2tan (problem 3.3.2)

Average Error: 36.6 → 15.9

Time: 7.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -4.01563370897399 \cdot 10^{-170}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \log \left(e^{\tan x \cdot \tan \varepsilon}\right)} - \tan x\\ \mathbf{elif}\;\varepsilon \le 1.20351147258237903 \cdot 10^{-60}:\\ \;\;\;\;\left(\varepsilon \cdot x\right) \cdot \left(x + \varepsilon\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{\log \left(e^{1 - \tan x \cdot \tan \varepsilon}\right)} - \tan x\\ \end{array}\]

C

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

↓

double code(double x, double eps) {
	double VAR;
	if ((eps <= -4.01563370897399e-170)) {
		VAR = ((double) (((double) (((double) (((double) tan(x)) + ((double) tan(eps)))) / ((double) (1.0 - ((double) log(((double) exp(((double) (((double) tan(x)) * ((double) tan(eps)))))))))))) - ((double) tan(x))));
	} else {
		double VAR_1;
		if ((eps <= 1.203511472582379e-60)) {
			VAR_1 = ((double) (((double) (((double) (eps * x)) * ((double) (x + eps)))) + eps));
		} else {
			VAR_1 = ((double) (((double) (((double) (((double) tan(x)) + ((double) tan(eps)))) / ((double) log(((double) exp(((double) (1.0 - ((double) (((double) tan(x)) * ((double) tan(eps)))))))))))) - ((double) tan(x))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	36.6
Target	15.2
Herbie	15.9

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

Derivation

1. Split input into 3 regimes

2. if eps < -4.01563370897399e-170

   1. Initial program 31.9

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

   3. Applied tan-sum 12.0

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
   4. Using strategy rm

   5. Applied add-log-exp 12.1

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\log \left(e^{\tan x \cdot \tan \varepsilon}\right)}} - \tan x\]

   if -4.01563370897399e-170 < eps < 1.203511472582379e-60

   1. Initial program 48.7

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

   2. Taylor expanded around 0 31.4

      \[\leadsto \color{blue}{x \cdot {\varepsilon}^{2} + \left(\varepsilon + {x}^{2} \cdot \varepsilon\right)}\]

   3. Simplified 31.2

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

   if 1.203511472582379e-60 < eps

   1. Initial program 30.3

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

   3. Applied tan-sum 5.2

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
   4. Using strategy rm

   5. Applied add-log-exp 5.3

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\log \left(e^{\tan x \cdot \tan \varepsilon}\right)}} - \tan x\]

   6. Applied add-log-exp 5.3

      \[\leadsto \frac{\tan x + \tan \varepsilon}{\color{blue}{\log \left(e^{1}\right)} - \log \left(e^{\tan x \cdot \tan \varepsilon}\right)} - \tan x\]

   7. Applied diff-log 5.4

      \[\leadsto \frac{\tan x + \tan \varepsilon}{\color{blue}{\log \left(\frac{e^{1}}{e^{\tan x \cdot \tan \varepsilon}}\right)}} - \tan x\]

   8. Simplified 5.3

      \[\leadsto \frac{\tan x + \tan \varepsilon}{\log \color{blue}{\left(e^{1 - \tan x \cdot \tan \varepsilon}\right)}} - \tan x\]
3. Recombined 3 regimes into one program.

4. Final simplification 15.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -4.01563370897399 \cdot 10^{-170}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \log \left(e^{\tan x \cdot \tan \varepsilon}\right)} - \tan x\\ \mathbf{elif}\;\varepsilon \le 1.20351147258237903 \cdot 10^{-60}:\\ \;\;\;\;\left(\varepsilon \cdot x\right) \cdot \left(x + \varepsilon\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{\log \left(e^{1 - \tan x \cdot \tan \varepsilon}\right)} - \tan x\\ \end{array}\]

Reproduce

herbie shell --seed 2020126 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :precision binary64

  :herbie-target
  (/ (sin eps) (* (cos x) (cos (+ x eps))))

  (- (tan (+ x eps)) (tan x)))