Average Error: 31.0 → 0.0
Time: 10.5s
Precision: binary64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -0.028423004362203453 \lor \neg \left(x \leq 0.027512049940656768\right):\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.225 - \left(0.5 + 0.009642857142857142 \cdot {x}^{4}\right)\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -0.028423004362203453 \lor \neg \left(x \leq 0.027512049940656768\right):\\
\;\;\;\;\frac{x - \sin x}{x - \tan x}\\

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

\end{array}
(FPCore (x) :precision binary64 (/ (- x (sin x)) (- x (tan x))))
↓
(FPCore (x)
 :precision binary64
 (if (or (<= x -0.028423004362203453) (not (<= x 0.027512049940656768)))
   (/ (- 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 <= -0.028423004362203453) || !(x <= 0.027512049940656768)) {
		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 < -0.028423004362203453 or 0.027512049940656768 < x

    1. Initial program 0.0

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

    if -0.028423004362203453 < x < 0.027512049940656768

    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

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.028423004362203453 \lor \neg \left(x \leq 0.027512049940656768\right):\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.225 - \left(0.5 + 0.009642857142857142 \cdot {x}^{4}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020253 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))