2tan (problem 3.3.2)

Average Error: 37.1 → 15.0

Time: 28.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -6.929244313528967253629797030059496249939 \cdot 10^{-77}:\\ \;\;\;\;\frac{\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon}{\left(1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\cos x \cdot \cos \varepsilon\right)} - \tan x\\ \mathbf{elif}\;\varepsilon \le 1.084923224249312910690492748285721451604 \cdot 10^{-19}:\\ \;\;\;\;\varepsilon + \left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \sin x \cdot \left(1 - \tan \varepsilon \cdot \tan x\right)}{\left(1 - \tan \varepsilon \cdot \tan x\right) \cdot \cos x}\\ \end{array}\]

C

double f(double x, double eps) {
        double r113754 = x;
        double r113755 = eps;
        double r113756 = r113754 + r113755;
        double r113757 = tan(r113756);
        double r113758 = tan(r113754);
        double r113759 = r113757 - r113758;
        return r113759;
}

↓

double f(double x, double eps) {
        double r113760 = eps;
        double r113761 = -6.929244313528967e-77;
        bool r113762 = r113760 <= r113761;
        double r113763 = x;
        double r113764 = sin(r113763);
        double r113765 = cos(r113760);
        double r113766 = r113764 * r113765;
        double r113767 = cos(r113763);
        double r113768 = sin(r113760);
        double r113769 = r113767 * r113768;
        double r113770 = r113766 + r113769;
        double r113771 = 1.0;
        double r113772 = tan(r113763);
        double r113773 = r113772 * r113768;
        double r113774 = r113773 / r113765;
        double r113775 = r113771 - r113774;
        double r113776 = r113767 * r113765;
        double r113777 = r113775 * r113776;
        double r113778 = r113770 / r113777;
        double r113779 = r113778 - r113772;
        double r113780 = 1.084923224249313e-19;
        bool r113781 = r113760 <= r113780;
        double r113782 = r113763 * r113760;
        double r113783 = r113760 + r113763;
        double r113784 = r113782 * r113783;
        double r113785 = r113760 + r113784;
        double r113786 = tan(r113760);
        double r113787 = r113772 + r113786;
        double r113788 = r113787 * r113767;
        double r113789 = r113786 * r113772;
        double r113790 = r113771 - r113789;
        double r113791 = r113764 * r113790;
        double r113792 = r113788 - r113791;
        double r113793 = r113790 * r113767;
        double r113794 = r113792 / r113793;
        double r113795 = r113781 ? r113785 : r113794;
        double r113796 = r113762 ? r113779 : r113795;
        return r113796;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.1
Target	15.0
Herbie	15.0

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

Derivation

1. Split input into 3 regimes

2. if eps < -6.929244313528967e-77

   1. Initial program 30.9

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

   3. Applied tan-sum 6.0

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

   5. Applied tan-quot 6.1

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

   6. Applied tan-quot 6.1

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

   7. Applied frac-add 6.1

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

   8. Applied associate-/l/ 6.1

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

   10. Applied tan-quot 6.1

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

   11. Applied associate-*r/ 6.1

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

   if -6.929244313528967e-77 < eps < 1.084923224249313e-19

   1. Initial program 46.6

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

   3. Applied tan-sum 46.6

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

   4. Taylor expanded around 0 30.1

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

   5. Simplified 29.9

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

   if 1.084923224249313e-19 < eps

   1. Initial program 29.0

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

   3. Applied tan-sum 1.2

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

   5. Applied add-cube-cbrt 1.4

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

   7. Applied tan-quot 1.4

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

   8. Applied frac-sub 1.4

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

   9. Simplified 1.6

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

   10. Simplified 1.3

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

4. Final simplification 15.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -6.929244313528967253629797030059496249939 \cdot 10^{-77}:\\ \;\;\;\;\frac{\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon}{\left(1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\cos x \cdot \cos \varepsilon\right)} - \tan x\\ \mathbf{elif}\;\varepsilon \le 1.084923224249312910690492748285721451604 \cdot 10^{-19}:\\ \;\;\;\;\varepsilon + \left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \sin x \cdot \left(1 - \tan \varepsilon \cdot \tan x\right)}{\left(1 - \tan \varepsilon \cdot \tan x\right) \cdot \cos x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(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)))