Average Error: 31.5 → 0.1
Time: 10.2s
Precision: 64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0245372772323286699 \lor \neg \left(x \le 0.0290270587105998616\right):\\ \;\;\;\;\left(x - \sin x\right) \cdot \frac{1}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{9}{40}, {x}^{2}, -\mathsf{fma}\left(\frac{27}{2800}, {x}^{4}, \frac{1}{2}\right)\right)\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0245372772323286699 \lor \neg \left(x \le 0.0290270587105998616\right):\\
\;\;\;\;\left(x - \sin x\right) \cdot \frac{1}{x - \tan x}\\

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

\end{array}
double code(double x) {
	return ((x - sin(x)) / (x - tan(x)));
}
↓
double code(double x) {
	double VAR;
	if (((x <= -0.02453727723232867) || !(x <= 0.02902705871059986))) {
		VAR = ((x - sin(x)) * (1.0 / (x - tan(x))));
	} else {
		VAR = fma(0.225, pow(x, 2.0), -fma(0.009642857142857142, pow(x, 4.0), 0.5));
	}
	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 < -0.02453727723232867 or 0.02902705871059986 < x

    1. Initial program 0.1

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

    3. Applied div-inv0.2

      \[\leadsto \color{blue}{\left(x - \sin x\right) \cdot \frac{1}{x - \tan x}}\]

    if -0.02453727723232867 < x < 0.02902705871059986

    1. Initial program 63.0

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

    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

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0245372772323286699 \lor \neg \left(x \le 0.0290270587105998616\right):\\ \;\;\;\;\left(x - \sin x\right) \cdot \frac{1}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{9}{40}, {x}^{2}, -\mathsf{fma}\left(\frac{27}{2800}, {x}^{4}, \frac{1}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 +o rules:numerics
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))