Average Error: 10.1 → 0.2
Time: 27.1s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -115.81600190160799:\\ \;\;\;\;\frac{2}{{x}^{3}} + \left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 109.21127232659:\\ \;\;\;\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{5}}\right) + \frac{\frac{1}{\frac{x}{\frac{2}{x}}}}{x}\\ \end{array}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -115.81600190160799:\\
\;\;\;\;\frac{2}{{x}^{3}} + \left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{5}}\right)\\

\mathbf{elif}\;x \le 109.21127232659:\\
\;\;\;\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{5}}\right) + \frac{\frac{1}{\frac{x}{\frac{2}{x}}}}{x}\\

\end{array}
double f(double x) {
        double r3330053 = 1.0;
        double r3330054 = x;
        double r3330055 = r3330054 + r3330053;
        double r3330056 = r3330053 / r3330055;
        double r3330057 = 2.0;
        double r3330058 = r3330057 / r3330054;
        double r3330059 = r3330056 - r3330058;
        double r3330060 = r3330054 - r3330053;
        double r3330061 = r3330053 / r3330060;
        double r3330062 = r3330059 + r3330061;
        return r3330062;
}
↓
double f(double x) {
        double r3330063 = x;
        double r3330064 = -115.81600190160799;
        bool r3330065 = r3330063 <= r3330064;
        double r3330066 = 2.0;
        double r3330067 = 3.0;
        double r3330068 = pow(r3330063, r3330067);
        double r3330069 = r3330066 / r3330068;
        double r3330070 = 7.0;
        double r3330071 = pow(r3330063, r3330070);
        double r3330072 = r3330066 / r3330071;
        double r3330073 = 5.0;
        double r3330074 = pow(r3330063, r3330073);
        double r3330075 = r3330066 / r3330074;
        double r3330076 = r3330072 + r3330075;
        double r3330077 = r3330069 + r3330076;
        double r3330078 = 109.21127232659;
        bool r3330079 = r3330063 <= r3330078;
        double r3330080 = 1.0;
        double r3330081 = r3330080 + r3330063;
        double r3330082 = r3330080 / r3330081;
        double r3330083 = r3330066 / r3330063;
        double r3330084 = r3330082 - r3330083;
        double r3330085 = r3330063 - r3330080;
        double r3330086 = r3330080 / r3330085;
        double r3330087 = r3330084 + r3330086;
        double r3330088 = r3330063 / r3330083;
        double r3330089 = r3330080 / r3330088;
        double r3330090 = r3330089 / r3330063;
        double r3330091 = r3330076 + r3330090;
        double r3330092 = r3330079 ? r3330087 : r3330091;
        double r3330093 = r3330065 ? r3330077 : r3330092;
        return r3330093;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Split input into 3 regimes

  2. if x < -115.81600190160799

    1. Initial program 20.4

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

    2. Taylor expanded around -inf 0.5

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

    3. Simplified0.5

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

    5. Applied pow10.5

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

    6. Applied pow10.5

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

    7. Applied pow-sqr0.5

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

    8. Applied pow10.5

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

    9. Applied pow-prod-up0.5

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

    10. Simplified0.5

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

    if -115.81600190160799 < x < 109.21127232659

    1. Initial program 0.0

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

    if 109.21127232659 < x

    1. Initial program 19.8

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

    2. Taylor expanded around -inf 0.4

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

    3. Simplified0.4

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

    4. Taylor expanded around inf 0.4

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

    5. Simplified0.1

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

    7. Applied associate-/r*0.1

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

    9. Applied clear-num0.2

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{x}{\frac{2}{x}}}}}{x} + \left(\frac{2}{{x}^{5}} + \frac{2}{{x}^{7}}\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -115.81600190160799:\\ \;\;\;\;\frac{2}{{x}^{3}} + \left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 109.21127232659:\\ \;\;\;\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{5}}\right) + \frac{\frac{1}{\frac{x}{\frac{2}{x}}}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019133 +o rules:numerics
(FPCore (x)
  :name "3frac (problem 3.3.3)"

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

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