Average Error: 15.1 → 0.1
Time: 1.8s
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 code(double x) {
	return (x / ((x * x) + 1.0));
}
↓
double code(double x) {
	return (1.0 / fma(1.0, (1.0 / x), 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-num15.1

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

  4. Simplified15.1

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

  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 2020091 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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