Average Error: 15.4 → 0.1
Time: 1.5s
Precision: binary64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\frac{1}{x + \frac{1}{x}}\]
\frac{x}{x \cdot x + 1}
↓
\frac{1}{x + \frac{1}{x}}
double code(double x) {
	return (x / ((double) (((double) (x * x)) + 1.0)));
}
↓
double code(double x) {
	return (1.0 / ((double) (x + (1.0 / x))));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original15.4
Target0.1
Herbie0.1
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Initial program Error: 15.4 bits

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

  3. Applied clear-numError: 15.4 bits

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

  4. Taylor expanded around 0 Error: 0.1 bits

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

  5. SimplifiedError: 0.1 bits

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

  6. Final simplificationError: 0.1 bits

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

Reproduce

herbie shell --seed 2020200 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

  :herbie-target
  (/ 1.0 (+ x (/ 1.0 x)))

  (/ x (+ (* x x) 1.0)))