Average Error: 31.2 → 0.0
Time: 10.4s
Precision: 64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.034878409292342524 \lor \neg \left(x \le 0.025887302823870001\right):\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{9}{40} \cdot {x}^{2}\right)}^{3} - {\left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}^{3}}{{x}^{4} \cdot \left(\frac{81}{1600} + \left(\frac{27}{5600} + {x}^{4} \cdot \frac{729}{7840000}\right)\right) + \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right) \cdot \left(\frac{1}{2} + \frac{9}{40} \cdot {x}^{2}\right)}\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.034878409292342524 \lor \neg \left(x \le 0.025887302823870001\right):\\
\;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\

\mathbf{else}:\\
\;\;\;\;\frac{{\left(\frac{9}{40} \cdot {x}^{2}\right)}^{3} - {\left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}^{3}}{{x}^{4} \cdot \left(\frac{81}{1600} + \left(\frac{27}{5600} + {x}^{4} \cdot \frac{729}{7840000}\right)\right) + \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right) \cdot \left(\frac{1}{2} + \frac{9}{40} \cdot {x}^{2}\right)}\\

\end{array}
double f(double x) {
        double r17579 = x;
        double r17580 = sin(r17579);
        double r17581 = r17579 - r17580;
        double r17582 = tan(r17579);
        double r17583 = r17579 - r17582;
        double r17584 = r17581 / r17583;
        return r17584;
}
↓
double f(double x) {
        double r17585 = x;
        double r17586 = -0.034878409292342524;
        bool r17587 = r17585 <= r17586;
        double r17588 = 0.02588730282387;
        bool r17589 = r17585 <= r17588;
        double r17590 = !r17589;
        bool r17591 = r17587 || r17590;
        double r17592 = tan(r17585);
        double r17593 = r17585 - r17592;
        double r17594 = r17585 / r17593;
        double r17595 = sin(r17585);
        double r17596 = r17595 / r17593;
        double r17597 = r17594 - r17596;
        double r17598 = 0.225;
        double r17599 = 2.0;
        double r17600 = pow(r17585, r17599);
        double r17601 = r17598 * r17600;
        double r17602 = 3.0;
        double r17603 = pow(r17601, r17602);
        double r17604 = 0.009642857142857142;
        double r17605 = 4.0;
        double r17606 = pow(r17585, r17605);
        double r17607 = r17604 * r17606;
        double r17608 = 0.5;
        double r17609 = r17607 + r17608;
        double r17610 = pow(r17609, r17602);
        double r17611 = r17603 - r17610;
        double r17612 = 0.050625;
        double r17613 = 0.004821428571428571;
        double r17614 = 9.298469387755101e-05;
        double r17615 = r17606 * r17614;
        double r17616 = r17613 + r17615;
        double r17617 = r17612 + r17616;
        double r17618 = r17606 * r17617;
        double r17619 = r17608 + r17601;
        double r17620 = r17609 * r17619;
        double r17621 = r17618 + r17620;
        double r17622 = r17611 / r17621;
        double r17623 = r17591 ? r17597 : r17622;
        return r17623;
}

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 < -0.034878409292342524 or 0.02588730282387 < x

    1. Initial program 0.1

      \[\frac{x - \sin x}{x - \tan x}\]
    2. Using strategy rm

    3. Applied div-sub0.1

      \[\leadsto \color{blue}{\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}}\]

    if -0.034878409292342524 < x < 0.02588730282387

    1. Initial program 63.3

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

    2. Taylor expanded around 0 0.0

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

    4. Applied flip3--0.0

      \[\leadsto \color{blue}{\frac{{\left(\frac{9}{40} \cdot {x}^{2}\right)}^{3} - {\left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}^{3}}{\left(\frac{9}{40} \cdot {x}^{2}\right) \cdot \left(\frac{9}{40} \cdot {x}^{2}\right) + \left(\left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right) \cdot \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right) + \left(\frac{9}{40} \cdot {x}^{2}\right) \cdot \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)\right)}}\]

    5. Simplified0.0

      \[\leadsto \frac{{\left(\frac{9}{40} \cdot {x}^{2}\right)}^{3} - {\left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}^{3}}{\color{blue}{{x}^{4} \cdot \left(\frac{81}{1600} + \left(\frac{27}{5600} + {x}^{4} \cdot \frac{729}{7840000}\right)\right) + \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right) \cdot \left(\frac{1}{2} + \frac{9}{40} \cdot {x}^{2}\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.034878409292342524 \lor \neg \left(x \le 0.025887302823870001\right):\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{9}{40} \cdot {x}^{2}\right)}^{3} - {\left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}^{3}}{{x}^{4} \cdot \left(\frac{81}{1600} + \left(\frac{27}{5600} + {x}^{4} \cdot \frac{729}{7840000}\right)\right) + \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right) \cdot \left(\frac{1}{2} + \frac{9}{40} \cdot {x}^{2}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))