Average Error: 31.4 → 0.0
Time: 9.2s
Precision: 64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0306713647544953581 \lor \neg \left(x \le 0.0300209026266443739\right):\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{81}{1600} \cdot {x}^{4} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right) \cdot \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)} \cdot \sqrt[3]{\frac{81}{1600} \cdot {x}^{4} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right) \cdot \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}}{\sqrt{\left(\frac{9}{40} \cdot {x}^{2} + \frac{27}{2800} \cdot {x}^{4}\right) + \frac{1}{2}}} \cdot \frac{\sqrt[3]{\frac{81}{1600} \cdot {x}^{4} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right) \cdot \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}}{\sqrt{\left(\frac{9}{40} \cdot {x}^{2} + \frac{27}{2800} \cdot {x}^{4}\right) + \frac{1}{2}}}\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0306713647544953581 \lor \neg \left(x \le 0.0300209026266443739\right):\\
\;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt[3]{\frac{81}{1600} \cdot {x}^{4} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right) \cdot \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)} \cdot \sqrt[3]{\frac{81}{1600} \cdot {x}^{4} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right) \cdot \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}}{\sqrt{\left(\frac{9}{40} \cdot {x}^{2} + \frac{27}{2800} \cdot {x}^{4}\right) + \frac{1}{2}}} \cdot \frac{\sqrt[3]{\frac{81}{1600} \cdot {x}^{4} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right) \cdot \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}}{\sqrt{\left(\frac{9}{40} \cdot {x}^{2} + \frac{27}{2800} \cdot {x}^{4}\right) + \frac{1}{2}}}\\

\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 <= -0.030671364754495358) || !(x <= 0.030020902626644374))) {
		VAR = ((double) (((double) (x / ((double) (x - ((double) tan(x)))))) - ((double) (((double) sin(x)) / ((double) (x - ((double) tan(x))))))));
	} else {
		VAR = ((double) (((double) (((double) (((double) cbrt(((double) (((double) (0.050625 * ((double) pow(x, 4.0)))) - ((double) (((double) (((double) (0.009642857142857142 * ((double) pow(x, 4.0)))) + 0.5)) * ((double) (((double) (0.009642857142857142 * ((double) pow(x, 4.0)))) + 0.5)))))))) * ((double) cbrt(((double) (((double) (0.050625 * ((double) pow(x, 4.0)))) - ((double) (((double) (((double) (0.009642857142857142 * ((double) pow(x, 4.0)))) + 0.5)) * ((double) (((double) (0.009642857142857142 * ((double) pow(x, 4.0)))) + 0.5)))))))))) / ((double) sqrt(((double) (((double) (((double) (0.225 * ((double) pow(x, 2.0)))) + ((double) (0.009642857142857142 * ((double) pow(x, 4.0)))))) + 0.5)))))) * ((double) (((double) cbrt(((double) (((double) (0.050625 * ((double) pow(x, 4.0)))) - ((double) (((double) (((double) (0.009642857142857142 * ((double) pow(x, 4.0)))) + 0.5)) * ((double) (((double) (0.009642857142857142 * ((double) pow(x, 4.0)))) + 0.5)))))))) / ((double) sqrt(((double) (((double) (((double) (0.225 * ((double) pow(x, 2.0)))) + ((double) (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 < -0.030671364754495358 or 0.030020902626644374 < x

    1. Initial program 0.0

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

    3. Applied div-sub0.0

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

    if -0.030671364754495358 < x < 0.030020902626644374

    1. Initial program 63.2

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

    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. Using strategy rm

    4. Applied add-log-exp0.0

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

    6. Applied flip--0.0

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

    7. Simplified0.0

      \[\leadsto \frac{\color{blue}{\frac{81}{1600} \cdot {x}^{4} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right) \cdot \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}}{\log \left(e^{\frac{9}{40} \cdot {x}^{2}}\right) + \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}\]

    8. Simplified0.0

      \[\leadsto \frac{\frac{81}{1600} \cdot {x}^{4} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right) \cdot \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}{\color{blue}{\left(\frac{9}{40} \cdot {x}^{2} + \frac{27}{2800} \cdot {x}^{4}\right) + \frac{1}{2}}}\]
    9. Using strategy rm

    10. Applied add-sqr-sqrt1.6

      \[\leadsto \frac{\frac{81}{1600} \cdot {x}^{4} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right) \cdot \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}{\color{blue}{\sqrt{\left(\frac{9}{40} \cdot {x}^{2} + \frac{27}{2800} \cdot {x}^{4}\right) + \frac{1}{2}} \cdot \sqrt{\left(\frac{9}{40} \cdot {x}^{2} + \frac{27}{2800} \cdot {x}^{4}\right) + \frac{1}{2}}}}\]

    11. Applied add-cube-cbrt1.6

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{81}{1600} \cdot {x}^{4} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right) \cdot \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)} \cdot \sqrt[3]{\frac{81}{1600} \cdot {x}^{4} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right) \cdot \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}\right) \cdot \sqrt[3]{\frac{81}{1600} \cdot {x}^{4} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right) \cdot \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}}}{\sqrt{\left(\frac{9}{40} \cdot {x}^{2} + \frac{27}{2800} \cdot {x}^{4}\right) + \frac{1}{2}} \cdot \sqrt{\left(\frac{9}{40} \cdot {x}^{2} + \frac{27}{2800} \cdot {x}^{4}\right) + \frac{1}{2}}}\]

    12. Applied times-frac0.0

      \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{81}{1600} \cdot {x}^{4} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right) \cdot \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)} \cdot \sqrt[3]{\frac{81}{1600} \cdot {x}^{4} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right) \cdot \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}}{\sqrt{\left(\frac{9}{40} \cdot {x}^{2} + \frac{27}{2800} \cdot {x}^{4}\right) + \frac{1}{2}}} \cdot \frac{\sqrt[3]{\frac{81}{1600} \cdot {x}^{4} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right) \cdot \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}}{\sqrt{\left(\frac{9}{40} \cdot {x}^{2} + \frac{27}{2800} \cdot {x}^{4}\right) + \frac{1}{2}}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0306713647544953581 \lor \neg \left(x \le 0.0300209026266443739\right):\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{81}{1600} \cdot {x}^{4} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right) \cdot \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)} \cdot \sqrt[3]{\frac{81}{1600} \cdot {x}^{4} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right) \cdot \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}}{\sqrt{\left(\frac{9}{40} \cdot {x}^{2} + \frac{27}{2800} \cdot {x}^{4}\right) + \frac{1}{2}}} \cdot \frac{\sqrt[3]{\frac{81}{1600} \cdot {x}^{4} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right) \cdot \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}}{\sqrt{\left(\frac{9}{40} \cdot {x}^{2} + \frac{27}{2800} \cdot {x}^{4}\right) + \frac{1}{2}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020140 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))