Average Error: 31.4 → 0.0
Time: 9.2s
Precision: binary64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -0.03062657705051721:\\ \;\;\;\;\sqrt[3]{{\left(\frac{x - \sin x}{x - \sqrt[3]{\tan x} \cdot \left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right)}\right)}^{3}}\\ \mathbf{elif}\;x \leq 0.028616561567871263:\\ \;\;\;\;\left(0.225 \cdot \left(x \cdot x\right) - 0.5\right) - 0.009642857142857142 \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{x - \sin x}{x - \tan x}\right)}^{3}}\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -0.03062657705051721:\\
\;\;\;\;\sqrt[3]{{\left(\frac{x - \sin x}{x - \sqrt[3]{\tan x} \cdot \left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right)}\right)}^{3}}\\

\mathbf{elif}\;x \leq 0.028616561567871263:\\
\;\;\;\;\left(0.225 \cdot \left(x \cdot x\right) - 0.5\right) - 0.009642857142857142 \cdot {x}^{4}\\

\mathbf{else}:\\
\;\;\;\;\sqrt[3]{{\left(\frac{x - \sin x}{x - \tan x}\right)}^{3}}\\

\end{array}
(FPCore (x) :precision binary64 (/ (- x (sin x)) (- x (tan x))))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -0.03062657705051721)
   (cbrt
    (pow
     (/
      (- x (sin x))
      (- x (* (cbrt (tan x)) (* (cbrt (tan x)) (cbrt (tan x))))))
     3.0))
   (if (<= x 0.028616561567871263)
     (- (- (* 0.225 (* x x)) 0.5) (* 0.009642857142857142 (pow x 4.0)))
     (cbrt (pow (/ (- x (sin x)) (- x (tan x))) 3.0)))))
double code(double x) {
	return (x - sin(x)) / (x - tan(x));
}
↓
double code(double x) {
	double tmp;
	if (x <= -0.03062657705051721) {
		tmp = cbrt(pow(((x - sin(x)) / (x - (cbrt(tan(x)) * (cbrt(tan(x)) * cbrt(tan(x)))))), 3.0));
	} else if (x <= 0.028616561567871263) {
		tmp = ((0.225 * (x * x)) - 0.5) - (0.009642857142857142 * pow(x, 4.0));
	} else {
		tmp = cbrt(pow(((x - sin(x)) / (x - tan(x))), 3.0));
	}
	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 3 regimes

  2. if x < -0.030626577050517211

    1. Initial program 0.0

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

    3. Applied add-cbrt-cube_binary640.1

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

    4. Simplified0.1

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

    6. Applied add-cube-cbrt_binary640.1

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

    7. Applied cancel-sign-sub-inv_binary640.1

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

    if -0.030626577050517211 < x < 0.028616561567871263

    1. Initial program 63.2

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

    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

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

    5. Applied associate--r+_binary640.0

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

    if 0.028616561567871263 < x

    1. Initial program 0.1

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

    3. Applied add-cbrt-cube_binary640.1

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

    4. Simplified0.1

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.03062657705051721:\\ \;\;\;\;\sqrt[3]{{\left(\frac{x - \sin x}{x - \sqrt[3]{\tan x} \cdot \left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right)}\right)}^{3}}\\ \mathbf{elif}\;x \leq 0.028616561567871263:\\ \;\;\;\;\left(0.225 \cdot \left(x \cdot x\right) - 0.5\right) - 0.009642857142857142 \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{x - \sin x}{x - \tan x}\right)}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021173 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))