Average Error: 37.5 → 16.2
Time: 25.0s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -5.246119333473003277498362100151355132534 \cdot 10^{-164} \lor \neg \left(\varepsilon \le 5.307417583108514168007488766475587468342 \cdot 10^{-118}\right):\\ \;\;\;\;\frac{\left(\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \sin x\right) - \left(-\sin x \cdot \frac{\sin x \cdot \tan \varepsilon}{\cos x}\right)}{\cos x + \left(-\cos x \cdot \frac{\sin x \cdot \tan \varepsilon}{\cos x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right)\\ \end{array}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -5.246119333473003277498362100151355132534 \cdot 10^{-164} \lor \neg \left(\varepsilon \le 5.307417583108514168007488766475587468342 \cdot 10^{-118}\right):\\
\;\;\;\;\frac{\left(\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \sin x\right) - \left(-\sin x \cdot \frac{\sin x \cdot \tan \varepsilon}{\cos x}\right)}{\cos x + \left(-\cos x \cdot \frac{\sin x \cdot \tan \varepsilon}{\cos x}\right)}\\

\mathbf{else}:\\
\;\;\;\;\varepsilon + \left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right)\\

\end{array}
double f(double x, double eps) {
        double r82935 = x;
        double r82936 = eps;
        double r82937 = r82935 + r82936;
        double r82938 = tan(r82937);
        double r82939 = tan(r82935);
        double r82940 = r82938 - r82939;
        return r82940;
}
↓
double f(double x, double eps) {
        double r82941 = eps;
        double r82942 = -5.246119333473003e-164;
        bool r82943 = r82941 <= r82942;
        double r82944 = 5.307417583108514e-118;
        bool r82945 = r82941 <= r82944;
        double r82946 = !r82945;
        bool r82947 = r82943 || r82946;
        double r82948 = x;
        double r82949 = tan(r82948);
        double r82950 = tan(r82941);
        double r82951 = r82949 + r82950;
        double r82952 = cos(r82948);
        double r82953 = r82951 * r82952;
        double r82954 = sin(r82948);
        double r82955 = r82953 - r82954;
        double r82956 = r82954 * r82950;
        double r82957 = r82956 / r82952;
        double r82958 = r82954 * r82957;
        double r82959 = -r82958;
        double r82960 = r82955 - r82959;
        double r82961 = r82952 * r82957;
        double r82962 = -r82961;
        double r82963 = r82952 + r82962;
        double r82964 = r82960 / r82963;
        double r82965 = r82948 * r82941;
        double r82966 = r82941 + r82948;
        double r82967 = r82965 * r82966;
        double r82968 = r82941 + r82967;
        double r82969 = r82947 ? r82964 : r82968;
        return r82969;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.5
Target15.4
Herbie16.2
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 2 regimes

  2. if eps < -5.246119333473003e-164 or 5.307417583108514e-118 < eps

    1. Initial program 32.4

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

    3. Applied tan-sum11.4

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

    5. Applied tan-quot11.4

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

    6. Applied associate-*l/11.4

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

    8. Applied add-cube-cbrt11.5

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

    10. Applied tan-quot11.5

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

    11. Applied frac-sub11.5

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

    12. Simplified10.7

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

    13. Simplified10.5

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

    if -5.246119333473003e-164 < eps < 5.307417583108514e-118

    1. Initial program 51.2

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

    3. Applied tan-sum51.2

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

    4. Taylor expanded around 0 31.9

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

    5. Simplified31.7

      \[\leadsto \color{blue}{\varepsilon + \left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification16.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -5.246119333473003277498362100151355132534 \cdot 10^{-164} \lor \neg \left(\varepsilon \le 5.307417583108514168007488766475587468342 \cdot 10^{-118}\right):\\ \;\;\;\;\frac{\left(\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \sin x\right) - \left(-\sin x \cdot \frac{\sin x \cdot \tan \varepsilon}{\cos x}\right)}{\cos x + \left(-\cos x \cdot \frac{\sin x \cdot \tan \varepsilon}{\cos x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019294 
(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)))