Average Error: 36.7 → 15.4
Time: 10.5s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.8367694074004902 \cdot 10^{-68}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 - \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \tan x\right) \cdot \tan \varepsilon}, 1 + \tan x \cdot \tan \varepsilon, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \le 1.476728953358803 \cdot 10^{-117}:\\ \;\;\;\;\mathsf{fma}\left({\varepsilon}^{2}, x, \mathsf{fma}\left(\varepsilon, {x}^{2}, \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\\ \end{array}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -1.8367694074004902 \cdot 10^{-68}:\\
\;\;\;\;\mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 - \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \tan x\right) \cdot \tan \varepsilon}, 1 + \tan x \cdot \tan \varepsilon, -\tan x\right)\\

\mathbf{elif}\;\varepsilon \le 1.476728953358803 \cdot 10^{-117}:\\
\;\;\;\;\mathsf{fma}\left({\varepsilon}^{2}, x, \mathsf{fma}\left(\varepsilon, {x}^{2}, \varepsilon\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\\

\end{array}
double f(double x, double eps) {
        double r120847 = x;
        double r120848 = eps;
        double r120849 = r120847 + r120848;
        double r120850 = tan(r120849);
        double r120851 = tan(r120847);
        double r120852 = r120850 - r120851;
        return r120852;
}
↓
double f(double x, double eps) {
        double r120853 = eps;
        double r120854 = -1.8367694074004902e-68;
        bool r120855 = r120853 <= r120854;
        double r120856 = x;
        double r120857 = tan(r120856);
        double r120858 = tan(r120853);
        double r120859 = r120857 + r120858;
        double r120860 = 1.0;
        double r120861 = r120857 * r120858;
        double r120862 = r120861 * r120857;
        double r120863 = r120862 * r120858;
        double r120864 = r120860 - r120863;
        double r120865 = r120859 / r120864;
        double r120866 = r120860 + r120861;
        double r120867 = -r120857;
        double r120868 = fma(r120865, r120866, r120867);
        double r120869 = 1.4767289533588029e-117;
        bool r120870 = r120853 <= r120869;
        double r120871 = 2.0;
        double r120872 = pow(r120853, r120871);
        double r120873 = pow(r120856, r120871);
        double r120874 = fma(r120853, r120873, r120853);
        double r120875 = fma(r120872, r120856, r120874);
        double r120876 = cos(r120856);
        double r120877 = r120859 * r120876;
        double r120878 = r120860 - r120861;
        double r120879 = sin(r120856);
        double r120880 = r120878 * r120879;
        double r120881 = r120877 - r120880;
        double r120882 = r120878 * r120876;
        double r120883 = r120881 / r120882;
        double r120884 = r120870 ? r120875 : r120883;
        double r120885 = r120855 ? r120868 : r120884;
        return r120885;
}

Error

Bits error versus x

Bits error versus eps

Target

Original36.7
Target15.1
Herbie15.4
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if eps < -1.8367694074004902e-68

    1. Initial program 30.1

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

    3. Applied tan-sum5.1

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
    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 associate-*r*5.2

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

    if -1.8367694074004902e-68 < eps < 1.4767289533588029e-117

    1. Initial program 48.6

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

    2. Taylor expanded around 0 31.1

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

    3. Simplified31.1

      \[\leadsto \color{blue}{\mathsf{fma}\left({\varepsilon}^{2}, x, \mathsf{fma}\left(\varepsilon, {x}^{2}, \varepsilon\right)\right)}\]

    if 1.4767289533588029e-117 < eps

    1. Initial program 30.6

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

    3. Applied tan-quot30.5

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

    4. Applied tan-sum8.9

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

    5. Applied frac-sub8.9

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

  4. Final simplification15.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.8367694074004902 \cdot 10^{-68}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 - \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \tan x\right) \cdot \tan \varepsilon}, 1 + \tan x \cdot \tan \varepsilon, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \le 1.476728953358803 \cdot 10^{-117}:\\ \;\;\;\;\mathsf{fma}\left({\varepsilon}^{2}, x, \mathsf{fma}\left(\varepsilon, {x}^{2}, \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 +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)))