Average Error: 31.2 → 0.2
Time: 9.6s
Precision: binary64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -0.10627318114190262:\\ \;\;\;\;\frac{x - \sqrt[3]{\sin x} \cdot \left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}\right)}{x - \tan x}\\ \mathbf{elif}\;x \leq 8.375998260749977:\\ \;\;\;\;\frac{{\left(0.225 \cdot \left(x \cdot x\right) - \left(0.5 + 0.009642857142857142 \cdot {x}^{4}\right)\right)}^{3} + {\left(0.00024107142857142857 \cdot {x}^{6}\right)}^{3}}{\left(0.225 \cdot \left(x \cdot x\right) - \left(0.5 + 0.009642857142857142 \cdot {x}^{4}\right)\right) \cdot \left(0.225 \cdot \left(x \cdot x\right) - \left(0.5 + 0.009642857142857142 \cdot {x}^{4}\right)\right) + \left(\left(0.00024107142857142857 \cdot {x}^{6}\right) \cdot \left(0.00024107142857142857 \cdot {x}^{6}\right) - \left(0.225 \cdot \left(x \cdot x\right) - \left(0.5 + 0.009642857142857142 \cdot {x}^{4}\right)\right) \cdot \left(0.00024107142857142857 \cdot {x}^{6}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(3 \cdot \frac{\sin x}{x \cdot \cos x} + 1\right) - 3 \cdot \frac{\sin x}{x}}\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -0.10627318114190262:\\
\;\;\;\;\frac{x - \sqrt[3]{\sin x} \cdot \left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}\right)}{x - \tan x}\\

\mathbf{elif}\;x \leq 8.375998260749977:\\
\;\;\;\;\frac{{\left(0.225 \cdot \left(x \cdot x\right) - \left(0.5 + 0.009642857142857142 \cdot {x}^{4}\right)\right)}^{3} + {\left(0.00024107142857142857 \cdot {x}^{6}\right)}^{3}}{\left(0.225 \cdot \left(x \cdot x\right) - \left(0.5 + 0.009642857142857142 \cdot {x}^{4}\right)\right) \cdot \left(0.225 \cdot \left(x \cdot x\right) - \left(0.5 + 0.009642857142857142 \cdot {x}^{4}\right)\right) + \left(\left(0.00024107142857142857 \cdot {x}^{6}\right) \cdot \left(0.00024107142857142857 \cdot {x}^{6}\right) - \left(0.225 \cdot \left(x \cdot x\right) - \left(0.5 + 0.009642857142857142 \cdot {x}^{4}\right)\right) \cdot \left(0.00024107142857142857 \cdot {x}^{6}\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;\sqrt[3]{\left(3 \cdot \frac{\sin x}{x \cdot \cos x} + 1\right) - 3 \cdot \frac{\sin x}{x}}\\

\end{array}
(FPCore (x) :precision binary64 (/ (- x (sin x)) (- x (tan x))))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -0.10627318114190262)
   (/ (- x (* (cbrt (sin x)) (* (cbrt (sin x)) (cbrt (sin x))))) (- x (tan x)))
   (if (<= x 8.375998260749977)
     (/
      (+
       (pow
        (- (* 0.225 (* x x)) (+ 0.5 (* 0.009642857142857142 (pow x 4.0))))
        3.0)
       (pow (* 0.00024107142857142857 (pow x 6.0)) 3.0))
      (+
       (*
        (- (* 0.225 (* x x)) (+ 0.5 (* 0.009642857142857142 (pow x 4.0))))
        (- (* 0.225 (* x x)) (+ 0.5 (* 0.009642857142857142 (pow x 4.0)))))
       (-
        (*
         (* 0.00024107142857142857 (pow x 6.0))
         (* 0.00024107142857142857 (pow x 6.0)))
        (*
         (- (* 0.225 (* x x)) (+ 0.5 (* 0.009642857142857142 (pow x 4.0))))
         (* 0.00024107142857142857 (pow x 6.0))))))
     (cbrt
      (- (+ (* 3.0 (/ (sin x) (* x (cos x)))) 1.0) (* 3.0 (/ (sin x) x)))))))
double code(double x) {
	return (x - sin(x)) / (x - tan(x));
}
↓
double code(double x) {
	double tmp;
	if (x <= -0.10627318114190262) {
		tmp = (x - (cbrt(sin(x)) * (cbrt(sin(x)) * cbrt(sin(x))))) / (x - tan(x));
	} else if (x <= 8.375998260749977) {
		tmp = (pow(((0.225 * (x * x)) - (0.5 + (0.009642857142857142 * pow(x, 4.0)))), 3.0) + pow((0.00024107142857142857 * pow(x, 6.0)), 3.0)) / ((((0.225 * (x * x)) - (0.5 + (0.009642857142857142 * pow(x, 4.0)))) * ((0.225 * (x * x)) - (0.5 + (0.009642857142857142 * pow(x, 4.0))))) + (((0.00024107142857142857 * pow(x, 6.0)) * (0.00024107142857142857 * pow(x, 6.0))) - (((0.225 * (x * x)) - (0.5 + (0.009642857142857142 * pow(x, 4.0)))) * (0.00024107142857142857 * pow(x, 6.0)))));
	} else {
		tmp = cbrt(((3.0 * (sin(x) / (x * cos(x)))) + 1.0) - (3.0 * (sin(x) / 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 < -0.10627318114190262

    1. Initial program 0.0

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

    3. Applied add-cube-cbrt_binary640.0

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

    4. Applied cancel-sign-sub-inv_binary640.0

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

    if -0.10627318114190262 < x < 8.3759982607499772

    1. Initial program 62.8

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

    2. Taylor expanded around 0 0.1

      \[\leadsto \color{blue}{\left(0.225 \cdot {x}^{2} + 0.00024107142857142857 \cdot {x}^{6}\right) - \left(0.009642857142857142 \cdot {x}^{4} + 0.5\right)}\]

    3. Simplified0.1

      \[\leadsto \color{blue}{\left(0.225 \cdot \left(x \cdot x\right) - \left(0.5 + 0.009642857142857142 \cdot {x}^{4}\right)\right) + 0.00024107142857142857 \cdot {x}^{6}}\]
    4. Using strategy rm

    5. Applied flip3-+_binary640.1

      \[\leadsto \color{blue}{\frac{{\left(0.225 \cdot \left(x \cdot x\right) - \left(0.5 + 0.009642857142857142 \cdot {x}^{4}\right)\right)}^{3} + {\left(0.00024107142857142857 \cdot {x}^{6}\right)}^{3}}{\left(0.225 \cdot \left(x \cdot x\right) - \left(0.5 + 0.009642857142857142 \cdot {x}^{4}\right)\right) \cdot \left(0.225 \cdot \left(x \cdot x\right) - \left(0.5 + 0.009642857142857142 \cdot {x}^{4}\right)\right) + \left(\left(0.00024107142857142857 \cdot {x}^{6}\right) \cdot \left(0.00024107142857142857 \cdot {x}^{6}\right) - \left(0.225 \cdot \left(x \cdot x\right) - \left(0.5 + 0.009642857142857142 \cdot {x}^{4}\right)\right) \cdot \left(0.00024107142857142857 \cdot {x}^{6}\right)\right)}}\]

    if 8.3759982607499772 < x

    1. Initial program 0.0

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

    3. Applied add-cbrt-cube_binary640.0

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

    4. Simplified0.0

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

    5. Taylor expanded around inf 0.7

      \[\leadsto \sqrt[3]{\color{blue}{\left(3 \cdot \frac{\sin x}{x \cdot \cos x} + 1\right) - 3 \cdot \frac{\sin x}{x}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.10627318114190262:\\ \;\;\;\;\frac{x - \sqrt[3]{\sin x} \cdot \left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}\right)}{x - \tan x}\\ \mathbf{elif}\;x \leq 8.375998260749977:\\ \;\;\;\;\frac{{\left(0.225 \cdot \left(x \cdot x\right) - \left(0.5 + 0.009642857142857142 \cdot {x}^{4}\right)\right)}^{3} + {\left(0.00024107142857142857 \cdot {x}^{6}\right)}^{3}}{\left(0.225 \cdot \left(x \cdot x\right) - \left(0.5 + 0.009642857142857142 \cdot {x}^{4}\right)\right) \cdot \left(0.225 \cdot \left(x \cdot x\right) - \left(0.5 + 0.009642857142857142 \cdot {x}^{4}\right)\right) + \left(\left(0.00024107142857142857 \cdot {x}^{6}\right) \cdot \left(0.00024107142857142857 \cdot {x}^{6}\right) - \left(0.225 \cdot \left(x \cdot x\right) - \left(0.5 + 0.009642857142857142 \cdot {x}^{4}\right)\right) \cdot \left(0.00024107142857142857 \cdot {x}^{6}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(3 \cdot \frac{\sin x}{x \cdot \cos x} + 1\right) - 3 \cdot \frac{\sin x}{x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021175 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))