Average Error: 31.2 → 0.3
Time: 9.1s
Precision: 64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.36088798636529651 \lor \neg \left(x \le 2.3734522517745011\right):\\ \;\;\;\;\frac{x - \sin x}{\left(\left(x + \frac{{\left(\sin x\right)}^{2}}{x \cdot {\left(\cos x\right)}^{2}}\right) - \frac{\sin x}{\cos x}\right) - \frac{\tan x \cdot \tan x}{x + \tan x}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{9}{40}, {x}^{2}, -\mathsf{fma}\left(\frac{27}{2800}, {x}^{4}, \frac{1}{2}\right)\right)\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -2.36088798636529651 \lor \neg \left(x \le 2.3734522517745011\right):\\
\;\;\;\;\frac{x - \sin x}{\left(\left(x + \frac{{\left(\sin x\right)}^{2}}{x \cdot {\left(\cos x\right)}^{2}}\right) - \frac{\sin x}{\cos x}\right) - \frac{\tan x \cdot \tan x}{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 <= -2.3608879863652965) || !(x <= 2.373452251774501))) {
		VAR = ((x - sin(x)) / (((x + (pow(sin(x), 2.0) / (x * pow(cos(x), 2.0)))) - (sin(x) / cos(x))) - ((tan(x) * tan(x)) / (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 < -2.3608879863652965 or 2.373452251774501 < x

    1. Initial program 0.0

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

    3. Applied flip--31.2

      \[\leadsto \frac{x - \sin x}{\color{blue}{\frac{x \cdot x - \tan x \cdot \tan x}{x + \tan x}}}\]
    4. Using strategy rm

    5. Applied div-sub31.2

      \[\leadsto \frac{x - \sin x}{\color{blue}{\frac{x \cdot x}{x + \tan x} - \frac{\tan x \cdot \tan x}{x + \tan x}}}\]

    6. Taylor expanded around inf 0.4

      \[\leadsto \frac{x - \sin x}{\color{blue}{\left(\left(x + \frac{{\left(\sin x\right)}^{2}}{x \cdot {\left(\cos x\right)}^{2}}\right) - \frac{\sin x}{\cos x}\right)} - \frac{\tan x \cdot \tan x}{x + \tan x}}\]

    if -2.3608879863652965 < x < 2.373452251774501

    1. Initial program 62.9

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

    2. Taylor expanded around 0 0.1

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

    3. Simplified0.1

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.36088798636529651 \lor \neg \left(x \le 2.3734522517745011\right):\\ \;\;\;\;\frac{x - \sin x}{\left(\left(x + \frac{{\left(\sin x\right)}^{2}}{x \cdot {\left(\cos x\right)}^{2}}\right) - \frac{\sin x}{\cos x}\right) - \frac{\tan x \cdot \tan x}{x + \tan x}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{9}{40}, {x}^{2}, -\mathsf{fma}\left(\frac{27}{2800}, {x}^{4}, \frac{1}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 +o rules:numerics
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))