Average Error: 31.9 → 0.0
Time: 9.8s
Precision: binary64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -0.027071839411171916 \lor \neg \left(x \leq 0.02695355388948877\right):\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot 0.225 - 0.5\right) - 0.009642857142857142 \cdot {x}^{4}\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -0.027071839411171916 \lor \neg \left(x \leq 0.02695355388948877\right):\\
\;\;\;\;\frac{x - \sin x}{x - \tan x}\\

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

\end{array}
(FPCore (x) :precision binary64 (/ (- x (sin x)) (- x (tan x))))
↓
(FPCore (x)
 :precision binary64
 (if (or (<= x -0.027071839411171916) (not (<= x 0.02695355388948877)))
   (/ (- x (sin x)) (- x (tan 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 <= -0.027071839411171916) || !(x <= 0.02695355388948877)) {
		tmp = (x - sin(x)) / (x - tan(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 < -0.027071839411171916 or 0.026953553889488769 < x

    1. Initial program 0.0

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

    if -0.027071839411171916 < x < 0.026953553889488769

    1. Initial program 63.2

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

    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

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

    5. Applied associate--r+_binary640.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.027071839411171916 \lor \neg \left(x \leq 0.02695355388948877\right):\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot 0.225 - 0.5\right) - 0.009642857142857142 \cdot {x}^{4}\\ \end{array}\]

Reproduce

herbie shell --seed 2020277 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))