Average Error: 15.2 → 0.0
Time: 2.8s
Precision: binary64
\[\frac{x}{x \cdot x + 1} \]
↓
\[\frac{\frac{x}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)} \]
\frac{x}{x \cdot x + 1}
↓
\frac{\frac{x}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}

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

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original15.2
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}} \]

Derivation

  1. Initial program 15.2

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

  2. Simplified15.2

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

  3. Applied egg-rr0.0

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

  4. Applied egg-rr0.0

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

  5. Final simplification0.0

    \[\leadsto \frac{\frac{x}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)} \]

Reproduce

herbie shell --seed 2022130 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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