Average Error: 9.8 → 0.0
Time: 12.6s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -118.866645563045708 \lor \neg \left(x \le 111.79023992954416\right):\\ \;\;\;\;2 \cdot \left(\left(\frac{1}{{x}^{5}} + {x}^{-3}\right) + \frac{1}{{x}^{7}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x \cdot x - 1 \cdot 1} \cdot \left(x + 1\right)\\ \end{array}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -118.866645563045708 \lor \neg \left(x \le 111.79023992954416\right):\\
\;\;\;\;2 \cdot \left(\left(\frac{1}{{x}^{5}} + {x}^{-3}\right) + \frac{1}{{x}^{7}}\right)\\

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

\end{array}
double f(double x) {
        double r106200 = 1.0;
        double r106201 = x;
        double r106202 = r106201 + r106200;
        double r106203 = r106200 / r106202;
        double r106204 = 2.0;
        double r106205 = r106204 / r106201;
        double r106206 = r106203 - r106205;
        double r106207 = r106201 - r106200;
        double r106208 = r106200 / r106207;
        double r106209 = r106206 + r106208;
        return r106209;
}
↓
double f(double x) {
        double r106210 = x;
        double r106211 = -118.86664556304571;
        bool r106212 = r106210 <= r106211;
        double r106213 = 111.79023992954416;
        bool r106214 = r106210 <= r106213;
        double r106215 = !r106214;
        bool r106216 = r106212 || r106215;
        double r106217 = 2.0;
        double r106218 = 1.0;
        double r106219 = 5.0;
        double r106220 = pow(r106210, r106219);
        double r106221 = r106218 / r106220;
        double r106222 = -3.0;
        double r106223 = pow(r106210, r106222);
        double r106224 = r106221 + r106223;
        double r106225 = 7.0;
        double r106226 = pow(r106210, r106225);
        double r106227 = r106218 / r106226;
        double r106228 = r106224 + r106227;
        double r106229 = r106217 * r106228;
        double r106230 = 1.0;
        double r106231 = r106210 + r106230;
        double r106232 = r106230 / r106231;
        double r106233 = r106217 / r106210;
        double r106234 = r106232 - r106233;
        double r106235 = r106210 * r106210;
        double r106236 = r106230 * r106230;
        double r106237 = r106235 - r106236;
        double r106238 = r106230 / r106237;
        double r106239 = r106238 * r106231;
        double r106240 = r106234 + r106239;
        double r106241 = r106216 ? r106229 : r106240;
        return r106241;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.8
Target0.3
Herbie0.0
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -118.86664556304571 or 111.79023992954416 < x

    1. Initial program 19.7

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

    3. Applied flip--49.8

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

    4. Applied associate-/r/52.8

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

    5. Taylor expanded around inf 0.5

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

    6. Simplified0.5

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

    8. Applied pow-flip0.0

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

    9. Simplified0.0

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

    if -118.86664556304571 < x < 111.79023992954416

    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--0.0

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

    4. Applied associate-/r/0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -118.866645563045708 \lor \neg \left(x \le 111.79023992954416\right):\\ \;\;\;\;2 \cdot \left(\left(\frac{1}{{x}^{5}} + {x}^{-3}\right) + \frac{1}{{x}^{7}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x \cdot x - 1 \cdot 1} \cdot \left(x + 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(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))))