Average Error: 9.8 → 0.1
Time: 28.2s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -103.0029928299844:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{\frac{2}{x}}{x \cdot x} + \frac{2}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 109.24087107751738:\\ \;\;\;\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{\frac{2}{x}}{x \cdot x} + \frac{2}{{x}^{5}}\right)\\ \end{array}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -103.0029928299844:\\
\;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{\frac{2}{x}}{x \cdot x} + \frac{2}{{x}^{5}}\right)\\

\mathbf{elif}\;x \le 109.24087107751738:\\
\;\;\;\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1}\\

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

\end{array}
double f(double x) {
        double r3052083 = 1.0;
        double r3052084 = x;
        double r3052085 = r3052084 + r3052083;
        double r3052086 = r3052083 / r3052085;
        double r3052087 = 2.0;
        double r3052088 = r3052087 / r3052084;
        double r3052089 = r3052086 - r3052088;
        double r3052090 = r3052084 - r3052083;
        double r3052091 = r3052083 / r3052090;
        double r3052092 = r3052089 + r3052091;
        return r3052092;
}
↓
double f(double x) {
        double r3052093 = x;
        double r3052094 = -103.0029928299844;
        bool r3052095 = r3052093 <= r3052094;
        double r3052096 = 2.0;
        double r3052097 = 7.0;
        double r3052098 = pow(r3052093, r3052097);
        double r3052099 = r3052096 / r3052098;
        double r3052100 = r3052096 / r3052093;
        double r3052101 = r3052093 * r3052093;
        double r3052102 = r3052100 / r3052101;
        double r3052103 = 5.0;
        double r3052104 = pow(r3052093, r3052103);
        double r3052105 = r3052096 / r3052104;
        double r3052106 = r3052102 + r3052105;
        double r3052107 = r3052099 + r3052106;
        double r3052108 = 109.24087107751738;
        bool r3052109 = r3052093 <= r3052108;
        double r3052110 = 1.0;
        double r3052111 = r3052110 + r3052093;
        double r3052112 = r3052110 / r3052111;
        double r3052113 = r3052112 - r3052100;
        double r3052114 = r3052093 - r3052110;
        double r3052115 = r3052110 / r3052114;
        double r3052116 = r3052113 + r3052115;
        double r3052117 = r3052109 ? r3052116 : r3052107;
        double r3052118 = r3052095 ? r3052107 : r3052117;
        return r3052118;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Split input into 2 regimes

  2. if x < -103.0029928299844 or 109.24087107751738 < x

    1. Initial program 19.7

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

    2. Taylor expanded around -inf 0.7

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

    3. Simplified0.1

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

    if -103.0029928299844 < x < 109.24087107751738

    1. Initial program 0.0

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

    3. Applied flip-+29.4

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

    5. Applied *-un-lft-identity29.4

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

    6. Applied *-un-lft-identity29.4

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

    7. Applied div-inv29.4

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

    8. Applied distribute-lft-out--29.4

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

    9. Applied distribute-lft-out--29.4

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

    10. Applied difference-of-squares29.4

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

    11. Applied times-frac0.0

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

    12. Simplified0.0

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

    13. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -103.0029928299844:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{\frac{2}{x}}{x \cdot x} + \frac{2}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 109.24087107751738:\\ \;\;\;\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{\frac{2}{x}}{x \cdot x} + \frac{2}{{x}^{5}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019132 
(FPCore (x)
  :name "3frac (problem 3.3.3)"

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

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