Average Error: 31.7 → 0.3
Time: 40.7s
Precision: 64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.483298028151702:\\ \;\;\;\;\left(\left(\frac{\frac{\sin x}{x}}{\cos x} - \frac{\sin x}{x}\right) - \frac{\frac{\sin x}{x} \cdot \frac{\sin x}{x}}{\cos x}\right) + \left(1 + \frac{\frac{\sin x}{x}}{\cos x} \cdot \frac{\frac{\sin x}{x}}{\cos x}\right)\\ \mathbf{elif}\;x \le 2.4050174326567912:\\ \;\;\;\;\left(x \cdot \left(x \cdot \frac{9}{40}\right) - {x}^{4} \cdot \frac{27}{2800}\right) - \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{\frac{\sin x}{x}}{\cos x} - \frac{\sin x}{x}\right) - \frac{\frac{\sin x}{x} \cdot \frac{\sin x}{x}}{\cos x}\right) + \left(1 + \frac{\frac{\sin x}{x}}{\cos x} \cdot \frac{\frac{\sin x}{x}}{\cos x}\right)\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -2.483298028151702:\\
\;\;\;\;\left(\left(\frac{\frac{\sin x}{x}}{\cos x} - \frac{\sin x}{x}\right) - \frac{\frac{\sin x}{x} \cdot \frac{\sin x}{x}}{\cos x}\right) + \left(1 + \frac{\frac{\sin x}{x}}{\cos x} \cdot \frac{\frac{\sin x}{x}}{\cos x}\right)\\

\mathbf{elif}\;x \le 2.4050174326567912:\\
\;\;\;\;\left(x \cdot \left(x \cdot \frac{9}{40}\right) - {x}^{4} \cdot \frac{27}{2800}\right) - \frac{1}{2}\\

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

\end{array}
double f(double x) {
        double r727041 = x;
        double r727042 = sin(r727041);
        double r727043 = r727041 - r727042;
        double r727044 = tan(r727041);
        double r727045 = r727041 - r727044;
        double r727046 = r727043 / r727045;
        return r727046;
}
↓
double f(double x) {
        double r727047 = x;
        double r727048 = -2.483298028151702;
        bool r727049 = r727047 <= r727048;
        double r727050 = sin(r727047);
        double r727051 = r727050 / r727047;
        double r727052 = cos(r727047);
        double r727053 = r727051 / r727052;
        double r727054 = r727053 - r727051;
        double r727055 = r727051 * r727051;
        double r727056 = r727055 / r727052;
        double r727057 = r727054 - r727056;
        double r727058 = 1.0;
        double r727059 = r727053 * r727053;
        double r727060 = r727058 + r727059;
        double r727061 = r727057 + r727060;
        double r727062 = 2.4050174326567912;
        bool r727063 = r727047 <= r727062;
        double r727064 = 0.225;
        double r727065 = r727047 * r727064;
        double r727066 = r727047 * r727065;
        double r727067 = 4.0;
        double r727068 = pow(r727047, r727067);
        double r727069 = 0.009642857142857142;
        double r727070 = r727068 * r727069;
        double r727071 = r727066 - r727070;
        double r727072 = 0.5;
        double r727073 = r727071 - r727072;
        double r727074 = r727063 ? r727073 : r727061;
        double r727075 = r727049 ? r727061 : r727074;
        return r727075;
}

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 < -2.483298028151702 or 2.4050174326567912 < 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

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

    if -2.483298028151702 < x < 2.4050174326567912

    1. Initial program 62.5

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

    2. Taylor expanded around 0 0.3

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

    3. Simplified0.3

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

    5. Applied fma-udef0.3

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

    6. Applied associate--r+0.3

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.483298028151702:\\ \;\;\;\;\left(\left(\frac{\frac{\sin x}{x}}{\cos x} - \frac{\sin x}{x}\right) - \frac{\frac{\sin x}{x} \cdot \frac{\sin x}{x}}{\cos x}\right) + \left(1 + \frac{\frac{\sin x}{x}}{\cos x} \cdot \frac{\frac{\sin x}{x}}{\cos x}\right)\\ \mathbf{elif}\;x \le 2.4050174326567912:\\ \;\;\;\;\left(x \cdot \left(x \cdot \frac{9}{40}\right) - {x}^{4} \cdot \frac{27}{2800}\right) - \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{\frac{\sin x}{x}}{\cos x} - \frac{\sin x}{x}\right) - \frac{\frac{\sin x}{x} \cdot \frac{\sin x}{x}}{\cos x}\right) + \left(1 + \frac{\frac{\sin x}{x}}{\cos x} \cdot \frac{\frac{\sin x}{x}}{\cos x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019134 +o rules:numerics
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))