Average Error: 31.0 → 0.2
Time: 12.3s
Precision: binary64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -4.728769858289019:\\ \;\;\;\;\left(1 + \frac{\sin x}{x \cdot \cos x}\right) - \frac{\sin x}{x}\\ \mathbf{elif}\;x \leq 0.02883492941991452:\\ \;\;\;\;\left(-0.5 + 0.225 \cdot \left(x \cdot x\right)\right) - 0.009642857142857142 \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -4.728769858289019:\\
\;\;\;\;\left(1 + \frac{\sin x}{x \cdot \cos x}\right) - \frac{\sin x}{x}\\

\mathbf{elif}\;x \leq 0.02883492941991452:\\
\;\;\;\;\left(-0.5 + 0.225 \cdot \left(x \cdot x\right)\right) - 0.009642857142857142 \cdot {x}^{4}\\

\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 -4.728769858289019)
   (- (+ 1.0 (/ (sin x) (* x (cos x)))) (/ (sin x) x))
   (if (<= x 0.02883492941991452)
     (- (+ -0.5 (* 0.225 (* x x))) (* 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 <= -4.728769858289019) {
		tmp = (1.0 + (sin(x) / (x * cos(x)))) - (sin(x) / x);
	} else if (x <= 0.02883492941991452) {
		tmp = (-0.5 + (0.225 * (x * x))) - (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 < -4.72876985828901919

    1. Initial program 0.0

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

    2. Taylor expanded around inf 0.6

      \[\leadsto \color{blue}{\left(\frac{\sin x}{x \cdot \cos x} + 1\right) - \frac{\sin x}{x}}\]

    3. Simplified0.6

      \[\leadsto \color{blue}{\left(1 + \frac{\sin x}{x \cdot \cos x}\right) - \frac{\sin x}{x}}\]

    if -4.72876985828901919 < x < 0.0288349294199145186

    1. Initial program 63.0

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

    2. Taylor expanded around 0 0.1

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

    3. Simplified0.1

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

    5. Applied associate--r+_binary640.1

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

    6. Simplified0.1

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

    if 0.0288349294199145186 < 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 simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.728769858289019:\\ \;\;\;\;\left(1 + \frac{\sin x}{x \cdot \cos x}\right) - \frac{\sin x}{x}\\ \mathbf{elif}\;x \leq 0.02883492941991452:\\ \;\;\;\;\left(-0.5 + 0.225 \cdot \left(x \cdot x\right)\right) - 0.009642857142857142 \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))