Average Error: 9.8 → 0.7
Time: 6.9s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \le \frac{-1596917391303941}{33554432} \lor \neg \left(\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \le \frac{1}{19807040628566084398385987584}\right):\\ \;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{1}{{x}^{7}} + \left(\frac{1}{{x}^{5}} + {x}^{\left(-3\right)}\right)\right)\\ \end{array}\]
\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 \frac{-1596917391303941}{33554432} \lor \neg \left(\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \le \frac{1}{19807040628566084398385987584}\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 r421965 = 1.0;
        double r421966 = x;
        double r421967 = r421966 + r421965;
        double r421968 = r421965 / r421967;
        double r421969 = 2.0;
        double r421970 = r421969 / r421966;
        double r421971 = r421968 - r421970;
        double r421972 = r421966 - r421965;
        double r421973 = r421965 / r421972;
        double r421974 = r421971 + r421973;
        return r421974;
}
↓
double f(double x) {
        double r421975 = 1.0;
        double r421976 = x;
        double r421977 = r421976 + r421975;
        double r421978 = r421975 / r421977;
        double r421979 = 2.0;
        double r421980 = r421979 / r421976;
        double r421981 = r421978 - r421980;
        double r421982 = r421976 - r421975;
        double r421983 = r421975 / r421982;
        double r421984 = r421981 + r421983;
        double r421985 = -1596917391303941.0;
        double r421986 = 33554432.0;
        double r421987 = r421985 / r421986;
        bool r421988 = r421984 <= r421987;
        double r421989 = 1.9807040628566084e+28;
        double r421990 = r421975 / r421989;
        bool r421991 = r421984 <= r421990;
        double r421992 = !r421991;
        bool r421993 = r421988 || r421992;
        double r421994 = 1.0;
        double r421995 = 7.0;
        double r421996 = pow(r421976, r421995);
        double r421997 = r421994 / r421996;
        double r421998 = 5.0;
        double r421999 = pow(r421976, r421998);
        double r422000 = r421994 / r421999;
        double r422001 = 3.0;
        double r422002 = -r422001;
        double r422003 = pow(r421976, r422002);
        double r422004 = r422000 + r422003;
        double r422005 = r421997 + r422004;
        double r422006 = r421979 * r422005;
        double r422007 = r421993 ? r421984 : r422006;
        return r422007;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.8
Target0.3
Herbie0.7
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))) < -47591846.92215744

    1. Initial program 0.0

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

    3. Applied frac-sub0.0

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

    if -47591846.92215744 < (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))) < 5.048709793414476e-29

    1. Initial program 19.4

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

    2. Taylor expanded around inf 1.4

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

    3. Simplified1.4

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

    5. Applied pow-flip0.9

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

    if 5.048709793414476e-29 < (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0)))

    1. Initial program 0.9

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \le \frac{-1596917391303941}{33554432} \lor \neg \left(\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \le \frac{1}{19807040628566084398385987584}\right):\\ \;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{1}{{x}^{7}} + \left(\frac{1}{{x}^{5}} + {x}^{\left(-3\right)}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 350497007 
(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))))