Average Error: 31.4 → 0.3
Time: 30.6s
Precision: 64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.5060118563807836:\\ \;\;\;\;\left(\frac{\sin x}{\cos x \cdot x} \cdot \frac{\sin x}{\cos x \cdot x} + \frac{\sin x}{\cos x \cdot x}\right) + \left(\left(1 - \frac{\sin x}{x}\right) - \frac{\frac{\sin x}{x} \cdot \frac{\sin x}{x}}{\cos x}\right)\\ \mathbf{elif}\;x \le 1.553788978235722:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \frac{9}{40} - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{27}{2800}\right) - \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sin x}{\cos x \cdot x} \cdot \frac{\sin x}{\cos x \cdot x} + \frac{\sin x}{\cos x \cdot x}\right) + \left(\left(1 - \frac{\sin x}{x}\right) - \frac{\frac{\sin x}{x} \cdot \frac{\sin x}{x}}{\cos x}\right)\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -2.5060118563807836:\\
\;\;\;\;\left(\frac{\sin x}{\cos x \cdot x} \cdot \frac{\sin x}{\cos x \cdot x} + \frac{\sin x}{\cos x \cdot x}\right) + \left(\left(1 - \frac{\sin x}{x}\right) - \frac{\frac{\sin x}{x} \cdot \frac{\sin x}{x}}{\cos x}\right)\\

\mathbf{elif}\;x \le 1.553788978235722:\\
\;\;\;\;\left(\left(x \cdot x\right) \cdot \frac{9}{40} - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{27}{2800}\right) - \frac{1}{2}\\

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

\end{array}
double f(double x) {
        double r554010 = x;
        double r554011 = sin(r554010);
        double r554012 = r554010 - r554011;
        double r554013 = tan(r554010);
        double r554014 = r554010 - r554013;
        double r554015 = r554012 / r554014;
        return r554015;
}
↓
double f(double x) {
        double r554016 = x;
        double r554017 = -2.5060118563807836;
        bool r554018 = r554016 <= r554017;
        double r554019 = sin(r554016);
        double r554020 = cos(r554016);
        double r554021 = r554020 * r554016;
        double r554022 = r554019 / r554021;
        double r554023 = r554022 * r554022;
        double r554024 = r554023 + r554022;
        double r554025 = 1.0;
        double r554026 = r554019 / r554016;
        double r554027 = r554025 - r554026;
        double r554028 = r554026 * r554026;
        double r554029 = r554028 / r554020;
        double r554030 = r554027 - r554029;
        double r554031 = r554024 + r554030;
        double r554032 = 1.553788978235722;
        bool r554033 = r554016 <= r554032;
        double r554034 = r554016 * r554016;
        double r554035 = 0.225;
        double r554036 = r554034 * r554035;
        double r554037 = r554034 * r554034;
        double r554038 = 0.009642857142857142;
        double r554039 = r554037 * r554038;
        double r554040 = r554036 - r554039;
        double r554041 = 0.5;
        double r554042 = r554040 - r554041;
        double r554043 = r554033 ? r554042 : r554031;
        double r554044 = r554018 ? r554031 : r554043;
        return r554044;
}

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 < -2.5060118563807836 or 1.553788978235722 < 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

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

    if -2.5060118563807836 < x < 1.553788978235722

    1. Initial program 62.3

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

    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

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.5060118563807836:\\ \;\;\;\;\left(\frac{\sin x}{\cos x \cdot x} \cdot \frac{\sin x}{\cos x \cdot x} + \frac{\sin x}{\cos x \cdot x}\right) + \left(\left(1 - \frac{\sin x}{x}\right) - \frac{\frac{\sin x}{x} \cdot \frac{\sin x}{x}}{\cos x}\right)\\ \mathbf{elif}\;x \le 1.553788978235722:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \frac{9}{40} - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{27}{2800}\right) - \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sin x}{\cos x \cdot x} \cdot \frac{\sin x}{\cos x \cdot x} + \frac{\sin x}{\cos x \cdot x}\right) + \left(\left(1 - \frac{\sin x}{x}\right) - \frac{\frac{\sin x}{x} \cdot \frac{\sin x}{x}}{\cos x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019130 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))