Average Error: 9.5 → 0.0
Time: 5.0s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -116.2424656228328245788361527957022190094 \lor \neg \left(x \le 120.1050191778223279470694251358509063721\right):\\ \;\;\;\;2 \cdot \left(\frac{1}{{x}^{7}} + \left(\frac{1}{{x}^{5}} + {x}^{\left(-3\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot x - \left(x + 1\right) \cdot 2}{\left(x + 1\right) \cdot x} + \frac{1}{x - 1}\\ \end{array}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -116.2424656228328245788361527957022190094 \lor \neg \left(x \le 120.1050191778223279470694251358509063721\right):\\
\;\;\;\;2 \cdot \left(\frac{1}{{x}^{7}} + \left(\frac{1}{{x}^{5}} + {x}^{\left(-3\right)}\right)\right)\\

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

\end{array}
double f(double x) {
        double r138021 = 1.0;
        double r138022 = x;
        double r138023 = r138022 + r138021;
        double r138024 = r138021 / r138023;
        double r138025 = 2.0;
        double r138026 = r138025 / r138022;
        double r138027 = r138024 - r138026;
        double r138028 = r138022 - r138021;
        double r138029 = r138021 / r138028;
        double r138030 = r138027 + r138029;
        return r138030;
}
↓
double f(double x) {
        double r138031 = x;
        double r138032 = -116.24246562283282;
        bool r138033 = r138031 <= r138032;
        double r138034 = 120.10501917782233;
        bool r138035 = r138031 <= r138034;
        double r138036 = !r138035;
        bool r138037 = r138033 || r138036;
        double r138038 = 2.0;
        double r138039 = 1.0;
        double r138040 = 7.0;
        double r138041 = pow(r138031, r138040);
        double r138042 = r138039 / r138041;
        double r138043 = 5.0;
        double r138044 = pow(r138031, r138043);
        double r138045 = r138039 / r138044;
        double r138046 = 3.0;
        double r138047 = -r138046;
        double r138048 = pow(r138031, r138047);
        double r138049 = r138045 + r138048;
        double r138050 = r138042 + r138049;
        double r138051 = r138038 * r138050;
        double r138052 = 1.0;
        double r138053 = r138052 * r138031;
        double r138054 = r138031 + r138052;
        double r138055 = r138054 * r138038;
        double r138056 = r138053 - r138055;
        double r138057 = r138054 * r138031;
        double r138058 = r138056 / r138057;
        double r138059 = r138031 - r138052;
        double r138060 = r138052 / r138059;
        double r138061 = r138058 + r138060;
        double r138062 = r138037 ? r138051 : r138061;
        return r138062;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.5
Target0.2
Herbie0.0
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -116.24246562283282 or 120.10501917782233 < x

    1. Initial program 19.4

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

    2. Taylor expanded around inf 0.4

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

    3. Simplified0.4

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

    5. Applied pow-flip0.0

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

    if -116.24246562283282 < x < 120.10501917782233

    1. Initial program 0.1

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

    3. Applied frac-sub0.1

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -116.2424656228328245788361527957022190094 \lor \neg \left(x \le 120.1050191778223279470694251358509063721\right):\\ \;\;\;\;2 \cdot \left(\frac{1}{{x}^{7}} + \left(\frac{1}{{x}^{5}} + {x}^{\left(-3\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot x - \left(x + 1\right) \cdot 2}{\left(x + 1\right) \cdot x} + \frac{1}{x - 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019346 
(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))))