Average Error: 31.8 → 0.1
Time: 9.1s
Precision: binary64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -0.0055547518031396894:\\ \;\;\;\;\log \left(e^{\frac{x - \sin x}{x - \tan x}}\right)\\ \mathbf{elif}\;x \leq 0.004611415987775008:\\ \;\;\;\;-0.5 + \left(x \cdot x\right) \cdot 0.225\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -0.0055547518031396894:\\
\;\;\;\;\log \left(e^{\frac{x - \sin x}{x - \tan x}}\right)\\

\mathbf{elif}\;x \leq 0.004611415987775008:\\
\;\;\;\;-0.5 + \left(x \cdot x\right) \cdot 0.225\\

\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 -0.0055547518031396894)
   (log (exp (/ (- x (sin x)) (- x (tan x)))))
   (if (<= x 0.004611415987775008)
     (+ -0.5 (* (* x x) 0.225))
     (- (/ 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 <= -0.0055547518031396894) {
		tmp = log(exp((x - sin(x)) / (x - tan(x))));
	} else if (x <= 0.004611415987775008) {
		tmp = -0.5 + ((x * x) * 0.225);
	} 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 < -0.00555475180313968944

    1. Initial program 0.1

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

    3. Applied add-log-exp_binary640.1

      \[\leadsto \color{blue}{\log \left(e^{\frac{x - \sin x}{x - \tan x}}\right)}\]

    if -0.00555475180313968944 < x < 0.004611415987775008

    1. Initial program 63.2

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

    2. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{0.225 \cdot {x}^{2} - 0.5}\]

    3. Simplified0.0

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

    if 0.004611415987775008 < x

    1. Initial program 0.1

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

    3. Applied div-sub_binary640.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0055547518031396894:\\ \;\;\;\;\log \left(e^{\frac{x - \sin x}{x - \tan x}}\right)\\ \mathbf{elif}\;x \leq 0.004611415987775008:\\ \;\;\;\;-0.5 + \left(x \cdot x\right) \cdot 0.225\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020344 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))