2tan (problem 3.3.2)

Average Error: 37.2 → 13.2

Time: 6.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.841647685826717 \cdot 10^{-15} \lor \neg \left(\varepsilon \leq 7.872211594490747 \cdot 10^{-39}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \log \left({\left({e}^{\left(\tan x\right)}\right)}^{\left(\tan \varepsilon\right)}\right)} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \left(\varepsilon \cdot \log 1\right) \cdot \left(\varepsilon + x\right)\\ \end{array}\]

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 <= -5.841647685826717e-15) || !(eps <= 7.872211594490747e-39))) {
		VAR = ((double) ((((double) (((double) tan(x)) + ((double) tan(eps)))) / ((double) (1.0 - ((double) log(((double) pow(((double) pow(((double) M_E), ((double) tan(x)))), ((double) tan(eps))))))))) - ((double) tan(x))));
	} else {
		VAR = ((double) (eps + ((double) (((double) (eps * ((double) log(1.0)))) * ((double) (eps + x))))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.2
Target	15.5
Herbie	13.2

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

Derivation

1. Split input into 2 regimes

2. if eps < -5.84164768582671696e-15 or 7.87221159449074657e-39 < eps

   1. Initial program 30.5

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

   3. Applied tan-sum 1.8

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

   5. Applied add-log-exp 1.9

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

   6. Simplified 2.0

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \log \color{blue}{\left({\left(e^{\tan x}\right)}^{\left(\tan \varepsilon\right)}\right)}} - \tan x\]
   7. Using strategy rm

   8. Applied *-un-lft-identity 2.0

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

   9. Applied exp-prod 2.0

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

   10. Simplified 2.0

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

   if -5.84164768582671696e-15 < eps < 7.87221159449074657e-39

   1. Initial program 45.1

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

   3. Applied tan-sum 45.1

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

   5. Applied add-log-exp 45.1

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

   6. Simplified 45.1

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

   7. Taylor expanded around 0 26.4

      \[\leadsto \color{blue}{\log 1 \cdot {\varepsilon}^{2} + \left(\varepsilon + \log 1 \cdot \left(x \cdot \varepsilon\right)\right)}\]

   8. Simplified 26.4

      \[\leadsto \color{blue}{\varepsilon + \left(\varepsilon \cdot \log 1\right) \cdot \left(\varepsilon + x\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 13.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.841647685826717 \cdot 10^{-15} \lor \neg \left(\varepsilon \leq 7.872211594490747 \cdot 10^{-39}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \log \left({\left({e}^{\left(\tan x\right)}\right)}^{\left(\tan \varepsilon\right)}\right)} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \left(\varepsilon \cdot \log 1\right) \cdot \left(\varepsilon + x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020196 
(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)))