Average Error: 31.2 → 0.0
Time: 31.2s
Precision: 64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.03021843902660959271799789860324381152168:\\ \;\;\;\;\sqrt[3]{\left(\frac{x - \sin x}{x - \tan x} \cdot \frac{x - \sin x}{x - \tan x}\right) \cdot \frac{x - \sin x}{x - \tan x}}\\ \mathbf{elif}\;x \le 0.02793749748564295504182908302936994004995:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\frac{9}{40} - \frac{27}{2800} \cdot \left(x \cdot x\right)\right) + \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\frac{x - \sin x}{x - \tan x} \cdot \frac{x - \sin x}{x - \tan x}\right) \cdot \frac{x - \sin x}{x - \tan x}}\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.03021843902660959271799789860324381152168:\\
\;\;\;\;\sqrt[3]{\left(\frac{x - \sin x}{x - \tan x} \cdot \frac{x - \sin x}{x - \tan x}\right) \cdot \frac{x - \sin x}{x - \tan x}}\\

\mathbf{elif}\;x \le 0.02793749748564295504182908302936994004995:\\
\;\;\;\;\left(x \cdot x\right) \cdot \left(\frac{9}{40} - \frac{27}{2800} \cdot \left(x \cdot x\right)\right) + \frac{-1}{2}\\

\mathbf{else}:\\
\;\;\;\;\sqrt[3]{\left(\frac{x - \sin x}{x - \tan x} \cdot \frac{x - \sin x}{x - \tan x}\right) \cdot \frac{x - \sin x}{x - \tan x}}\\

\end{array}
double f(double x) {
        double r2074243 = x;
        double r2074244 = sin(r2074243);
        double r2074245 = r2074243 - r2074244;
        double r2074246 = tan(r2074243);
        double r2074247 = r2074243 - r2074246;
        double r2074248 = r2074245 / r2074247;
        return r2074248;
}
↓
double f(double x) {
        double r2074249 = x;
        double r2074250 = -0.030218439026609593;
        bool r2074251 = r2074249 <= r2074250;
        double r2074252 = sin(r2074249);
        double r2074253 = r2074249 - r2074252;
        double r2074254 = tan(r2074249);
        double r2074255 = r2074249 - r2074254;
        double r2074256 = r2074253 / r2074255;
        double r2074257 = r2074256 * r2074256;
        double r2074258 = r2074257 * r2074256;
        double r2074259 = cbrt(r2074258);
        double r2074260 = 0.027937497485642955;
        bool r2074261 = r2074249 <= r2074260;
        double r2074262 = r2074249 * r2074249;
        double r2074263 = 0.225;
        double r2074264 = 0.009642857142857142;
        double r2074265 = r2074264 * r2074262;
        double r2074266 = r2074263 - r2074265;
        double r2074267 = r2074262 * r2074266;
        double r2074268 = -0.5;
        double r2074269 = r2074267 + r2074268;
        double r2074270 = r2074261 ? r2074269 : r2074259;
        double r2074271 = r2074251 ? r2074259 : r2074270;
        return r2074271;
}

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 < -0.030218439026609593 or 0.027937497485642955 < x

    1. Initial program 0.1

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

    3. Applied add-cbrt-cube41.5

      \[\leadsto \frac{x - \sin x}{\color{blue}{\sqrt[3]{\left(\left(x - \tan x\right) \cdot \left(x - \tan x\right)\right) \cdot \left(x - \tan x\right)}}}\]

    4. Applied add-cbrt-cube42.7

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(x - \sin x\right) \cdot \left(x - \sin x\right)\right) \cdot \left(x - \sin x\right)}}}{\sqrt[3]{\left(\left(x - \tan x\right) \cdot \left(x - \tan x\right)\right) \cdot \left(x - \tan x\right)}}\]

    5. Applied cbrt-undiv42.7

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\left(\left(x - \sin x\right) \cdot \left(x - \sin x\right)\right) \cdot \left(x - \sin x\right)}{\left(\left(x - \tan x\right) \cdot \left(x - \tan x\right)\right) \cdot \left(x - \tan x\right)}}}\]

    6. Simplified0.1

      \[\leadsto \sqrt[3]{\color{blue}{\left(\frac{x - \sin x}{x - \tan x} \cdot \frac{x - \sin x}{x - \tan x}\right) \cdot \frac{x - \sin x}{x - \tan x}}}\]

    if -0.030218439026609593 < x < 0.027937497485642955

    1. Initial program 63.2

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

    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

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

    4. Taylor expanded around 0 0.0

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

    5. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.03021843902660959271799789860324381152168:\\ \;\;\;\;\sqrt[3]{\left(\frac{x - \sin x}{x - \tan x} \cdot \frac{x - \sin x}{x - \tan x}\right) \cdot \frac{x - \sin x}{x - \tan x}}\\ \mathbf{elif}\;x \le 0.02793749748564295504182908302936994004995:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\frac{9}{40} - \frac{27}{2800} \cdot \left(x \cdot x\right)\right) + \frac{-1}{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\frac{x - \sin x}{x - \tan x} \cdot \frac{x - \sin x}{x - \tan x}\right) \cdot \frac{x - \sin x}{x - \tan x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019173 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))