2tan (problem 3.3.2)

Average Error: 36.9 → 15.1

Time: 32.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.538105143778487014527009869176991195373 \cdot 10^{-19}:\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \sqrt[3]{\left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}} - \tan x\\ \mathbf{elif}\;\varepsilon \le 2.35543875174405530005124676624834885103 \cdot 10^{-56}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \left(x \cdot \left(x + \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \frac{\tan \varepsilon \cdot \sin x}{\cos x}} - \tan x\\ \end{array}\]

C

double f(double x, double eps) {
        double r5832137 = x;
        double r5832138 = eps;
        double r5832139 = r5832137 + r5832138;
        double r5832140 = tan(r5832139);
        double r5832141 = tan(r5832137);
        double r5832142 = r5832140 - r5832141;
        return r5832142;
}

↓

double f(double x, double eps) {
        double r5832143 = eps;
        double r5832144 = -1.538105143778487e-19;
        bool r5832145 = r5832143 <= r5832144;
        double r5832146 = tan(r5832143);
        double r5832147 = x;
        double r5832148 = tan(r5832147);
        double r5832149 = r5832146 + r5832148;
        double r5832150 = 1.0;
        double r5832151 = r5832148 * r5832146;
        double r5832152 = r5832151 * r5832151;
        double r5832153 = r5832152 * r5832151;
        double r5832154 = cbrt(r5832153);
        double r5832155 = r5832150 - r5832154;
        double r5832156 = r5832149 / r5832155;
        double r5832157 = r5832156 - r5832148;
        double r5832158 = 2.3554387517440553e-56;
        bool r5832159 = r5832143 <= r5832158;
        double r5832160 = r5832147 + r5832143;
        double r5832161 = r5832147 * r5832160;
        double r5832162 = r5832143 * r5832161;
        double r5832163 = r5832143 + r5832162;
        double r5832164 = sin(r5832147);
        double r5832165 = r5832146 * r5832164;
        double r5832166 = cos(r5832147);
        double r5832167 = r5832165 / r5832166;
        double r5832168 = r5832150 - r5832167;
        double r5832169 = r5832149 / r5832168;
        double r5832170 = r5832169 - r5832148;
        double r5832171 = r5832159 ? r5832163 : r5832170;
        double r5832172 = r5832145 ? r5832157 : r5832171;
        return r5832172;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	36.9
Target	15.1
Herbie	15.1

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

Derivation

1. Split input into 3 regimes

2. if eps < -1.538105143778487e-19

   1. Initial program 29.6

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

   3. Applied tan-sum 1.1

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

   5. Applied add-cbrt-cube 1.2

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\sqrt[3]{\left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}}} - \tan x\]

   if -1.538105143778487e-19 < eps < 2.3554387517440553e-56

   1. Initial program 45.5

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

   2. Taylor expanded around 0 30.6

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

   3. Simplified 30.6

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

   if 2.3554387517440553e-56 < eps

   1. Initial program 30.6

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

   3. Applied tan-sum 4.6

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

   5. Applied add-cbrt-cube 4.7

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\sqrt[3]{\left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}}} - \tan x\]
   6. Using strategy rm

   7. Applied tan-quot 4.7

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \sqrt[3]{\left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right) \cdot \left(\color{blue}{\frac{\sin x}{\cos x}} \cdot \tan \varepsilon\right)}} - \tan x\]

   8. Applied associate-*l/ 4.7

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \sqrt[3]{\left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right) \cdot \color{blue}{\frac{\sin x \cdot \tan \varepsilon}{\cos x}}}} - \tan x\]

   9. Applied tan-quot 4.7

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \sqrt[3]{\left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\color{blue}{\frac{\sin x}{\cos x}} \cdot \tan \varepsilon\right)\right) \cdot \frac{\sin x \cdot \tan \varepsilon}{\cos x}}} - \tan x\]

   10. Applied associate-*l/ 4.7

       \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \sqrt[3]{\left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \color{blue}{\frac{\sin x \cdot \tan \varepsilon}{\cos x}}\right) \cdot \frac{\sin x \cdot \tan \varepsilon}{\cos x}}} - \tan x\]

   11. Applied tan-quot 4.7

       \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \sqrt[3]{\left(\left(\color{blue}{\frac{\sin x}{\cos x}} \cdot \tan \varepsilon\right) \cdot \frac{\sin x \cdot \tan \varepsilon}{\cos x}\right) \cdot \frac{\sin x \cdot \tan \varepsilon}{\cos x}}} - \tan x\]

   12. Applied associate-*l/ 4.7

       \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \sqrt[3]{\left(\color{blue}{\frac{\sin x \cdot \tan \varepsilon}{\cos x}} \cdot \frac{\sin x \cdot \tan \varepsilon}{\cos x}\right) \cdot \frac{\sin x \cdot \tan \varepsilon}{\cos x}}} - \tan x\]

   13. Applied frac-times 4.7

       \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \sqrt[3]{\color{blue}{\frac{\left(\sin x \cdot \tan \varepsilon\right) \cdot \left(\sin x \cdot \tan \varepsilon\right)}{\cos x \cdot \cos x}} \cdot \frac{\sin x \cdot \tan \varepsilon}{\cos x}}} - \tan x\]

   14. Applied frac-times 4.7

       \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \sqrt[3]{\color{blue}{\frac{\left(\left(\sin x \cdot \tan \varepsilon\right) \cdot \left(\sin x \cdot \tan \varepsilon\right)\right) \cdot \left(\sin x \cdot \tan \varepsilon\right)}{\left(\cos x \cdot \cos x\right) \cdot \cos x}}}} - \tan x\]

   15. Applied cbrt-div 4.7

       \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{\sqrt[3]{\left(\left(\sin x \cdot \tan \varepsilon\right) \cdot \left(\sin x \cdot \tan \varepsilon\right)\right) \cdot \left(\sin x \cdot \tan \varepsilon\right)}}{\sqrt[3]{\left(\cos x \cdot \cos x\right) \cdot \cos x}}}} - \tan x\]

   16. Simplified 4.7

       \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \frac{\color{blue}{\tan \varepsilon \cdot \sin x}}{\sqrt[3]{\left(\cos x \cdot \cos x\right) \cdot \cos x}}} - \tan x\]

   17. Simplified 4.6

       \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \frac{\tan \varepsilon \cdot \sin x}{\color{blue}{\cos x}}} - \tan x\]
3. Recombined 3 regimes into one program.

4. Final simplification 15.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.538105143778487014527009869176991195373 \cdot 10^{-19}:\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \sqrt[3]{\left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}} - \tan x\\ \mathbf{elif}\;\varepsilon \le 2.35543875174405530005124676624834885103 \cdot 10^{-56}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \left(x \cdot \left(x + \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \frac{\tan \varepsilon \cdot \sin x}{\cos x}} - \tan x\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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