Average Error: 37.1 → 15.0
Time: 26.6s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -6.929244313528967253629797030059496249939 \cdot 10^{-77}:\\ \;\;\;\;\frac{\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon}{\left(1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\cos x \cdot \cos \varepsilon\right)} - \tan x\\ \mathbf{elif}\;\varepsilon \le 1.084923224249312910690492748285721451604 \cdot 10^{-19}:\\ \;\;\;\;\varepsilon + \left(\varepsilon \cdot x\right) \cdot \left(x + \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\\ \end{array}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -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(\varepsilon \cdot x\right) \cdot \left(x + \varepsilon\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 r87170 = x;
        double r87171 = eps;
        double r87172 = r87170 + r87171;
        double r87173 = tan(r87172);
        double r87174 = tan(r87170);
        double r87175 = r87173 - r87174;
        return r87175;
}
↓
double f(double x, double eps) {
        double r87176 = eps;
        double r87177 = -6.929244313528967e-77;
        bool r87178 = r87176 <= r87177;
        double r87179 = x;
        double r87180 = sin(r87179);
        double r87181 = cos(r87176);
        double r87182 = r87180 * r87181;
        double r87183 = cos(r87179);
        double r87184 = sin(r87176);
        double r87185 = r87183 * r87184;
        double r87186 = r87182 + r87185;
        double r87187 = 1.0;
        double r87188 = tan(r87179);
        double r87189 = r87188 * r87184;
        double r87190 = r87189 / r87181;
        double r87191 = r87187 - r87190;
        double r87192 = r87183 * r87181;
        double r87193 = r87191 * r87192;
        double r87194 = r87186 / r87193;
        double r87195 = r87194 - r87188;
        double r87196 = 1.084923224249313e-19;
        bool r87197 = r87176 <= r87196;
        double r87198 = r87176 * r87179;
        double r87199 = r87179 + r87176;
        double r87200 = r87198 * r87199;
        double r87201 = r87176 + r87200;
        double r87202 = tan(r87176);
        double r87203 = r87188 + r87202;
        double r87204 = r87203 * r87183;
        double r87205 = r87188 * r87202;
        double r87206 = r87187 - r87205;
        double r87207 = r87206 * r87180;
        double r87208 = r87204 - r87207;
        double r87209 = r87206 * r87183;
        double r87210 = r87208 / r87209;
        double r87211 = r87197 ? r87201 : r87210;
        double r87212 = r87178 ? r87195 : r87211;
        return r87212;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.1
Target15.0
Herbie15.0
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

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-sum6.0

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

    5. Applied tan-quot6.1

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

    6. Applied tan-quot6.1

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

    7. Applied frac-add6.1

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

    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-quot6.1

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

    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-sum46.6

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

    5. Applied add-cube-cbrt46.6

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

    6. Taylor expanded around 0 30.1

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

    7. Simplified29.9

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

    if 1.084923224249313e-19 < eps

    1. Initial program 29.0

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

    3. Applied tan-quot28.8

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

    4. Applied tan-sum1.2

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

    5. Applied frac-sub1.3

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

  4. Final simplification15.0

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

Reproduce

herbie shell --seed 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)))