Average Error: 31.4 → 0.3
Time: 24.4s
Precision: 64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.412246929618039992249123315559700131416:\\ \;\;\;\;\left(\left(\frac{\frac{\sin x}{x}}{\cos x} - \frac{\sin x}{x}\right) + \left(\frac{\frac{\sin x}{\cos x}}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x} + 1\right)\right) - \frac{\sin x}{x \cdot x} \cdot \frac{\sin x}{\cos x}\\ \mathbf{elif}\;x \le 2.424314253224005177855815418297424912453:\\ \;\;\;\;\left(\left(x \cdot \frac{9}{40}\right) \cdot x - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{27}{2800}\right) - \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{\frac{\sin x}{x}}{\cos x} - \frac{\sin x}{x}\right) + \left(\frac{\frac{\sin x}{\cos x}}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x} + 1\right)\right) - \frac{\sin x}{x \cdot x} \cdot \frac{\sin x}{\cos x}\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -2.412246929618039992249123315559700131416:\\
\;\;\;\;\left(\left(\frac{\frac{\sin x}{x}}{\cos x} - \frac{\sin x}{x}\right) + \left(\frac{\frac{\sin x}{\cos x}}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x} + 1\right)\right) - \frac{\sin x}{x \cdot x} \cdot \frac{\sin x}{\cos x}\\

\mathbf{elif}\;x \le 2.424314253224005177855815418297424912453:\\
\;\;\;\;\left(\left(x \cdot \frac{9}{40}\right) \cdot x - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{27}{2800}\right) - \frac{1}{2}\\

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

\end{array}
double f(double x) {
        double r562116 = x;
        double r562117 = sin(r562116);
        double r562118 = r562116 - r562117;
        double r562119 = tan(r562116);
        double r562120 = r562116 - r562119;
        double r562121 = r562118 / r562120;
        return r562121;
}
↓
double f(double x) {
        double r562122 = x;
        double r562123 = -2.41224692961804;
        bool r562124 = r562122 <= r562123;
        double r562125 = sin(r562122);
        double r562126 = r562125 / r562122;
        double r562127 = cos(r562122);
        double r562128 = r562126 / r562127;
        double r562129 = r562128 - r562126;
        double r562130 = r562125 / r562127;
        double r562131 = r562130 / r562122;
        double r562132 = r562131 * r562131;
        double r562133 = 1.0;
        double r562134 = r562132 + r562133;
        double r562135 = r562129 + r562134;
        double r562136 = r562122 * r562122;
        double r562137 = r562125 / r562136;
        double r562138 = r562137 * r562130;
        double r562139 = r562135 - r562138;
        double r562140 = 2.424314253224005;
        bool r562141 = r562122 <= r562140;
        double r562142 = 0.225;
        double r562143 = r562122 * r562142;
        double r562144 = r562143 * r562122;
        double r562145 = r562136 * r562136;
        double r562146 = 0.009642857142857142;
        double r562147 = r562145 * r562146;
        double r562148 = r562144 - r562147;
        double r562149 = 0.5;
        double r562150 = r562148 - r562149;
        double r562151 = r562141 ? r562150 : r562139;
        double r562152 = r562124 ? r562139 : r562151;
        return r562152;
}

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 < -2.41224692961804 or 2.424314253224005 < 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

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

    if -2.41224692961804 < x < 2.424314253224005

    1. Initial program 63.0

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

    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

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.412246929618039992249123315559700131416:\\ \;\;\;\;\left(\left(\frac{\frac{\sin x}{x}}{\cos x} - \frac{\sin x}{x}\right) + \left(\frac{\frac{\sin x}{\cos x}}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x} + 1\right)\right) - \frac{\sin x}{x \cdot x} \cdot \frac{\sin x}{\cos x}\\ \mathbf{elif}\;x \le 2.424314253224005177855815418297424912453:\\ \;\;\;\;\left(\left(x \cdot \frac{9}{40}\right) \cdot x - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{27}{2800}\right) - \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{\frac{\sin x}{x}}{\cos x} - \frac{\sin x}{x}\right) + \left(\frac{\frac{\sin x}{\cos x}}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x} + 1\right)\right) - \frac{\sin x}{x \cdot x} \cdot \frac{\sin x}{\cos x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))