Average Error: 31.2 → 0.5
Time: 11.1s
Precision: binary64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x - \sin x}{x - \tan x} \leq 1.0000001359929322:\\ \;\;\;\;\log \left(e^{\frac{x - \sin x}{x - \tan x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.225 - \left(0.5 + 0.009642857142857142 \cdot {x}^{4}\right)\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x - \sin x}{x - \tan x} \leq 1.0000001359929322:\\
\;\;\;\;\log \left(e^{\frac{x - \sin x}{x - \tan x}}\right)\\

\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 (<= (/ (- x (sin x)) (- x (tan x))) 1.0000001359929322)
   (log (exp (/ (- 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 - sin(x)) / (x - tan(x))) <= 1.0000001359929322) {
		tmp = log(exp((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 (/.f64 (-.f64 x (sin.f64 x)) (-.f64 x (tan.f64 x))) < 1.00000013599293225

    1. Initial program 0.5

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

    3. Applied add-log-exp_binary640.5

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

    if 1.00000013599293225 < (/.f64 (-.f64 x (sin.f64 x)) (-.f64 x (tan.f64 x)))

    1. Initial program 63.5

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

    2. Taylor expanded around 0 0.5

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

    3. Simplified0.5

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \sin x}{x - \tan x} \leq 1.0000001359929322:\\ \;\;\;\;\log \left(e^{\frac{x - \sin x}{x - \tan x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.225 - \left(0.5 + 0.009642857142857142 \cdot {x}^{4}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020288 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))