Average Error: 31.7 → 0.0
Time: 10.1s
Precision: binary64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.026930065898567369 \lor \neg \left(x \le 0.031007281596359057\right):\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{9}{40}\right) + \left(\frac{-1}{2} + {x}^{4} \cdot \frac{-27}{2800}\right)\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.026930065898567369 \lor \neg \left(x \le 0.031007281596359057\right):\\
\;\;\;\;\frac{x - \sin x}{x - \tan x}\\

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

\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 <= -0.02693006589856737) || !(x <= 0.031007281596359057))) {
		VAR = ((double) (((double) (x - ((double) sin(x)))) / ((double) (x - ((double) tan(x))))));
	} else {
		VAR = ((double) (((double) (x * ((double) (x * 0.225)))) + ((double) (-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 < -0.026930065898567369 or 0.031007281596359057 < x

    1. Initial program 0.0

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

    if -0.026930065898567369 < x < 0.031007281596359057

    1. Initial program 63.3

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

    2. Taylor expanded around 0 62.8

      \[\leadsto \frac{x - \sin x}{\color{blue}{-\left(\frac{17}{315} \cdot {x}^{7} + \left(\frac{1}{3} \cdot {x}^{3} + \frac{2}{15} \cdot {x}^{5}\right)\right)}}\]

    3. Simplified62.8

      \[\leadsto \frac{x - \sin x}{\color{blue}{{x}^{7} \cdot \frac{-17}{315} + \left({x}^{5} \cdot \frac{-2}{15} + {x}^{3} \cdot \frac{-1}{3}\right)}}\]

    4. Taylor expanded around 0 0.0

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

    5. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.026930065898567369 \lor \neg \left(x \le 0.031007281596359057\right):\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{9}{40}\right) + \left(\frac{-1}{2} + {x}^{4} \cdot \frac{-27}{2800}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020185 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))