Average Error: 29.7 → 0.2
Time: 2.6s
Precision: binary64
\[\sqrt{x + 1} - \sqrt{x} \]
↓
\[\frac{-1}{\left(-\sqrt{1 + x}\right) - \sqrt{x}} \]
(FPCore (x) :precision binary64 (- (sqrt (+ x 1.0)) (sqrt x)))
↓
(FPCore (x) :precision binary64 (/ -1.0 (- (- (sqrt (+ 1.0 x))) (sqrt x))))
double code(double x) {
	return sqrt((x + 1.0)) - sqrt(x);
}
↓
double code(double x) {
	return -1.0 / (-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) / (-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 / (-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 / (-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(-1.0 / Float64(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 / (-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[(-1.0 / N[((-N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]) - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.7
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}} \]

Derivation

  1. Initial program 29.7

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

  2. Applied egg-rr29.1

    \[\leadsto \color{blue}{\frac{x + \left(1 - x\right)}{\sqrt{x + 1} + \sqrt{x}}} \]

  3. Taylor expanded in x around 0 0.2

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

  4. Applied egg-rr0.2

    \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\sqrt{1 + x} + \sqrt{x}}\right)\right)} \]

  5. Applied egg-rr0.2

    \[\leadsto \color{blue}{\frac{-1}{-\left(\sqrt{1 + x} + \sqrt{x}\right)}} \]

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2022160 
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64

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

  (- (sqrt (+ x 1.0)) (sqrt x)))