2cos (problem 3.3.5)

Average Error: 39.6 → 16.1

Time: 7.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -2.9500525747086803 \cdot 10^{-33}:\\ \;\;\;\;\log \left(e^{\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon}\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 4.69434538371788253 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{24}, {\varepsilon}^{4}, -\mathsf{fma}\left(x, \varepsilon, \frac{1}{2} \cdot {\varepsilon}^{2}\right)\right) \cdot \left(\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x\right)}{\left(\cos x \cdot \cos \varepsilon - \mathsf{expm1}\left(\mathsf{log1p}\left(\sin x \cdot \sin \varepsilon\right)\right)\right) + \cos x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos \varepsilon, \cos x, -\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right) \cdot \left(\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x\right)}{\left(\cos x \cdot \cos \varepsilon - \mathsf{expm1}\left(\mathsf{log1p}\left(\sin x \cdot \sin \varepsilon\right)\right)\right) + \cos x}\\ \end{array}\]

C

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

↓

double code(double x, double eps) {
	double VAR;
	if ((eps <= -2.9500525747086803e-33)) {
		VAR = (log(exp(((cos(x) * cos(eps)) - (sin(x) * sin(eps))))) - cos(x));
	} else {
		double VAR_1;
		if ((eps <= 4.6943453837178825e-06)) {
			VAR_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)) - expm1(log1p((sin(x) * sin(eps))))) + cos(x)));
		} else {
			VAR_1 = ((fma(cos(eps), cos(x), -fma(sin(x), sin(eps), cos(x))) * (((cos(x) * cos(eps)) - (sin(x) * sin(eps))) + cos(x))) / (((cos(x) * cos(eps)) - expm1(log1p((sin(x) * sin(eps))))) + cos(x)));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus eps

Derivation

1. Split input into 3 regimes

2. if eps < -2.9500525747086803e-33

   1. Initial program 33.0

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

   3. Applied cos-sum 4.9

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

   5. Applied add-log-exp 5.1

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

   6. Applied add-log-exp 5.2

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

   7. Applied diff-log 5.2

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

   8. Simplified 5.1

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

   if -2.9500525747086803e-33 < eps < 4.6943453837178825e-06

   1. Initial program 48.6

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

   3. Applied cos-sum 48.3

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

   5. Applied flip-- 48.3

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

   6. Simplified 48.3

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

   8. Applied expm1-log1p-u 48.3

      \[\leadsto \frac{\mathsf{fma}\left(\cos \varepsilon, \cos x, -\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right) \cdot \left(\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x\right)}{\left(\cos x \cdot \cos \varepsilon - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin x \cdot \sin \varepsilon\right)\right)}\right) + \cos x}\]

   9. Taylor expanded around 0 30.9

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{24} \cdot {\varepsilon}^{4} - \left(x \cdot \varepsilon + \frac{1}{2} \cdot {\varepsilon}^{2}\right)\right)} \cdot \left(\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x\right)}{\left(\cos x \cdot \cos \varepsilon - \mathsf{expm1}\left(\mathsf{log1p}\left(\sin x \cdot \sin \varepsilon\right)\right)\right) + \cos x}\]

   10. Simplified 30.9

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {\varepsilon}^{4}, -\mathsf{fma}\left(x, \varepsilon, \frac{1}{2} \cdot {\varepsilon}^{2}\right)\right)} \cdot \left(\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x\right)}{\left(\cos x \cdot \cos \varepsilon - \mathsf{expm1}\left(\mathsf{log1p}\left(\sin x \cdot \sin \varepsilon\right)\right)\right) + \cos x}\]

   if 4.6943453837178825e-06 < eps

   1. Initial program 30.4

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

   3. Applied cos-sum 0.9

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

   5. Applied flip-- 1.4

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

   6. Simplified 1.0

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

   8. Applied expm1-log1p-u 1.0

      \[\leadsto \frac{\mathsf{fma}\left(\cos \varepsilon, \cos x, -\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right) \cdot \left(\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x\right)}{\left(\cos x \cdot \cos \varepsilon - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin x \cdot \sin \varepsilon\right)\right)}\right) + \cos x}\]
3. Recombined 3 regimes into one program.

4. Final simplification 16.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -2.9500525747086803 \cdot 10^{-33}:\\ \;\;\;\;\log \left(e^{\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon}\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 4.69434538371788253 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{24}, {\varepsilon}^{4}, -\mathsf{fma}\left(x, \varepsilon, \frac{1}{2} \cdot {\varepsilon}^{2}\right)\right) \cdot \left(\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x\right)}{\left(\cos x \cdot \cos \varepsilon - \mathsf{expm1}\left(\mathsf{log1p}\left(\sin x \cdot \sin \varepsilon\right)\right)\right) + \cos x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos \varepsilon, \cos x, -\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right) \cdot \left(\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x\right)}{\left(\cos x \cdot \cos \varepsilon - \mathsf{expm1}\left(\mathsf{log1p}\left(\sin x \cdot \sin \varepsilon\right)\right)\right) + \cos x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))