Average Error: 14.7 → 0.1
Time: 1.1s
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 r58991 = x;
        double r58992 = r58991 * r58991;
        double r58993 = 1.0;
        double r58994 = r58992 + r58993;
        double r58995 = r58991 / r58994;
        return r58995;
}
↓
double f(double x) {
        double r58996 = 1.0;
        double r58997 = 1.0;
        double r58998 = x;
        double r58999 = r58996 / r58998;
        double r59000 = fma(r58997, r58999, r58998);
        double r59001 = r58996 / r59000;
        return r59001;
}

Error

Bits error versus x

Target

Original14.7
Target0.1
Herbie0.1
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Initial program 14.7

    \[\frac{x}{x \cdot x + 1}\]
  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 2020003 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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