Average Error: 31.1 → 0.3
Time: 2.2m
Precision: 64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.4965764062960356:\\ \;\;\;\;\left(\left(\frac{\frac{\sin x}{\cos x}}{x} - \frac{\sin x}{x}\right) - \left(-1 + \frac{\sin x}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x}\right)\right) + \frac{\frac{\sin x}{\cos x}}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x}\\ \mathbf{elif}\;x \le 2.4905165381526166:\\ \;\;\;\;\frac{9}{40} \cdot \left(x \cdot x\right) - \left(\frac{1}{2} + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{27}{2800}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{\frac{\sin x}{\cos x}}{x} - \frac{\sin x}{x}\right) - \left(-1 + \frac{\sin x}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x}\right)\right) + \frac{\frac{\sin x}{\cos x}}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x}\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -2.4965764062960356:\\
\;\;\;\;\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 2.4905165381526166:\\
\;\;\;\;\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 r6474533 = x;
        double r6474534 = sin(r6474533);
        double r6474535 = r6474533 - r6474534;
        double r6474536 = tan(r6474533);
        double r6474537 = r6474533 - r6474536;
        double r6474538 = r6474535 / r6474537;
        return r6474538;
}
↓
double f(double x) {
        double r6474539 = x;
        double r6474540 = -2.4965764062960356;
        bool r6474541 = r6474539 <= r6474540;
        double r6474542 = sin(r6474539);
        double r6474543 = cos(r6474539);
        double r6474544 = r6474542 / r6474543;
        double r6474545 = r6474544 / r6474539;
        double r6474546 = r6474542 / r6474539;
        double r6474547 = r6474545 - r6474546;
        double r6474548 = -1.0;
        double r6474549 = r6474546 * r6474545;
        double r6474550 = r6474548 + r6474549;
        double r6474551 = r6474547 - r6474550;
        double r6474552 = r6474545 * r6474545;
        double r6474553 = r6474551 + r6474552;
        double r6474554 = 2.4905165381526166;
        bool r6474555 = r6474539 <= r6474554;
        double r6474556 = 0.225;
        double r6474557 = r6474539 * r6474539;
        double r6474558 = r6474556 * r6474557;
        double r6474559 = 0.5;
        double r6474560 = r6474557 * r6474557;
        double r6474561 = 0.009642857142857142;
        double r6474562 = r6474560 * r6474561;
        double r6474563 = r6474559 + r6474562;
        double r6474564 = r6474558 - r6474563;
        double r6474565 = r6474555 ? r6474564 : r6474553;
        double r6474566 = r6474541 ? r6474553 : r6474565;
        return r6474566;
}

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 < -2.4965764062960356 or 2.4905165381526166 < x

    1. Initial program 0.0

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

    2. Taylor expanded around -inf 0.4

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

    3. Simplified0.4

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

    if -2.4965764062960356 < x < 2.4905165381526166

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.4965764062960356:\\ \;\;\;\;\left(\left(\frac{\frac{\sin x}{\cos x}}{x} - \frac{\sin x}{x}\right) - \left(-1 + \frac{\sin x}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x}\right)\right) + \frac{\frac{\sin x}{\cos x}}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x}\\ \mathbf{elif}\;x \le 2.4905165381526166:\\ \;\;\;\;\frac{9}{40} \cdot \left(x \cdot x\right) - \left(\frac{1}{2} + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{27}{2800}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{\frac{\sin x}{\cos x}}{x} - \frac{\sin x}{x}\right) - \left(-1 + \frac{\sin x}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x}\right)\right) + \frac{\frac{\sin x}{\cos x}}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019104 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))