Average Error: 15.0 → 0.1
Time: 16.0s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\frac{1}{x + \frac{1}{x}}\]
\frac{x}{x \cdot x + 1}
↓
\frac{1}{x + \frac{1}{x}}
double f(double x) {
        double r35988 = x;
        double r35989 = r35988 * r35988;
        double r35990 = 1.0;
        double r35991 = r35989 + r35990;
        double r35992 = r35988 / r35991;
        return r35992;
}
↓
double f(double x) {
        double r35993 = 1.0;
        double r35994 = x;
        double r35995 = 1.0;
        double r35996 = r35995 / r35994;
        double r35997 = r35994 + r35996;
        double r35998 = r35993 / r35997;
        return r35998;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original15.0
Target0.1
Herbie0.1
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Initial program 15.0

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

  3. Applied clear-num15.0

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

  4. Simplified15.0

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

  5. Taylor expanded around 0 0.1

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

  6. Simplified0.1

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

  7. Final simplification0.1

    \[\leadsto \frac{1}{x + \frac{1}{x}}\]

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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