Average Error: 9.9 → 0.2
Time: 10.5s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -128.46375794518042:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{1}{x \cdot x} \cdot \frac{2}{x} + \frac{2}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 110.41034748839138:\\ \;\;\;\;\frac{x - \left(1 + x\right) \cdot 2}{x \cdot \left(1 + x\right)} + \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{x \cdot \left(x \cdot x\right)} + \frac{2}{{x}^{5}}\right) + \frac{2}{{x}^{7}}\\ \end{array}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -128.46375794518042:\\
\;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{1}{x \cdot x} \cdot \frac{2}{x} + \frac{2}{{x}^{5}}\right)\\

\mathbf{elif}\;x \le 110.41034748839138:\\
\;\;\;\;\frac{x - \left(1 + x\right) \cdot 2}{x \cdot \left(1 + x\right)} + \frac{1}{x - 1}\\

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

\end{array}
double f(double x) {
        double r1541716 = 1.0;
        double r1541717 = x;
        double r1541718 = r1541717 + r1541716;
        double r1541719 = r1541716 / r1541718;
        double r1541720 = 2.0;
        double r1541721 = r1541720 / r1541717;
        double r1541722 = r1541719 - r1541721;
        double r1541723 = r1541717 - r1541716;
        double r1541724 = r1541716 / r1541723;
        double r1541725 = r1541722 + r1541724;
        return r1541725;
}
↓
double f(double x) {
        double r1541726 = x;
        double r1541727 = -128.46375794518042;
        bool r1541728 = r1541726 <= r1541727;
        double r1541729 = 2.0;
        double r1541730 = 7.0;
        double r1541731 = pow(r1541726, r1541730);
        double r1541732 = r1541729 / r1541731;
        double r1541733 = 1.0;
        double r1541734 = r1541726 * r1541726;
        double r1541735 = r1541733 / r1541734;
        double r1541736 = r1541729 / r1541726;
        double r1541737 = r1541735 * r1541736;
        double r1541738 = 5.0;
        double r1541739 = pow(r1541726, r1541738);
        double r1541740 = r1541729 / r1541739;
        double r1541741 = r1541737 + r1541740;
        double r1541742 = r1541732 + r1541741;
        double r1541743 = 110.41034748839138;
        bool r1541744 = r1541726 <= r1541743;
        double r1541745 = r1541733 + r1541726;
        double r1541746 = r1541745 * r1541729;
        double r1541747 = r1541726 - r1541746;
        double r1541748 = r1541726 * r1541745;
        double r1541749 = r1541747 / r1541748;
        double r1541750 = r1541726 - r1541733;
        double r1541751 = r1541733 / r1541750;
        double r1541752 = r1541749 + r1541751;
        double r1541753 = r1541726 * r1541734;
        double r1541754 = r1541729 / r1541753;
        double r1541755 = r1541754 + r1541740;
        double r1541756 = r1541755 + r1541732;
        double r1541757 = r1541744 ? r1541752 : r1541756;
        double r1541758 = r1541728 ? r1541742 : r1541757;
        return r1541758;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.9
Target0.3
Herbie0.2
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -128.46375794518042

    1. Initial program 19.2

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

    3. Applied frac-sub52.7

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

    4. Taylor expanded around inf 0.6

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

    5. Simplified0.6

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

    7. Applied *-un-lft-identity0.6

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

    8. Applied times-frac0.1

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

    if -128.46375794518042 < x < 110.41034748839138

    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-sub0.1

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

    if 110.41034748839138 < x

    1. Initial program 20.4

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

    3. Applied frac-sub53.1

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

    4. Taylor expanded around inf 0.5

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

    5. Simplified0.5

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -128.46375794518042:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{1}{x \cdot x} \cdot \frac{2}{x} + \frac{2}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 110.41034748839138:\\ \;\;\;\;\frac{x - \left(1 + x\right) \cdot 2}{x \cdot \left(1 + x\right)} + \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{x \cdot \left(x \cdot x\right)} + \frac{2}{{x}^{5}}\right) + \frac{2}{{x}^{7}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 
(FPCore (x)
  :name "3frac (problem 3.3.3)"

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

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