Average Error: 31.6 → 0.0
Time: 8.3s
Precision: 64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0304666040013462057 \lor \neg \left(x \le 0.0255064208901601384\right):\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{9}{40} \cdot {x}^{2} - \frac{1}{2}\right) - \frac{27}{2800} \cdot {x}^{4}\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0304666040013462057 \lor \neg \left(x \le 0.0255064208901601384\right):\\
\;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\

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

\end{array}
double code(double x) {
	return ((x - sin(x)) / (x - tan(x)));
}
↓
double code(double x) {
	double VAR;
	if (((x <= -0.030466604001346206) || !(x <= 0.02550642089016014))) {
		VAR = ((x / (x - tan(x))) - (sin(x) / (x - tan(x))));
	} else {
		VAR = (((0.225 * pow(x, 2.0)) - 0.5) - (0.009642857142857142 * pow(x, 4.0)));
	}
	return VAR;
}

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 < -0.030466604001346206 or 0.02550642089016014 < x

    1. Initial program 0.0

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

    3. Applied div-sub0.0

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

    if -0.030466604001346206 < x < 0.02550642089016014

    1. Initial program 63.1

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

    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. Using strategy rm

    4. Applied +-commutative0.0

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

    5. Applied associate--r+0.0

      \[\leadsto \color{blue}{\left(\frac{9}{40} \cdot {x}^{2} - \frac{1}{2}\right) - \frac{27}{2800} \cdot {x}^{4}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0304666040013462057 \lor \neg \left(x \le 0.0255064208901601384\right):\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{9}{40} \cdot {x}^{2} - \frac{1}{2}\right) - \frac{27}{2800} \cdot {x}^{4}\\ \end{array}\]

Reproduce

herbie shell --seed 2020071 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))