Average Error: 31.7 → 0.2
Time: 30.5s
Precision: 64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.483298028151702:\\ \;\;\;\;\left(\left(\frac{\sin x}{\cos x \cdot x} \cdot \frac{\sin x}{\cos x \cdot x} + \frac{\sin x}{\cos x \cdot x}\right) - \frac{\sin x}{x}\right) - \left(\frac{\frac{\sin x}{x} \cdot \frac{\sin x}{x}}{\cos x} - 1\right)\\ \mathbf{elif}\;x \le 0.02327314422661118:\\ \;\;\;\;\left(x \cdot \left(x \cdot \frac{9}{40}\right) - \frac{1}{2}\right) - {x}^{4} \cdot \frac{27}{2800}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -2.483298028151702:\\
\;\;\;\;\left(\left(\frac{\sin x}{\cos x \cdot x} \cdot \frac{\sin x}{\cos x \cdot x} + \frac{\sin x}{\cos x \cdot x}\right) - \frac{\sin x}{x}\right) - \left(\frac{\frac{\sin x}{x} \cdot \frac{\sin x}{x}}{\cos x} - 1\right)\\

\mathbf{elif}\;x \le 0.02327314422661118:\\
\;\;\;\;\left(x \cdot \left(x \cdot \frac{9}{40}\right) - \frac{1}{2}\right) - {x}^{4} \cdot \frac{27}{2800}\\

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

\end{array}
double f(double x) {
        double r890230 = x;
        double r890231 = sin(r890230);
        double r890232 = r890230 - r890231;
        double r890233 = tan(r890230);
        double r890234 = r890230 - r890233;
        double r890235 = r890232 / r890234;
        return r890235;
}
↓
double f(double x) {
        double r890236 = x;
        double r890237 = -2.483298028151702;
        bool r890238 = r890236 <= r890237;
        double r890239 = sin(r890236);
        double r890240 = cos(r890236);
        double r890241 = r890240 * r890236;
        double r890242 = r890239 / r890241;
        double r890243 = r890242 * r890242;
        double r890244 = r890243 + r890242;
        double r890245 = r890239 / r890236;
        double r890246 = r890244 - r890245;
        double r890247 = r890245 * r890245;
        double r890248 = r890247 / r890240;
        double r890249 = 1.0;
        double r890250 = r890248 - r890249;
        double r890251 = r890246 - r890250;
        double r890252 = 0.02327314422661118;
        bool r890253 = r890236 <= r890252;
        double r890254 = 0.225;
        double r890255 = r890236 * r890254;
        double r890256 = r890236 * r890255;
        double r890257 = 0.5;
        double r890258 = r890256 - r890257;
        double r890259 = 4.0;
        double r890260 = pow(r890236, r890259);
        double r890261 = 0.009642857142857142;
        double r890262 = r890260 * r890261;
        double r890263 = r890258 - r890262;
        double r890264 = r890236 - r890239;
        double r890265 = tan(r890236);
        double r890266 = r890236 - r890265;
        double r890267 = r890264 / r890266;
        double r890268 = r890253 ? r890263 : r890267;
        double r890269 = r890238 ? r890251 : r890268;
        return r890269;
}

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 < -2.483298028151702

    1. Initial program 0.0

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

    2. Taylor expanded around -inf 0.0

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

    3. Taylor expanded around inf 0.4

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

    4. Simplified0.4

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

    if -2.483298028151702 < x < 0.02327314422661118

    1. Initial program 62.7

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

    2. Taylor expanded around -inf 62.7

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

    3. Taylor expanded around 0 0.1

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

    4. Simplified0.1

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

    if 0.02327314422661118 < x

    1. Initial program 0.1

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

    2. Taylor expanded around -inf 0.1

      \[\leadsto \frac{\color{blue}{x - \sin x}}{x - \tan x}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.483298028151702:\\ \;\;\;\;\left(\left(\frac{\sin x}{\cos x \cdot x} \cdot \frac{\sin x}{\cos x \cdot x} + \frac{\sin x}{\cos x \cdot x}\right) - \frac{\sin x}{x}\right) - \left(\frac{\frac{\sin x}{x} \cdot \frac{\sin x}{x}}{\cos x} - 1\right)\\ \mathbf{elif}\;x \le 0.02327314422661118:\\ \;\;\;\;\left(x \cdot \left(x \cdot \frac{9}{40}\right) - \frac{1}{2}\right) - {x}^{4} \cdot \frac{27}{2800}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019134 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))