Average Error: 31.5 → 0.1
Time: 14.9s
Precision: binary64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -1.5604285908945796:\\ \;\;\;\;\sqrt{\frac{x - \sin x}{x - \tan x}} \cdot \sqrt{\frac{x - \sin x}{x - \tan x}}\\ \mathbf{elif}\;x \leq 0.0363653530648353:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.225 - \left(0.5 + 0.009642857142857142 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -1.5604285908945796:\\
\;\;\;\;\sqrt{\frac{x - \sin x}{x - \tan x}} \cdot \sqrt{\frac{x - \sin x}{x - \tan x}}\\

\mathbf{elif}\;x \leq 0.0363653530648353:\\
\;\;\;\;\left(x \cdot x\right) \cdot 0.225 - \left(0.5 + 0.009642857142857142 \cdot {x}^{4}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\

\end{array}
(FPCore (x) :precision binary64 (/ (- x (sin x)) (- x (tan x))))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -1.5604285908945796)
   (*
    (sqrt (/ (- x (sin x)) (- x (tan x))))
    (sqrt (/ (- x (sin x)) (- x (tan x)))))
   (if (<= x 0.0363653530648353)
     (- (* (* x x) 0.225) (+ 0.5 (* 0.009642857142857142 (pow x 4.0))))
     (- (/ x (- x (tan 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 <= -1.5604285908945796) {
		tmp = sqrt((x - sin(x)) / (x - tan(x))) * sqrt((x - sin(x)) / (x - tan(x)));
	} else if (x <= 0.0363653530648353) {
		tmp = ((x * x) * 0.225) - (0.5 + (0.009642857142857142 * pow(x, 4.0)));
	} else {
		tmp = (x / (x - tan(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 < -1.56042859089457964

    1. Initial program 0.0

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

    3. Applied add-sqr-sqrt_binary640.0

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

    if -1.56042859089457964 < x < 0.0363653530648352999

    1. Initial program 63.1

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

    2. Taylor expanded around 0 0.1

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

    3. Simplified0.1

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

    if 0.0363653530648352999 < x

    1. Initial program 0.0

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

    3. Applied div-sub_binary640.0

      \[\leadsto \color{blue}{\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5604285908945796:\\ \;\;\;\;\sqrt{\frac{x - \sin x}{x - \tan x}} \cdot \sqrt{\frac{x - \sin x}{x - \tan x}}\\ \mathbf{elif}\;x \leq 0.0363653530648353:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.225 - \left(0.5 + 0.009642857142857142 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020295 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))