Average Error: 31.5 → 0.2
Time: 13.2s
Precision: 64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.566653604556901724365047812170814722776:\\ \;\;\;\;\sqrt{\frac{x - \sin x}{x - \tan x}} \cdot \sqrt{\frac{x - \sin x}{x - \tan x}}\\ \mathbf{elif}\;x \le 2.419818388459178848393094085622578859329:\\ \;\;\;\;\left(\frac{9}{40} \cdot {x}^{2} - \frac{27}{2800} \cdot {x}^{4}\right) - \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sin x}{x \cdot \cos x} + \left(\frac{{\left(\sin x\right)}^{2}}{{x}^{2} \cdot {\left(\cos x\right)}^{2}} + 1\right)\right) - \left(\frac{\sin x}{x} + \frac{{\left(\sin x\right)}^{2}}{{x}^{2} \cdot \cos x}\right)\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.566653604556901724365047812170814722776:\\
\;\;\;\;\sqrt{\frac{x - \sin x}{x - \tan x}} \cdot \sqrt{\frac{x - \sin x}{x - \tan x}}\\

\mathbf{elif}\;x \le 2.419818388459178848393094085622578859329:\\
\;\;\;\;\left(\frac{9}{40} \cdot {x}^{2} - \frac{27}{2800} \cdot {x}^{4}\right) - \frac{1}{2}\\

\mathbf{else}:\\
\;\;\;\;\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)\\

\end{array}
double f(double x) {
        double r19053 = x;
        double r19054 = sin(r19053);
        double r19055 = r19053 - r19054;
        double r19056 = tan(r19053);
        double r19057 = r19053 - r19056;
        double r19058 = r19055 / r19057;
        return r19058;
}
↓
double f(double x) {
        double r19059 = x;
        double r19060 = -1.5666536045569017;
        bool r19061 = r19059 <= r19060;
        double r19062 = sin(r19059);
        double r19063 = r19059 - r19062;
        double r19064 = tan(r19059);
        double r19065 = r19059 - r19064;
        double r19066 = r19063 / r19065;
        double r19067 = sqrt(r19066);
        double r19068 = r19067 * r19067;
        double r19069 = 2.419818388459179;
        bool r19070 = r19059 <= r19069;
        double r19071 = 0.225;
        double r19072 = 2.0;
        double r19073 = pow(r19059, r19072);
        double r19074 = r19071 * r19073;
        double r19075 = 0.009642857142857142;
        double r19076 = 4.0;
        double r19077 = pow(r19059, r19076);
        double r19078 = r19075 * r19077;
        double r19079 = r19074 - r19078;
        double r19080 = 0.5;
        double r19081 = r19079 - r19080;
        double r19082 = cos(r19059);
        double r19083 = r19059 * r19082;
        double r19084 = r19062 / r19083;
        double r19085 = pow(r19062, r19072);
        double r19086 = pow(r19082, r19072);
        double r19087 = r19073 * r19086;
        double r19088 = r19085 / r19087;
        double r19089 = 1.0;
        double r19090 = r19088 + r19089;
        double r19091 = r19084 + r19090;
        double r19092 = r19062 / r19059;
        double r19093 = r19073 * r19082;
        double r19094 = r19085 / r19093;
        double r19095 = r19092 + r19094;
        double r19096 = r19091 - r19095;
        double r19097 = r19070 ? r19081 : r19096;
        double r19098 = r19061 ? r19068 : r19097;
        return r19098;
}

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 < -1.5666536045569017

    1. Initial program 0.0

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

    3. Applied add-sqr-sqrt0.0

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

    if -1.5666536045569017 < x < 2.419818388459179

    1. Initial program 62.7

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

    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

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

    if 2.419818388459179 < x

    1. Initial program 0.0

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

    2. Taylor expanded around inf 0.4

      \[\leadsto \color{blue}{\left(\frac{\sin x}{x \cdot \cos x} + \left(\frac{{\left(\sin x\right)}^{2}}{{x}^{2} \cdot {\left(\cos x\right)}^{2}} + 1\right)\right) - \left(\frac{\sin x}{x} + \frac{{\left(\sin x\right)}^{2}}{{x}^{2} \cdot \cos x}\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.566653604556901724365047812170814722776:\\ \;\;\;\;\sqrt{\frac{x - \sin x}{x - \tan x}} \cdot \sqrt{\frac{x - \sin x}{x - \tan x}}\\ \mathbf{elif}\;x \le 2.419818388459178848393094085622578859329:\\ \;\;\;\;\left(\frac{9}{40} \cdot {x}^{2} - \frac{27}{2800} \cdot {x}^{4}\right) - \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sin x}{x \cdot \cos x} + \left(\frac{{\left(\sin x\right)}^{2}}{{x}^{2} \cdot {\left(\cos x\right)}^{2}} + 1\right)\right) - \left(\frac{\sin x}{x} + \frac{{\left(\sin x\right)}^{2}}{{x}^{2} \cdot \cos x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019318 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))