Average Error: 36.8 → 15.1
Time: 10.4s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -9.832752395103017500583649272309788342243 \cdot 10^{-18} \lor \neg \left(\varepsilon \le 1.938370975879928752703824162828496810488 \cdot 10^{-31}\right):\\ \;\;\;\;\frac{\tan x \cdot \tan x - \tan \varepsilon \cdot \tan \varepsilon}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x - \tan \varepsilon\right)} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right) + \varepsilon\\ \end{array}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -9.832752395103017500583649272309788342243 \cdot 10^{-18} \lor \neg \left(\varepsilon \le 1.938370975879928752703824162828496810488 \cdot 10^{-31}\right):\\
\;\;\;\;\frac{\tan x \cdot \tan x - \tan \varepsilon \cdot \tan \varepsilon}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x - \tan \varepsilon\right)} - \tan x\\

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

\end{array}
double f(double x, double eps) {
        double r125616 = x;
        double r125617 = eps;
        double r125618 = r125616 + r125617;
        double r125619 = tan(r125618);
        double r125620 = tan(r125616);
        double r125621 = r125619 - r125620;
        return r125621;
}
↓
double f(double x, double eps) {
        double r125622 = eps;
        double r125623 = -9.832752395103018e-18;
        bool r125624 = r125622 <= r125623;
        double r125625 = 1.9383709758799288e-31;
        bool r125626 = r125622 <= r125625;
        double r125627 = !r125626;
        bool r125628 = r125624 || r125627;
        double r125629 = x;
        double r125630 = tan(r125629);
        double r125631 = r125630 * r125630;
        double r125632 = tan(r125622);
        double r125633 = r125632 * r125632;
        double r125634 = r125631 - r125633;
        double r125635 = 1.0;
        double r125636 = r125630 * r125632;
        double r125637 = r125635 - r125636;
        double r125638 = r125630 - r125632;
        double r125639 = r125637 * r125638;
        double r125640 = r125634 / r125639;
        double r125641 = r125640 - r125630;
        double r125642 = r125629 * r125622;
        double r125643 = r125622 + r125629;
        double r125644 = r125642 * r125643;
        double r125645 = r125644 + r125622;
        double r125646 = r125628 ? r125641 : r125645;
        return r125646;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.8
Target15.2
Herbie15.1
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 2 regimes

  2. if eps < -9.832752395103018e-18 or 1.9383709758799288e-31 < eps

    1. Initial program 29.9

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

    3. Applied tan-sum1.7

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

    5. Applied flip-+1.8

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

    6. Applied associate-/l/1.8

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

    if -9.832752395103018e-18 < eps < 1.9383709758799288e-31

    1. Initial program 45.0

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

    3. Applied tan-sum45.0

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

    4. Taylor expanded around 0 31.0

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

    5. Simplified30.8

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

  4. Final simplification15.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -9.832752395103017500583649272309788342243 \cdot 10^{-18} \lor \neg \left(\varepsilon \le 1.938370975879928752703824162828496810488 \cdot 10^{-31}\right):\\ \;\;\;\;\frac{\tan x \cdot \tan x - \tan \varepsilon \cdot \tan \varepsilon}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x - \tan \varepsilon\right)} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right) + \varepsilon\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 
(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)))