Average Error: 36.8 → 15.1
Time: 10.8s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -9.832752395103017500583649272309788342243 \cdot 10^{-18} \lor \neg \left(\varepsilon \le 1.938370975879928752703824162828496810488 \cdot 10^{-31}\right):\\ \;\;\;\;\frac{\tan x \cdot \tan x - \tan \varepsilon \cdot \tan \varepsilon}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x - \tan \varepsilon\right)} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot x\right) \cdot \left(x + \varepsilon\right) + \varepsilon\\ \end{array}\]
\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(\varepsilon \cdot x\right) \cdot \left(x + \varepsilon\right) + \varepsilon\\

\end{array}
double f(double x, double eps) {
        double r143675 = x;
        double r143676 = eps;
        double r143677 = r143675 + r143676;
        double r143678 = tan(r143677);
        double r143679 = tan(r143675);
        double r143680 = r143678 - r143679;
        return r143680;
}
↓
double f(double x, double eps) {
        double r143681 = eps;
        double r143682 = -9.832752395103018e-18;
        bool r143683 = r143681 <= r143682;
        double r143684 = 1.9383709758799288e-31;
        bool r143685 = r143681 <= r143684;
        double r143686 = !r143685;
        bool r143687 = r143683 || r143686;
        double r143688 = x;
        double r143689 = tan(r143688);
        double r143690 = r143689 * r143689;
        double r143691 = tan(r143681);
        double r143692 = r143691 * r143691;
        double r143693 = r143690 - r143692;
        double r143694 = 1.0;
        double r143695 = r143689 * r143691;
        double r143696 = r143694 - r143695;
        double r143697 = r143689 - r143691;
        double r143698 = r143696 * r143697;
        double r143699 = r143693 / r143698;
        double r143700 = r143699 - r143689;
        double r143701 = r143681 * r143688;
        double r143702 = r143688 + r143681;
        double r143703 = r143701 * r143702;
        double r143704 = r143703 + r143681;
        double r143705 = r143687 ? r143700 : r143704;
        return r143705;
}

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. Taylor expanded around 0 31.0

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

    3. Simplified30.8

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

  4. Final simplification15.1

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

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)))