Average Error: 10.1 → 0.3
Time: 4.6s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -133358930.129605 \lor \neg \left(x \le 449.06402820041797\right):\\ \;\;\;\;2 \cdot \left(\frac{1}{{x}^{7}} + \left(\frac{1}{{x}^{5}} + \frac{1}{{x}^{3}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 \cdot x - \left(x + 1\right) \cdot 2\right) \cdot \left(x - 1\right) + \left(\left(x + 1\right) \cdot x\right) \cdot 1}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}\\ \end{array}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -133358930.129605 \lor \neg \left(x \le 449.06402820041797\right):\\
\;\;\;\;2 \cdot \left(\frac{1}{{x}^{7}} + \left(\frac{1}{{x}^{5}} + \frac{1}{{x}^{3}}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(1 \cdot x - \left(x + 1\right) \cdot 2\right) \cdot \left(x - 1\right) + \left(\left(x + 1\right) \cdot x\right) \cdot 1}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}\\

\end{array}
double f(double x) {
        double r162000 = 1.0;
        double r162001 = x;
        double r162002 = r162001 + r162000;
        double r162003 = r162000 / r162002;
        double r162004 = 2.0;
        double r162005 = r162004 / r162001;
        double r162006 = r162003 - r162005;
        double r162007 = r162001 - r162000;
        double r162008 = r162000 / r162007;
        double r162009 = r162006 + r162008;
        return r162009;
}
↓
double f(double x) {
        double r162010 = x;
        double r162011 = -133358930.129605;
        bool r162012 = r162010 <= r162011;
        double r162013 = 449.06402820041797;
        bool r162014 = r162010 <= r162013;
        double r162015 = !r162014;
        bool r162016 = r162012 || r162015;
        double r162017 = 2.0;
        double r162018 = 1.0;
        double r162019 = 7.0;
        double r162020 = pow(r162010, r162019);
        double r162021 = r162018 / r162020;
        double r162022 = 5.0;
        double r162023 = pow(r162010, r162022);
        double r162024 = r162018 / r162023;
        double r162025 = 3.0;
        double r162026 = pow(r162010, r162025);
        double r162027 = r162018 / r162026;
        double r162028 = r162024 + r162027;
        double r162029 = r162021 + r162028;
        double r162030 = r162017 * r162029;
        double r162031 = 1.0;
        double r162032 = r162031 * r162010;
        double r162033 = r162010 + r162031;
        double r162034 = r162033 * r162017;
        double r162035 = r162032 - r162034;
        double r162036 = r162010 - r162031;
        double r162037 = r162035 * r162036;
        double r162038 = r162033 * r162010;
        double r162039 = r162038 * r162031;
        double r162040 = r162037 + r162039;
        double r162041 = r162038 * r162036;
        double r162042 = r162040 / r162041;
        double r162043 = r162016 ? r162030 : r162042;
        return r162043;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original10.1
Target0.3
Herbie0.3
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -133358930.129605 or 449.06402820041797 < x

    1. Initial program 20.0

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]

    2. Taylor expanded around inf 0.6

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{x}^{7}} + \left(2 \cdot \frac{1}{{x}^{5}} + 2 \cdot \frac{1}{{x}^{3}}\right)}\]

    3. Simplified0.6

      \[\leadsto \color{blue}{2 \cdot \left(\frac{1}{{x}^{7}} + \left(\frac{1}{{x}^{5}} + \frac{1}{{x}^{3}}\right)\right)}\]

    if -133358930.129605 < x < 449.06402820041797

    1. Initial program 0.5

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
    2. Using strategy rm

    3. Applied frac-sub0.5

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

    4. Applied frac-add0.0

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -133358930.129605 \lor \neg \left(x \le 449.06402820041797\right):\\ \;\;\;\;2 \cdot \left(\frac{1}{{x}^{7}} + \left(\frac{1}{{x}^{5}} + \frac{1}{{x}^{3}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 \cdot x - \left(x + 1\right) \cdot 2\right) \cdot \left(x - 1\right) + \left(\left(x + 1\right) \cdot x\right) \cdot 1}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64

  :herbie-target
  (/ 2 (* x (- (* x x) 1)))

  (+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1))))