos \varepsilon, \cos x, -\color{blue}{\sqrt[3]{\left(\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right) \cdot \mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right) \cdot \mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)}}\right)}\right)\]

   13. Simplified 1.6

       \[\leadsto \log \left(e^{\mathsf{fma}\left(\cos \varepsilon, \cos x, -\sqrt[3]{\color{blue}{{\left(\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right)}^{3}}}\right)}\right)\]

   if -8.54115634291998e-10 < eps < 1.1618945885559161e-07

   1. Initial program 49.1

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

   3. Applied cos-sum 48.7

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

   5. Applied flip-- 48.7

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

   6. Simplified 48.7

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

   7. Taylor expanded around 0 30.7

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

   8. Simplified 30.7

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

   if 1.1618945885559161e-07 < eps

   1. Initial program 31.2

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

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

   6. Simplified 1.3

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

4. Final simplification 15.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -8.5411563429199795 \cdot 10^{-10}:\\ \;\;\;\;\log \left(e^{\mathsf{fma}\left(\cos \varepsilon, \cos x, -\sqrt[3]{{\left(\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right)}^{3}}\right)}\right)\\ \mathbf{elif}\;\varepsilon \le 1.1618945885559161 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{24}, {\varepsilon}^{4}, -\mathsf{fma}\left(x, \varepsilon, \frac{1}{2} \cdot {\varepsilon}^{2}\right)\right) \cdot \left(\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x\right)}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} - {\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{\mathsf{fma}\left(\sin x \cdot \sin \varepsilon, \mathsf{fma}\left(\cos x, \cos \varepsilon, \sin x \cdot \sin \varepsilon\right), \left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\cos x \cdot \cos \varepsilon\right)\right)} - \cos x\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))