Average Error: 37.1 → 13.9
Time: 10.6s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -3.335377231362400299868570199811322820453 \cdot 10^{-22}:\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}\right) \cdot \sin x}{\left(1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}\right) \cdot \cos x}\\ \mathbf{elif}\;\varepsilon \le 1.349170156454840996446574643298394842499 \cdot 10^{-17}:\\ \;\;\;\;{\varepsilon}^{2} \cdot \left(\varepsilon \cdot \frac{1}{3} + x\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin \varepsilon}{\cos \varepsilon} + \log \left(e^{\frac{\sin x}{\cos x}}\right)}{1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}} - \tan x\\ \end{array}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -3.335377231362400299868570199811322820453 \cdot 10^{-22}:\\
\;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}\right) \cdot \sin x}{\left(1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}\right) \cdot \cos x}\\

\mathbf{elif}\;\varepsilon \le 1.349170156454840996446574643298394842499 \cdot 10^{-17}:\\
\;\;\;\;{\varepsilon}^{2} \cdot \left(\varepsilon \cdot \frac{1}{3} + x\right) + \varepsilon\\

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

\end{array}
double f(double x, double eps) {
        double r130000 = x;
        double r130001 = eps;
        double r130002 = r130000 + r130001;
        double r130003 = tan(r130002);
        double r130004 = tan(r130000);
        double r130005 = r130003 - r130004;
        return r130005;
}
↓
double f(double x, double eps) {
        double r130006 = eps;
        double r130007 = -3.3353772313624003e-22;
        bool r130008 = r130006 <= r130007;
        double r130009 = x;
        double r130010 = tan(r130009);
        double r130011 = tan(r130006);
        double r130012 = r130010 + r130011;
        double r130013 = cos(r130009);
        double r130014 = r130012 * r130013;
        double r130015 = 1.0;
        double r130016 = sin(r130009);
        double r130017 = r130016 * r130011;
        double r130018 = r130017 / r130013;
        double r130019 = r130015 - r130018;
        double r130020 = r130019 * r130016;
        double r130021 = r130014 - r130020;
        double r130022 = r130019 * r130013;
        double r130023 = r130021 / r130022;
        double r130024 = 1.349170156454841e-17;
        bool r130025 = r130006 <= r130024;
        double r130026 = 2.0;
        double r130027 = pow(r130006, r130026);
        double r130028 = 0.3333333333333333;
        double r130029 = r130006 * r130028;
        double r130030 = r130029 + r130009;
        double r130031 = r130027 * r130030;
        double r130032 = r130031 + r130006;
        double r130033 = sin(r130006);
        double r130034 = cos(r130006);
        double r130035 = r130033 / r130034;
        double r130036 = r130016 / r130013;
        double r130037 = exp(r130036);
        double r130038 = log(r130037);
        double r130039 = r130035 + r130038;
        double r130040 = r130039 / r130019;
        double r130041 = r130040 - r130010;
        double r130042 = r130025 ? r130032 : r130041;
        double r130043 = r130008 ? r130023 : r130042;
        return r130043;
}

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
Herbie13.9
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if eps < -3.3353772313624003e-22

    1. Initial program 29.6

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

    3. Applied tan-sum1.4

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

    5. Applied tan-quot1.5

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

    6. Applied associate-*l/1.4

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

    8. Applied tan-quot1.5

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

    9. Applied frac-sub1.5

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

    if -3.3353772313624003e-22 < eps < 1.349170156454841e-17

    1. Initial program 45.3

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

    3. Applied tan-sum45.3

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

    5. Applied tan-quot45.3

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

    6. Applied associate-*l/45.3

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

    7. Taylor expanded around inf 45.5

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

    8. Taylor expanded around 0 27.9

      \[\leadsto \color{blue}{x \cdot {\varepsilon}^{2} + \left(\frac{1}{3} \cdot {\varepsilon}^{3} + \varepsilon\right)}\]

    9. Simplified27.9

      \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot \left(\varepsilon \cdot \frac{1}{3} + x\right) + \varepsilon}\]

    if 1.349170156454841e-17 < eps

    1. Initial program 29.9

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

    3. Applied tan-sum0.9

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

    5. Applied tan-quot0.9

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

    6. Applied associate-*l/0.9

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

    7. Taylor expanded around inf 1.1

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

    9. Applied add-log-exp1.2

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

  4. Final simplification13.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -3.335377231362400299868570199811322820453 \cdot 10^{-22}:\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}\right) \cdot \sin x}{\left(1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}\right) \cdot \cos x}\\ \mathbf{elif}\;\varepsilon \le 1.349170156454840996446574643298394842499 \cdot 10^{-17}:\\ \;\;\;\;{\varepsilon}^{2} \cdot \left(\varepsilon \cdot \frac{1}{3} + x\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin \varepsilon}{\cos \varepsilon} + \log \left(e^{\frac{\sin x}{\cos x}}\right)}{1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}} - \tan x\\ \end{array}\]

Reproduce

herbie shell --seed 2019352 
(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)))