2tan (problem 3.3.2)

Average Error: 37.4 → 13.8

Time: 34.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -5.135442363002646 \cdot 10^{-18}:\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \left(\tan x \cdot \left(\tan \varepsilon \cdot \tan x\right)\right)\right)} \cdot \left(\left(\tan \varepsilon \cdot \tan x + \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) + 1\right) - \tan x\\ \mathbf{elif}\;\varepsilon \le 2.106612565673929 \cdot 10^{-51}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \frac{1}{3}\right) + \left(x \cdot \left(\varepsilon \cdot \varepsilon\right) + \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\tan \varepsilon + \tan x}{1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \left(\tan x \cdot \left(\tan \varepsilon \cdot \tan x\right)\right)\right)} \cdot \left(\left(\tan \varepsilon \cdot \tan x + \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) + 1\right)\right) \cdot \left(\frac{\tan \varepsilon + \tan x}{1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \left(\tan x \cdot \left(\tan \varepsilon \cdot \tan x\right)\right)\right)} \cdot \left(\left(\tan \varepsilon \cdot \tan x + \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) + 1\right)\right) - \tan x \cdot \tan x}{\frac{\tan \varepsilon + \tan x}{1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \left(\tan x \cdot \left(\tan \varepsilon \cdot \tan x\right)\right)\right)} \cdot \left(\left(\tan \varepsilon \cdot \tan x + \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) + 1\right) + \tan x}\\ \end{array}\]

C

double f(double x, double eps) {
        double r4191883 = x;
        double r4191884 = eps;
        double r4191885 = r4191883 + r4191884;
        double r4191886 = tan(r4191885);
        double r4191887 = tan(r4191883);
        double r4191888 = r4191886 - r4191887;
        return r4191888;
}

↓

double f(double x, double eps) {
        double r4191889 = eps;
        double r4191890 = -5.135442363002646e-18;
        bool r4191891 = r4191889 <= r4191890;
        double r4191892 = tan(r4191889);
        double r4191893 = x;
        double r4191894 = tan(r4191893);
        double r4191895 = r4191892 + r4191894;
        double r4191896 = 1.0;
        double r4191897 = r4191892 * r4191894;
        double r4191898 = r4191894 * r4191897;
        double r4191899 = r4191892 * r4191898;
        double r4191900 = r4191897 * r4191899;
        double r4191901 = r4191896 - r4191900;
        double r4191902 = r4191895 / r4191901;
        double r4191903 = r4191897 * r4191897;
        double r4191904 = r4191897 + r4191903;
        double r4191905 = r4191904 + r4191896;
        double r4191906 = r4191902 * r4191905;
        double r4191907 = r4191906 - r4191894;
        double r4191908 = 2.106612565673929e-51;
        bool r4191909 = r4191889 <= r4191908;
        double r4191910 = r4191889 * r4191889;
        double r4191911 = 0.3333333333333333;
        double r4191912 = r4191910 * r4191911;
        double r4191913 = r4191889 * r4191912;
        double r4191914 = r4191893 * r4191910;
        double r4191915 = r4191914 + r4191889;
        double r4191916 = r4191913 + r4191915;
        double r4191917 = r4191906 * r4191906;
        double r4191918 = r4191894 * r4191894;
        double r4191919 = r4191917 - r4191918;
        double r4191920 = r4191906 + r4191894;
        double r4191921 = r4191919 / r4191920;
        double r4191922 = r4191909 ? r4191916 : r4191921;
        double r4191923 = r4191891 ? r4191907 : r4191922;
        return r4191923;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.4
Target	15.6
Herbie	13.8

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

Derivation

1. Split input into 3 regimes

2. if eps < -5.135442363002646e-18

   1. Initial program 30.7

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

   3. Applied tan-sum 0.9

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

   5. Applied flip3-- 1.0

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

   6. Applied associate-/r/ 1.0

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

   7. Simplified 1.0

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

   9. Applied associate-*l* 1.0

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

   if -5.135442363002646e-18 < eps < 2.106612565673929e-51

   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. Using strategy rm

   5. Applied flip3-- 45.7

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

   6. Applied associate-/r/ 45.7

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

   7. Simplified 45.7

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

   8. Taylor expanded around 0 27.7

      \[\leadsto \color{blue}{x \cdot {\varepsilon}^{2} + \left(\frac{1}{3} \cdot {\varepsilon}^{3} + \varepsilon\right)}\]

   9. Simplified 27.7

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

   if 2.106612565673929e-51 < eps

   1. Initial program 30.8

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

   3. Applied tan-sum 4.0

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

   5. Applied flip3-- 4.1

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

   6. Applied associate-/r/ 4.1

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

   7. Simplified 4.1

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

   9. Applied associate-*l* 4.1

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

   11. Applied flip-- 4.2

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

4. Final simplification 13.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -5.135442363002646 \cdot 10^{-18}:\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \left(\tan x \cdot \left(\tan \varepsilon \cdot \tan x\right)\right)\right)} \cdot \left(\left(\tan \varepsilon \cdot \tan x + \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) + 1\right) - \tan x\\ \mathbf{elif}\;\varepsilon \le 2.106612565673929 \cdot 10^{-51}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \frac{1}{3}\right) + \left(x \cdot \left(\varepsilon \cdot \varepsilon\right) + \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\tan \varepsilon + \tan x}{1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \left(\tan x \cdot \left(\tan \varepsilon \cdot \tan x\right)\right)\right)} \cdot \left(\left(\tan \varepsilon \cdot \tan x + \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) + 1\right)\right) \cdot \left(\frac{\tan \varepsilon + \tan x}{1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \left(\tan x \cdot \left(\tan \varepsilon \cdot \tan x\right)\right)\right)} \cdot \left(\left(\tan \varepsilon \cdot \tan x + \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) + 1\right)\right) - \tan x \cdot \tan x}{\frac{\tan \varepsilon + \tan x}{1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \left(\tan x \cdot \left(\tan \varepsilon \cdot \tan x\right)\right)\right)} \cdot \left(\left(\tan \varepsilon \cdot \tan x + \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) + 1\right) + \tan x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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