Average Error: 31.2 → 0.0
Time: 10.0s
Precision: binary64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -0.0302404303590001:\\ \;\;\;\;\sqrt[3]{\frac{x - \sin x}{x - \tan x}} \cdot \left(\sqrt[3]{\frac{x - \sin x}{x - \tan x}} \cdot \sqrt[3]{\frac{x - \sin x}{x - \tan x}}\right)\\ \mathbf{elif}\;x \leq 0.026672942658527293:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.225 - \left(0.5 + 0.009642857142857142 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{x - \sin x}{x - \tan x}}\right)\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -0.0302404303590001:\\
\;\;\;\;\sqrt[3]{\frac{x - \sin x}{x - \tan x}} \cdot \left(\sqrt[3]{\frac{x - \sin x}{x - \tan x}} \cdot \sqrt[3]{\frac{x - \sin x}{x - \tan x}}\right)\\

\mathbf{elif}\;x \leq 0.026672942658527293:\\
\;\;\;\;\left(x \cdot x\right) \cdot 0.225 - \left(0.5 + 0.009642857142857142 \cdot {x}^{4}\right)\\

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

\end{array}
(FPCore (x) :precision binary64 (/ (- x (sin x)) (- x (tan x))))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -0.0302404303590001)
   (*
    (cbrt (/ (- x (sin x)) (- x (tan x))))
    (*
     (cbrt (/ (- x (sin x)) (- x (tan x))))
     (cbrt (/ (- x (sin x)) (- x (tan x))))))
   (if (<= x 0.026672942658527293)
     (- (* (* x x) 0.225) (+ 0.5 (* 0.009642857142857142 (pow x 4.0))))
     (log (exp (/ (- x (sin x)) (- x (tan x))))))))
double code(double x) {
	return (x - sin(x)) / (x - tan(x));
}
↓
double code(double x) {
	double tmp;
	if (x <= -0.0302404303590001) {
		tmp = cbrt((x - sin(x)) / (x - tan(x))) * (cbrt((x - sin(x)) / (x - tan(x))) * cbrt((x - sin(x)) / (x - tan(x))));
	} else if (x <= 0.026672942658527293) {
		tmp = ((x * x) * 0.225) - (0.5 + (0.009642857142857142 * pow(x, 4.0)));
	} else {
		tmp = log(exp((x - sin(x)) / (x - tan(x))));
	}
	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 3 regimes

  2. if x < -0.030240430359000099

    1. Initial program 0.0

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

    3. Applied add-cube-cbrt_binary640.1

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{x - \sin x}{x - \tan x}} \cdot \sqrt[3]{\frac{x - \sin x}{x - \tan x}}\right) \cdot \sqrt[3]{\frac{x - \sin x}{x - \tan x}}}\]

    if -0.030240430359000099 < x < 0.026672942658527293

    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

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

    if 0.026672942658527293 < x

    1. Initial program 0.1

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

    3. Applied add-log-exp_binary640.1

      \[\leadsto \color{blue}{\log \left(e^{\frac{x - \sin x}{x - \tan x}}\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0302404303590001:\\ \;\;\;\;\sqrt[3]{\frac{x - \sin x}{x - \tan x}} \cdot \left(\sqrt[3]{\frac{x - \sin x}{x - \tan x}} \cdot \sqrt[3]{\frac{x - \sin x}{x - \tan x}}\right)\\ \mathbf{elif}\;x \leq 0.026672942658527293:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.225 - \left(0.5 + 0.009642857142857142 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{x - \sin x}{x - \tan x}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020280 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))