Average Error: 31.5 → 0.1
Time: 10.3s
Precision: 64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.01665811831216180982639940566514269448817:\\ \;\;\;\;\frac{1}{\frac{x}{x - \sin x} - \frac{\tan x}{x - \sin x}}\\ \mathbf{elif}\;x \le 0.01529116643609437255213467921066694543697:\\ \;\;\;\;\frac{1}{\left(-{x}^{4} \cdot \frac{81}{100}\right) + \left(-\left(-\left(\frac{513}{1400} \cdot {x}^{4} + 2\right) \cdot \left(\frac{513}{1400} \cdot {x}^{4} + 2\right)\right)\right)} \cdot \left(\frac{9}{10} \cdot {x}^{2} - \left(\frac{513}{1400} \cdot {x}^{4} + 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x - \tan x}{x - \sin x}}\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.01665811831216180982639940566514269448817:\\
\;\;\;\;\frac{1}{\frac{x}{x - \sin x} - \frac{\tan x}{x - \sin x}}\\

\mathbf{elif}\;x \le 0.01529116643609437255213467921066694543697:\\
\;\;\;\;\frac{1}{\left(-{x}^{4} \cdot \frac{81}{100}\right) + \left(-\left(-\left(\frac{513}{1400} \cdot {x}^{4} + 2\right) \cdot \left(\frac{513}{1400} \cdot {x}^{4} + 2\right)\right)\right)} \cdot \left(\frac{9}{10} \cdot {x}^{2} - \left(\frac{513}{1400} \cdot {x}^{4} + 2\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\frac{x - \tan x}{x - \sin x}}\\

\end{array}
double f(double x) {
        double r13549 = x;
        double r13550 = sin(r13549);
        double r13551 = r13549 - r13550;
        double r13552 = tan(r13549);
        double r13553 = r13549 - r13552;
        double r13554 = r13551 / r13553;
        return r13554;
}
↓
double f(double x) {
        double r13555 = x;
        double r13556 = -0.01665811831216181;
        bool r13557 = r13555 <= r13556;
        double r13558 = 1.0;
        double r13559 = sin(r13555);
        double r13560 = r13555 - r13559;
        double r13561 = r13555 / r13560;
        double r13562 = tan(r13555);
        double r13563 = r13562 / r13560;
        double r13564 = r13561 - r13563;
        double r13565 = r13558 / r13564;
        double r13566 = 0.015291166436094373;
        bool r13567 = r13555 <= r13566;
        double r13568 = 4.0;
        double r13569 = pow(r13555, r13568);
        double r13570 = 0.81;
        double r13571 = r13569 * r13570;
        double r13572 = -r13571;
        double r13573 = 0.36642857142857144;
        double r13574 = r13573 * r13569;
        double r13575 = 2.0;
        double r13576 = r13574 + r13575;
        double r13577 = r13576 * r13576;
        double r13578 = -r13577;
        double r13579 = -r13578;
        double r13580 = r13572 + r13579;
        double r13581 = r13558 / r13580;
        double r13582 = 0.9;
        double r13583 = pow(r13555, r13575);
        double r13584 = r13582 * r13583;
        double r13585 = r13584 - r13576;
        double r13586 = r13581 * r13585;
        double r13587 = r13555 - r13562;
        double r13588 = r13587 / r13560;
        double r13589 = r13558 / r13588;
        double r13590 = r13567 ? r13586 : r13589;
        double r13591 = r13557 ? r13565 : r13590;
        return r13591;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if x < -0.01665811831216181

    1. Initial program 0.1

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

    3. Applied clear-num0.1

      \[\leadsto \color{blue}{\frac{1}{\frac{x - \tan x}{x - \sin x}}}\]
    4. Using strategy rm

    5. Applied div-sub0.1

      \[\leadsto \frac{1}{\color{blue}{\frac{x}{x - \sin x} - \frac{\tan x}{x - \sin x}}}\]

    if -0.01665811831216181 < x < 0.015291166436094373

    1. Initial program 63.2

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

    3. Applied clear-num63.2

      \[\leadsto \color{blue}{\frac{1}{\frac{x - \tan x}{x - \sin x}}}\]

    4. Taylor expanded around 0 0.0

      \[\leadsto \frac{1}{\color{blue}{-\left(\frac{9}{10} \cdot {x}^{2} + \left(\frac{513}{1400} \cdot {x}^{4} + 2\right)\right)}}\]
    5. Using strategy rm

    6. Applied flip-+0.0

      \[\leadsto \frac{1}{-\color{blue}{\frac{\left(\frac{9}{10} \cdot {x}^{2}\right) \cdot \left(\frac{9}{10} \cdot {x}^{2}\right) - \left(\frac{513}{1400} \cdot {x}^{4} + 2\right) \cdot \left(\frac{513}{1400} \cdot {x}^{4} + 2\right)}{\frac{9}{10} \cdot {x}^{2} - \left(\frac{513}{1400} \cdot {x}^{4} + 2\right)}}}\]

    7. Applied distribute-neg-frac0.0

      \[\leadsto \frac{1}{\color{blue}{\frac{-\left(\left(\frac{9}{10} \cdot {x}^{2}\right) \cdot \left(\frac{9}{10} \cdot {x}^{2}\right) - \left(\frac{513}{1400} \cdot {x}^{4} + 2\right) \cdot \left(\frac{513}{1400} \cdot {x}^{4} + 2\right)\right)}{\frac{9}{10} \cdot {x}^{2} - \left(\frac{513}{1400} \cdot {x}^{4} + 2\right)}}}\]

    8. Applied associate-/r/0.0

      \[\leadsto \color{blue}{\frac{1}{-\left(\left(\frac{9}{10} \cdot {x}^{2}\right) \cdot \left(\frac{9}{10} \cdot {x}^{2}\right) - \left(\frac{513}{1400} \cdot {x}^{4} + 2\right) \cdot \left(\frac{513}{1400} \cdot {x}^{4} + 2\right)\right)} \cdot \left(\frac{9}{10} \cdot {x}^{2} - \left(\frac{513}{1400} \cdot {x}^{4} + 2\right)\right)}\]

    9. Simplified0.0

      \[\leadsto \color{blue}{\frac{1}{\left(-{x}^{4} \cdot \frac{81}{100}\right) + \left(-\left(-\left(\frac{513}{1400} \cdot {x}^{4} + 2\right) \cdot \left(\frac{513}{1400} \cdot {x}^{4} + 2\right)\right)\right)}} \cdot \left(\frac{9}{10} \cdot {x}^{2} - \left(\frac{513}{1400} \cdot {x}^{4} + 2\right)\right)\]

    if 0.015291166436094373 < x

    1. Initial program 0.1

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

    3. Applied clear-num0.1

      \[\leadsto \color{blue}{\frac{1}{\frac{x - \tan x}{x - \sin x}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.01665811831216180982639940566514269448817:\\ \;\;\;\;\frac{1}{\frac{x}{x - \sin x} - \frac{\tan x}{x - \sin x}}\\ \mathbf{elif}\;x \le 0.01529116643609437255213467921066694543697:\\ \;\;\;\;\frac{1}{\left(-{x}^{4} \cdot \frac{81}{100}\right) + \left(-\left(-\left(\frac{513}{1400} \cdot {x}^{4} + 2\right) \cdot \left(\frac{513}{1400} \cdot {x}^{4} + 2\right)\right)\right)} \cdot \left(\frac{9}{10} \cdot {x}^{2} - \left(\frac{513}{1400} \cdot {x}^{4} + 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x - \tan x}{x - \sin x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))