Average Error: 0.0 → 0.0
Time: 4.1m
Precision: 64
\[\frac{1}{x - 1} + \frac{x}{x + 1}\]
↓
\[\frac{\mathsf{fma}\left(\frac{x}{1 + x} \cdot \frac{x}{1 + x}, \frac{x}{1 + x}, \frac{\frac{1}{x - 1}}{\frac{x - 1}{\frac{1}{x - 1}}}\right)}{\mathsf{fma}\left(\frac{1}{x - 1}, \frac{1}{x - 1}, \left(\frac{x}{1 + x} - \frac{1}{x - 1}\right) \cdot \frac{x}{1 + x}\right)}\]
\frac{1}{x - 1} + \frac{x}{x + 1}
↓
\frac{\mathsf{fma}\left(\frac{x}{1 + x} \cdot \frac{x}{1 + x}, \frac{x}{1 + x}, \frac{\frac{1}{x - 1}}{\frac{x - 1}{\frac{1}{x - 1}}}\right)}{\mathsf{fma}\left(\frac{1}{x - 1}, \frac{1}{x - 1}, \left(\frac{x}{1 + x} - \frac{1}{x - 1}\right) \cdot \frac{x}{1 + x}\right)}
double f(double x) {
        double r6469412 = 1.0;
        double r6469413 = x;
        double r6469414 = r6469413 - r6469412;
        double r6469415 = r6469412 / r6469414;
        double r6469416 = r6469413 + r6469412;
        double r6469417 = r6469413 / r6469416;
        double r6469418 = r6469415 + r6469417;
        return r6469418;
}
↓
double f(double x) {
        double r6469419 = x;
        double r6469420 = 1.0;
        double r6469421 = r6469420 + r6469419;
        double r6469422 = r6469419 / r6469421;
        double r6469423 = r6469422 * r6469422;
        double r6469424 = r6469419 - r6469420;
        double r6469425 = r6469420 / r6469424;
        double r6469426 = r6469424 / r6469425;
        double r6469427 = r6469425 / r6469426;
        double r6469428 = fma(r6469423, r6469422, r6469427);
        double r6469429 = r6469422 - r6469425;
        double r6469430 = r6469429 * r6469422;
        double r6469431 = fma(r6469425, r6469425, r6469430);
        double r6469432 = r6469428 / r6469431;
        return r6469432;
}

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

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

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