Average Error: 15.1 → 0.1
Time: 2.2s
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}}

(FPCore (x) :precision binary64 (/ x (+ (* x x) 1.0)))
↓
(FPCore (x) :precision binary64 (/ 1.0 (+ x (/ 1.0 x))))
double code(double x) {
	return x / ((x * x) + 1.0);
}
↓
double code(double x) {
	return 1.0 / (x + (1.0 / x));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 15.1

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

  3. Applied clear-num_binary6415.2

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

  4. Simplified15.2

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

  5. Taylor expanded around 0 0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2021190 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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