Average Error: 31.2 → 0.0
Time: 39.6s
Precision: 64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.02687365489421502:\\ \;\;\;\;\log \left(\sqrt{e^{\frac{x - \sin x}{x - \tan x}}}\right) + \log \left(\sqrt{e^{\frac{x - \sin x}{x - \tan x}}}\right)\\ \mathbf{elif}\;x \le 0.026232161789625133:\\ \;\;\;\;\frac{9}{40} \cdot \left(x \cdot x\right) - \left(\frac{1}{2} + \frac{27}{2800} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{e^{\frac{x - \sin x}{x - \tan x}}}\right) + \log \left(\sqrt{e^{\frac{x - \sin x}{x - \tan x}}}\right)\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.02687365489421502:\\
\;\;\;\;\log \left(\sqrt{e^{\frac{x - \sin x}{x - \tan x}}}\right) + \log \left(\sqrt{e^{\frac{x - \sin x}{x - \tan x}}}\right)\\

\mathbf{elif}\;x \le 0.026232161789625133:\\
\;\;\;\;\frac{9}{40} \cdot \left(x \cdot x\right) - \left(\frac{1}{2} + \frac{27}{2800} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(\sqrt{e^{\frac{x - \sin x}{x - \tan x}}}\right) + \log \left(\sqrt{e^{\frac{x - \sin x}{x - \tan x}}}\right)\\

\end{array}
double f(double x) {
        double r889526 = x;
        double r889527 = sin(r889526);
        double r889528 = r889526 - r889527;
        double r889529 = tan(r889526);
        double r889530 = r889526 - r889529;
        double r889531 = r889528 / r889530;
        return r889531;
}
↓
double f(double x) {
        double r889532 = x;
        double r889533 = -0.02687365489421502;
        bool r889534 = r889532 <= r889533;
        double r889535 = sin(r889532);
        double r889536 = r889532 - r889535;
        double r889537 = tan(r889532);
        double r889538 = r889532 - r889537;
        double r889539 = r889536 / r889538;
        double r889540 = exp(r889539);
        double r889541 = sqrt(r889540);
        double r889542 = log(r889541);
        double r889543 = r889542 + r889542;
        double r889544 = 0.026232161789625133;
        bool r889545 = r889532 <= r889544;
        double r889546 = 0.225;
        double r889547 = r889532 * r889532;
        double r889548 = r889546 * r889547;
        double r889549 = 0.5;
        double r889550 = 0.009642857142857142;
        double r889551 = r889547 * r889547;
        double r889552 = r889550 * r889551;
        double r889553 = r889549 + r889552;
        double r889554 = r889548 - r889553;
        double r889555 = r889545 ? r889554 : r889543;
        double r889556 = r889534 ? r889543 : r889555;
        return r889556;
}

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 < -0.02687365489421502 or 0.026232161789625133 < x

    1. Initial program 0.0

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

    3. Applied add-log-exp0.1

      \[\leadsto \color{blue}{\log \left(e^{\frac{x - \sin x}{x - \tan x}}\right)}\]
    4. Using strategy rm

    5. Applied add-sqr-sqrt0.1

      \[\leadsto \log \color{blue}{\left(\sqrt{e^{\frac{x - \sin x}{x - \tan x}}} \cdot \sqrt{e^{\frac{x - \sin x}{x - \tan x}}}\right)}\]

    6. Applied log-prod0.1

      \[\leadsto \color{blue}{\log \left(\sqrt{e^{\frac{x - \sin x}{x - \tan x}}}\right) + \log \left(\sqrt{e^{\frac{x - \sin x}{x - \tan x}}}\right)}\]

    if -0.02687365489421502 < x < 0.026232161789625133

    1. Initial program 62.8

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

    2. Taylor expanded around 0 0.0

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

    3. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.02687365489421502:\\ \;\;\;\;\log \left(\sqrt{e^{\frac{x - \sin x}{x - \tan x}}}\right) + \log \left(\sqrt{e^{\frac{x - \sin x}{x - \tan x}}}\right)\\ \mathbf{elif}\;x \le 0.026232161789625133:\\ \;\;\;\;\frac{9}{40} \cdot \left(x \cdot x\right) - \left(\frac{1}{2} + \frac{27}{2800} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{e^{\frac{x - \sin x}{x - \tan x}}}\right) + \log \left(\sqrt{e^{\frac{x - \sin x}{x - \tan x}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019121 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))