Average Error: 36.7 → 15.5
Time: 27.5s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.757922927393181967231988612780291928877 \cdot 10^{-27} \lor \neg \left(\varepsilon \le 1.760250544683835501772756411202209200384 \cdot 10^{-79}\right):\\ \;\;\;\;\frac{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} \cdot \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x \cdot \tan x}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \tan x}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right) + \varepsilon\\ \end{array}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -1.757922927393181967231988612780291928877 \cdot 10^{-27} \lor \neg \left(\varepsilon \le 1.760250544683835501772756411202209200384 \cdot 10^{-79}\right):\\
\;\;\;\;\frac{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} \cdot \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x \cdot \tan x}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \tan x}\\

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

\end{array}
double f(double x, double eps) {
        double r92501 = x;
        double r92502 = eps;
        double r92503 = r92501 + r92502;
        double r92504 = tan(r92503);
        double r92505 = tan(r92501);
        double r92506 = r92504 - r92505;
        return r92506;
}
↓
double f(double x, double eps) {
        double r92507 = eps;
        double r92508 = -1.757922927393182e-27;
        bool r92509 = r92507 <= r92508;
        double r92510 = 1.7602505446838355e-79;
        bool r92511 = r92507 <= r92510;
        double r92512 = !r92511;
        bool r92513 = r92509 || r92512;
        double r92514 = x;
        double r92515 = tan(r92514);
        double r92516 = tan(r92507);
        double r92517 = r92515 + r92516;
        double r92518 = 1.0;
        double r92519 = r92515 * r92516;
        double r92520 = r92518 - r92519;
        double r92521 = r92517 / r92520;
        double r92522 = r92521 * r92521;
        double r92523 = r92515 * r92515;
        double r92524 = r92522 - r92523;
        double r92525 = r92521 + r92515;
        double r92526 = r92524 / r92525;
        double r92527 = r92514 * r92507;
        double r92528 = r92507 + r92514;
        double r92529 = r92527 * r92528;
        double r92530 = r92529 + r92507;
        double r92531 = r92513 ? r92526 : r92530;
        return r92531;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.7
Target14.9
Herbie15.5
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 2 regimes

  2. if eps < -1.757922927393182e-27 or 1.7602505446838355e-79 < eps

    1. Initial program 29.6

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

    3. Applied tan-sum4.5

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

    5. Applied flip--4.6

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

    if -1.757922927393182e-27 < eps < 1.7602505446838355e-79

    1. Initial program 46.8

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

    3. Applied tan-sum46.8

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

    4. Taylor expanded around 0 31.4

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

    5. Simplified31.2

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

  4. Final simplification15.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.757922927393181967231988612780291928877 \cdot 10^{-27} \lor \neg \left(\varepsilon \le 1.760250544683835501772756411202209200384 \cdot 10^{-79}\right):\\ \;\;\;\;\frac{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} \cdot \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x \cdot \tan x}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \tan x}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right) + \varepsilon\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(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)))