Average Error: 31.7 → 0.0
Time: 9.1s
Precision: 64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.030047705489404199 \lor \neg \left(x \le 0.0286594947398211321\right):\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;\frac{9}{40} \cdot {x}^{2} - \left(\log \left(e^{\frac{27}{2800} \cdot {x}^{4}}\right) + \frac{1}{2}\right)\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.030047705489404199 \lor \neg \left(x \le 0.0286594947398211321\right):\\
\;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\

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

\end{array}
double code(double x) {
	return ((x - sin(x)) / (x - tan(x)));
}
↓
double code(double x) {
	double VAR;
	if (((x <= -0.0300477054894042) || !(x <= 0.028659494739821132))) {
		VAR = ((x / (x - tan(x))) - (sin(x) / (x - tan(x))));
	} else {
		VAR = ((0.225 * pow(x, 2.0)) - (log(exp((0.009642857142857142 * pow(x, 4.0)))) + 0.5));
	}
	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 < -0.0300477054894042 or 0.028659494739821132 < x

    1. Initial program 0.0

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

    3. Applied div-sub0.1

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

    if -0.0300477054894042 < x < 0.028659494739821132

    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 add-log-exp0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.030047705489404199 \lor \neg \left(x \le 0.0286594947398211321\right):\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;\frac{9}{40} \cdot {x}^{2} - \left(\log \left(e^{\frac{27}{2800} \cdot {x}^{4}}\right) + \frac{1}{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020092 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))