Average Error: 0.0 → 0.0
Time: 1.8m
Precision: 64
\[\frac{1}{x - 1} + \frac{x}{x + 1}\]
↓
\[\sqrt[3]{\frac{(\left(\frac{x}{1 + x}\right) \cdot \left(\frac{x}{1 + x} \cdot \frac{x}{1 + x}\right) + \left(\frac{\frac{1}{x - 1}}{\left(x - 1\right) \cdot \left(x - 1\right)}\right))_* \cdot \left(\left(\frac{1}{x - 1} \cdot \frac{1}{x - 1} - \frac{x}{1 + x} \cdot \frac{x}{1 + x}\right) \cdot (\left(\frac{x}{1 + x}\right) \cdot \left(\frac{x}{1 + x} \cdot \frac{x}{1 + x}\right) + \left(\frac{\frac{1}{x - 1}}{\left(x - 1\right) \cdot \left(x - 1\right)}\right))_*\right)}{(\left(\frac{x}{1 + x}\right) \cdot \left(\frac{x}{1 + x}\right) + \left(\left(\frac{1}{x - 1} - \frac{x}{1 + x}\right) \cdot \frac{1}{x - 1}\right))_* \cdot \left(\left(\frac{1}{x - 1} - \frac{x}{1 + x}\right) \cdot (\left(\frac{x}{1 + x}\right) \cdot \left(\frac{x}{1 + x}\right) + \left(\left(\frac{1}{x - 1} - \frac{x}{1 + x}\right) \cdot \frac{1}{x - 1}\right))_*\right)}}\]
\frac{1}{x - 1} + \frac{x}{x + 1}
↓
\sqrt[3]{\frac{(\left(\frac{x}{1 + x}\right) \cdot \left(\frac{x}{1 + x} \cdot \frac{x}{1 + x}\right) + \left(\frac{\frac{1}{x - 1}}{\left(x - 1\right) \cdot \left(x - 1\right)}\right))_* \cdot \left(\left(\frac{1}{x - 1} \cdot \frac{1}{x - 1} - \frac{x}{1 + x} \cdot \frac{x}{1 + x}\right) \cdot (\left(\frac{x}{1 + x}\right) \cdot \left(\frac{x}{1 + x} \cdot \frac{x}{1 + x}\right) + \left(\frac{\frac{1}{x - 1}}{\left(x - 1\right) \cdot \left(x - 1\right)}\right))_*\right)}{(\left(\frac{x}{1 + x}\right) \cdot \left(\frac{x}{1 + x}\right) + \left(\left(\frac{1}{x - 1} - \frac{x}{1 + x}\right) \cdot \frac{1}{x - 1}\right))_* \cdot \left(\left(\frac{1}{x - 1} - \frac{x}{1 + x}\right) \cdot (\left(\frac{x}{1 + x}\right) \cdot \left(\frac{x}{1 + x}\right) + \left(\left(\frac{1}{x - 1} - \frac{x}{1 + x}\right) \cdot \frac{1}{x - 1}\right))_*\right)}}
double f(double x) {
        double r27318312 = 1.0;
        double r27318313 = x;
        double r27318314 = r27318313 - r27318312;
        double r27318315 = r27318312 / r27318314;
        double r27318316 = r27318313 + r27318312;
        double r27318317 = r27318313 / r27318316;
        double r27318318 = r27318315 + r27318317;
        return r27318318;
}
↓
double f(double x) {
        double r27318319 = x;
        double r27318320 = 1.0;
        double r27318321 = r27318320 + r27318319;
        double r27318322 = r27318319 / r27318321;
        double r27318323 = r27318322 * r27318322;
        double r27318324 = r27318319 - r27318320;
        double r27318325 = r27318320 / r27318324;
        double r27318326 = r27318324 * r27318324;
        double r27318327 = r27318325 / r27318326;
        double r27318328 = fma(r27318322, r27318323, r27318327);
        double r27318329 = r27318325 * r27318325;
        double r27318330 = r27318329 - r27318323;
        double r27318331 = r27318330 * r27318328;
        double r27318332 = r27318328 * r27318331;
        double r27318333 = r27318325 - r27318322;
        double r27318334 = r27318333 * r27318325;
        double r27318335 = fma(r27318322, r27318322, r27318334);
        double r27318336 = r27318333 * r27318335;
        double r27318337 = r27318335 * r27318336;
        double r27318338 = r27318332 / r27318337;
        double r27318339 = cbrt(r27318338);
        return r27318339;
}

Error

Bits error versus x

Derivation

  1. Initial program 0.0

    \[\frac{1}{x - 1} + \frac{x}{x + 1}\]
  2. Using strategy rm

  3. Applied add-cbrt-cube0.0

    \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\frac{1}{x - 1} + \frac{x}{x + 1}\right) \cdot \left(\frac{1}{x - 1} + \frac{x}{x + 1}\right)\right) \cdot \left(\frac{1}{x - 1} + \frac{x}{x + 1}\right)}}\]
  4. Using strategy rm

  5. Applied flip3-+0.0

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

  6. Applied flip3-+0.0

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

  7. Applied flip-+0.0

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

  8. Applied frac-times0.0

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

  9. Applied frac-times0.0

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

  10. Simplified0.0

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

  11. Simplified0.0

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

  12. Final simplification0.0

    \[\leadsto \sqrt[3]{\frac{(\left(\frac{x}{1 + x}\right) \cdot \left(\frac{x}{1 + x} \cdot \frac{x}{1 + x}\right) + \left(\frac{\frac{1}{x - 1}}{\left(x - 1\right) \cdot \left(x - 1\right)}\right))_* \cdot \left(\left(\frac{1}{x - 1} \cdot \frac{1}{x - 1} - \frac{x}{1 + x} \cdot \frac{x}{1 + x}\right) \cdot (\left(\frac{x}{1 + x}\right) \cdot \left(\frac{x}{1 + x} \cdot \frac{x}{1 + x}\right) + \left(\frac{\frac{1}{x - 1}}{\left(x - 1\right) \cdot \left(x - 1\right)}\right))_*\right)}{(\left(\frac{x}{1 + x}\right) \cdot \left(\frac{x}{1 + x}\right) + \left(\left(\frac{1}{x - 1} - \frac{x}{1 + x}\right) \cdot \frac{1}{x - 1}\right))_* \cdot \left(\left(\frac{1}{x - 1} - \frac{x}{1 + x}\right) \cdot (\left(\frac{x}{1 + x}\right) \cdot \left(\frac{x}{1 + x}\right) + \left(\left(\frac{1}{x - 1} - \frac{x}{1 + x}\right) \cdot \frac{1}{x - 1}\right))_*\right)}}\]

Reproduce

herbie shell --seed 2019119 +o rules:numerics
(FPCore (x)
  :name "Asymptote B"
  (+ (/ 1 (- x 1)) (/ x (+ x 1))))