# ?

Average Accuracy: 99.3% → 99.9%
Time: 5.3s
Precision: binary64
Cost: 13504

# ?

$0 \leq x \land x \leq 1000000000$
$\sqrt{x + 1} - \sqrt{x}$
$\frac{\left(1 + x\right) - x}{\sqrt{1 + x} + \sqrt{x}}$
(FPCore (x) :precision binary64 (- (sqrt (+ x 1.0)) (sqrt x)))
(FPCore (x)
:precision binary64
(/ (- (+ 1.0 x) x) (+ (sqrt (+ 1.0 x)) (sqrt x))))
double code(double x) {
return sqrt((x + 1.0)) - sqrt(x);
}

double code(double x) {
return ((1.0 + x) - x) / (sqrt((1.0 + x)) + sqrt(x));
}

real(8) function code(x)
real(8), intent (in) :: x
code = sqrt((x + 1.0d0)) - sqrt(x)
end function

real(8) function code(x)
real(8), intent (in) :: x
code = ((1.0d0 + x) - x) / (sqrt((1.0d0 + x)) + sqrt(x))
end function

public static double code(double x) {
return Math.sqrt((x + 1.0)) - Math.sqrt(x);
}

public static double code(double x) {
return ((1.0 + x) - x) / (Math.sqrt((1.0 + x)) + Math.sqrt(x));
}

def code(x):
return math.sqrt((x + 1.0)) - math.sqrt(x)

def code(x):
return ((1.0 + x) - x) / (math.sqrt((1.0 + x)) + math.sqrt(x))

function code(x)
return Float64(sqrt(Float64(x + 1.0)) - sqrt(x))
end

function code(x)
return Float64(Float64(Float64(1.0 + x) - x) / Float64(sqrt(Float64(1.0 + x)) + sqrt(x)))
end

function tmp = code(x)
tmp = sqrt((x + 1.0)) - sqrt(x);
end

function tmp = code(x)
tmp = ((1.0 + x) - x) / (sqrt((1.0 + x)) + sqrt(x));
end

code[x_] := N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]],$MachinePrecision] - N[Sqrt[x], $MachinePrecision]),$MachinePrecision]

code[x_] := N[(N[(N[(1.0 + x), $MachinePrecision] - x),$MachinePrecision] / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]],$MachinePrecision] + N[Sqrt[x], $MachinePrecision]),$MachinePrecision]), \$MachinePrecision]

\sqrt{x + 1} - \sqrt{x}

\frac{\left(1 + x\right) - x}{\sqrt{1 + x} + \sqrt{x}}


# Try it out?

Results

 In Out
Enter valid numbers for all inputs

# Derivation?

1. Initial program 99.3%

$\sqrt{x + 1} - \sqrt{x}$
2. Applied egg-rr99.9%

$\leadsto \color{blue}{\left(x + \left(1 - x\right)\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}}$
Proof
[Start]99.3 $\sqrt{x + 1} - \sqrt{x}$ $\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}}$ $\color{blue}{\left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}}$ $\left(\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}$ $\left(\left(x + 1\right) - \color{blue}{x}\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}$ $\color{blue}{\left(x + \left(1 - x\right)\right)} \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}$
3. Simplified99.9%

$\leadsto \color{blue}{\frac{\left(1 + x\right) - x}{\sqrt{1 + x} + \sqrt{x}}}$
Proof
[Start]99.9 $\left(x + \left(1 - x\right)\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}$ $\color{blue}{\frac{\left(x + \left(1 - x\right)\right) \cdot 1}{\sqrt{x + 1} + \sqrt{x}}}$ $\frac{\color{blue}{x + \left(1 - x\right)}}{\sqrt{x + 1} + \sqrt{x}}$ $\frac{\color{blue}{\left(x + 1\right) - x}}{\sqrt{x + 1} + \sqrt{x}}$ $\frac{\color{blue}{\left(1 + x\right)} - x}{\sqrt{x + 1} + \sqrt{x}}$ $\frac{\left(1 + x\right) - x}{\sqrt{\color{blue}{1 + x}} + \sqrt{x}}$
4. Final simplification99.9%

$\leadsto \frac{\left(1 + x\right) - x}{\sqrt{1 + x} + \sqrt{x}}$

# Alternatives

Alternative 1
Accuracy99.9%
Cost13248
$\frac{1}{\sqrt{1 + x} + \sqrt{x}}$
Alternative 2
Accuracy99.3%
Cost13120
$\sqrt{1 + x} - \sqrt{x}$
Alternative 3
Accuracy96.5%
Cost6976
$\frac{1}{\sqrt{x} + \left(1 + x \cdot 0.5\right)}$
Alternative 4
Accuracy96.5%
Cost6848
$x \cdot 0.5 + \left(1 - \sqrt{x}\right)$
Alternative 5
Accuracy93.4%
Cost64
$1$

# Reproduce?

herbie shell --seed 1
(FPCore (x)
:name "sqrt(x+1) - sqrt(x)"
:precision binary64
:pre (and (<= 0.0 x) (<= x 1000000000.0))
(- (sqrt (+ x 1.0)) (sqrt x)))