?

Average Error: 0.2 → 0.1
Time: 8.9s
Precision: binary64
Cost: 7232

?

\[\frac{x}{1 + \sqrt{x + 1}} \]
\[\frac{x}{1 + \frac{1}{\sqrt{x + 1}} \cdot \left(1 + x\right)} \]
(FPCore (x) :precision binary64 (/ x (+ 1.0 (sqrt (+ x 1.0)))))
(FPCore (x)
 :precision binary64
 (/ x (+ 1.0 (* (/ 1.0 (sqrt (+ x 1.0))) (+ 1.0 x)))))
double code(double x) {
	return x / (1.0 + sqrt((x + 1.0)));
}
double code(double x) {
	return x / (1.0 + ((1.0 / sqrt((x + 1.0))) * (1.0 + x)));
}
real(8) function code(x)
    real(8), intent (in) :: x
    code = x / (1.0d0 + sqrt((x + 1.0d0)))
end function
real(8) function code(x)
    real(8), intent (in) :: x
    code = x / (1.0d0 + ((1.0d0 / sqrt((x + 1.0d0))) * (1.0d0 + x)))
end function
public static double code(double x) {
	return x / (1.0 + Math.sqrt((x + 1.0)));
}
public static double code(double x) {
	return x / (1.0 + ((1.0 / Math.sqrt((x + 1.0))) * (1.0 + x)));
}
def code(x):
	return x / (1.0 + math.sqrt((x + 1.0)))
def code(x):
	return x / (1.0 + ((1.0 / math.sqrt((x + 1.0))) * (1.0 + x)))
function code(x)
	return Float64(x / Float64(1.0 + sqrt(Float64(x + 1.0))))
end
function code(x)
	return Float64(x / Float64(1.0 + Float64(Float64(1.0 / sqrt(Float64(x + 1.0))) * Float64(1.0 + x))))
end
function tmp = code(x)
	tmp = x / (1.0 + sqrt((x + 1.0)));
end
function tmp = code(x)
	tmp = x / (1.0 + ((1.0 / sqrt((x + 1.0))) * (1.0 + x)));
end
code[x_] := N[(x / N[(1.0 + N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
code[x_] := N[(x / N[(1.0 + N[(N[(1.0 / N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
\frac{x}{1 + \sqrt{x + 1}}
\frac{x}{1 + \frac{1}{\sqrt{x + 1}} \cdot \left(1 + x\right)}

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 0.2

    \[\frac{x}{1 + \sqrt{x + 1}} \]
  2. Applied egg-rr10.5

    \[\leadsto \frac{x}{1 + \color{blue}{\frac{1}{\sqrt{x + 1}} \cdot \sqrt{\left(x + 1\right) \cdot \left(x + 1\right)}}} \]
  3. Taylor expanded in x around 0 0.1

    \[\leadsto \frac{x}{1 + \frac{1}{\sqrt{x + 1}} \cdot \color{blue}{\left(1 + x\right)}} \]
  4. Final simplification0.1

    \[\leadsto \frac{x}{1 + \frac{1}{\sqrt{x + 1}} \cdot \left(1 + x\right)} \]

Alternatives

Alternative 1
Error0.2
Cost6848
\[\frac{x}{1 + \sqrt{x + 1}} \]
Alternative 2
Error20.3
Cost448
\[\frac{x}{0.5 \cdot x + 2} \]
Alternative 3
Error54.0
Cost192
\[x \cdot 0.3333333333333333 \]
Alternative 4
Error20.7
Cost192
\[\frac{x}{2} \]
Alternative 5
Error60.9
Cost64
\[2 \]
Alternative 6
Error54.5
Cost64
\[x \]

Error

Reproduce?

herbie shell --seed 2023077 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1.0 (sqrt (+ x 1.0)))))