Average Error: 31.1 → 0.4
Time: 8.9s
Precision: binary64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -0.09296178271628956:\\ \;\;\;\;\frac{1}{\frac{x - \tan x}{x - \sin x}}\\ \mathbf{elif}\;x \leq 4.527337310895144:\\ \;\;\;\;0.00024107142857142857 \cdot {x}^{6} + \left(\left(0.225 \cdot \left(x \cdot x\right) - 0.5\right) - 0.009642857142857142 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -0.09296178271628956:\\
\;\;\;\;\frac{1}{\frac{x - \tan x}{x - \sin x}}\\

\mathbf{elif}\;x \leq 4.527337310895144:\\
\;\;\;\;0.00024107142857142857 \cdot {x}^{6} + \left(\left(0.225 \cdot \left(x \cdot x\right) - 0.5\right) - 0.009642857142857142 \cdot {x}^{4}\right)\\

\mathbf{else}:\\
\;\;\;\;1\\

\end{array}
(FPCore (x) :precision binary64 (/ (- x (sin x)) (- x (tan x))))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -0.09296178271628956)
   (/ 1.0 (/ (- x (tan x)) (- x (sin x))))
   (if (<= x 4.527337310895144)
     (+
      (* 0.00024107142857142857 (pow x 6.0))
      (- (- (* 0.225 (* x x)) 0.5) (* 0.009642857142857142 (pow x 4.0))))
     1.0)))
double code(double x) {
	return (x - sin(x)) / (x - tan(x));
}
↓
double code(double x) {
	double tmp;
	if (x <= -0.09296178271628956) {
		tmp = 1.0 / ((x - tan(x)) / (x - sin(x)));
	} else if (x <= 4.527337310895144) {
		tmp = (0.00024107142857142857 * pow(x, 6.0)) + (((0.225 * (x * x)) - 0.5) - (0.009642857142857142 * pow(x, 4.0)));
	} else {
		tmp = 1.0;
	}
	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 3 regimes

  2. if x < -0.0929617827162895621

    1. Initial program 0.0

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

    3. Applied clear-num_binary640.1

      \[\leadsto \color{blue}{\frac{1}{\frac{x - \tan x}{x - \sin x}}}\]

    if -0.0929617827162895621 < x < 4.5273373108951436

    1. Initial program 62.9

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

    2. Taylor expanded around 0 0.1

      \[\leadsto \color{blue}{\left(0.225 \cdot {x}^{2} + 0.00024107142857142857 \cdot {x}^{6}\right) - \left(0.009642857142857142 \cdot {x}^{4} + 0.5\right)}\]

    3. Simplified0.1

      \[\leadsto \color{blue}{0.00024107142857142857 \cdot {x}^{6} + \left(0.225 \cdot \left(x \cdot x\right) - \left(0.5 + 0.009642857142857142 \cdot {x}^{4}\right)\right)}\]
    4. Using strategy rm

    5. Applied associate--r+_binary640.1

      \[\leadsto 0.00024107142857142857 \cdot {x}^{6} + \color{blue}{\left(\left(0.225 \cdot \left(x \cdot x\right) - 0.5\right) - 0.009642857142857142 \cdot {x}^{4}\right)}\]

    if 4.5273373108951436 < x

    1. Initial program 0.0

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

    2. Taylor expanded around inf 1.1

      \[\leadsto \color{blue}{1}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.09296178271628956:\\ \;\;\;\;\frac{1}{\frac{x - \tan x}{x - \sin x}}\\ \mathbf{elif}\;x \leq 4.527337310895144:\\ \;\;\;\;0.00024107142857142857 \cdot {x}^{6} + \left(\left(0.225 \cdot \left(x \cdot x\right) - 0.5\right) - 0.009642857142857142 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Reproduce

herbie shell --seed 2021176 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))