Average Error: 19.9 → 0.4
Time: 4.1s
Precision: binary64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
↓
\[\begin{array}{l} t_0 := \sqrt{x + 1}\\ \frac{\frac{{\left(\sqrt{x} + t_0\right)}^{-1}}{\sqrt{x}}}{t_0} \end{array} \]
(FPCore (x) :precision binary64 (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ x 1.0))))
   (/ (/ (pow (+ (sqrt x) t_0) -1.0) (sqrt x)) t_0)))
double code(double x) {
	return (1.0 / sqrt(x)) - (1.0 / sqrt((x + 1.0)));
}
↓
double code(double x) {
	double t_0 = sqrt((x + 1.0));
	return (pow((sqrt(x) + t_0), -1.0) / sqrt(x)) / t_0;
}

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
    real(8) :: t_0
    t_0 = sqrt((x + 1.0d0))
    code = (((sqrt(x) + t_0) ** (-1.0d0)) / sqrt(x)) / t_0
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) {
	double t_0 = Math.sqrt((x + 1.0));
	return (Math.pow((Math.sqrt(x) + t_0), -1.0) / Math.sqrt(x)) / t_0;
}

def code(x):
	return (1.0 / math.sqrt(x)) - (1.0 / math.sqrt((x + 1.0)))
↓
def code(x):
	t_0 = math.sqrt((x + 1.0))
	return (math.pow((math.sqrt(x) + t_0), -1.0) / math.sqrt(x)) / t_0

function code(x)
	return Float64(Float64(1.0 / sqrt(x)) - Float64(1.0 / sqrt(Float64(x + 1.0))))
end
↓
function code(x)
	t_0 = sqrt(Float64(x + 1.0))
	return Float64(Float64((Float64(sqrt(x) + t_0) ^ -1.0) / sqrt(x)) / t_0)
end

function tmp = code(x)
	tmp = (1.0 / sqrt(x)) - (1.0 / sqrt((x + 1.0)));
end
↓
function tmp = code(x)
	t_0 = sqrt((x + 1.0));
	tmp = (((sqrt(x) + t_0) ^ -1.0) / sqrt(x)) / t_0;
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_] := Block[{t$95$0 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[Power[N[(N[Sqrt[x], $MachinePrecision] + t$95$0), $MachinePrecision], -1.0], $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]

\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\begin{array}{l}
t_0 := \sqrt{x + 1}\\
\frac{\frac{{\left(\sqrt{x} + t_0\right)}^{-1}}{\sqrt{x}}}{t_0}
\end{array}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.9
Target0.7
Herbie0.4
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}} \]

Derivation

  1. Initial program 19.9

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

  2. Applied egg-rr19.9

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

  3. Applied egg-rr0.4

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

  4. Applied egg-rr0.6

    \[\leadsto \frac{\frac{\frac{1 + \left(x - x\right)}{\color{blue}{{\left(\sqrt[3]{\mathsf{hypot}\left(1, \sqrt{x}\right) + \sqrt{x}}\right)}^{3}}}}{\sqrt{x}}}{\sqrt{1 + x}} \]

  5. Applied egg-rr0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2022159 
(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)))))