2tan (problem 3.3.2)

Average Error: 37.3 → 15.2

Time: 26.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -3.607294612619752330986954992098521721906 \cdot 10^{-18} \lor \neg \left(\varepsilon \le 2.462633582693938122893310300874585297069 \cdot 10^{-42}\right):\\ \;\;\;\;\frac{1}{\frac{1 - \tan x \cdot \tan \varepsilon}{\tan x + \tan \varepsilon}} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right)\\ \end{array}\]

C

double f(double x, double eps) {
        double r96755 = x;
        double r96756 = eps;
        double r96757 = r96755 + r96756;
        double r96758 = tan(r96757);
        double r96759 = tan(r96755);
        double r96760 = r96758 - r96759;
        return r96760;
}

↓

double f(double x, double eps) {
        double r96761 = eps;
        double r96762 = -3.607294612619752e-18;
        bool r96763 = r96761 <= r96762;
        double r96764 = 2.462633582693938e-42;
        bool r96765 = r96761 <= r96764;
        double r96766 = !r96765;
        bool r96767 = r96763 || r96766;
        double r96768 = 1.0;
        double r96769 = x;
        double r96770 = tan(r96769);
        double r96771 = tan(r96761);
        double r96772 = r96770 * r96771;
        double r96773 = r96768 - r96772;
        double r96774 = r96770 + r96771;
        double r96775 = r96773 / r96774;
        double r96776 = r96768 / r96775;
        double r96777 = r96776 - r96770;
        double r96778 = r96769 * r96761;
        double r96779 = r96761 + r96769;
        double r96780 = r96778 * r96779;
        double r96781 = r96761 + r96780;
        double r96782 = r96767 ? r96777 : r96781;
        return r96782;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.3
Target	15.2
Herbie	15.2

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

Derivation

1. Split input into 2 regimes

2. if eps < -3.607294612619752e-18 or 2.462633582693938e-42 < eps

   1. Initial program 30.3

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

   3. Applied tan-sum 2.3

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

   5. Applied clear-num 2.4

      \[\leadsto \color{blue}{\frac{1}{\frac{1 - \tan x \cdot \tan \varepsilon}{\tan x + \tan \varepsilon}}} - \tan x\]

   if -3.607294612619752e-18 < eps < 2.462633582693938e-42

   1. Initial program 45.7

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

   3. Applied tan-sum 45.7

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]

   4. Taylor expanded around 0 31.0

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

   5. Simplified 30.8

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

4. Final simplification 15.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -3.607294612619752330986954992098521721906 \cdot 10^{-18} \lor \neg \left(\varepsilon \le 2.462633582693938122893310300874585297069 \cdot 10^{-42}\right):\\ \;\;\;\;\frac{1}{\frac{1 - \tan x \cdot \tan \varepsilon}{\tan x + \tan \varepsilon}} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019297 
(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)))