Average Error: 31.1 → 0.0
Time: 10.5s
Precision: binary64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.033049280051316229 \lor \neg \left(x \le 0.029172484020573174\right):\\ \;\;\;\;\sqrt[3]{{\left(\frac{x - \sin x}{x - \tan x}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot \frac{9}{40}\right) + \frac{-1}{2}\right) + {x}^{4} \cdot \frac{-27}{2800}\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.033049280051316229 \lor \neg \left(x \le 0.029172484020573174\right):\\
\;\;\;\;\sqrt[3]{{\left(\frac{x - \sin x}{x - \tan x}\right)}^{3}}\\

\mathbf{else}:\\
\;\;\;\;\left(x \cdot \left(x \cdot \frac{9}{40}\right) + \frac{-1}{2}\right) + {x}^{4} \cdot \frac{-27}{2800}\\

\end{array}
double code(double x) {
	return ((double) (((double) (x - ((double) sin(x)))) / ((double) (x - ((double) tan(x))))));
}
↓
double code(double x) {
	double VAR;
	if (((x <= -0.03304928005131623) || !(x <= 0.029172484020573174))) {
		VAR = ((double) cbrt(((double) pow(((double) (((double) (x - ((double) sin(x)))) / ((double) (x - ((double) tan(x)))))), 3.0))));
	} else {
		VAR = ((double) (((double) (((double) (x * ((double) (x * 0.225)))) + -0.5)) + ((double) (((double) pow(x, 4.0)) * -0.009642857142857142))));
	}
	return VAR;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < -0.033049280051316229 or 0.029172484020573174 < x

    1. Initial program 0.0

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

    3. Applied add-cbrt-cube41.2

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

    4. Applied add-cbrt-cube41.0

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

    5. Applied cbrt-undiv41.0

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

    6. Simplified0.1

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

    if -0.033049280051316229 < x < 0.029172484020573174

    1. Initial program 62.8

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

    2. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\frac{9}{40} \cdot {x}^{2} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}\]

    3. Simplified0.0

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \frac{9}{40}\right) + \left(\frac{-1}{2} + {x}^{4} \cdot \frac{-27}{2800}\right)}\]
    4. Using strategy rm

    5. Applied associate-+r+0.0

      \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot \frac{9}{40}\right) + \frac{-1}{2}\right) + {x}^{4} \cdot \frac{-27}{2800}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.033049280051316229 \lor \neg \left(x \le 0.029172484020573174\right):\\ \;\;\;\;\sqrt[3]{{\left(\frac{x - \sin x}{x - \tan x}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot \frac{9}{40}\right) + \frac{-1}{2}\right) + {x}^{4} \cdot \frac{-27}{2800}\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))