Average Error: 31.9 → 0.0
Time: 9.7s
Precision: binary64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -0.023947268971373566:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{elif}\;x \leq 0.029551278348766707:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot 0.225 - 0.5\right) - 0.009642857142857142 \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -0.023947268971373566:\\
\;\;\;\;\frac{x - \sin x}{x - \tan x}\\

\mathbf{elif}\;x \leq 0.029551278348766707:\\
\;\;\;\;\left(\left(x \cdot x\right) \cdot 0.225 - 0.5\right) - 0.009642857142857142 \cdot {x}^{4}\\

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

\end{array}
(FPCore (x) :precision binary64 (/ (- x (sin x)) (- x (tan x))))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -0.023947268971373566)
   (/ (- x (sin x)) (- x (tan x)))
   (if (<= x 0.029551278348766707)
     (- (- (* (* x x) 0.225) 0.5) (* 0.009642857142857142 (pow x 4.0)))
     (/ (- 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 <= -0.023947268971373566) {
		tmp = (x - sin(x)) / (x - tan(x));
	} else if (x <= 0.029551278348766707) {
		tmp = (((x * x) * 0.225) - 0.5) - (0.009642857142857142 * pow(x, 4.0));
	} else {
		tmp = (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 2 regimes

  2. if x < -0.023947268971373566 or 0.0295512783487667068 < x

    1. Initial program 0.0

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

    if -0.023947268971373566 < x < 0.0295512783487667068

    1. Initial program 63.3

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.023947268971373566:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{elif}\;x \leq 0.029551278348766707:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot 0.225 - 0.5\right) - 0.009642857142857142 \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020273 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))