Average Error: 31.6 → 0.1
Time: 9.5s
Precision: binary64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -1.5678631412367456 \lor \neg \left(x \leq 1.580616986681567\right):\\ \;\;\;\;\sqrt{\frac{x - \sin x}{x - \tan x}} \cdot \sqrt{\frac{x - \sin x}{x - \tan x}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.225 - \left(0.009642857142857142 \cdot {x}^{4} + 0.5\right)\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -1.5678631412367456 \lor \neg \left(x \leq 1.580616986681567\right):\\
\;\;\;\;\sqrt{\frac{x - \sin x}{x - \tan x}} \cdot \sqrt{\frac{x - \sin x}{x - \tan x}}\\

\mathbf{else}:\\
\;\;\;\;\left(x \cdot x\right) \cdot 0.225 - \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 -1.5678631412367456) (not (<= x 1.580616986681567)))
   (*
    (sqrt (/ (- x (sin x)) (- x (tan x))))
    (sqrt (/ (- x (sin x)) (- x (tan x)))))
   (- (* (* x x) 0.225) (+ (* 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 <= -1.5678631412367456) || !(x <= 1.580616986681567)) {
		tmp = sqrt((x - sin(x)) / (x - tan(x))) * sqrt((x - sin(x)) / (x - tan(x)));
	} else {
		tmp = ((x * x) * 0.225) - ((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 < -1.56786314123674564 or 1.58061698668156692 < x

    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 -1.56786314123674564 < x < 1.58061698668156692

    1. Initial program 62.8

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

    2. Taylor expanded around 0 0.2

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

    3. Simplified0.2

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5678631412367456 \lor \neg \left(x \leq 1.580616986681567\right):\\ \;\;\;\;\sqrt{\frac{x - \sin x}{x - \tan x}} \cdot \sqrt{\frac{x - \sin x}{x - \tan x}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.225 - \left(0.009642857142857142 \cdot {x}^{4} + 0.5\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020224 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))