Average Error: 9.7 → 0.1
Time: 18.6s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -105.47328403477972:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{\frac{\frac{2}{x}}{x}}{x} + \frac{2}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 124.52448790004959:\\ \;\;\;\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{\frac{\frac{2}{x}}{x}}{x} + \frac{2}{{x}^{5}}\right)\\ \end{array}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -105.47328403477972:\\
\;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{\frac{\frac{2}{x}}{x}}{x} + \frac{2}{{x}^{5}}\right)\\

\mathbf{elif}\;x \le 124.52448790004959:\\
\;\;\;\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1}\\

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

\end{array}
double f(double x) {
        double r3186967 = 1.0;
        double r3186968 = x;
        double r3186969 = r3186968 + r3186967;
        double r3186970 = r3186967 / r3186969;
        double r3186971 = 2.0;
        double r3186972 = r3186971 / r3186968;
        double r3186973 = r3186970 - r3186972;
        double r3186974 = r3186968 - r3186967;
        double r3186975 = r3186967 / r3186974;
        double r3186976 = r3186973 + r3186975;
        return r3186976;
}
↓
double f(double x) {
        double r3186977 = x;
        double r3186978 = -105.47328403477972;
        bool r3186979 = r3186977 <= r3186978;
        double r3186980 = 2.0;
        double r3186981 = 7.0;
        double r3186982 = pow(r3186977, r3186981);
        double r3186983 = r3186980 / r3186982;
        double r3186984 = r3186980 / r3186977;
        double r3186985 = r3186984 / r3186977;
        double r3186986 = r3186985 / r3186977;
        double r3186987 = 5.0;
        double r3186988 = pow(r3186977, r3186987);
        double r3186989 = r3186980 / r3186988;
        double r3186990 = r3186986 + r3186989;
        double r3186991 = r3186983 + r3186990;
        double r3186992 = 124.52448790004959;
        bool r3186993 = r3186977 <= r3186992;
        double r3186994 = 1.0;
        double r3186995 = r3186994 + r3186977;
        double r3186996 = r3186994 / r3186995;
        double r3186997 = r3186996 - r3186984;
        double r3186998 = r3186977 - r3186994;
        double r3186999 = r3186994 / r3186998;
        double r3187000 = r3186997 + r3186999;
        double r3187001 = r3186993 ? r3187000 : r3186991;
        double r3187002 = r3186979 ? r3186991 : r3187001;
        return r3187002;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.7
Target0.3
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -105.47328403477972 or 124.52448790004959 < x

    1. Initial program 19.9

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

    2. Taylor expanded around inf 0.6

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

    3. Simplified0.1

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

    5. Applied associate-/r*0.1

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

    if -105.47328403477972 < x < 124.52448790004959

    1. Initial program 0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -105.47328403477972:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{\frac{\frac{2}{x}}{x}}{x} + \frac{2}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 124.52448790004959:\\ \;\;\;\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{\frac{\frac{2}{x}}{x}}{x} + \frac{2}{{x}^{5}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019137 
(FPCore (x)
  :name "3frac (problem 3.3.3)"

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

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