Average Error: 9.5 → 0.0
Time: 7.6s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -93.412061148281623 \lor \neg \left(x \le 98.642881580886154\right):\\ \;\;\;\;2 \cdot \left(\frac{1}{{x}^{7}} + \left(\frac{1}{{x}^{5}} + {x}^{\left(-3\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{\frac{1}{\sqrt[3]{x - 1} \cdot \sqrt[3]{x - 1}}}{\sqrt[3]{x - 1}}\\ \end{array}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -93.412061148281623 \lor \neg \left(x \le 98.642881580886154\right):\\
\;\;\;\;2 \cdot \left(\frac{1}{{x}^{7}} + \left(\frac{1}{{x}^{5}} + {x}^{\left(-3\right)}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{\frac{1}{\sqrt[3]{x - 1} \cdot \sqrt[3]{x - 1}}}{\sqrt[3]{x - 1}}\\

\end{array}
double f(double x) {
        double r144250 = 1.0;
        double r144251 = x;
        double r144252 = r144251 + r144250;
        double r144253 = r144250 / r144252;
        double r144254 = 2.0;
        double r144255 = r144254 / r144251;
        double r144256 = r144253 - r144255;
        double r144257 = r144251 - r144250;
        double r144258 = r144250 / r144257;
        double r144259 = r144256 + r144258;
        return r144259;
}
↓
double f(double x) {
        double r144260 = x;
        double r144261 = -93.41206114828162;
        bool r144262 = r144260 <= r144261;
        double r144263 = 98.64288158088615;
        bool r144264 = r144260 <= r144263;
        double r144265 = !r144264;
        bool r144266 = r144262 || r144265;
        double r144267 = 2.0;
        double r144268 = 1.0;
        double r144269 = 7.0;
        double r144270 = pow(r144260, r144269);
        double r144271 = r144268 / r144270;
        double r144272 = 5.0;
        double r144273 = pow(r144260, r144272);
        double r144274 = r144268 / r144273;
        double r144275 = 3.0;
        double r144276 = -r144275;
        double r144277 = pow(r144260, r144276);
        double r144278 = r144274 + r144277;
        double r144279 = r144271 + r144278;
        double r144280 = r144267 * r144279;
        double r144281 = 1.0;
        double r144282 = r144260 + r144281;
        double r144283 = r144281 / r144282;
        double r144284 = r144267 / r144260;
        double r144285 = r144283 - r144284;
        double r144286 = r144260 - r144281;
        double r144287 = cbrt(r144286);
        double r144288 = r144287 * r144287;
        double r144289 = r144281 / r144288;
        double r144290 = r144289 / r144287;
        double r144291 = r144285 + r144290;
        double r144292 = r144266 ? r144280 : r144291;
        return r144292;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.5
Target0.3
Herbie0.0
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -93.41206114828162 or 98.64288158088615 < x

    1. Initial program 19.1

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

    2. Taylor expanded around inf 0.5

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

    3. Simplified0.5

      \[\leadsto \color{blue}{2 \cdot \left(\frac{1}{{x}^{7}} + \left(\frac{1}{{x}^{5}} + \frac{1}{{x}^{3}}\right)\right)}\]
    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 -93.41206114828162 < x < 98.64288158088615

    1. Initial program 0.0

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

    3. Applied add-cube-cbrt0.1

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

    4. Applied associate-/r*0.1

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -93.412061148281623 \lor \neg \left(x \le 98.642881580886154\right):\\ \;\;\;\;2 \cdot \left(\frac{1}{{x}^{7}} + \left(\frac{1}{{x}^{5}} + {x}^{\left(-3\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{\frac{1}{\sqrt[3]{x - 1} \cdot \sqrt[3]{x - 1}}}{\sqrt[3]{x - 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 
(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))))