Average Error: 15.0 → 0.1
Time: 14.6s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\frac{1}{\frac{1}{x} + x}\]
\frac{x}{x \cdot x + 1}
↓
\frac{1}{\frac{1}{x} + x}
double f(double x) {
        double r40758 = x;
        double r40759 = r40758 * r40758;
        double r40760 = 1.0;
        double r40761 = r40759 + r40760;
        double r40762 = r40758 / r40761;
        return r40762;
}
↓
double f(double x) {
        double r40763 = 1.0;
        double r40764 = 1.0;
        double r40765 = x;
        double r40766 = r40764 / r40765;
        double r40767 = r40766 + r40765;
        double r40768 = r40763 / r40767;
        return r40768;
}

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

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

  5. Taylor expanded around 0 0.1

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

  6. Simplified0.1

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

  7. Final simplification0.1

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

Reproduce

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

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

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