Average Error: 31.1 → 0.2
Time: 29.5s
Precision: 64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.030161987993530287:\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \mathbf{elif}\;x \le 2.4087276690935244:\\ \;\;\;\;\frac{-27}{2800} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) + \left(x \cdot \left(x \cdot \frac{9}{40}\right) + \frac{-1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin x}{\cos x}}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x} + \left(\left(1 + \frac{\frac{\sin x}{x}}{\cos x}\right) - \left(\frac{\frac{\sin x}{x}}{\cos x} \cdot \frac{\sin x}{x} + \frac{\sin x}{x}\right)\right)\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.030161987993530287:\\
\;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\

\mathbf{elif}\;x \le 2.4087276690935244:\\
\;\;\;\;\frac{-27}{2800} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) + \left(x \cdot \left(x \cdot \frac{9}{40}\right) + \frac{-1}{2}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\sin x}{\cos x}}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x} + \left(\left(1 + \frac{\frac{\sin x}{x}}{\cos x}\right) - \left(\frac{\frac{\sin x}{x}}{\cos x} \cdot \frac{\sin x}{x} + \frac{\sin x}{x}\right)\right)\\

\end{array}
double f(double x) {
        double r937352 = x;
        double r937353 = sin(r937352);
        double r937354 = r937352 - r937353;
        double r937355 = tan(r937352);
        double r937356 = r937352 - r937355;
        double r937357 = r937354 / r937356;
        return r937357;
}
↓
double f(double x) {
        double r937358 = x;
        double r937359 = -0.030161987993530287;
        bool r937360 = r937358 <= r937359;
        double r937361 = tan(r937358);
        double r937362 = r937358 - r937361;
        double r937363 = r937358 / r937362;
        double r937364 = sin(r937358);
        double r937365 = r937364 / r937362;
        double r937366 = r937363 - r937365;
        double r937367 = 2.4087276690935244;
        bool r937368 = r937358 <= r937367;
        double r937369 = -0.009642857142857142;
        double r937370 = r937358 * r937358;
        double r937371 = r937370 * r937370;
        double r937372 = r937369 * r937371;
        double r937373 = 0.225;
        double r937374 = r937358 * r937373;
        double r937375 = r937358 * r937374;
        double r937376 = -0.5;
        double r937377 = r937375 + r937376;
        double r937378 = r937372 + r937377;
        double r937379 = cos(r937358);
        double r937380 = r937364 / r937379;
        double r937381 = r937380 / r937358;
        double r937382 = r937381 * r937381;
        double r937383 = 1.0;
        double r937384 = r937364 / r937358;
        double r937385 = r937384 / r937379;
        double r937386 = r937383 + r937385;
        double r937387 = r937385 * r937384;
        double r937388 = r937387 + r937384;
        double r937389 = r937386 - r937388;
        double r937390 = r937382 + r937389;
        double r937391 = r937368 ? r937378 : r937390;
        double r937392 = r937360 ? r937366 : r937391;
        return r937392;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if x < -0.030161987993530287

    1. Initial program 0.0

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

    3. Applied div-sub0.1

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

    if -0.030161987993530287 < x < 2.4087276690935244

    1. Initial program 62.6

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

    2. Taylor expanded around 0 0.2

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

    3. Simplified0.2

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

    if 2.4087276690935244 < x

    1. Initial program 0.0

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

    3. Applied div-sub0.0

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

    4. Taylor expanded around inf 0.4

      \[\leadsto \color{blue}{\left(1 + \left(\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {x}^{2}} + \frac{\sin x}{\cos x \cdot x}\right)\right) - \left(\frac{\sin x}{x} + \frac{{\left(\sin x\right)}^{2}}{\cos x \cdot {x}^{2}}\right)}\]

    5. Simplified0.4

      \[\leadsto \color{blue}{\frac{\frac{\sin x}{\cos x}}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x} + \left(\left(1 + \frac{\frac{\sin x}{x}}{\cos x}\right) - \left(\frac{\sin x}{x} + \frac{\sin x}{x} \cdot \frac{\frac{\sin x}{x}}{\cos x}\right)\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.030161987993530287:\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \mathbf{elif}\;x \le 2.4087276690935244:\\ \;\;\;\;\frac{-27}{2800} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) + \left(x \cdot \left(x \cdot \frac{9}{40}\right) + \frac{-1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin x}{\cos x}}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x} + \left(\left(1 + \frac{\frac{\sin x}{x}}{\cos x}\right) - \left(\frac{\frac{\sin x}{x}}{\cos x} \cdot \frac{\sin x}{x} + \frac{\sin x}{x}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))