Average Error: 0.0 → 0.0
Time: 16.6s
Precision: 64
\[\frac{1}{x - 1} + \frac{x}{x + 1}\]
↓
\[\frac{\mathsf{fma}\left(\frac{x}{1 + x}, \frac{x}{1 + x} \cdot \frac{x}{1 + x}, \frac{\frac{1}{x - 1} \cdot \frac{1}{x - 1}}{x - 1}\right)}{\mathsf{fma}\left(\frac{x}{1 + x}, \frac{x}{1 + x} - \frac{1}{x - 1}, \frac{1}{x - 1} \cdot \frac{1}{x - 1}\right)}\]
\frac{1}{x - 1} + \frac{x}{x + 1}
↓
\frac{\mathsf{fma}\left(\frac{x}{1 + x}, \frac{x}{1 + x} \cdot \frac{x}{1 + x}, \frac{\frac{1}{x - 1} \cdot \frac{1}{x - 1}}{x - 1}\right)}{\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 r2387329 = 1.0;
        double r2387330 = x;
        double r2387331 = r2387330 - r2387329;
        double r2387332 = r2387329 / r2387331;
        double r2387333 = r2387330 + r2387329;
        double r2387334 = r2387330 / r2387333;
        double r2387335 = r2387332 + r2387334;
        return r2387335;
}
↓
double f(double x) {
        double r2387336 = x;
        double r2387337 = 1.0;
        double r2387338 = r2387337 + r2387336;
        double r2387339 = r2387336 / r2387338;
        double r2387340 = r2387339 * r2387339;
        double r2387341 = r2387336 - r2387337;
        double r2387342 = r2387337 / r2387341;
        double r2387343 = r2387342 * r2387342;
        double r2387344 = r2387343 / r2387341;
        double r2387345 = fma(r2387339, r2387340, r2387344);
        double r2387346 = r2387339 - r2387342;
        double r2387347 = fma(r2387339, r2387346, r2387343);
        double r2387348 = r2387345 / r2387347;
        return r2387348;
}

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 flip3-+0.0

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

  4. Simplified0.0

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

  5. Simplified0.0

    \[\leadsto \frac{\mathsf{fma}\left(\frac{x}{1 + x}, \frac{x}{1 + x} \cdot \frac{x}{1 + x}, \frac{1}{x - 1} \cdot \left(\frac{1}{x - 1} \cdot \frac{1}{x - 1}\right)\right)}{\color{blue}{\mathsf{fma}\left(\frac{x}{1 + x}, \frac{x}{1 + x} - \frac{1}{x - 1}, \frac{1}{x - 1} \cdot \frac{1}{x - 1}\right)}}\]
  6. Using strategy rm

  7. Applied un-div-inv0.0

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

  8. Applied associate-*r/0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (x)
  :name "Asymptote B"
  (+ (/ 1 (- x 1)) (/ x (+ x 1))))