Average Error: 32.0 → 0.0
Time: 32.1s
Precision: 64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.03252464094224265:\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \mathbf{elif}\;x \le 0.03271128832128569:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \frac{9}{40} - \frac{27}{2800} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) - \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.03252464094224265:\\
\;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\

\mathbf{elif}\;x \le 0.03271128832128569:\\
\;\;\;\;\left(\left(x \cdot x\right) \cdot \frac{9}{40} - \frac{27}{2800} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) - \frac{1}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\

\end{array}
double f(double x) {
        double r500302 = x;
        double r500303 = sin(r500302);
        double r500304 = r500302 - r500303;
        double r500305 = tan(r500302);
        double r500306 = r500302 - r500305;
        double r500307 = r500304 / r500306;
        return r500307;
}
↓
double f(double x) {
        double r500308 = x;
        double r500309 = -0.03252464094224265;
        bool r500310 = r500308 <= r500309;
        double r500311 = tan(r500308);
        double r500312 = r500308 - r500311;
        double r500313 = r500308 / r500312;
        double r500314 = sin(r500308);
        double r500315 = r500314 / r500312;
        double r500316 = r500313 - r500315;
        double r500317 = 0.03271128832128569;
        bool r500318 = r500308 <= r500317;
        double r500319 = r500308 * r500308;
        double r500320 = 0.225;
        double r500321 = r500319 * r500320;
        double r500322 = 0.009642857142857142;
        double r500323 = r500319 * r500319;
        double r500324 = r500322 * r500323;
        double r500325 = r500321 - r500324;
        double r500326 = 0.5;
        double r500327 = r500325 - r500326;
        double r500328 = r500318 ? r500327 : r500316;
        double r500329 = r500310 ? r500316 : r500328;
        return r500329;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < -0.03252464094224265 or 0.03271128832128569 < x

    1. Initial program 0.0

      \[\frac{x - \sin x}{x - \tan x}\]
    2. Using strategy rm

    3. Applied div-sub0.0

      \[\leadsto \color{blue}{\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}}\]

    if -0.03252464094224265 < x < 0.03271128832128569

    1. Initial program 62.9

      \[\frac{x - \sin x}{x - \tan x}\]

    2. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\frac{9}{40} \cdot {x}^{2} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}\]

    3. Simplified0.0

      \[\leadsto \color{blue}{\left(\left(x \cdot x\right) \cdot \frac{9}{40} - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{27}{2800}\right) - \frac{1}{2}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.03252464094224265:\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \mathbf{elif}\;x \le 0.03271128832128569:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \frac{9}{40} - \frac{27}{2800} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) - \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019137 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))