Average Error: 19.9 → 0.2
Time: 4.1s
Precision: binary64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
↓
\[{\left(1 + x\right)}^{-0.5} \cdot \frac{{x}^{-0.5}}{\sqrt{x} + \sqrt{1 + x}} \]
(FPCore (x) :precision binary64 (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (* (pow (+ 1.0 x) -0.5) (/ (pow x -0.5) (+ (sqrt x) (sqrt (+ 1.0 x))))))
double code(double x) {
	return (1.0 / sqrt(x)) - (1.0 / sqrt((x + 1.0)));
}
↓
double code(double x) {
	return pow((1.0 + x), -0.5) * (pow(x, -0.5) / (sqrt(x) + sqrt((1.0 + x))));
}

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / sqrt(x)) - (1.0d0 / sqrt((x + 1.0d0)))
end function
↓
real(8) function code(x)
    real(8), intent (in) :: x
    code = ((1.0d0 + x) ** (-0.5d0)) * ((x ** (-0.5d0)) / (sqrt(x) + sqrt((1.0d0 + x))))
end function

public static double code(double x) {
	return (1.0 / Math.sqrt(x)) - (1.0 / Math.sqrt((x + 1.0)));
}
↓
public static double code(double x) {
	return Math.pow((1.0 + x), -0.5) * (Math.pow(x, -0.5) / (Math.sqrt(x) + Math.sqrt((1.0 + x))));
}

def code(x):
	return (1.0 / math.sqrt(x)) - (1.0 / math.sqrt((x + 1.0)))
↓
def code(x):
	return math.pow((1.0 + x), -0.5) * (math.pow(x, -0.5) / (math.sqrt(x) + math.sqrt((1.0 + x))))

function code(x)
	return Float64(Float64(1.0 / sqrt(x)) - Float64(1.0 / sqrt(Float64(x + 1.0))))
end
↓
function code(x)
	return Float64((Float64(1.0 + x) ^ -0.5) * Float64((x ^ -0.5) / Float64(sqrt(x) + sqrt(Float64(1.0 + x)))))
end

function tmp = code(x)
	tmp = (1.0 / sqrt(x)) - (1.0 / sqrt((x + 1.0)));
end
↓
function tmp = code(x)
	tmp = ((1.0 + x) ^ -0.5) * ((x ^ -0.5) / (sqrt(x) + sqrt((1.0 + x))));
end

code[x_] := N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[x_] := N[(N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision] * N[(N[Power[x, -0.5], $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
{\left(1 + x\right)}^{-0.5} \cdot \frac{{x}^{-0.5}}{\sqrt{x} + \sqrt{1 + x}}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.9
Target0.6
Herbie0.2
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}} \]

Derivation

  1. Initial program 19.9

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

  2. Applied egg-rr19.9

    \[\leadsto \color{blue}{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{\sqrt{x}}} \]

  3. Applied egg-rr0.3

    \[\leadsto \color{blue}{\frac{\left(1 + \left(x - x\right)\right) \cdot \frac{{\left(1 + x\right)}^{-0.5}}{\sqrt{x}}}{\sqrt{1 + x} + \sqrt{x}}} \]

  4. Applied egg-rr0.2

    \[\leadsto \color{blue}{\frac{{\left(1 + x\right)}^{-0.5}}{1} \cdot \frac{{x}^{-0.5}}{\sqrt{x} + \sqrt{1 + x}}} \]

  5. Final simplification0.2

    \[\leadsto {\left(1 + x\right)}^{-0.5} \cdot \frac{{x}^{-0.5}}{\sqrt{x} + \sqrt{1 + x}} \]

Reproduce

herbie shell --seed 2022153 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64

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

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