Average Error: 37.2 → 15.9
Time: 11.2s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -5.2890647743663734 \cdot 10^{-65} \lor \neg \left(\varepsilon \le 5.600552306151778 \cdot 10^{-65}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 - \frac{\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \sin \varepsilon\right)}{\cos \varepsilon}}, 1 + \left(\tan x \cdot \left(\sqrt[3]{\tan \varepsilon} \cdot \sqrt[3]{\tan \varepsilon}\right)\right) \cdot \sqrt[3]{\tan \varepsilon}, -\tan x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({\varepsilon}^{2}, x, \mathsf{fma}\left(\varepsilon, {x}^{2}, \varepsilon\right)\right)\\ \end{array}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -5.2890647743663734 \cdot 10^{-65} \lor \neg \left(\varepsilon \le 5.600552306151778 \cdot 10^{-65}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 - \frac{\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \sin \varepsilon\right)}{\cos \varepsilon}}, 1 + \left(\tan x \cdot \left(\sqrt[3]{\tan \varepsilon} \cdot \sqrt[3]{\tan \varepsilon}\right)\right) \cdot \sqrt[3]{\tan \varepsilon}, -\tan x\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left({\varepsilon}^{2}, x, \mathsf{fma}\left(\varepsilon, {x}^{2}, \varepsilon\right)\right)\\

\end{array}
double f(double x, double eps) {
        double r108815 = x;
        double r108816 = eps;
        double r108817 = r108815 + r108816;
        double r108818 = tan(r108817);
        double r108819 = tan(r108815);
        double r108820 = r108818 - r108819;
        return r108820;
}
↓
double f(double x, double eps) {
        double r108821 = eps;
        double r108822 = -5.289064774366373e-65;
        bool r108823 = r108821 <= r108822;
        double r108824 = 5.600552306151778e-65;
        bool r108825 = r108821 <= r108824;
        double r108826 = !r108825;
        bool r108827 = r108823 || r108826;
        double r108828 = x;
        double r108829 = tan(r108828);
        double r108830 = tan(r108821);
        double r108831 = r108829 + r108830;
        double r108832 = 1.0;
        double r108833 = r108829 * r108830;
        double r108834 = sin(r108821);
        double r108835 = r108829 * r108834;
        double r108836 = r108833 * r108835;
        double r108837 = cos(r108821);
        double r108838 = r108836 / r108837;
        double r108839 = r108832 - r108838;
        double r108840 = r108831 / r108839;
        double r108841 = cbrt(r108830);
        double r108842 = r108841 * r108841;
        double r108843 = r108829 * r108842;
        double r108844 = r108843 * r108841;
        double r108845 = r108832 + r108844;
        double r108846 = -r108829;
        double r108847 = fma(r108840, r108845, r108846);
        double r108848 = 2.0;
        double r108849 = pow(r108821, r108848);
        double r108850 = pow(r108828, r108848);
        double r108851 = fma(r108821, r108850, r108821);
        double r108852 = fma(r108849, r108828, r108851);
        double r108853 = r108827 ? r108847 : r108852;
        return r108853;
}

Error

Bits error versus x

Bits error versus eps

Target

Original37.2
Target15.1
Herbie15.9
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 2 regimes

  2. if eps < -5.289064774366373e-65 or 5.600552306151778e-65 < eps

    1. Initial program 30.5

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

    3. Applied tan-sum5.2

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

    5. Applied flip--5.2

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

    6. Applied associate-/r/5.2

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

    7. Applied fma-neg5.2

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

    9. Applied tan-quot5.2

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

    10. Applied associate-*r/5.2

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

    11. Applied associate-*r/5.2

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

    13. Applied add-cube-cbrt5.3

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

    14. Applied associate-*r*5.3

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

    if -5.289064774366373e-65 < eps < 5.600552306151778e-65

    1. Initial program 47.3

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

    2. Taylor expanded around 0 31.9

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

    3. Simplified31.9

      \[\leadsto \color{blue}{\mathsf{fma}\left({\varepsilon}^{2}, x, \mathsf{fma}\left(\varepsilon, {x}^{2}, \varepsilon\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification15.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -5.2890647743663734 \cdot 10^{-65} \lor \neg \left(\varepsilon \le 5.600552306151778 \cdot 10^{-65}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 - \frac{\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \sin \varepsilon\right)}{\cos \varepsilon}}, 1 + \left(\tan x \cdot \left(\sqrt[3]{\tan \varepsilon} \cdot \sqrt[3]{\tan \varepsilon}\right)\right) \cdot \sqrt[3]{\tan \varepsilon}, -\tan x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({\varepsilon}^{2}, x, \mathsf{fma}\left(\varepsilon, {x}^{2}, \varepsilon\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 +o rules:numerics
(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)))