Average Error: 0.0 → 0.0
Time: 22.2s
Precision: 64
\[\frac{1}{x - 1} + \frac{x}{x + 1}\]
↓
\[\sqrt[3]{\frac{\mathsf{fma}\left(\frac{x}{1 + x}, \frac{x}{1 + x} \cdot \frac{x}{1 + x}, \left(\frac{1}{x - 1} \cdot \frac{1}{x - 1}\right) \cdot \frac{1}{x - 1}\right) \cdot \left(\mathsf{fma}\left(\frac{x}{1 + x}, \frac{x}{1 + x} \cdot \frac{x}{1 + x}, \left(\frac{1}{x - 1} \cdot \frac{1}{x - 1}\right) \cdot \frac{1}{x - 1}\right) \cdot \mathsf{fma}\left(\frac{x}{1 + x}, \frac{x}{1 + x} \cdot \frac{x}{1 + x}, \left(\frac{1}{x - 1} \cdot \frac{1}{x - 1}\right) \cdot \frac{1}{x - 1}\right)\right)}{\left(\mathsf{fma}\left(\frac{x}{1 + x}, \frac{x}{1 + x} - \frac{1}{x - 1}, \frac{1}{x - 1} \cdot \frac{1}{x - 1}\right) \cdot \mathsf{fma}\left(\frac{x}{1 + x}, \frac{x}{1 + x} - \frac{1}{x - 1}, \frac{1}{x - 1} \cdot \frac{1}{x - 1}\right)\right) \cdot \mathsf{fma}\left(\frac{x}{1 + x}, \frac{x}{1 + x} - \frac{1}{x - 1}, \frac{1}{x - 1} \cdot \frac{1}{x - 1}\right)}}\]
\frac{1}{x - 1} + \frac{x}{x + 1}
↓
\sqrt[3]{\frac{\mathsf{fma}\left(\frac{x}{1 + x}, \frac{x}{1 + x} \cdot \frac{x}{1 + x}, \left(\frac{1}{x - 1} \cdot \frac{1}{x - 1}\right) \cdot \frac{1}{x - 1}\right) \cdot \left(\mathsf{fma}\left(\frac{x}{1 + x}, \frac{x}{1 + x} \cdot \frac{x}{1 + x}, \left(\frac{1}{x - 1} \cdot \frac{1}{x - 1}\right) \cdot \frac{1}{x - 1}\right) \cdot \mathsf{fma}\left(\frac{x}{1 + x}, \frac{x}{1 + x} \cdot \frac{x}{1 + x}, \left(\frac{1}{x - 1} \cdot \frac{1}{x - 1}\right) \cdot \frac{1}{x - 1}\right)\right)}{\left(\mathsf{fma}\left(\frac{x}{1 + x}, \frac{x}{1 + x} - \frac{1}{x - 1}, \frac{1}{x - 1} \cdot \frac{1}{x - 1}\right) \cdot \mathsf{fma}\left(\frac{x}{1 + x}, \frac{x}{1 + x} - \frac{1}{x - 1}, \frac{1}{x - 1} \cdot \frac{1}{x - 1}\right)\right) \cdot \mathsf{fma}\left(\frac{x}{1 + x}, \frac{x}{1 + x} - \frac{1}{x - 1}, \frac{1}{x - 1} \cdot \frac{1}{x - 1}\right)}}
double f(double x) {
        double r3243376 = 1.0;
        double r3243377 = x;
        double r3243378 = r3243377 - r3243376;
        double r3243379 = r3243376 / r3243378;
        double r3243380 = r3243377 + r3243376;
        double r3243381 = r3243377 / r3243380;
        double r3243382 = r3243379 + r3243381;
        return r3243382;
}
↓
double f(double x) {
        double r3243383 = x;
        double r3243384 = 1.0;
        double r3243385 = r3243384 + r3243383;
        double r3243386 = r3243383 / r3243385;
        double r3243387 = r3243386 * r3243386;
        double r3243388 = r3243383 - r3243384;
        double r3243389 = r3243384 / r3243388;
        double r3243390 = r3243389 * r3243389;
        double r3243391 = r3243390 * r3243389;
        double r3243392 = fma(r3243386, r3243387, r3243391);
        double r3243393 = r3243392 * r3243392;
        double r3243394 = r3243392 * r3243393;
        double r3243395 = r3243386 - r3243389;
        double r3243396 = fma(r3243386, r3243395, r3243390);
        double r3243397 = r3243396 * r3243396;
        double r3243398 = r3243397 * r3243396;
        double r3243399 = r3243394 / r3243398;
        double r3243400 = cbrt(r3243399);
        return r3243400;
}

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 flip3-+0.0

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

  8. Applied frac-times0.0

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

  9. Applied frac-times0.0

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

  10. Simplified0.0

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

  11. Simplified0.0

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

  12. Final simplification0.0

    \[\leadsto \sqrt[3]{\frac{\mathsf{fma}\left(\frac{x}{1 + x}, \frac{x}{1 + x} \cdot \frac{x}{1 + x}, \left(\frac{1}{x - 1} \cdot \frac{1}{x - 1}\right) \cdot \frac{1}{x - 1}\right) \cdot \left(\mathsf{fma}\left(\frac{x}{1 + x}, \frac{x}{1 + x} \cdot \frac{x}{1 + x}, \left(\frac{1}{x - 1} \cdot \frac{1}{x - 1}\right) \cdot \frac{1}{x - 1}\right) \cdot \mathsf{fma}\left(\frac{x}{1 + x}, \frac{x}{1 + x} \cdot \frac{x}{1 + x}, \left(\frac{1}{x - 1} \cdot \frac{1}{x - 1}\right) \cdot \frac{1}{x - 1}\right)\right)}{\left(\mathsf{fma}\left(\frac{x}{1 + x}, \frac{x}{1 + x} - \frac{1}{x - 1}, \frac{1}{x - 1} \cdot \frac{1}{x - 1}\right) \cdot \mathsf{fma}\left(\frac{x}{1 + x}, \frac{x}{1 + x} - \frac{1}{x - 1}, \frac{1}{x - 1} \cdot \frac{1}{x - 1}\right)\right) \cdot \mathsf{fma}\left(\frac{x}{1 + x}, \frac{x}{1 + x} - \frac{1}{x - 1}, \frac{1}{x - 1} \cdot \frac{1}{x - 1}\right)}}\]

Reproduce

herbie shell --seed 2019134 +o rules:numerics
(FPCore (x)
  :name "Asymptote B"
  (+ (/ 1 (- x 1)) (/ x (+ x 1))))