Average Error: 20.0 → 0.6
Time: 10.7s
Precision: binary64
Cost: 26948
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
↓
\[\begin{array}{l} t_0 := \sqrt{x + 1}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{t_0} \leq 10^{-20}:\\ \;\;\;\;\frac{\frac{1}{x + 0.5}}{\sqrt{x} + t_0}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\ \end{array} \]
(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))))
   (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 t_0)) 1e-20)
     (/ (/ 1.0 (+ x 0.5)) (+ (sqrt x) t_0))
     (- (pow x -0.5) (pow (+ x 1.0) -0.5)))))
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));
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / t_0)) <= 1e-20) {
		tmp = (1.0 / (x + 0.5)) / (sqrt(x) + t_0);
	} else {
		tmp = pow(x, -0.5) - pow((x + 1.0), -0.5);
	}
	return tmp;
}

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
    real(8) :: tmp
    t_0 = sqrt((x + 1.0d0))
    if (((1.0d0 / sqrt(x)) + ((-1.0d0) / t_0)) <= 1d-20) then
        tmp = (1.0d0 / (x + 0.5d0)) / (sqrt(x) + t_0)
    else
        tmp = (x ** (-0.5d0)) - ((x + 1.0d0) ** (-0.5d0))
    end if
    code = tmp
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));
	double tmp;
	if (((1.0 / Math.sqrt(x)) + (-1.0 / t_0)) <= 1e-20) {
		tmp = (1.0 / (x + 0.5)) / (Math.sqrt(x) + t_0);
	} else {
		tmp = Math.pow(x, -0.5) - Math.pow((x + 1.0), -0.5);
	}
	return tmp;
}

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))
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / t_0)) <= 1e-20:
		tmp = (1.0 / (x + 0.5)) / (math.sqrt(x) + t_0)
	else:
		tmp = math.pow(x, -0.5) - math.pow((x + 1.0), -0.5)
	return tmp

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))
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / t_0)) <= 1e-20)
		tmp = Float64(Float64(1.0 / Float64(x + 0.5)) / Float64(sqrt(x) + t_0));
	else
		tmp = Float64((x ^ -0.5) - (Float64(x + 1.0) ^ -0.5));
	end
	return tmp
end

function tmp = code(x)
	tmp = (1.0 / sqrt(x)) - (1.0 / sqrt((x + 1.0)));
end
↓
function tmp_2 = code(x)
	t_0 = sqrt((x + 1.0));
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + (-1.0 / t_0)) <= 1e-20)
		tmp = (1.0 / (x + 0.5)) / (sqrt(x) + t_0);
	else
		tmp = (x ^ -0.5) - ((x + 1.0) ^ -0.5);
	end
	tmp_2 = tmp;
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]}, If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], 1e-20], N[(N[(1.0 / N[(x + 0.5), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]]

\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\begin{array}{l}
t_0 := \sqrt{x + 1}\\
\mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{t_0} \leq 10^{-20}:\\
\;\;\;\;\frac{\frac{1}{x + 0.5}}{\sqrt{x} + t_0}\\

\mathbf{else}:\\
\;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\


\end{array}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.0
Target0.6
Herbie0.6
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}} \]

Derivation

  1. Split input into 2 regimes

  2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 9.99999999999999945e-21

    1. Initial program 40.2

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

    2. Applied egg-rr11.4

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

    3. Simplified11.4

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

      [Start]11.4

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

      associate-/r* [=>]11.4

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

      +-commutative [=>]11.4

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

      +-inverses [=>]11.4

      \[ \frac{\frac{\color{blue}{0} + 1}{\sqrt{x + x \cdot x}}}{\sqrt{x} + \sqrt{x + 1}} \]

      metadata-eval [=>]11.4

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

      +-commutative [=>]11.4

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

    4. Taylor expanded in x around inf 0.2

      \[\leadsto \frac{\frac{1}{\color{blue}{0.5 + x}}}{\sqrt{x} + \sqrt{1 + x}} \]

    5. Simplified0.2

      \[\leadsto \frac{\frac{1}{\color{blue}{x + 0.5}}}{\sqrt{x} + \sqrt{1 + x}} \]
      Proof

      [Start]0.2

      \[ \frac{\frac{1}{0.5 + x}}{\sqrt{x} + \sqrt{1 + x}} \]

      +-commutative [=>]0.2

      \[ \frac{\frac{1}{\color{blue}{x + 0.5}}}{\sqrt{x} + \sqrt{1 + x}} \]

    if 9.99999999999999945e-21 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

    1. Initial program 1.2

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

    2. Applied egg-rr1.2

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

    3. Applied egg-rr0.9

      \[\leadsto \color{blue}{{x}^{-0.5}} - {\left(x + 1\right)}^{-0.5} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{x + 1}} \leq 10^{-20}:\\ \;\;\;\;\frac{\frac{1}{x + 0.5}}{\sqrt{x} + \sqrt{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\ \end{array} \]

Alternatives

Alternative 1
Error0.5
Cost39296
\[\frac{{x}^{-0.25} \cdot \frac{{x}^{-0.25}}{\mathsf{hypot}\left(1, \sqrt{x}\right)}}{\sqrt{x} + \sqrt{x + 1}} \]
Alternative 2
Error0.5
Cost33088
\[\begin{array}{l} t_0 := \sqrt{x + 1}\\ \frac{\frac{1}{{x}^{0.25} \cdot \left({x}^{0.25} \cdot t_0\right)}}{\sqrt{x} + t_0} \end{array} \]
Alternative 3
Error0.8
Cost13508
\[\begin{array}{l} \mathbf{if}\;x \leq 64000000:\\ \;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(\sqrt{x} + \sqrt{x + 1}\right)}\\ \end{array} \]
Alternative 4
Error0.4
Cost13508
\[\begin{array}{l} \mathbf{if}\;x \leq 64000000:\\ \;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{\sqrt{x} + \sqrt{x + 1}}\\ \end{array} \]
Alternative 5
Error19.7
Cost7232
\[\frac{1}{x + \sqrt{\left(x + 1\right) \cdot \left(x + x \cdot x\right)}} \]
Alternative 6
Error21.5
Cost6848
\[\frac{1}{\sqrt{x + x \cdot x}} \]
Alternative 7
Error30.1
Cost6788
\[\begin{array}{l} \mathbf{if}\;x \leq 0.16:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + 0.5}\\ \end{array} \]
Alternative 8
Error59.3
Cost320
\[\frac{1}{x + 0.5} \]
Alternative 9
Error59.3
Cost192
\[\frac{1}{x} \]
Alternative 10
Error60.3
Cost64
\[2 \]

Error

Reproduce

herbie shell --seed 2023011 
(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)))))