Average Error: 31.7 → 0.0
Time: 13.3s
Precision: binary64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -0.09316693960348439 \lor \neg \left(x \leq 0.10427973266118423\right):\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;0.00024107142857142857 \cdot {x}^{6} + \left(0.225 \cdot \left(x \cdot x\right) - \left(0.5 + 0.009642857142857142 \cdot {x}^{4}\right)\right)\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -0.09316693960348439 \lor \neg \left(x \leq 0.10427973266118423\right):\\
\;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\

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

\end{array}
(FPCore (x) :precision binary64 (/ (- x (sin x)) (- x (tan x))))
↓
(FPCore (x)
 :precision binary64
 (if (or (<= x -0.09316693960348439) (not (<= x 0.10427973266118423)))
   (- (/ x (- x (tan x))) (/ (sin x) (- x (tan x))))
   (+
    (* 0.00024107142857142857 (pow x 6.0))
    (- (* 0.225 (* x x)) (+ 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 <= -0.09316693960348439) || !(x <= 0.10427973266118423)) {
		tmp = (x / (x - tan(x))) - (sin(x) / (x - tan(x)));
	} else {
		tmp = (0.00024107142857142857 * pow(x, 6.0)) + ((0.225 * (x * x)) - (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 < -0.0931669396034843916 or 0.10427973266118423 < x

    1. Initial program 0.0

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

    3. Applied div-sub_binary640.0

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

    if -0.0931669396034843916 < x < 0.10427973266118423

    1. Initial program 63.0

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

    2. Taylor expanded around 0 0.0

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

    3. Simplified0.0

      \[\leadsto \color{blue}{0.00024107142857142857 \cdot {x}^{6} + \left(0.225 \cdot \left(x \cdot x\right) - \left(0.5 + 0.009642857142857142 \cdot {x}^{4}\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.09316693960348439 \lor \neg \left(x \leq 0.10427973266118423\right):\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;0.00024107142857142857 \cdot {x}^{6} + \left(0.225 \cdot \left(x \cdot x\right) - \left(0.5 + 0.009642857142857142 \cdot {x}^{4}\right)\right)\\ \end{array}\]

Alternatives

Reproduce

herbie shell --seed 2021098 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))