Average Error: 31.5 → 0.0
Time: 10.2s
Precision: binary64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -0.028939001950817302:\\ \;\;\;\;\sqrt[3]{{\left(\frac{x - \sin x}{x - \tan x}\right)}^{3}}\\ \mathbf{elif}\;x \leq 0.02800543700756853:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.225 - \left(0.009642857142857142 \cdot {x}^{4} + 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x - \tan x}{x - \sin x}}\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -0.028939001950817302:\\
\;\;\;\;\sqrt[3]{{\left(\frac{x - \sin x}{x - \tan x}\right)}^{3}}\\

\mathbf{elif}\;x \leq 0.02800543700756853:\\
\;\;\;\;\left(x \cdot x\right) \cdot 0.225 - \left(0.009642857142857142 \cdot {x}^{4} + 0.5\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\frac{x - \tan x}{x - \sin x}}\\

\end{array}
(FPCore (x) :precision binary64 (/ (- x (sin x)) (- x (tan x))))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -0.028939001950817302)
   (cbrt (pow (/ (- x (sin x)) (- x (tan x))) 3.0))
   (if (<= x 0.02800543700756853)
     (- (* (* x x) 0.225) (+ (* 0.009642857142857142 (pow x 4.0)) 0.5))
     (/ 1.0 (/ (- x (tan x)) (- x (sin x)))))))
double code(double x) {
	return (((double) (x - ((double) sin(x)))) / ((double) (x - ((double) tan(x)))));
}
↓
double code(double x) {
	double tmp;
	if ((x <= -0.028939001950817302)) {
		tmp = ((double) cbrt(((double) pow((((double) (x - ((double) sin(x)))) / ((double) (x - ((double) tan(x))))), 3.0))));
	} else {
		double tmp_1;
		if ((x <= 0.02800543700756853)) {
			tmp_1 = ((double) (((double) (((double) (x * x)) * 0.225)) - ((double) (((double) (0.009642857142857142 * ((double) pow(x, 4.0)))) + 0.5))));
		} else {
			tmp_1 = (1.0 / (((double) (x - ((double) tan(x)))) / ((double) (x - ((double) sin(x))))));
		}
		tmp = tmp_1;
	}
	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 < -0.0289390019508173023

    1. Initial program 0.1

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

    3. Applied add-cbrt-cube_binary6441.4

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

    4. Applied add-cbrt-cube_binary6442.6

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

    5. Applied cbrt-undiv_binary6442.6

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

    6. Simplified0.1

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

    if -0.0289390019508173023 < x < 0.02800543700756853

    1. Initial program 63.1

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

    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

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

    if 0.02800543700756853 < x

    1. Initial program 0.0

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

    3. Applied clear-num_binary640.0

      \[\leadsto \color{blue}{\frac{1}{\frac{x - \tan x}{x - \sin x}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.028939001950817302:\\ \;\;\;\;\sqrt[3]{{\left(\frac{x - \sin x}{x - \tan x}\right)}^{3}}\\ \mathbf{elif}\;x \leq 0.02800543700756853:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.225 - \left(0.009642857142857142 \cdot {x}^{4} + 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x - \tan x}{x - \sin x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020219 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))