Average Error: 30.9 → 0.0
Time: 11.0s
Precision: binary64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -0.029010316284022238 \lor \neg \left(x \leq 0.027954977951543752\right):\\ \;\;\;\;\frac{1}{\frac{x}{x - \sin x} - \frac{\tan x}{x - \sin x}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.225 - \left(0.5 + 0.009642857142857142 \cdot {x}^{4}\right)\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -0.029010316284022238 \lor \neg \left(x \leq 0.027954977951543752\right):\\
\;\;\;\;\frac{1}{\frac{x}{x - \sin x} - \frac{\tan x}{x - \sin x}}\\

\mathbf{else}:\\
\;\;\;\;\left(x \cdot x\right) \cdot 0.225 - \left(0.5 + 0.009642857142857142 \cdot {x}^{4}\right)\\

\end{array}
(FPCore (x) :precision binary64 (/ (- x (sin x)) (- x (tan x))))
↓
(FPCore (x)
 :precision binary64
 (if (or (<= x -0.029010316284022238) (not (<= x 0.027954977951543752)))
   (/ 1.0 (- (/ x (- x (sin x))) (/ (tan x) (- x (sin x)))))
   (- (* (* x x) 0.225) (+ 0.5 (* 0.009642857142857142 (pow x 4.0))))))
double code(double x) {
	return (x - sin(x)) / (x - tan(x));
}
↓
double code(double x) {
	double tmp;
	if ((x <= -0.029010316284022238) || !(x <= 0.027954977951543752)) {
		tmp = 1.0 / ((x / (x - sin(x))) - (tan(x) / (x - sin(x))));
	} else {
		tmp = ((x * x) * 0.225) - (0.5 + (0.009642857142857142 * pow(x, 4.0)));
	}
	return tmp;
}

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 < -0.029010316284022238 or 0.027954977951543752 < x

    1. Initial program 0.0

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

    3. Applied clear-num_binary640.0

      \[\leadsto \color{blue}{\frac{1}{\frac{x - \tan x}{x - \sin x}}}\]
    4. Using strategy rm

    5. Applied div-sub_binary640.0

      \[\leadsto \frac{1}{\color{blue}{\frac{x}{x - \sin x} - \frac{\tan x}{x - \sin x}}}\]

    if -0.029010316284022238 < x < 0.027954977951543752

    1. Initial program 63.1

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

    2. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{0.225 \cdot {x}^{2} - \left(0.5 + 0.009642857142857142 \cdot {x}^{4}\right)}\]

    3. Simplified0.0

      \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot 0.225 - \left(0.5 + 0.009642857142857142 \cdot {x}^{4}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.029010316284022238 \lor \neg \left(x \leq 0.027954977951543752\right):\\ \;\;\;\;\frac{1}{\frac{x}{x - \sin x} - \frac{\tan x}{x - \sin x}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.225 - \left(0.5 + 0.009642857142857142 \cdot {x}^{4}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020281 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))