Average Error: 31.3 → 0.3
Time: 9.4s
Precision: 64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.4354142091382851 \lor \neg \left(x \le 2.4300372653342581\right):\\ \;\;\;\;\left(\frac{\sin x}{x \cdot \cos x} + \left(\frac{{\left(\sin x\right)}^{2}}{{x}^{2} \cdot {\left(\cos x\right)}^{2}} + 1\right)\right) - \left(\frac{\sin x}{x} + \frac{{\left(\sin x\right)}^{2}}{{x}^{2} \cdot \cos x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{9}{40} \cdot {x}^{2} - \frac{27}{2800} \cdot {x}^{4}\right) - \frac{1}{2}\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -2.4354142091382851 \lor \neg \left(x \le 2.4300372653342581\right):\\
\;\;\;\;\left(\frac{\sin x}{x \cdot \cos x} + \left(\frac{{\left(\sin x\right)}^{2}}{{x}^{2} \cdot {\left(\cos x\right)}^{2}} + 1\right)\right) - \left(\frac{\sin x}{x} + \frac{{\left(\sin x\right)}^{2}}{{x}^{2} \cdot \cos x}\right)\\

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

\end{array}
double f(double x) {
        double r11242 = x;
        double r11243 = sin(r11242);
        double r11244 = r11242 - r11243;
        double r11245 = tan(r11242);
        double r11246 = r11242 - r11245;
        double r11247 = r11244 / r11246;
        return r11247;
}
↓
double f(double x) {
        double r11248 = x;
        double r11249 = -2.435414209138285;
        bool r11250 = r11248 <= r11249;
        double r11251 = 2.430037265334258;
        bool r11252 = r11248 <= r11251;
        double r11253 = !r11252;
        bool r11254 = r11250 || r11253;
        double r11255 = sin(r11248);
        double r11256 = cos(r11248);
        double r11257 = r11248 * r11256;
        double r11258 = r11255 / r11257;
        double r11259 = 2.0;
        double r11260 = pow(r11255, r11259);
        double r11261 = pow(r11248, r11259);
        double r11262 = pow(r11256, r11259);
        double r11263 = r11261 * r11262;
        double r11264 = r11260 / r11263;
        double r11265 = 1.0;
        double r11266 = r11264 + r11265;
        double r11267 = r11258 + r11266;
        double r11268 = r11255 / r11248;
        double r11269 = r11261 * r11256;
        double r11270 = r11260 / r11269;
        double r11271 = r11268 + r11270;
        double r11272 = r11267 - r11271;
        double r11273 = 0.225;
        double r11274 = r11273 * r11261;
        double r11275 = 0.009642857142857142;
        double r11276 = 4.0;
        double r11277 = pow(r11248, r11276);
        double r11278 = r11275 * r11277;
        double r11279 = r11274 - r11278;
        double r11280 = 0.5;
        double r11281 = r11279 - r11280;
        double r11282 = r11254 ? r11272 : r11281;
        return r11282;
}

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 < -2.435414209138285 or 2.430037265334258 < x

    1. Initial program 0.0

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

    2. Taylor expanded around inf 0.4

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

    if -2.435414209138285 < x < 2.430037265334258

    1. Initial program 62.8

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

    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. Using strategy rm

    4. Applied associate--r+0.2

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.4354142091382851 \lor \neg \left(x \le 2.4300372653342581\right):\\ \;\;\;\;\left(\frac{\sin x}{x \cdot \cos x} + \left(\frac{{\left(\sin x\right)}^{2}}{{x}^{2} \cdot {\left(\cos x\right)}^{2}} + 1\right)\right) - \left(\frac{\sin x}{x} + \frac{{\left(\sin x\right)}^{2}}{{x}^{2} \cdot \cos x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{9}{40} \cdot {x}^{2} - \frac{27}{2800} \cdot {x}^{4}\right) - \frac{1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))