Average Error: 31.7 → 0.2
Time: 16.6s
Precision: binary64
\[\frac{x - \sin x}{x - \tan x} \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -2.208632844132881:\\ \;\;\;\;\left(\left(1 + \frac{\sin x}{x \cdot \cos x}\right) + \frac{\sin x}{x \cdot x} \cdot \left(\frac{\sin x}{{\cos x}^{2}} - \frac{\sin x}{\cos x}\right)\right) - \frac{\sin x}{x}\\ \mathbf{elif}\;x \leq 0.003965314230899635:\\ \;\;\;\;0.225 \cdot {x}^{2} - 0.5\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{x - \sin x}{x - \tan x}}\right)\\ \end{array} \]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -2.208632844132881:\\
\;\;\;\;\left(\left(1 + \frac{\sin x}{x \cdot \cos x}\right) + \frac{\sin x}{x \cdot x} \cdot \left(\frac{\sin x}{{\cos x}^{2}} - \frac{\sin x}{\cos x}\right)\right) - \frac{\sin x}{x}\\

\mathbf{elif}\;x \leq 0.003965314230899635:\\
\;\;\;\;0.225 \cdot {x}^{2} - 0.5\\

\mathbf{else}:\\
\;\;\;\;\log \left(e^{\frac{x - \sin x}{x - \tan x}}\right)\\


\end{array}

(FPCore (x) :precision binary64 (/ (- x (sin x)) (- x (tan x))))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -2.208632844132881)
   (-
    (+
     (+ 1.0 (/ (sin x) (* x (cos x))))
     (*
      (/ (sin x) (* x x))
      (- (/ (sin x) (pow (cos x) 2.0)) (/ (sin x) (cos x)))))
    (/ (sin x) x))
   (if (<= x 0.003965314230899635)
     (- (* 0.225 (pow x 2.0)) 0.5)
     (log (exp (/ (- 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 <= -2.208632844132881) {
		tmp = ((1.0 + (sin(x) / (x * cos(x)))) + ((sin(x) / (x * x)) * ((sin(x) / pow(cos(x), 2.0)) - (sin(x) / cos(x))))) - (sin(x) / x);
	} else if (x <= 0.003965314230899635) {
		tmp = (0.225 * pow(x, 2.0)) - 0.5;
	} else {
		tmp = log(exp((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 < -2.20863284413288108

    1. Initial program 0.0

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

    2. Taylor expanded in x around inf 0.4

      \[\leadsto \color{blue}{\left(\frac{\sin x}{\cos x \cdot x} + \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2} \cdot {x}^{2}}\right)\right) - \left(\frac{{\sin x}^{2}}{\cos x \cdot {x}^{2}} + \frac{\sin x}{x}\right)} \]

    3. Simplified0.4

      \[\leadsto \color{blue}{\left(\left(1 + \frac{\sin x}{x \cdot \cos x}\right) + \frac{\sin x}{x \cdot x} \cdot \left(\frac{\sin x}{{\cos x}^{2}} - \frac{\sin x}{\cos x}\right)\right) - \frac{\sin x}{x}} \]

    if -2.20863284413288108 < x < 0.00396531423089963465

    1. Initial program 63.1

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

    2. Taylor expanded in x around 0 0.1

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

    if 0.00396531423089963465 < x

    1. Initial program 0.1

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

    2. Applied add-log-exp_binary640.1

      \[\leadsto \color{blue}{\log \left(e^{\frac{x - \sin x}{x - \tan x}}\right)} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.208632844132881:\\ \;\;\;\;\left(\left(1 + \frac{\sin x}{x \cdot \cos x}\right) + \frac{\sin x}{x \cdot x} \cdot \left(\frac{\sin x}{{\cos x}^{2}} - \frac{\sin x}{\cos x}\right)\right) - \frac{\sin x}{x}\\ \mathbf{elif}\;x \leq 0.003965314230899635:\\ \;\;\;\;0.225 \cdot {x}^{2} - 0.5\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{x - \sin x}{x - \tan x}}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021215 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))