Average Error: 30.0 → 0.1
Time: 6.5s
Precision: binary64
Cost: 13248
$0 \leq x \land x \leq 1.79 \cdot 10^{+308}$
$\sqrt{x + 1.002} - \sqrt{x}$
$\frac{1.002}{\sqrt{1.002 + x} + \sqrt{x}}$
(FPCore (x) :precision binary64 (- (sqrt (+ x 1.002)) (sqrt x)))
(FPCore (x) :precision binary64 (/ 1.002 (+ (sqrt (+ 1.002 x)) (sqrt x))))
double code(double x) {
return sqrt((x + 1.002)) - sqrt(x);
}

double code(double x) {
return 1.002 / (sqrt((1.002 + x)) + sqrt(x));
}

real(8) function code(x)
real(8), intent (in) :: x
code = sqrt((x + 1.002d0)) - sqrt(x)
end function

real(8) function code(x)
real(8), intent (in) :: x
code = 1.002d0 / (sqrt((1.002d0 + x)) + sqrt(x))
end function

public static double code(double x) {
return Math.sqrt((x + 1.002)) - Math.sqrt(x);
}

public static double code(double x) {
return 1.002 / (Math.sqrt((1.002 + x)) + Math.sqrt(x));
}

def code(x):
return math.sqrt((x + 1.002)) - math.sqrt(x)

def code(x):
return 1.002 / (math.sqrt((1.002 + x)) + math.sqrt(x))

function code(x)
return Float64(sqrt(Float64(x + 1.002)) - sqrt(x))
end

function code(x)
return Float64(1.002 / Float64(sqrt(Float64(1.002 + x)) + sqrt(x)))
end

function tmp = code(x)
tmp = sqrt((x + 1.002)) - sqrt(x);
end

function tmp = code(x)
tmp = 1.002 / (sqrt((1.002 + x)) + sqrt(x));
end

code[x_] := N[(N[Sqrt[N[(x + 1.002), $MachinePrecision]],$MachinePrecision] - N[Sqrt[x], $MachinePrecision]),$MachinePrecision]

code[x_] := N[(1.002 / N[(N[Sqrt[N[(1.002 + x), $MachinePrecision]],$MachinePrecision] + N[Sqrt[x], $MachinePrecision]),$MachinePrecision]), \$MachinePrecision]

\sqrt{x + 1.002} - \sqrt{x}

\frac{1.002}{\sqrt{1.002 + x} + \sqrt{x}}


# Try it out

Results

 In Out
Enter valid numbers for all inputs

# Derivation

1. Initial program 30.0

$\sqrt{x + 1.002} - \sqrt{x}$
2. Applied egg-rr30.0

$\leadsto \color{blue}{\frac{x + \left(1.002 - x\right)}{\sqrt{x + 1.002} + \sqrt{x}}}$
3. Taylor expanded in x around 0 0.1

$\leadsto \frac{\color{blue}{1.002}}{\sqrt{x + 1.002} + \sqrt{x}}$
4. Final simplification0.1

$\leadsto \frac{1.002}{\sqrt{1.002 + x} + \sqrt{x}}$

# Alternatives

Alternative 1
Error30.0
Cost13184
${\left(1.002 + x\right)}^{0.5} - \sqrt{x}$
Alternative 2
Error30.0
Cost13120
$\sqrt{1.002 + x} - \sqrt{x}$
Alternative 3
Error31.2
Cost6464
$\sqrt{1.002}$

# Reproduce

herbie shell --seed 1
(FPCore (x)
:name "sqrt(x + 1.002) - sqrt(x)"
:precision binary64
:pre (and (<= 0.0 x) (<= x 1.79e+308))
(- (sqrt (+ x 1.002)) (sqrt x)))