Average Error: 31.2 → 0.0
Time: 9.3s
Precision: binary64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -0.033413605152308526 \lor \neg \left(x \leq 0.03654414618884699\right):\\ \;\;\;\;\frac{x - \sin x}{x - \sqrt[3]{\tan x} \cdot \left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.225 \cdot {x}^{2} - \left(0.009642857142857142 \cdot {x}^{4} + 0.5\right)\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -0.033413605152308526 \lor \neg \left(x \leq 0.03654414618884699\right):\\
\;\;\;\;\frac{x - \sin x}{x - \sqrt[3]{\tan x} \cdot \left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right)}\\

\mathbf{else}:\\
\;\;\;\;0.225 \cdot {x}^{2} - \left(0.009642857142857142 \cdot {x}^{4} + 0.5\right)\\

\end{array}
(FPCore (x) :precision binary64 (/ (- x (sin x)) (- x (tan x))))
↓
(FPCore (x)
 :precision binary64
 (if (or (<= x -0.033413605152308526) (not (<= x 0.03654414618884699)))
   (/ (- x (sin x)) (- x (* (cbrt (tan x)) (* (cbrt (tan x)) (cbrt (tan x))))))
   (- (* 0.225 (pow x 2.0)) (+ (* 0.009642857142857142 (pow x 4.0)) 0.5))))
double code(double x) {
	return (x - sin(x)) / (x - tan(x));
}
↓
double code(double x) {
	double tmp;
	if ((x <= -0.033413605152308526) || !(x <= 0.03654414618884699)) {
		tmp = (x - sin(x)) / (x - (cbrt(tan(x)) * (cbrt(tan(x)) * cbrt(tan(x)))));
	} else {
		tmp = (0.225 * pow(x, 2.0)) - ((0.009642857142857142 * pow(x, 4.0)) + 0.5);
	}
	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 < -0.033413605152308526 or 0.036544146188846993 < x

    1. Initial program 0.0

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

    3. Applied add-cube-cbrt_binary640.1

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

    4. Applied cancel-sign-sub-inv_binary640.1

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

    if -0.033413605152308526 < x < 0.036544146188846993

    1. Initial program 63.1

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

    2. Taylor expanded around 0 0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.033413605152308526 \lor \neg \left(x \leq 0.03654414618884699\right):\\ \;\;\;\;\frac{x - \sin x}{x - \sqrt[3]{\tan x} \cdot \left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.225 \cdot {x}^{2} - \left(0.009642857142857142 \cdot {x}^{4} + 0.5\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021015 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))