Average Error: 31.9 → 0.3
Time: 31.0s
Precision: 64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.4051902967603365:\\ \;\;\;\;\left(\frac{\frac{\sin x}{\cos x}}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x} + \left(\left(1 - \frac{\sin x}{x}\right) + \frac{\frac{\sin x}{x}}{\cos x}\right)\right) - \frac{\sin x}{x \cdot x} \cdot \frac{\sin x}{\cos x}\\ \mathbf{elif}\;x \le 7.937565998741664:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\frac{9}{40} - \left(x \cdot x\right) \cdot \frac{27}{2800}\right) - \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\sin x}{\cos x}}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x} + \left(\left(1 - \frac{\sin x}{x}\right) + \frac{\frac{\sin x}{x}}{\cos x}\right)\right) - \frac{\sin x}{x \cdot x} \cdot \frac{\sin x}{\cos x}\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -2.4051902967603365:\\
\;\;\;\;\left(\frac{\frac{\sin x}{\cos x}}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x} + \left(\left(1 - \frac{\sin x}{x}\right) + \frac{\frac{\sin x}{x}}{\cos x}\right)\right) - \frac{\sin x}{x \cdot x} \cdot \frac{\sin x}{\cos x}\\

\mathbf{elif}\;x \le 7.937565998741664:\\
\;\;\;\;\left(x \cdot x\right) \cdot \left(\frac{9}{40} - \left(x \cdot x\right) \cdot \frac{27}{2800}\right) - \frac{1}{2}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{\frac{\sin x}{\cos x}}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x} + \left(\left(1 - \frac{\sin x}{x}\right) + \frac{\frac{\sin x}{x}}{\cos x}\right)\right) - \frac{\sin x}{x \cdot x} \cdot \frac{\sin x}{\cos x}\\

\end{array}
double f(double x) {
        double r619973 = x;
        double r619974 = sin(r619973);
        double r619975 = r619973 - r619974;
        double r619976 = tan(r619973);
        double r619977 = r619973 - r619976;
        double r619978 = r619975 / r619977;
        return r619978;
}
↓
double f(double x) {
        double r619979 = x;
        double r619980 = -2.4051902967603365;
        bool r619981 = r619979 <= r619980;
        double r619982 = sin(r619979);
        double r619983 = cos(r619979);
        double r619984 = r619982 / r619983;
        double r619985 = r619984 / r619979;
        double r619986 = r619985 * r619985;
        double r619987 = 1.0;
        double r619988 = r619982 / r619979;
        double r619989 = r619987 - r619988;
        double r619990 = r619988 / r619983;
        double r619991 = r619989 + r619990;
        double r619992 = r619986 + r619991;
        double r619993 = r619979 * r619979;
        double r619994 = r619982 / r619993;
        double r619995 = r619994 * r619984;
        double r619996 = r619992 - r619995;
        double r619997 = 7.937565998741664;
        bool r619998 = r619979 <= r619997;
        double r619999 = 0.225;
        double r620000 = 0.009642857142857142;
        double r620001 = r619993 * r620000;
        double r620002 = r619999 - r620001;
        double r620003 = r619993 * r620002;
        double r620004 = 0.5;
        double r620005 = r620003 - r620004;
        double r620006 = r619998 ? r620005 : r619996;
        double r620007 = r619981 ? r619996 : r620006;
        return r620007;
}

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 < -2.4051902967603365 or 7.937565998741664 < x

    1. Initial program 0.0

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

    2. Taylor expanded around -inf 0.4

      \[\leadsto \color{blue}{\left(1 + \left(\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {x}^{2}} + \frac{\sin x}{\cos x \cdot x}\right)\right) - \left(\frac{\sin x}{x} + \frac{{\left(\sin x\right)}^{2}}{\cos x \cdot {x}^{2}}\right)}\]

    3. Simplified0.4

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

    if -2.4051902967603365 < x < 7.937565998741664

    1. Initial program 62.5

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

    2. Taylor expanded around 0 0.2

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

    3. Simplified0.2

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.4051902967603365:\\ \;\;\;\;\left(\frac{\frac{\sin x}{\cos x}}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x} + \left(\left(1 - \frac{\sin x}{x}\right) + \frac{\frac{\sin x}{x}}{\cos x}\right)\right) - \frac{\sin x}{x \cdot x} \cdot \frac{\sin x}{\cos x}\\ \mathbf{elif}\;x \le 7.937565998741664:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\frac{9}{40} - \left(x \cdot x\right) \cdot \frac{27}{2800}\right) - \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\sin x}{\cos x}}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x} + \left(\left(1 - \frac{\sin x}{x}\right) + \frac{\frac{\sin x}{x}}{\cos x}\right)\right) - \frac{\sin x}{x \cdot x} \cdot \frac{\sin x}{\cos x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019142 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))