Average Error: 31.1 → 0.0
Time: 1.1m
Precision: 64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.027235330385386256:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x - \sin x}{x - \tan x}}}\\ \mathbf{elif}\;x \le 0.030173252547550686:\\ \;\;\;\;(\left(x \cdot x\right) \cdot \frac{9}{40} + \left((\frac{-27}{2800} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) + \frac{-1}{2})_*\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x - \sin x}{x - \tan x}}}\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.027235330385386256:\\
\;\;\;\;\frac{1}{\frac{1}{\frac{x - \sin x}{x - \tan x}}}\\

\mathbf{elif}\;x \le 0.030173252547550686:\\
\;\;\;\;(\left(x \cdot x\right) \cdot \frac{9}{40} + \left((\frac{-27}{2800} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) + \frac{-1}{2})_*\right))_*\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\frac{1}{\frac{x - \sin x}{x - \tan x}}}\\

\end{array}
double f(double x) {
        double r1421315 = x;
        double r1421316 = sin(r1421315);
        double r1421317 = r1421315 - r1421316;
        double r1421318 = tan(r1421315);
        double r1421319 = r1421315 - r1421318;
        double r1421320 = r1421317 / r1421319;
        return r1421320;
}
↓
double f(double x) {
        double r1421321 = x;
        double r1421322 = -0.027235330385386256;
        bool r1421323 = r1421321 <= r1421322;
        double r1421324 = 1.0;
        double r1421325 = sin(r1421321);
        double r1421326 = r1421321 - r1421325;
        double r1421327 = tan(r1421321);
        double r1421328 = r1421321 - r1421327;
        double r1421329 = r1421326 / r1421328;
        double r1421330 = r1421324 / r1421329;
        double r1421331 = r1421324 / r1421330;
        double r1421332 = 0.030173252547550686;
        bool r1421333 = r1421321 <= r1421332;
        double r1421334 = r1421321 * r1421321;
        double r1421335 = 0.225;
        double r1421336 = -0.009642857142857142;
        double r1421337 = r1421334 * r1421334;
        double r1421338 = -0.5;
        double r1421339 = fma(r1421336, r1421337, r1421338);
        double r1421340 = fma(r1421334, r1421335, r1421339);
        double r1421341 = r1421333 ? r1421340 : r1421331;
        double r1421342 = r1421323 ? r1421331 : r1421341;
        return r1421342;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -0.027235330385386256 or 0.030173252547550686 < x

    1. Initial program 0.0

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

    3. Applied *-un-lft-identity0.0

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

    4. Applied associate-/l*0.0

      \[\leadsto \color{blue}{\frac{1}{\frac{x - \tan x}{x - \sin x}}}\]
    5. Using strategy rm

    6. Applied clear-num0.1

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

    if -0.027235330385386256 < x < 0.030173252547550686

    1. Initial program 62.8

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

    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

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.027235330385386256:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x - \sin x}{x - \tan x}}}\\ \mathbf{elif}\;x \le 0.030173252547550686:\\ \;\;\;\;(\left(x \cdot x\right) \cdot \frac{9}{40} + \left((\frac{-27}{2800} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) + \frac{-1}{2})_*\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x - \sin x}{x - \tan x}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019104 +o rules:numerics
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))