Average Error: 31.2 → 0.3
Time: 50.4s
Precision: 64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.4749055755197364:\\ \;\;\;\;\left(\left(\frac{\frac{\sin x}{\cos x}}{x} - \frac{\sin x}{x}\right) - \left(-1 + \frac{\sin x}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x}\right)\right) + \frac{\frac{\sin x}{\cos x}}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x}\\ \mathbf{elif}\;x \le 2.4545514823208507:\\ \;\;\;\;\frac{9}{40} \cdot \left(x \cdot x\right) - \left(\frac{1}{2} + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{27}{2800}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{\frac{\sin x}{\cos x}}{x} - \frac{\sin x}{x}\right) - \left(-1 + \frac{\sin x}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x}\right)\right) + \frac{\frac{\sin x}{\cos x}}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x}\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -2.4749055755197364:\\
\;\;\;\;\left(\left(\frac{\frac{\sin x}{\cos x}}{x} - \frac{\sin x}{x}\right) - \left(-1 + \frac{\sin x}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x}\right)\right) + \frac{\frac{\sin x}{\cos x}}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x}\\

\mathbf{elif}\;x \le 2.4545514823208507:\\
\;\;\;\;\frac{9}{40} \cdot \left(x \cdot x\right) - \left(\frac{1}{2} + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{27}{2800}\right)\\

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

\end{array}
double f(double x) {
        double r1288258 = x;
        double r1288259 = sin(r1288258);
        double r1288260 = r1288258 - r1288259;
        double r1288261 = tan(r1288258);
        double r1288262 = r1288258 - r1288261;
        double r1288263 = r1288260 / r1288262;
        return r1288263;
}
↓
double f(double x) {
        double r1288264 = x;
        double r1288265 = -2.4749055755197364;
        bool r1288266 = r1288264 <= r1288265;
        double r1288267 = sin(r1288264);
        double r1288268 = cos(r1288264);
        double r1288269 = r1288267 / r1288268;
        double r1288270 = r1288269 / r1288264;
        double r1288271 = r1288267 / r1288264;
        double r1288272 = r1288270 - r1288271;
        double r1288273 = -1.0;
        double r1288274 = r1288271 * r1288270;
        double r1288275 = r1288273 + r1288274;
        double r1288276 = r1288272 - r1288275;
        double r1288277 = r1288270 * r1288270;
        double r1288278 = r1288276 + r1288277;
        double r1288279 = 2.4545514823208507;
        bool r1288280 = r1288264 <= r1288279;
        double r1288281 = 0.225;
        double r1288282 = r1288264 * r1288264;
        double r1288283 = r1288281 * r1288282;
        double r1288284 = 0.5;
        double r1288285 = r1288282 * r1288282;
        double r1288286 = 0.009642857142857142;
        double r1288287 = r1288285 * r1288286;
        double r1288288 = r1288284 + r1288287;
        double r1288289 = r1288283 - r1288288;
        double r1288290 = r1288280 ? r1288289 : r1288278;
        double r1288291 = r1288266 ? r1288278 : r1288290;
        return r1288291;
}

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 < -2.4749055755197364 or 2.4545514823208507 < x

    1. Initial program 0.0

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.3

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

    if -2.4749055755197364 < x < 2.4545514823208507

    1. Initial program 62.5

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

    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

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.4749055755197364:\\ \;\;\;\;\left(\left(\frac{\frac{\sin x}{\cos x}}{x} - \frac{\sin x}{x}\right) - \left(-1 + \frac{\sin x}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x}\right)\right) + \frac{\frac{\sin x}{\cos x}}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x}\\ \mathbf{elif}\;x \le 2.4545514823208507:\\ \;\;\;\;\frac{9}{40} \cdot \left(x \cdot x\right) - \left(\frac{1}{2} + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{27}{2800}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{\frac{\sin x}{\cos x}}{x} - \frac{\sin x}{x}\right) - \left(-1 + \frac{\sin x}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x}\right)\right) + \frac{\frac{\sin x}{\cos x}}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019119 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))