2isqrt (example 3.6)

Average Accuracy: 68.9% → 99.5%

Time: 15.4s

Precision: binary64

Cost: 27076

Math

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

↓

\[\begin{array}{l} t_0 := \frac{-1}{\sqrt{1 + x}}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + t_0 \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{1}{x}}{\sqrt{\frac{1}{x}} \cdot 1.5 + \sqrt{x} \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} + t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ -1.0 (sqrt (+ 1.0 x)))))
   (if (<= (+ (/ 1.0 (sqrt x)) t_0) 5e-14)
     (/ (/ 1.0 x) (+ (* (sqrt (/ 1.0 x)) 1.5) (* (sqrt x) 2.0)))
     (+ (pow x -0.5) t_0))))

C

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

↓

double code(double x) {
	double t_0 = -1.0 / sqrt((1.0 + x));
	double tmp;
	if (((1.0 / sqrt(x)) + t_0) <= 5e-14) {
		tmp = (1.0 / x) / ((sqrt((1.0 / x)) * 1.5) + (sqrt(x) * 2.0));
	} else {
		tmp = pow(x, -0.5) + t_0;
	}
	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 = (-1.0d0) / sqrt((1.0d0 + x))
    if (((1.0d0 / sqrt(x)) + t_0) <= 5d-14) then
        tmp = (1.0d0 / x) / ((sqrt((1.0d0 / x)) * 1.5d0) + (sqrt(x) * 2.0d0))
    else
        tmp = (x ** (-0.5d0)) + t_0
    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 = -1.0 / Math.sqrt((1.0 + x));
	double tmp;
	if (((1.0 / Math.sqrt(x)) + t_0) <= 5e-14) {
		tmp = (1.0 / x) / ((Math.sqrt((1.0 / x)) * 1.5) + (Math.sqrt(x) * 2.0));
	} else {
		tmp = Math.pow(x, -0.5) + t_0;
	}
	return tmp;
}

Python

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

↓

def code(x):
	t_0 = -1.0 / math.sqrt((1.0 + x))
	tmp = 0
	if ((1.0 / math.sqrt(x)) + t_0) <= 5e-14:
		tmp = (1.0 / x) / ((math.sqrt((1.0 / x)) * 1.5) + (math.sqrt(x) * 2.0))
	else:
		tmp = math.pow(x, -0.5) + t_0
	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 = Float64(-1.0 / sqrt(Float64(1.0 + x)))
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + t_0) <= 5e-14)
		tmp = Float64(Float64(1.0 / x) / Float64(Float64(sqrt(Float64(1.0 / x)) * 1.5) + Float64(sqrt(x) * 2.0)));
	else
		tmp = Float64((x ^ -0.5) + t_0);
	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 = -1.0 / sqrt((1.0 + x));
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + t_0) <= 5e-14)
		tmp = (1.0 / x) / ((sqrt((1.0 / x)) * 1.5) + (sqrt(x) * 2.0));
	else
		tmp = (x ^ -0.5) + t_0;
	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[(-1.0 / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], 5e-14], N[(N[(1.0 / x), $MachinePrecision] / N[(N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * 1.5), $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] + t$95$0), $MachinePrecision]]]

Target

Original	68.9%
Target	99.0%
Herbie	99.5%

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 5.0000000000000002e-14

   1. Initial program 37.6%

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

   2. Applied egg-rr 37.7%

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

      [Start] 37.6	\[ \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
      flip-- [=>] 37.6	\[ \color{blue}{\frac{\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \cdot \frac{1}{\sqrt{x + 1}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}} \]
      div-inv [=>] 37.6	\[ \color{blue}{\left(\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \cdot \frac{1}{\sqrt{x + 1}}\right) \cdot \frac{1}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}} \]

   3. Simplified 37.7%

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

      [Start] 37.7	\[ \left(\frac{1}{x} + \frac{-1}{1 + x}\right) \cdot \frac{1}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]
      associate-*r/ [=>] 37.7	\[ \color{blue}{\frac{\left(\frac{1}{x} + \frac{-1}{1 + x}\right) \cdot 1}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}} \]
      *-rgt-identity [=>] 37.7	\[ \frac{\color{blue}{\frac{1}{x} + \frac{-1}{1 + x}}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]

   4. Applied egg-rr 39.1%

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

      [Start] 37.7	\[ \frac{\frac{1}{x} + \frac{-1}{1 + x}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]
      frac-2neg [=>] 37.7	\[ \frac{\frac{1}{x} + \color{blue}{\frac{--1}{-\left(1 + x\right)}}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]
      metadata-eval [=>] 37.7	\[ \frac{\frac{1}{x} + \frac{\color{blue}{1}}{-\left(1 + x\right)}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]
      frac-add [=>] 39.1	\[ \frac{\color{blue}{\frac{1 \cdot \left(-\left(1 + x\right)\right) + x \cdot 1}{x \cdot \left(-\left(1 + x\right)\right)}}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]
      *-un-lft-identity [<=] 39.1	\[ \frac{\frac{\color{blue}{\left(-\left(1 + x\right)\right)} + x \cdot 1}{x \cdot \left(-\left(1 + x\right)\right)}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]
      *-commutative [<=] 39.1	\[ \frac{\frac{\left(-\left(1 + x\right)\right) + \color{blue}{1 \cdot x}}{x \cdot \left(-\left(1 + x\right)\right)}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]
      *-un-lft-identity [<=] 39.1	\[ \frac{\frac{\left(-\left(1 + x\right)\right) + \color{blue}{x}}{x \cdot \left(-\left(1 + x\right)\right)}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]
      +-commutative [=>] 39.1	\[ \frac{\frac{\color{blue}{x + \left(-\left(1 + x\right)\right)}}{x \cdot \left(-\left(1 + x\right)\right)}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]
      neg-sub0 [=>] 39.1	\[ \frac{\frac{x + \color{blue}{\left(0 - \left(1 + x\right)\right)}}{x \cdot \left(-\left(1 + x\right)\right)}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]
      metadata-eval [<=] 39.1	\[ \frac{\frac{x + \left(\color{blue}{\log 1} - \left(1 + x\right)\right)}{x \cdot \left(-\left(1 + x\right)\right)}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]
      associate--r+ [=>] 39.1	\[ \frac{\frac{x + \color{blue}{\left(\left(\log 1 - 1\right) - x\right)}}{x \cdot \left(-\left(1 + x\right)\right)}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]
      metadata-eval [=>] 39.1	\[ \frac{\frac{x + \left(\left(\color{blue}{0} - 1\right) - x\right)}{x \cdot \left(-\left(1 + x\right)\right)}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]
      metadata-eval [=>] 39.1	\[ \frac{\frac{x + \left(\color{blue}{-1} - x\right)}{x \cdot \left(-\left(1 + x\right)\right)}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]
      neg-sub0 [=>] 39.1	\[ \frac{\frac{x + \left(-1 - x\right)}{x \cdot \color{blue}{\left(0 - \left(1 + x\right)\right)}}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]
      metadata-eval [<=] 39.1	\[ \frac{\frac{x + \left(-1 - x\right)}{x \cdot \left(\color{blue}{\log 1} - \left(1 + x\right)\right)}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]
      associate--r+ [=>] 39.1	\[ \frac{\frac{x + \left(-1 - x\right)}{x \cdot \color{blue}{\left(\left(\log 1 - 1\right) - x\right)}}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]
      metadata-eval [=>] 39.1	\[ \frac{\frac{x + \left(-1 - x\right)}{x \cdot \left(\left(\color{blue}{0} - 1\right) - x\right)}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]
      metadata-eval [=>] 39.1	\[ \frac{\frac{x + \left(-1 - x\right)}{x \cdot \left(\color{blue}{-1} - x\right)}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]

   5. Simplified 84.7%

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

      [Start] 39.1	\[ \frac{\frac{x + \left(-1 - x\right)}{x \cdot \left(-1 - x\right)}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]
      associate-/r* [=>] 39.1	\[ \frac{\color{blue}{\frac{\frac{x + \left(-1 - x\right)}{x}}{-1 - x}}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]
      +-commutative [=>] 39.1	\[ \frac{\frac{\frac{\color{blue}{\left(-1 - x\right) + x}}{x}}{-1 - x}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]
      associate-+l- [=>] 84.7	\[ \frac{\frac{\frac{\color{blue}{-1 - \left(x - x\right)}}{x}}{-1 - x}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]
      +-inverses [=>] 84.7	\[ \frac{\frac{\frac{-1 - \color{blue}{0}}{x}}{-1 - x}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]
      metadata-eval [=>] 84.7	\[ \frac{\frac{\frac{\color{blue}{-1}}{x}}{-1 - x}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]

   6. Applied egg-rr 99.3%

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

      [Start] 84.7	\[ \frac{\frac{\frac{-1}{x}}{-1 - x}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]
      div-inv [=>] 84.7	\[ \color{blue}{\frac{\frac{-1}{x}}{-1 - x} \cdot \frac{1}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}} \]
      div-inv [=>] 84.6	\[ \color{blue}{\left(\frac{-1}{x} \cdot \frac{1}{-1 - x}\right)} \cdot \frac{1}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]
      associate-*l* [=>] 99.3	\[ \color{blue}{\frac{-1}{x} \cdot \left(\frac{1}{-1 - x} \cdot \frac{1}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}\right)} \]
      frac-2neg [=>] 99.3	\[ \frac{-1}{x} \cdot \left(\color{blue}{\frac{-1}{-\left(-1 - x\right)}} \cdot \frac{1}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}\right) \]
      metadata-eval [=>] 99.3	\[ \frac{-1}{x} \cdot \left(\frac{\color{blue}{-1}}{-\left(-1 - x\right)} \cdot \frac{1}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}\right) \]
      neg-sub0 [=>] 99.3	\[ \frac{-1}{x} \cdot \left(\frac{-1}{\color{blue}{0 - \left(-1 - x\right)}} \cdot \frac{1}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}\right) \]
      metadata-eval [<=] 99.3	\[ \frac{-1}{x} \cdot \left(\frac{-1}{\color{blue}{\log 1} - \left(-1 - x\right)} \cdot \frac{1}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}\right) \]
      associate--r- [=>] 99.3	\[ \frac{-1}{x} \cdot \left(\frac{-1}{\color{blue}{\left(\log 1 - -1\right) + x}} \cdot \frac{1}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}\right) \]
      metadata-eval [=>] 99.3	\[ \frac{-1}{x} \cdot \left(\frac{-1}{\left(\color{blue}{0} - -1\right) + x} \cdot \frac{1}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}\right) \]
      metadata-eval [=>] 99.3	\[ \frac{-1}{x} \cdot \left(\frac{-1}{\color{blue}{1} + x} \cdot \frac{1}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}\right) \]
      +-commutative [=>] 99.3	\[ \frac{-1}{x} \cdot \left(\frac{-1}{\color{blue}{x + 1}} \cdot \frac{1}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}\right) \]
      +-commutative [=>] 99.3	\[ \frac{-1}{x} \cdot \left(\frac{-1}{x + 1} \cdot \frac{1}{{x}^{-0.5} + {\color{blue}{\left(x + 1\right)}}^{-0.5}}\right) \]

   7. Simplified 99.6%

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

      [Start] 99.3	\[ \frac{-1}{x} \cdot \left(\frac{-1}{x + 1} \cdot \frac{1}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}}\right) \]
      *-commutative [=>] 99.3	\[ \color{blue}{\left(\frac{-1}{x + 1} \cdot \frac{1}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}}\right) \cdot \frac{-1}{x}} \]
      associate-*l/ [=>] 99.5	\[ \color{blue}{\frac{-1 \cdot \frac{1}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}}}{x + 1}} \cdot \frac{-1}{x} \]
      associate-*r/ [=>] 99.5	\[ \frac{\color{blue}{\frac{-1 \cdot 1}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}}}}{x + 1} \cdot \frac{-1}{x} \]
      metadata-eval [=>] 99.5	\[ \frac{\frac{\color{blue}{-1}}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}}}{x + 1} \cdot \frac{-1}{x} \]
      associate-/l/ [=>] 99.5	\[ \color{blue}{\frac{-1}{\left(x + 1\right) \cdot \left({x}^{-0.5} + {\left(x + 1\right)}^{-0.5}\right)}} \cdot \frac{-1}{x} \]
      associate-*l/ [=>] 99.6	\[ \color{blue}{\frac{-1 \cdot \frac{-1}{x}}{\left(x + 1\right) \cdot \left({x}^{-0.5} + {\left(x + 1\right)}^{-0.5}\right)}} \]
      associate-*r/ [=>] 99.6	\[ \frac{\color{blue}{\frac{-1 \cdot -1}{x}}}{\left(x + 1\right) \cdot \left({x}^{-0.5} + {\left(x + 1\right)}^{-0.5}\right)} \]
      metadata-eval [=>] 99.6	\[ \frac{\frac{\color{blue}{1}}{x}}{\left(x + 1\right) \cdot \left({x}^{-0.5} + {\left(x + 1\right)}^{-0.5}\right)} \]
      +-commutative [=>] 99.6	\[ \frac{\frac{1}{x}}{\color{blue}{\left(1 + x\right)} \cdot \left({x}^{-0.5} + {\left(x + 1\right)}^{-0.5}\right)} \]
      +-commutative [=>] 99.6	\[ \frac{\frac{1}{x}}{\left(1 + x\right) \cdot \left({x}^{-0.5} + {\color{blue}{\left(1 + x\right)}}^{-0.5}\right)} \]

   8. Taylor expanded in x around inf 99.6%

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

   9. Simplified 99.6%

      \[\leadsto \frac{\frac{1}{x}}{\color{blue}{\sqrt{\frac{1}{x}} \cdot 1.5 + 2 \cdot \sqrt{x}}} \]
      Proof

      [Start] 99.6	\[ \frac{\frac{1}{x}}{-0.5 \cdot \sqrt{\frac{1}{x}} + \left(2 \cdot \sqrt{x} + 2 \cdot \sqrt{\frac{1}{x}}\right)} \]
      +-commutative [=>] 99.6	\[ \frac{\frac{1}{x}}{-0.5 \cdot \sqrt{\frac{1}{x}} + \color{blue}{\left(2 \cdot \sqrt{\frac{1}{x}} + 2 \cdot \sqrt{x}\right)}} \]
      associate-+r+ [=>] 99.6	\[ \frac{\frac{1}{x}}{\color{blue}{\left(-0.5 \cdot \sqrt{\frac{1}{x}} + 2 \cdot \sqrt{\frac{1}{x}}\right) + 2 \cdot \sqrt{x}}} \]
      distribute-rgt-out [=>] 99.6	\[ \frac{\frac{1}{x}}{\color{blue}{\sqrt{\frac{1}{x}} \cdot \left(-0.5 + 2\right)} + 2 \cdot \sqrt{x}} \]
      metadata-eval [=>] 99.6	\[ \frac{\frac{1}{x}}{\sqrt{\frac{1}{x}} \cdot \color{blue}{1.5} + 2 \cdot \sqrt{x}} \]

   if 5.0000000000000002e-14 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 99.0%

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

   2. Applied egg-rr 91.9%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-0.5}\right)} - 1\right)} - \frac{1}{\sqrt{x + 1}} \]
      Proof

      [Start] 99.0	\[ \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
      expm1-log1p-u [=>] 92.2	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{x}}\right)\right)} - \frac{1}{\sqrt{x + 1}} \]
      expm1-udef [=>] 91.9	\[ \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{x}}\right)} - 1\right)} - \frac{1}{\sqrt{x + 1}} \]
      pow1/2 [=>] 91.9	\[ \left(e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{{x}^{0.5}}}\right)} - 1\right) - \frac{1}{\sqrt{x + 1}} \]
      pow-flip [=>] 91.9	\[ \left(e^{\mathsf{log1p}\left(\color{blue}{{x}^{\left(-0.5\right)}}\right)} - 1\right) - \frac{1}{\sqrt{x + 1}} \]
      metadata-eval [=>] 91.9	\[ \left(e^{\mathsf{log1p}\left({x}^{\color{blue}{-0.5}}\right)} - 1\right) - \frac{1}{\sqrt{x + 1}} \]

   3. Simplified 99.4%

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

      [Start] 91.9	\[ \left(e^{\mathsf{log1p}\left({x}^{-0.5}\right)} - 1\right) - \frac{1}{\sqrt{x + 1}} \]
      expm1-def [=>] 92.2	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-0.5}\right)\right)} - \frac{1}{\sqrt{x + 1}} \]
      expm1-log1p [=>] 99.4	\[ \color{blue}{{x}^{-0.5}} - \frac{1}{\sqrt{x + 1}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{1}{x}}{\sqrt{\frac{1}{x}} \cdot 1.5 + \sqrt{x} \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{\sqrt{1 + x}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.3%
Cost	26756

\[\begin{array}{l} t_0 := \frac{-1}{\sqrt{1 + x}}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + t_0 \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + x\right) \cdot \left(\sqrt{\frac{1}{x}} \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} + t_0\\ \end{array} \]

Alternative 2
Accuracy	99.4%
Cost	13696

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

Alternative 3
Accuracy	99.3%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;x \leq 112000000:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + x\right) \cdot \left(\sqrt{\frac{1}{x}} \cdot 2\right)}\\ \end{array} \]

Alternative 4
Accuracy	98.4%
Cost	7364

\[\begin{array}{l} \mathbf{if}\;x \leq 0.66:\\ \;\;\;\;{x}^{-0.5} + \left(-1 + x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + x\right) \cdot \left(\sqrt{\frac{1}{x}} \cdot 2\right)}\\ \end{array} \]

Alternative 5
Accuracy	98.3%
Cost	7044

\[\begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;{x}^{-0.5} + \left(-1 + x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{\sqrt{x} \cdot 2}\\ \end{array} \]

Alternative 6
Accuracy	98.0%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;x \leq 0.66:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{\sqrt{x} \cdot 2}\\ \end{array} \]

Alternative 7
Accuracy	67.4%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;x \leq 4.6 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{x + \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-1}{x}}{-1 - x}}{x \cdot -0.5}\\ \end{array} \]

Alternative 8
Accuracy	65.8%
Cost	6784

\[\left(1 + {x}^{-0.5}\right) + -1 \]

Alternative 9
Accuracy	66.1%
Cost	6660

\[\begin{array}{l} \mathbf{if}\;x \leq 8.4 \cdot 10^{+122}:\\ \;\;\;\;{x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-1}{x}}{-1 - x}}{x \cdot -0.5}\\ \end{array} \]

Alternative 10
Accuracy	22.6%
Cost	836

\[\begin{array}{l} \mathbf{if}\;x \leq 4.6 \cdot 10^{+153}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{x + 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-1}{x}}{-1 - x}}{x \cdot -0.5}\\ \end{array} \]

Alternative 11
Accuracy	7.5%
Cost	576

\[\frac{1 + \left(x - x\right)}{x + 0.5} \]

Alternative 12
Accuracy	7.4%
Cost	192

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

Alternative 13
Accuracy	1.9%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023133 
(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)))))