Average Error: 30.8 → 0.3
Time: 50.6s
Precision: 64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.508948866519329:\\ \;\;\;\;\left(\frac{\frac{\sin x}{x}}{\cos x} \cdot \frac{\frac{\sin x}{x}}{\cos x} + \left(\frac{\frac{\sin x}{x}}{\cos x} - \frac{\sin x}{x}\right)\right) - \left(\frac{\frac{\sin x \cdot \sin x}{\cos x}}{x \cdot x} - 1\right)\\ \mathbf{elif}\;x \le 2.4203399319729506:\\ \;\;\;\;x \cdot \left(x \cdot \frac{9}{40}\right) - \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{27}{2800}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\sin x}{x}}{\cos x} \cdot \frac{\frac{\sin x}{x}}{\cos x} + \left(\frac{\frac{\sin x}{x}}{\cos x} - \frac{\sin x}{x}\right)\right) - \left(\frac{\frac{\sin x \cdot \sin x}{\cos x}}{x \cdot x} - 1\right)\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -2.508948866519329:\\
\;\;\;\;\left(\frac{\frac{\sin x}{x}}{\cos x} \cdot \frac{\frac{\sin x}{x}}{\cos x} + \left(\frac{\frac{\sin x}{x}}{\cos x} - \frac{\sin x}{x}\right)\right) - \left(\frac{\frac{\sin x \cdot \sin x}{\cos x}}{x \cdot x} - 1\right)\\

\mathbf{elif}\;x \le 2.4203399319729506:\\
\;\;\;\;x \cdot \left(x \cdot \frac{9}{40}\right) - \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{27}{2800}\right)\right)\\

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

\end{array}
double f(double x) {
        double r891266 = x;
        double r891267 = sin(r891266);
        double r891268 = r891266 - r891267;
        double r891269 = tan(r891266);
        double r891270 = r891266 - r891269;
        double r891271 = r891268 / r891270;
        return r891271;
}
↓
double f(double x) {
        double r891272 = x;
        double r891273 = -2.508948866519329;
        bool r891274 = r891272 <= r891273;
        double r891275 = sin(r891272);
        double r891276 = r891275 / r891272;
        double r891277 = cos(r891272);
        double r891278 = r891276 / r891277;
        double r891279 = r891278 * r891278;
        double r891280 = r891278 - r891276;
        double r891281 = r891279 + r891280;
        double r891282 = r891275 * r891275;
        double r891283 = r891282 / r891277;
        double r891284 = r891272 * r891272;
        double r891285 = r891283 / r891284;
        double r891286 = 1.0;
        double r891287 = r891285 - r891286;
        double r891288 = r891281 - r891287;
        double r891289 = 2.4203399319729506;
        bool r891290 = r891272 <= r891289;
        double r891291 = 0.225;
        double r891292 = r891272 * r891291;
        double r891293 = r891272 * r891292;
        double r891294 = 0.5;
        double r891295 = 0.009642857142857142;
        double r891296 = r891284 * r891295;
        double r891297 = r891284 * r891296;
        double r891298 = r891294 + r891297;
        double r891299 = r891293 - r891298;
        double r891300 = r891290 ? r891299 : r891288;
        double r891301 = r891274 ? r891288 : r891300;
        return r891301;
}

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 < -2.508948866519329 or 2.4203399319729506 < 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

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

    if -2.508948866519329 < x < 2.4203399319729506

    1. Initial program 62.6

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

    2. Taylor expanded around 0 0.3

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

    3. Simplified0.3

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.508948866519329:\\ \;\;\;\;\left(\frac{\frac{\sin x}{x}}{\cos x} \cdot \frac{\frac{\sin x}{x}}{\cos x} + \left(\frac{\frac{\sin x}{x}}{\cos x} - \frac{\sin x}{x}\right)\right) - \left(\frac{\frac{\sin x \cdot \sin x}{\cos x}}{x \cdot x} - 1\right)\\ \mathbf{elif}\;x \le 2.4203399319729506:\\ \;\;\;\;x \cdot \left(x \cdot \frac{9}{40}\right) - \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{27}{2800}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\sin x}{x}}{\cos x} \cdot \frac{\frac{\sin x}{x}}{\cos x} + \left(\frac{\frac{\sin x}{x}}{\cos x} - \frac{\sin x}{x}\right)\right) - \left(\frac{\frac{\sin x \cdot \sin x}{\cos x}}{x \cdot x} - 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))