Average Error: 10.3 → 0.0
Time: 4.0s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -24271205.0909876414 \lor \neg \left(x \le 43997267.71868038\right):\\ \;\;\;\;2 \cdot \left(\frac{1}{{x}^{7}} + \left(\frac{1}{{x}^{5}} + {x}^{\left(-3\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 \cdot x - \left(x + 1\right) \cdot 2\right) \cdot \left(x - 1\right) + \left(\left(x + 1\right) \cdot x\right) \cdot 1}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}\\ \end{array}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -24271205.0909876414 \lor \neg \left(x \le 43997267.71868038\right):\\
\;\;\;\;2 \cdot \left(\frac{1}{{x}^{7}} + \left(\frac{1}{{x}^{5}} + {x}^{\left(-3\right)}\right)\right)\\

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

\end{array}
double f(double x) {
        double r112917 = 1.0;
        double r112918 = x;
        double r112919 = r112918 + r112917;
        double r112920 = r112917 / r112919;
        double r112921 = 2.0;
        double r112922 = r112921 / r112918;
        double r112923 = r112920 - r112922;
        double r112924 = r112918 - r112917;
        double r112925 = r112917 / r112924;
        double r112926 = r112923 + r112925;
        return r112926;
}
↓
double f(double x) {
        double r112927 = x;
        double r112928 = -24271205.09098764;
        bool r112929 = r112927 <= r112928;
        double r112930 = 43997267.71868038;
        bool r112931 = r112927 <= r112930;
        double r112932 = !r112931;
        bool r112933 = r112929 || r112932;
        double r112934 = 2.0;
        double r112935 = 1.0;
        double r112936 = 7.0;
        double r112937 = pow(r112927, r112936);
        double r112938 = r112935 / r112937;
        double r112939 = 5.0;
        double r112940 = pow(r112927, r112939);
        double r112941 = r112935 / r112940;
        double r112942 = 3.0;
        double r112943 = -r112942;
        double r112944 = pow(r112927, r112943);
        double r112945 = r112941 + r112944;
        double r112946 = r112938 + r112945;
        double r112947 = r112934 * r112946;
        double r112948 = 1.0;
        double r112949 = r112948 * r112927;
        double r112950 = r112927 + r112948;
        double r112951 = r112950 * r112934;
        double r112952 = r112949 - r112951;
        double r112953 = r112927 - r112948;
        double r112954 = r112952 * r112953;
        double r112955 = r112950 * r112927;
        double r112956 = r112955 * r112948;
        double r112957 = r112954 + r112956;
        double r112958 = r112955 * r112953;
        double r112959 = r112957 / r112958;
        double r112960 = r112933 ? r112947 : r112959;
        return r112960;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original10.3
Target0.3
Herbie0.0
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -24271205.09098764 or 43997267.71868038 < x

    1. Initial program 20.2

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

    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-flip0.0

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

    if -24271205.09098764 < x < 43997267.71868038

    1. Initial program 0.7

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

    3. Applied frac-sub0.7

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

    4. Applied frac-add0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -24271205.0909876414 \lor \neg \left(x \le 43997267.71868038\right):\\ \;\;\;\;2 \cdot \left(\frac{1}{{x}^{7}} + \left(\frac{1}{{x}^{5}} + {x}^{\left(-3\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 \cdot x - \left(x + 1\right) \cdot 2\right) \cdot \left(x - 1\right) + \left(\left(x + 1\right) \cdot x\right) \cdot 1}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 
(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))))