Average Error: 31.6 → 0.0
Time: 10.7s
Precision: binary64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -0.036947823895944455:\\ \;\;\;\;\frac{x - \sin x}{x - \sqrt[3]{\tan x} \cdot \left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right)}\\ \mathbf{elif}\;x \leq 0.03205460313101732:\\ \;\;\;\;\log \left(e^{0.225 \cdot \left(x \cdot x\right)}\right) - \left(0.5 + 0.009642857142857142 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -0.036947823895944455:\\
\;\;\;\;\frac{x - \sin x}{x - \sqrt[3]{\tan x} \cdot \left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right)}\\

\mathbf{elif}\;x \leq 0.03205460313101732:\\
\;\;\;\;\log \left(e^{0.225 \cdot \left(x \cdot x\right)}\right) - \left(0.5 + 0.009642857142857142 \cdot {x}^{4}\right)\\

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

\end{array}
(FPCore (x) :precision binary64 (/ (- x (sin x)) (- x (tan x))))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -0.036947823895944455)
   (/ (- x (sin x)) (- x (* (cbrt (tan x)) (* (cbrt (tan x)) (cbrt (tan x))))))
   (if (<= x 0.03205460313101732)
     (-
      (log (exp (* 0.225 (* x x))))
      (+ 0.5 (* 0.009642857142857142 (pow x 4.0))))
     (- (/ x (- x (tan 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 <= -0.036947823895944455) {
		tmp = (x - sin(x)) / (x - (cbrt(tan(x)) * (cbrt(tan(x)) * cbrt(tan(x)))));
	} else if (x <= 0.03205460313101732) {
		tmp = log(exp(0.225 * (x * x))) - (0.5 + (0.009642857142857142 * pow(x, 4.0)));
	} else {
		tmp = (x / (x - tan(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 3 regimes

  2. if x < -0.0369478238959444552

    1. Initial program 0.1

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

    3. Applied add-cube-cbrt_binary640.1

      \[\leadsto \frac{x - \sin x}{x - \color{blue}{\left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right) \cdot \sqrt[3]{\tan x}}}\]

    4. Applied cancel-sign-sub-inv_binary640.1

      \[\leadsto \frac{x - \sin x}{\color{blue}{x + \left(-\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right) \cdot \sqrt[3]{\tan x}}}\]

    if -0.0369478238959444552 < x < 0.032054603131017323

    1. Initial program 63.0

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

    2. Taylor expanded around 0 0.0

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

    3. Simplified0.0

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

    5. Applied add-log-exp_binary640.0

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

    if 0.032054603131017323 < x

    1. Initial program 0.0

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

    3. Applied div-sub_binary640.0

      \[\leadsto \color{blue}{\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.036947823895944455:\\ \;\;\;\;\frac{x - \sin x}{x - \sqrt[3]{\tan x} \cdot \left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right)}\\ \mathbf{elif}\;x \leq 0.03205460313101732:\\ \;\;\;\;\log \left(e^{0.225 \cdot \left(x \cdot x\right)}\right) - \left(0.5 + 0.009642857142857142 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \end{array}\]

Reproduce

herbie shell --seed 2021045 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))