Average Error: 14.6 → 0.1
Time: 1.6s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\frac{1}{\mathsf{fma}\left(1, \frac{1}{x}, x\right)}\]
\frac{x}{x \cdot x + 1}
↓
\frac{1}{\mathsf{fma}\left(1, \frac{1}{x}, x\right)}
double f(double x) {
        double r82977 = x;
        double r82978 = r82977 * r82977;
        double r82979 = 1.0;
        double r82980 = r82978 + r82979;
        double r82981 = r82977 / r82980;
        return r82981;
}
↓
double f(double x) {
        double r82982 = 1.0;
        double r82983 = 1.0;
        double r82984 = x;
        double r82985 = r82982 / r82984;
        double r82986 = fma(r82983, r82985, r82984);
        double r82987 = r82982 / r82986;
        return r82987;
}

Error

Bits error versus x

Target

Original14.6
Target0.1
Herbie0.1
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Initial program 14.6

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

  3. Applied clear-num14.7

    \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + 1}{x}}}\]

  4. Simplified14.7

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

  5. Taylor expanded around 0 0.1

    \[\leadsto \frac{1}{\color{blue}{x + 1 \cdot \frac{1}{x}}}\]

  6. Simplified0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2020064 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

  :herbie-target
  (/ 1 (+ x (/ 1 x)))

  (/ x (+ (* x x) 1)))