2tan (problem 3.3.2)

Average Error: 37.8 → 15.9

Time: 18.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le \frac{-5784497869627675}{79228162514264337593543950336}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} \cdot 1 + \tan x \cdot \left(\tan \varepsilon \cdot \frac{\tan x + \tan \varepsilon}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} - 1\right)\\ \mathbf{elif}\;\varepsilon \le \frac{5930776181004985}{21267647932558653966460912964485513216}:\\ \;\;\;\;x \cdot {\varepsilon}^{2} + \left(\varepsilon + {x}^{2} \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \sqrt[3]{{\left(\tan x \cdot \tan \varepsilon\right)}^{3}}} - \tan x\\ \end{array}\]

C

double f(double x, double eps) {
        double r104851 = x;
        double r104852 = eps;
        double r104853 = r104851 + r104852;
        double r104854 = tan(r104853);
        double r104855 = tan(r104851);
        double r104856 = r104854 - r104855;
        return r104856;
}

↓

double f(double x, double eps) {
        double r104857 = eps;
        double r104858 = -5784497869627675.0;
        double r104859 = 7.922816251426434e+28;
        double r104860 = r104858 / r104859;
        bool r104861 = r104857 <= r104860;
        double r104862 = x;
        double r104863 = tan(r104862);
        double r104864 = tan(r104857);
        double r104865 = r104863 + r104864;
        double r104866 = 1.0;
        double r104867 = r104863 * r104864;
        double r104868 = r104867 * r104867;
        double r104869 = r104866 - r104868;
        double r104870 = r104865 / r104869;
        double r104871 = r104870 * r104866;
        double r104872 = r104864 * r104870;
        double r104873 = r104872 - r104866;
        double r104874 = r104863 * r104873;
        double r104875 = r104871 + r104874;
        double r104876 = 5930776181004985.0;
        double r104877 = 2.1267647932558654e+37;
        double r104878 = r104876 / r104877;
        bool r104879 = r104857 <= r104878;
        double r104880 = 2.0;
        double r104881 = pow(r104857, r104880);
        double r104882 = r104862 * r104881;
        double r104883 = pow(r104862, r104880);
        double r104884 = r104883 * r104857;
        double r104885 = r104857 + r104884;
        double r104886 = r104882 + r104885;
        double r104887 = 3.0;
        double r104888 = pow(r104867, r104887);
        double r104889 = cbrt(r104888);
        double r104890 = r104866 - r104889;
        double r104891 = r104865 / r104890;
        double r104892 = r104891 - r104863;
        double r104893 = r104879 ? r104886 : r104892;
        double r104894 = r104861 ? r104875 : r104893;
        return r104894;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.8
Target	15.5
Herbie	15.9

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

Derivation

1. Split input into 3 regimes

2. if eps < -7.301062761093603e-14

   1. Initial program 31.7

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

   3. Applied tan-sum 0.8

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

   5. Applied flip-- 0.8

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

   6. Applied associate-/r/ 0.8

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

   7. Simplified 0.8

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

   9. Applied distribute-lft-in 0.8

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

   10. Applied associate--l+ 0.8

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

   11. Simplified 0.9

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

   if -7.301062761093603e-14 < eps < 2.788637558705096e-22

   1. Initial program 45.9

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

   3. Applied tan-sum 45.8

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

   5. Applied flip-- 45.9

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

   6. Applied associate-/r/ 45.9

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

   7. Simplified 45.9

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

   8. Taylor expanded around 0 32.4

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

   if 2.788637558705096e-22 < eps

   1. Initial program 29.5

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

   3. Applied tan-sum 1.3

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

   5. Applied add-cbrt-cube 1.4

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

   6. Applied add-cbrt-cube 1.4

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

   7. Applied cbrt-unprod 1.4

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

   8. Simplified 1.4

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

4. Final simplification 15.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le \frac{-5784497869627675}{79228162514264337593543950336}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} \cdot 1 + \tan x \cdot \left(\tan \varepsilon \cdot \frac{\tan x + \tan \varepsilon}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} - 1\right)\\ \mathbf{elif}\;\varepsilon \le \frac{5930776181004985}{21267647932558653966460912964485513216}:\\ \;\;\;\;x \cdot {\varepsilon}^{2} + \left(\varepsilon + {x}^{2} \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \sqrt[3]{{\left(\tan x \cdot \tan \varepsilon\right)}^{3}}} - \tan x\\ \end{array}\]

Reproduce

herbie shell --seed 197574269 
(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)))