Average Error: 14.7 → 0.0
Time: 1.6s
Precision: binary64
\[\frac{x}{x \cdot x + 1} \]
↓
\[\frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{x}{\mathsf{hypot}\left(1, x\right)} \]
(FPCore (x) :precision binary64 (/ x (+ (* x x) 1.0)))
↓
(FPCore (x) :precision binary64 (* (/ 1.0 (hypot 1.0 x)) (/ x (hypot 1.0 x))))
double code(double x) {
	return x / ((x * x) + 1.0);
}
↓
double code(double x) {
	return (1.0 / hypot(1.0, x)) * (x / hypot(1.0, x));
}

public static double code(double x) {
	return x / ((x * x) + 1.0);
}
↓
public static double code(double x) {
	return (1.0 / Math.hypot(1.0, x)) * (x / Math.hypot(1.0, x));
}

def code(x):
	return x / ((x * x) + 1.0)
↓
def code(x):
	return (1.0 / math.hypot(1.0, x)) * (x / math.hypot(1.0, x))

function code(x)
	return Float64(x / Float64(Float64(x * x) + 1.0))
end
↓
function code(x)
	return Float64(Float64(1.0 / hypot(1.0, x)) * Float64(x / hypot(1.0, x)))
end

function tmp = code(x)
	tmp = x / ((x * x) + 1.0);
end
↓
function tmp = code(x)
	tmp = (1.0 / hypot(1.0, x)) * (x / hypot(1.0, x));
end

code[x_] := N[(x / N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]
↓
code[x_] := N[(N[(1.0 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\frac{x}{x \cdot x + 1}
↓
\frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{x}{\mathsf{hypot}\left(1, x\right)}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.7
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}} \]

Derivation

  1. Initial program 14.7

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

  2. Applied add-sqr-sqrt_binary6414.7

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

  3. Applied *-un-lft-identity_binary6414.7

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

  4. Applied times-frac_binary6414.6

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

  5. Simplified14.6

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

  6. Simplified0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2022129 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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