Average Error: 31.1 → 0.3
Time: 27.7s
Precision: 64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.421622929505549137729758513160049915314 \lor \neg \left(x \le 2.430594957022066626706191527773626148701\right):\\ \;\;\;\;\left(\frac{\sin x}{x \cdot \cos x} + \left(\frac{{\left(\sin x\right)}^{2}}{{x}^{2} \cdot {\left(\cos x\right)}^{2}} + 1\right)\right) - \left(\frac{\sin x}{x} + \frac{{\left(\sin x\right)}^{2}}{{x}^{2} \cdot \cos x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{9}{40} \cdot {x}^{2} - \mathsf{fma}\left(\frac{27}{2800}, {x}^{4}, \frac{1}{2}\right)\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -2.421622929505549137729758513160049915314 \lor \neg \left(x \le 2.430594957022066626706191527773626148701\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}:\\
\;\;\;\;\frac{9}{40} \cdot {x}^{2} - \mathsf{fma}\left(\frac{27}{2800}, {x}^{4}, \frac{1}{2}\right)\\

\end{array}
double f(double x) {
        double r24003 = x;
        double r24004 = sin(r24003);
        double r24005 = r24003 - r24004;
        double r24006 = tan(r24003);
        double r24007 = r24003 - r24006;
        double r24008 = r24005 / r24007;
        return r24008;
}
↓
double f(double x) {
        double r24009 = x;
        double r24010 = -2.421622929505549;
        bool r24011 = r24009 <= r24010;
        double r24012 = 2.4305949570220666;
        bool r24013 = r24009 <= r24012;
        double r24014 = !r24013;
        bool r24015 = r24011 || r24014;
        double r24016 = sin(r24009);
        double r24017 = cos(r24009);
        double r24018 = r24009 * r24017;
        double r24019 = r24016 / r24018;
        double r24020 = 2.0;
        double r24021 = pow(r24016, r24020);
        double r24022 = pow(r24009, r24020);
        double r24023 = pow(r24017, r24020);
        double r24024 = r24022 * r24023;
        double r24025 = r24021 / r24024;
        double r24026 = 1.0;
        double r24027 = r24025 + r24026;
        double r24028 = r24019 + r24027;
        double r24029 = r24016 / r24009;
        double r24030 = r24022 * r24017;
        double r24031 = r24021 / r24030;
        double r24032 = r24029 + r24031;
        double r24033 = r24028 - r24032;
        double r24034 = 0.225;
        double r24035 = r24034 * r24022;
        double r24036 = 0.009642857142857142;
        double r24037 = 4.0;
        double r24038 = pow(r24009, r24037);
        double r24039 = 0.5;
        double r24040 = fma(r24036, r24038, r24039);
        double r24041 = r24035 - r24040;
        double r24042 = r24015 ? r24033 : r24041;
        return r24042;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -2.421622929505549 or 2.4305949570220666 < 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 -2.421622929505549 < x < 2.4305949570220666

    1. Initial program 62.8

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

    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

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.421622929505549137729758513160049915314 \lor \neg \left(x \le 2.430594957022066626706191527773626148701\right):\\ \;\;\;\;\left(\frac{\sin x}{x \cdot \cos x} + \left(\frac{{\left(\sin x\right)}^{2}}{{x}^{2} \cdot {\left(\cos x\right)}^{2}} + 1\right)\right) - \left(\frac{\sin x}{x} + \frac{{\left(\sin x\right)}^{2}}{{x}^{2} \cdot \cos x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{9}{40} \cdot {x}^{2} - \mathsf{fma}\left(\frac{27}{2800}, {x}^{4}, \frac{1}{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 +o rules:numerics
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))