2isqrt (example 3.6)

Average Accuracy: 68.8% → 99.8%

Time: 13.0s

Precision: binary64

Cost: 27588

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{t_0} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{1}{t_0 + \sqrt{x}}}{\left(x + \left(0.5 + \frac{0.0625}{x \cdot x}\right)\right) - \frac{0.125}{x}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(1 - \sqrt{\frac{x}{1 + x}}\right)\\ \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ 1.0 x))))
   (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 t_0)) 5e-7)
     (/
      (/ 1.0 (+ t_0 (sqrt x)))
      (- (+ x (+ 0.5 (/ 0.0625 (* x x)))) (/ 0.125 x)))
     (* (pow x -0.5) (- 1.0 (sqrt (/ x (+ 1.0 x))))))))

C

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

↓

double code(double x) {
	double t_0 = sqrt((1.0 + x));
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / t_0)) <= 5e-7) {
		tmp = (1.0 / (t_0 + sqrt(x))) / ((x + (0.5 + (0.0625 / (x * x)))) - (0.125 / x));
	} else {
		tmp = pow(x, -0.5) * (1.0 - sqrt((x / (1.0 + x))));
	}
	return tmp;
}

Fortran

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((1.0d0 + x))
    if (((1.0d0 / sqrt(x)) + ((-1.0d0) / t_0)) <= 5d-7) then
        tmp = (1.0d0 / (t_0 + sqrt(x))) / ((x + (0.5d0 + (0.0625d0 / (x * x)))) - (0.125d0 / x))
    else
        tmp = (x ** (-0.5d0)) * (1.0d0 - sqrt((x / (1.0d0 + x))))
    end if
    code = tmp
end function

Java

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((1.0 + x));
	double tmp;
	if (((1.0 / Math.sqrt(x)) + (-1.0 / t_0)) <= 5e-7) {
		tmp = (1.0 / (t_0 + Math.sqrt(x))) / ((x + (0.5 + (0.0625 / (x * x)))) - (0.125 / x));
	} else {
		tmp = Math.pow(x, -0.5) * (1.0 - Math.sqrt((x / (1.0 + x))));
	}
	return tmp;
}

Python

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

↓

def code(x):
	t_0 = math.sqrt((1.0 + x))
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / t_0)) <= 5e-7:
		tmp = (1.0 / (t_0 + math.sqrt(x))) / ((x + (0.5 + (0.0625 / (x * x)))) - (0.125 / x))
	else:
		tmp = math.pow(x, -0.5) * (1.0 - math.sqrt((x / (1.0 + x))))
	return tmp

Julia

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(1.0 + x))
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / t_0)) <= 5e-7)
		tmp = Float64(Float64(1.0 / Float64(t_0 + sqrt(x))) / Float64(Float64(x + Float64(0.5 + Float64(0.0625 / Float64(x * x)))) - Float64(0.125 / x)));
	else
		tmp = Float64((x ^ -0.5) * Float64(1.0 - sqrt(Float64(x / Float64(1.0 + x)))));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x)
	t_0 = sqrt((1.0 + x));
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + (-1.0 / t_0)) <= 5e-7)
		tmp = (1.0 / (t_0 + sqrt(x))) / ((x + (0.5 + (0.0625 / (x * x)))) - (0.125 / x));
	else
		tmp = (x ^ -0.5) * (1.0 - sqrt((x / (1.0 + x))));
	end
	tmp_2 = tmp;
end

Wolfram

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[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], 5e-7], N[(N[(1.0 / N[(t$95$0 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x + N[(0.5 + N[(0.0625 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.125 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] * N[(1.0 - N[Sqrt[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	68.8%
Target	98.9%
Herbie	99.8%

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 4.99999999999999977e-7

   1. Initial program 35.7%

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

   2. Applied egg-rr 35.7%

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

      [Start] 35.7	\[ \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
      frac-sub [=>] 35.7	\[ \color{blue}{\frac{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt{x + 1}}} \]
      *-un-lft-identity [<=] 35.7	\[ \frac{\color{blue}{\sqrt{x + 1}} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt{x + 1}} \]
      +-commutative [=>] 35.7	\[ \frac{\sqrt{\color{blue}{1 + x}} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt{x + 1}} \]
      *-rgt-identity [=>] 35.7	\[ \frac{\sqrt{1 + x} - \color{blue}{\sqrt{x}}}{\sqrt{x} \cdot \sqrt{x + 1}} \]
      sqrt-unprod [=>] 35.7	\[ \frac{\sqrt{1 + x} - \sqrt{x}}{\color{blue}{\sqrt{x \cdot \left(x + 1\right)}}} \]
      +-commutative [=>] 35.7	\[ \frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{x \cdot \color{blue}{\left(1 + x\right)}}} \]

   3. Applied egg-rr 82.7%

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

      [Start] 35.7	\[ \frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{x \cdot \left(1 + x\right)}} \]
      flip-- [=>] 35.8	\[ \frac{\color{blue}{\frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}}}}{\sqrt{x \cdot \left(1 + x\right)}} \]
      div-inv [=>] 35.8	\[ \frac{\color{blue}{\left(\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}}}{\sqrt{x \cdot \left(1 + x\right)}} \]
      add-sqr-sqrt [<=] 36.7	\[ \frac{\left(\color{blue}{\left(1 + x\right)} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}}{\sqrt{x \cdot \left(1 + x\right)}} \]
      add-sqr-sqrt [<=] 36.3	\[ \frac{\left(\left(1 + x\right) - \color{blue}{x}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}}{\sqrt{x \cdot \left(1 + x\right)}} \]
      associate--l+ [=>] 82.7	\[ \frac{\color{blue}{\left(1 + \left(x - x\right)\right)} \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}}{\sqrt{x \cdot \left(1 + x\right)}} \]

   4. Simplified 82.7%

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

      [Start] 82.7	\[ \frac{\left(1 + \left(x - x\right)\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}}{\sqrt{x \cdot \left(1 + x\right)}} \]
      +-inverses [=>] 82.7	\[ \frac{\left(1 + \color{blue}{0}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}}{\sqrt{x \cdot \left(1 + x\right)}} \]
      metadata-eval [=>] 82.7	\[ \frac{\color{blue}{1} \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}}{\sqrt{x \cdot \left(1 + x\right)}} \]
      *-lft-identity [=>] 82.7	\[ \frac{\color{blue}{\frac{1}{\sqrt{1 + x} + \sqrt{x}}}}{\sqrt{x \cdot \left(1 + x\right)}} \]

   5. Taylor expanded in x around inf 99.6%

      \[\leadsto \frac{\frac{1}{\sqrt{1 + x} + \sqrt{x}}}{\color{blue}{\left(0.5 + \left(0.0625 \cdot \frac{1}{{x}^{2}} + x\right)\right) - 0.125 \cdot \frac{1}{x}}} \]

   6. Simplified 99.6%

      \[\leadsto \frac{\frac{1}{\sqrt{1 + x} + \sqrt{x}}}{\color{blue}{\left(\left(0.5 + \frac{0.0625}{x \cdot x}\right) + x\right) - \frac{0.125}{x}}} \]
      Step-by-step derivation

      [Start] 99.6	\[ \frac{\frac{1}{\sqrt{1 + x} + \sqrt{x}}}{\left(0.5 + \left(0.0625 \cdot \frac{1}{{x}^{2}} + x\right)\right) - 0.125 \cdot \frac{1}{x}} \]
      associate-+r+ [=>] 99.6	\[ \frac{\frac{1}{\sqrt{1 + x} + \sqrt{x}}}{\color{blue}{\left(\left(0.5 + 0.0625 \cdot \frac{1}{{x}^{2}}\right) + x\right)} - 0.125 \cdot \frac{1}{x}} \]
      unpow2 [=>] 99.6	\[ \frac{\frac{1}{\sqrt{1 + x} + \sqrt{x}}}{\left(\left(0.5 + 0.0625 \cdot \frac{1}{\color{blue}{x \cdot x}}\right) + x\right) - 0.125 \cdot \frac{1}{x}} \]
      associate-*r/ [=>] 99.6	\[ \frac{\frac{1}{\sqrt{1 + x} + \sqrt{x}}}{\left(\left(0.5 + \color{blue}{\frac{0.0625 \cdot 1}{x \cdot x}}\right) + x\right) - 0.125 \cdot \frac{1}{x}} \]
      metadata-eval [=>] 99.6	\[ \frac{\frac{1}{\sqrt{1 + x} + \sqrt{x}}}{\left(\left(0.5 + \frac{\color{blue}{0.0625}}{x \cdot x}\right) + x\right) - 0.125 \cdot \frac{1}{x}} \]
      associate-*r/ [=>] 99.6	\[ \frac{\frac{1}{\sqrt{1 + x} + \sqrt{x}}}{\left(\left(0.5 + \frac{0.0625}{x \cdot x}\right) + x\right) - \color{blue}{\frac{0.125 \cdot 1}{x}}} \]
      metadata-eval [=>] 99.6	\[ \frac{\frac{1}{\sqrt{1 + x} + \sqrt{x}}}{\left(\left(0.5 + \frac{0.0625}{x \cdot x}\right) + x\right) - \frac{\color{blue}{0.125}}{x}} \]

   if 4.99999999999999977e-7 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 99.4%

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

   2. Applied egg-rr 99.9%

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

      [Start] 99.4	\[ \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
      frac-sub [=>] 99.5	\[ \color{blue}{\frac{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt{x + 1}}} \]
      div-inv [=>] 99.5	\[ \color{blue}{\left(1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}}} \]
      *-un-lft-identity [<=] 99.5	\[ \left(\color{blue}{\sqrt{x + 1}} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]
      +-commutative [=>] 99.5	\[ \left(\sqrt{\color{blue}{1 + x}} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]
      *-rgt-identity [=>] 99.5	\[ \left(\sqrt{1 + x} - \color{blue}{\sqrt{x}}\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]
      metadata-eval [<=] 99.5	\[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\color{blue}{1 \cdot 1}}{\sqrt{x} \cdot \sqrt{x + 1}} \]
      frac-times [<=] 99.5	\[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \color{blue}{\left(\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x + 1}}\right)} \]
      un-div-inv [=>] 99.5	\[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \color{blue}{\frac{\frac{1}{\sqrt{x}}}{\sqrt{x + 1}}} \]
      pow1/2 [=>] 99.5	\[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\frac{1}{\color{blue}{{x}^{0.5}}}}{\sqrt{x + 1}} \]
      pow-flip [=>] 99.9	\[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\color{blue}{{x}^{\left(-0.5\right)}}}{\sqrt{x + 1}} \]
      metadata-eval [=>] 99.9	\[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{x}^{\color{blue}{-0.5}}}{\sqrt{x + 1}} \]
      +-commutative [=>] 99.9	\[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{x}^{-0.5}}{\sqrt{\color{blue}{1 + x}}} \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\left(-1 - \frac{\sqrt{x}}{-\mathsf{hypot}\left(1, \sqrt{x}\right)}\right) \cdot \frac{-1}{\sqrt{x}}} \]
      Step-by-step derivation

      [Start] 99.9	\[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{x}^{-0.5}}{\sqrt{1 + x}} \]
      associate-*r/ [=>] 99.8	\[ \color{blue}{\frac{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot {x}^{-0.5}}{\sqrt{1 + x}}} \]
      remove-double-neg [<=] 99.8	\[ \frac{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot {x}^{-0.5}}{\color{blue}{-\left(-\sqrt{1 + x}\right)}} \]
      neg-mul-1 [=>] 99.8	\[ \frac{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot {x}^{-0.5}}{\color{blue}{-1 \cdot \left(-\sqrt{1 + x}\right)}} \]
      *-commutative [=>] 99.8	\[ \frac{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot {x}^{-0.5}}{\color{blue}{\left(-\sqrt{1 + x}\right) \cdot -1}} \]
      times-frac [=>] 99.8	\[ \color{blue}{\frac{\sqrt{1 + x} - \sqrt{x}}{-\sqrt{1 + x}} \cdot \frac{{x}^{-0.5}}{-1}} \]

   4. Applied egg-rr 99.8%

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

      [Start] 99.5	\[ \left(-1 - \frac{\sqrt{x}}{-\mathsf{hypot}\left(1, \sqrt{x}\right)}\right) \cdot \frac{-1}{\sqrt{x}} \]
      *-commutative [=>] 99.5	\[ \color{blue}{\frac{-1}{\sqrt{x}} \cdot \left(-1 - \frac{\sqrt{x}}{-\mathsf{hypot}\left(1, \sqrt{x}\right)}\right)} \]
      sub-neg [=>] 99.5	\[ \frac{-1}{\sqrt{x}} \cdot \color{blue}{\left(-1 + \left(-\frac{\sqrt{x}}{-\mathsf{hypot}\left(1, \sqrt{x}\right)}\right)\right)} \]
      neg-mul-1 [=>] 99.5	\[ \frac{-1}{\sqrt{x}} \cdot \left(-1 + \color{blue}{-1 \cdot \frac{\sqrt{x}}{-\mathsf{hypot}\left(1, \sqrt{x}\right)}}\right) \]
      distribute-lft-in [=>] 99.4	\[ \color{blue}{\frac{-1}{\sqrt{x}} \cdot -1 + \frac{-1}{\sqrt{x}} \cdot \left(-1 \cdot \frac{\sqrt{x}}{-\mathsf{hypot}\left(1, \sqrt{x}\right)}\right)} \]
      div-inv [=>] 99.4	\[ \color{blue}{\left(-1 \cdot \frac{1}{\sqrt{x}}\right)} \cdot -1 + \frac{-1}{\sqrt{x}} \cdot \left(-1 \cdot \frac{\sqrt{x}}{-\mathsf{hypot}\left(1, \sqrt{x}\right)}\right) \]
      mul-1-neg [=>] 99.4	\[ \color{blue}{\left(-\frac{1}{\sqrt{x}}\right)} \cdot -1 + \frac{-1}{\sqrt{x}} \cdot \left(-1 \cdot \frac{\sqrt{x}}{-\mathsf{hypot}\left(1, \sqrt{x}\right)}\right) \]
      pow1/2 [=>] 99.4	\[ \left(-\frac{1}{\color{blue}{{x}^{0.5}}}\right) \cdot -1 + \frac{-1}{\sqrt{x}} \cdot \left(-1 \cdot \frac{\sqrt{x}}{-\mathsf{hypot}\left(1, \sqrt{x}\right)}\right) \]
      pow-flip [=>] 99.8	\[ \left(-\color{blue}{{x}^{\left(-0.5\right)}}\right) \cdot -1 + \frac{-1}{\sqrt{x}} \cdot \left(-1 \cdot \frac{\sqrt{x}}{-\mathsf{hypot}\left(1, \sqrt{x}\right)}\right) \]
      metadata-eval [=>] 99.8	\[ \left(-{x}^{\color{blue}{-0.5}}\right) \cdot -1 + \frac{-1}{\sqrt{x}} \cdot \left(-1 \cdot \frac{\sqrt{x}}{-\mathsf{hypot}\left(1, \sqrt{x}\right)}\right) \]
      div-inv [=>] 99.8	\[ \left(-{x}^{-0.5}\right) \cdot -1 + \color{blue}{\left(-1 \cdot \frac{1}{\sqrt{x}}\right)} \cdot \left(-1 \cdot \frac{\sqrt{x}}{-\mathsf{hypot}\left(1, \sqrt{x}\right)}\right) \]
      mul-1-neg [=>] 99.8	\[ \left(-{x}^{-0.5}\right) \cdot -1 + \color{blue}{\left(-\frac{1}{\sqrt{x}}\right)} \cdot \left(-1 \cdot \frac{\sqrt{x}}{-\mathsf{hypot}\left(1, \sqrt{x}\right)}\right) \]
      pow1/2 [=>] 99.8	\[ \left(-{x}^{-0.5}\right) \cdot -1 + \left(-\frac{1}{\color{blue}{{x}^{0.5}}}\right) \cdot \left(-1 \cdot \frac{\sqrt{x}}{-\mathsf{hypot}\left(1, \sqrt{x}\right)}\right) \]
      pow-flip [=>] 99.8	\[ \left(-{x}^{-0.5}\right) \cdot -1 + \left(-\color{blue}{{x}^{\left(-0.5\right)}}\right) \cdot \left(-1 \cdot \frac{\sqrt{x}}{-\mathsf{hypot}\left(1, \sqrt{x}\right)}\right) \]
      metadata-eval [=>] 99.8	\[ \left(-{x}^{-0.5}\right) \cdot -1 + \left(-{x}^{\color{blue}{-0.5}}\right) \cdot \left(-1 \cdot \frac{\sqrt{x}}{-\mathsf{hypot}\left(1, \sqrt{x}\right)}\right) \]
      associate-*r/ [=>] 99.8	\[ \left(-{x}^{-0.5}\right) \cdot -1 + \left(-{x}^{-0.5}\right) \cdot \color{blue}{\frac{-1 \cdot \sqrt{x}}{-\mathsf{hypot}\left(1, \sqrt{x}\right)}} \]
      neg-mul-1 [<=] 99.8	\[ \left(-{x}^{-0.5}\right) \cdot -1 + \left(-{x}^{-0.5}\right) \cdot \frac{\color{blue}{-\sqrt{x}}}{-\mathsf{hypot}\left(1, \sqrt{x}\right)} \]

   5. Simplified 99.8%

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

      [Start] 99.8	\[ \left(-{x}^{-0.5}\right) \cdot -1 + \left(-{x}^{-0.5}\right) \cdot \sqrt{\frac{x}{x + 1}} \]
      distribute-lft-out [=>] 99.8	\[ \color{blue}{\left(-{x}^{-0.5}\right) \cdot \left(-1 + \sqrt{\frac{x}{x + 1}}\right)} \]
      neg-mul-1 [=>] 99.8	\[ \color{blue}{\left(-1 \cdot {x}^{-0.5}\right)} \cdot \left(-1 + \sqrt{\frac{x}{x + 1}}\right) \]
      *-commutative [=>] 99.8	\[ \color{blue}{\left({x}^{-0.5} \cdot -1\right)} \cdot \left(-1 + \sqrt{\frac{x}{x + 1}}\right) \]
      associate-*l* [=>] 99.8	\[ \color{blue}{{x}^{-0.5} \cdot \left(-1 \cdot \left(-1 + \sqrt{\frac{x}{x + 1}}\right)\right)} \]
      neg-mul-1 [<=] 99.8	\[ {x}^{-0.5} \cdot \color{blue}{\left(-\left(-1 + \sqrt{\frac{x}{x + 1}}\right)\right)} \]
      distribute-neg-in [=>] 99.8	\[ {x}^{-0.5} \cdot \color{blue}{\left(\left(--1\right) + \left(-\sqrt{\frac{x}{x + 1}}\right)\right)} \]
      metadata-eval [=>] 99.8	\[ {x}^{-0.5} \cdot \left(\color{blue}{1} + \left(-\sqrt{\frac{x}{x + 1}}\right)\right) \]
      unsub-neg [=>] 99.8	\[ {x}^{-0.5} \cdot \color{blue}{\left(1 - \sqrt{\frac{x}{x + 1}}\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{1 + x} + \sqrt{x}}}{\left(x + \left(0.5 + \frac{0.0625}{x \cdot x}\right)\right) - \frac{0.125}{x}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(1 - \sqrt{\frac{x}{1 + x}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.5%
Cost	52224

\[\begin{array}{l} t_0 := {\left(\sqrt{1 + x} + \sqrt{x}\right)}^{-0.5}\\ t_0 \cdot \frac{t_0}{\mathsf{hypot}\left(x, \sqrt{x}\right)} \end{array} \]

Alternative 2
Accuracy	99.3%
Cost	26756

\[\begin{array}{l} t_0 := \frac{-1}{\sqrt{1 + x}}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + t_0 \leq 5 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{0.5}{x} - \frac{0.375}{x \cdot x}}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} + t_0\\ \end{array} \]

Alternative 3
Accuracy	98.9%
Cost	26240

\[\frac{1}{\left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \mathsf{hypot}\left(x, \sqrt{x}\right)} \]

Alternative 4
Accuracy	99.8%
Cost	13892

\[\begin{array}{l} \mathbf{if}\;x \leq 7300:\\ \;\;\;\;{x}^{-0.5} \cdot \left(1 - \sqrt{\frac{x}{1 + x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{1 + x} + \sqrt{x}}}{\left(x + 0.5\right) + \frac{-0.125}{x}}\\ \end{array} \]

Alternative 5
Accuracy	99.7%
Cost	13572

\[\begin{array}{l} \mathbf{if}\;x \leq 470000:\\ \;\;\;\;{x}^{-0.5} \cdot \left(1 - \sqrt{\frac{x}{1 + x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{x} - \frac{0.375}{x \cdot x}}{\sqrt{x}}\\ \end{array} \]

Alternative 6
Accuracy	99.7%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;x \leq 205000:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{x} - \frac{0.375}{x \cdot x}}{\sqrt{x}}\\ \end{array} \]

Alternative 7
Accuracy	98.9%
Cost	7236

\[\begin{array}{l} \mathbf{if}\;x \leq 1.1:\\ \;\;\;\;{x}^{-0.5} - \left(1 + x \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{x} - \frac{0.375}{x \cdot x}}{\sqrt{x}}\\ \end{array} \]

Alternative 8
Accuracy	98.5%
Cost	7044

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

Alternative 9
Accuracy	98.2%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;x \leq 0.68:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{x}}{\sqrt{x}}\\ \end{array} \]

Alternative 10
Accuracy	52.8%
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 0.16:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + 0.5}\\ \end{array} \]

Alternative 11
Accuracy	50.3%
Cost	6528

\[{x}^{-0.5} \]

Alternative 12
Accuracy	7.4%
Cost	320

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

Alternative 13
Accuracy	7.4%
Cost	192

\[\frac{1}{x} \]

Alternative 14
Accuracy	1.9%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023157 
(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)))))