Average Error: 31.2 → 0.3
Time: 24.0s
Precision: 64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.773716110019835:\\ \;\;\;\;\left(\left(\frac{\frac{\sin x}{\cos x}}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x} + \frac{\frac{\sin x}{\cos x}}{x}\right) - \left(\frac{\sin x \cdot \sin x}{\cos x \cdot \left(x \cdot x\right)} + \frac{\sin x}{x}\right)\right) + 1\\ \mathbf{elif}\;x \le 2.412135160853593:\\ \;\;\;\;\left(\frac{9}{40} \cdot \left(x \cdot x\right) - \frac{1}{2}\right) - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{27}{2800}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{\frac{\sin x}{\cos x}}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x} + \frac{\frac{\sin x}{\cos x}}{x}\right) - \left(\frac{\sin x \cdot \sin x}{\cos x \cdot \left(x \cdot x\right)} + \frac{\sin x}{x}\right)\right) + 1\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -4.773716110019835:\\
\;\;\;\;\left(\left(\frac{\frac{\sin x}{\cos x}}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x} + \frac{\frac{\sin x}{\cos x}}{x}\right) - \left(\frac{\sin x \cdot \sin x}{\cos x \cdot \left(x \cdot x\right)} + \frac{\sin x}{x}\right)\right) + 1\\

\mathbf{elif}\;x \le 2.412135160853593:\\
\;\;\;\;\left(\frac{9}{40} \cdot \left(x \cdot x\right) - \frac{1}{2}\right) - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{27}{2800}\\

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

\end{array}
double f(double x) {
        double r321396 = x;
        double r321397 = sin(r321396);
        double r321398 = r321396 - r321397;
        double r321399 = tan(r321396);
        double r321400 = r321396 - r321399;
        double r321401 = r321398 / r321400;
        return r321401;
}
↓
double f(double x) {
        double r321402 = x;
        double r321403 = -4.773716110019835;
        bool r321404 = r321402 <= r321403;
        double r321405 = sin(r321402);
        double r321406 = cos(r321402);
        double r321407 = r321405 / r321406;
        double r321408 = r321407 / r321402;
        double r321409 = r321408 * r321408;
        double r321410 = r321409 + r321408;
        double r321411 = r321405 * r321405;
        double r321412 = r321402 * r321402;
        double r321413 = r321406 * r321412;
        double r321414 = r321411 / r321413;
        double r321415 = r321405 / r321402;
        double r321416 = r321414 + r321415;
        double r321417 = r321410 - r321416;
        double r321418 = 1.0;
        double r321419 = r321417 + r321418;
        double r321420 = 2.412135160853593;
        bool r321421 = r321402 <= r321420;
        double r321422 = 0.225;
        double r321423 = r321422 * r321412;
        double r321424 = 0.5;
        double r321425 = r321423 - r321424;
        double r321426 = r321412 * r321412;
        double r321427 = 0.009642857142857142;
        double r321428 = r321426 * r321427;
        double r321429 = r321425 - r321428;
        double r321430 = r321421 ? r321429 : r321419;
        double r321431 = r321404 ? r321419 : r321430;
        return r321431;
}

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 < -4.773716110019835 or 2.412135160853593 < x

    1. Initial program 0.0

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

    3. Applied div-inv0.2

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

    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

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

    if -4.773716110019835 < x < 2.412135160853593

    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

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.773716110019835:\\ \;\;\;\;\left(\left(\frac{\frac{\sin x}{\cos x}}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x} + \frac{\frac{\sin x}{\cos x}}{x}\right) - \left(\frac{\sin x \cdot \sin x}{\cos x \cdot \left(x \cdot x\right)} + \frac{\sin x}{x}\right)\right) + 1\\ \mathbf{elif}\;x \le 2.412135160853593:\\ \;\;\;\;\left(\frac{9}{40} \cdot \left(x \cdot x\right) - \frac{1}{2}\right) - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{27}{2800}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{\frac{\sin x}{\cos x}}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x} + \frac{\frac{\sin x}{\cos x}}{x}\right) - \left(\frac{\sin x \cdot \sin x}{\cos x \cdot \left(x \cdot x\right)} + \frac{\sin x}{x}\right)\right) + 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))