Average Error: 31.6 → 0.1
Time: 9.5s
Precision: 64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.028761646689302346 \lor \neg \left(x \le 0.031844168701184422\right):\\ \;\;\;\;\frac{\sqrt[3]{x - \sin x} \cdot \sqrt[3]{x - \sin x}}{\sqrt[3]{x - \tan x} \cdot \sqrt[3]{x - \tan x}} \cdot \frac{\sqrt[3]{x - \sin x}}{\sqrt[3]{x - \tan x}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{9}{40}, {x}^{2}, -\mathsf{fma}\left(\frac{27}{2800}, {x}^{4}, \frac{1}{2}\right)\right)\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.028761646689302346 \lor \neg \left(x \le 0.031844168701184422\right):\\
\;\;\;\;\frac{\sqrt[3]{x - \sin x} \cdot \sqrt[3]{x - \sin x}}{\sqrt[3]{x - \tan x} \cdot \sqrt[3]{x - \tan x}} \cdot \frac{\sqrt[3]{x - \sin x}}{\sqrt[3]{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 ((double) (((double) (x - ((double) sin(x)))) / ((double) (x - ((double) tan(x))))));
}
↓
double code(double x) {
	double VAR;
	if (((x <= -0.028761646689302346) || !(x <= 0.03184416870118442))) {
		VAR = ((double) (((double) (((double) (((double) cbrt(((double) (x - ((double) sin(x)))))) * ((double) cbrt(((double) (x - ((double) sin(x)))))))) / ((double) (((double) cbrt(((double) (x - ((double) tan(x)))))) * ((double) cbrt(((double) (x - ((double) tan(x)))))))))) * ((double) (((double) cbrt(((double) (x - ((double) sin(x)))))) / ((double) cbrt(((double) (x - ((double) tan(x))))))))));
	} else {
		VAR = ((double) fma(0.225, ((double) pow(x, 2.0)), ((double) -(((double) fma(0.009642857142857142, ((double) 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 < -0.028761646689302346 or 0.03184416870118442 < x

    1. Initial program 0.0

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

    3. Applied add-cube-cbrt1.4

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

    4. Applied add-cube-cbrt0.1

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

    5. Applied times-frac0.1

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

    if -0.028761646689302346 < x < 0.03184416870118442

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.028761646689302346 \lor \neg \left(x \le 0.031844168701184422\right):\\ \;\;\;\;\frac{\sqrt[3]{x - \sin x} \cdot \sqrt[3]{x - \sin x}}{\sqrt[3]{x - \tan x} \cdot \sqrt[3]{x - \tan x}} \cdot \frac{\sqrt[3]{x - \sin x}}{\sqrt[3]{x - \tan x}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{9}{40}, {x}^{2}, -\mathsf{fma}\left(\frac{27}{2800}, {x}^{4}, \frac{1}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020121 +o rules:numerics
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))