Average Error: 9.9 → 0.2
Time: 3.7s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \le -11.0263644470328099 \lor \neg \left(\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \le 2.31611 \cdot 10^{-23}\right):\\ \;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{1}{{x}^{7}} + \left(\frac{1}{{x}^{5}} + {x}^{\left(-3\right)}\right)\right)\\ \end{array}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \le -11.0263644470328099 \lor \neg \left(\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \le 2.31611 \cdot 10^{-23}\right):\\
\;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\\

\mathbf{else}:\\
\;\;\;\;2 \cdot \left(\frac{1}{{x}^{7}} + \left(\frac{1}{{x}^{5}} + {x}^{\left(-3\right)}\right)\right)\\

\end{array}
double f(double x) {
        double r137961 = 1.0;
        double r137962 = x;
        double r137963 = r137962 + r137961;
        double r137964 = r137961 / r137963;
        double r137965 = 2.0;
        double r137966 = r137965 / r137962;
        double r137967 = r137964 - r137966;
        double r137968 = r137962 - r137961;
        double r137969 = r137961 / r137968;
        double r137970 = r137967 + r137969;
        return r137970;
}
↓
double f(double x) {
        double r137971 = 1.0;
        double r137972 = x;
        double r137973 = r137972 + r137971;
        double r137974 = r137971 / r137973;
        double r137975 = 2.0;
        double r137976 = r137975 / r137972;
        double r137977 = r137974 - r137976;
        double r137978 = r137972 - r137971;
        double r137979 = r137971 / r137978;
        double r137980 = r137977 + r137979;
        double r137981 = -11.02636444703281;
        bool r137982 = r137980 <= r137981;
        double r137983 = 2.3161057151484775e-23;
        bool r137984 = r137980 <= r137983;
        double r137985 = !r137984;
        bool r137986 = r137982 || r137985;
        double r137987 = 1.0;
        double r137988 = 7.0;
        double r137989 = pow(r137972, r137988);
        double r137990 = r137987 / r137989;
        double r137991 = 5.0;
        double r137992 = pow(r137972, r137991);
        double r137993 = r137987 / r137992;
        double r137994 = 3.0;
        double r137995 = -r137994;
        double r137996 = pow(r137972, r137995);
        double r137997 = r137993 + r137996;
        double r137998 = r137990 + r137997;
        double r137999 = r137975 * r137998;
        double r138000 = r137986 ? r137980 : r137999;
        return r138000;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.9
Target0.2
Herbie0.2
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Split input into 2 regimes

  2. if (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))) < -11.02636444703281 or 2.3161057151484775e-23 < (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0)))

    1. Initial program 0.3

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

    if -11.02636444703281 < (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))) < 2.3161057151484775e-23

    1. Initial program 19.5

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

    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

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

    5. Applied pow-flip0.2

      \[\leadsto 2 \cdot \left(\frac{1}{{x}^{7}} + \left(\frac{1}{{x}^{5}} + \color{blue}{{x}^{\left(-3\right)}}\right)\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \le -11.0263644470328099 \lor \neg \left(\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \le 2.31611 \cdot 10^{-23}\right):\\ \;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{1}{{x}^{7}} + \left(\frac{1}{{x}^{5}} + {x}^{\left(-3\right)}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 
(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))))