2isqrt (example 3.6)

Percentage Accurate: 69.3% → 99.8%

Time: 15.6s

Precision: binary64

Cost: 27012

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{x + 1}} \leq 0:\\ \;\;\;\;\frac{{x}^{-0.5}}{x + \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.5}}{\left(x + 1\right) + \sqrt{x + x \cdot x}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 (sqrt (+ x 1.0)))) 0.0)
   (/ (pow x -0.5) (+ x (+ x 1.0)))
   (/ (pow x -0.5) (+ (+ x 1.0) (sqrt (+ x (* x x)))))))

C

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

↓

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((x + 1.0)))) <= 0.0) {
		tmp = pow(x, -0.5) / (x + (x + 1.0));
	} else {
		tmp = pow(x, -0.5) / ((x + 1.0) + sqrt((x + (x * 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) :: tmp
    if (((1.0d0 / sqrt(x)) + ((-1.0d0) / sqrt((x + 1.0d0)))) <= 0.0d0) then
        tmp = (x ** (-0.5d0)) / (x + (x + 1.0d0))
    else
        tmp = (x ** (-0.5d0)) / ((x + 1.0d0) + sqrt((x + (x * 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 tmp;
	if (((1.0 / Math.sqrt(x)) + (-1.0 / Math.sqrt((x + 1.0)))) <= 0.0) {
		tmp = Math.pow(x, -0.5) / (x + (x + 1.0));
	} else {
		tmp = Math.pow(x, -0.5) / ((x + 1.0) + Math.sqrt((x + (x * x))));
	}
	return tmp;
}

Python

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

↓

def code(x):
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / math.sqrt((x + 1.0)))) <= 0.0:
		tmp = math.pow(x, -0.5) / (x + (x + 1.0))
	else:
		tmp = math.pow(x, -0.5) / ((x + 1.0) + math.sqrt((x + (x * 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)
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / sqrt(Float64(x + 1.0)))) <= 0.0)
		tmp = Float64((x ^ -0.5) / Float64(x + Float64(x + 1.0)));
	else
		tmp = Float64((x ^ -0.5) / Float64(Float64(x + 1.0) + sqrt(Float64(x + Float64(x * 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)
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((x + 1.0)))) <= 0.0)
		tmp = (x ^ -0.5) / (x + (x + 1.0));
	else
		tmp = (x ^ -0.5) / ((x + 1.0) + sqrt((x + (x * 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_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[Power[x, -0.5], $MachinePrecision] / N[(x + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] + N[Sqrt[N[(x + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	69.3%
Target	99.0%
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)))) < 0.0

   1. Initial program 30.3%

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

   2. Applied egg-rr 30.3%

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

      [Start] 30.3%	\[ \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
      frac-sub [=>] 30.3%	\[ \color{blue}{\frac{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt{x + 1}}} \]
      div-inv [=>] 30.3%	\[ \color{blue}{\left(1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}}} \]
      *-un-lft-identity [<=] 30.3%	\[ \left(\color{blue}{\sqrt{x + 1}} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]
      +-commutative [=>] 30.3%	\[ \left(\sqrt{\color{blue}{1 + x}} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]
      *-rgt-identity [=>] 30.3%	\[ \left(\sqrt{1 + x} - \color{blue}{\sqrt{x}}\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]
      metadata-eval [<=] 30.3%	\[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\color{blue}{1 \cdot 1}}{\sqrt{x} \cdot \sqrt{x + 1}} \]
      frac-times [<=] 30.3%	\[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \color{blue}{\left(\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x + 1}}\right)} \]
      un-div-inv [=>] 30.3%	\[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \color{blue}{\frac{\frac{1}{\sqrt{x}}}{\sqrt{x + 1}}} \]
      pow1/2 [=>] 30.3%	\[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\frac{1}{\color{blue}{{x}^{0.5}}}}{\sqrt{x + 1}} \]
      pow-flip [=>] 30.3%	\[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\color{blue}{{x}^{\left(-0.5\right)}}}{\sqrt{x + 1}} \]
      metadata-eval [=>] 30.3%	\[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{x}^{\color{blue}{-0.5}}}{\sqrt{x + 1}} \]
      +-commutative [=>] 30.3%	\[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{x}^{-0.5}}{\sqrt{\color{blue}{1 + x}}} \]

   3. Simplified 30.3%

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

      [Start] 30.3%	\[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{x}^{-0.5}}{\sqrt{1 + x}} \]
      associate-*r/ [=>] 30.3%	\[ \color{blue}{\frac{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot {x}^{-0.5}}{\sqrt{1 + x}}} \]

   4. Applied egg-rr 30.3%

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

      [Start] 30.3%	\[ \frac{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot {x}^{-0.5}}{\sqrt{1 + x}} \]
      flip-- [=>] 30.3%	\[ \frac{\color{blue}{\frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}}} \cdot {x}^{-0.5}}{\sqrt{1 + x}} \]
      add-sqr-sqrt [<=] 13.5%	\[ \frac{\frac{\color{blue}{\left(1 + x\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{-0.5}}{\sqrt{1 + x}} \]
      add-sqr-sqrt [<=] 30.3%	\[ \frac{\frac{\left(1 + x\right) - \color{blue}{x}}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{-0.5}}{\sqrt{1 + x}} \]

   5. Simplified 99.3%

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

      [Start] 30.3%	\[ \frac{\frac{\left(1 + x\right) - x}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{-0.5}}{\sqrt{1 + x}} \]
      associate--l+ [=>] 99.3%	\[ \frac{\frac{\color{blue}{1 + \left(x - x\right)}}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{-0.5}}{\sqrt{1 + x}} \]
      +-inverses [=>] 99.3%	\[ \frac{\frac{1 + \color{blue}{0}}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{-0.5}}{\sqrt{1 + x}} \]
      metadata-eval [=>] 99.3%	\[ \frac{\frac{\color{blue}{1}}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{-0.5}}{\sqrt{1 + x}} \]

   6. Applied egg-rr 30.3%

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

      [Start] 99.3%	\[ \frac{\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{-0.5}}{\sqrt{1 + x}} \]
      expm1-log1p-u [=>] 99.3%	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{-0.5}}{\sqrt{1 + x}}\right)\right)} \]
      expm1-udef [=>] 30.3%	\[ \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{-0.5}}{\sqrt{1 + x}}\right)} - 1} \]
      associate-*l/ [=>] 30.3%	\[ e^{\mathsf{log1p}\left(\frac{\color{blue}{\frac{1 \cdot {x}^{-0.5}}{\sqrt{1 + x} + \sqrt{x}}}}{\sqrt{1 + x}}\right)} - 1 \]
      *-un-lft-identity [<=] 30.3%	\[ e^{\mathsf{log1p}\left(\frac{\frac{\color{blue}{{x}^{-0.5}}}{\sqrt{1 + x} + \sqrt{x}}}{\sqrt{1 + x}}\right)} - 1 \]

   7. Simplified 99.7%

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

      [Start] 30.3%	\[ e^{\mathsf{log1p}\left(\frac{\frac{{x}^{-0.5}}{\sqrt{1 + x} + \sqrt{x}}}{\sqrt{1 + x}}\right)} - 1 \]
      expm1-def [=>] 99.4%	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{{x}^{-0.5}}{\sqrt{1 + x} + \sqrt{x}}}{\sqrt{1 + x}}\right)\right)} \]
      expm1-log1p [=>] 99.4%	\[ \color{blue}{\frac{\frac{{x}^{-0.5}}{\sqrt{1 + x} + \sqrt{x}}}{\sqrt{1 + x}}} \]
      associate-/l/ [=>] 99.4%	\[ \color{blue}{\frac{{x}^{-0.5}}{\sqrt{1 + x} \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)}} \]
      distribute-lft-in [=>] 99.4%	\[ \frac{{x}^{-0.5}}{\color{blue}{\sqrt{1 + x} \cdot \sqrt{1 + x} + \sqrt{1 + x} \cdot \sqrt{x}}} \]
      rem-square-sqrt [=>] 99.7%	\[ \frac{{x}^{-0.5}}{\color{blue}{\left(1 + x\right)} + \sqrt{1 + x} \cdot \sqrt{x}} \]
      +-commutative [=>] 99.7%	\[ \frac{{x}^{-0.5}}{\color{blue}{\left(x + 1\right)} + \sqrt{1 + x} \cdot \sqrt{x}} \]
      +-commutative [=>] 99.7%	\[ \frac{{x}^{-0.5}}{\left(x + 1\right) + \sqrt{\color{blue}{x + 1}} \cdot \sqrt{x}} \]

   8. Taylor expanded in x around inf 99.8%

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

   if 0.0 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 96.2%

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

   2. Applied egg-rr 96.7%

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

      [Start] 96.2%	\[ \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
      frac-sub [=>] 96.3%	\[ \color{blue}{\frac{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt{x + 1}}} \]
      div-inv [=>] 96.3%	\[ \color{blue}{\left(1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}}} \]
      *-un-lft-identity [<=] 96.3%	\[ \left(\color{blue}{\sqrt{x + 1}} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]
      +-commutative [=>] 96.3%	\[ \left(\sqrt{\color{blue}{1 + x}} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]
      *-rgt-identity [=>] 96.3%	\[ \left(\sqrt{1 + x} - \color{blue}{\sqrt{x}}\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]
      metadata-eval [<=] 96.3%	\[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\color{blue}{1 \cdot 1}}{\sqrt{x} \cdot \sqrt{x + 1}} \]
      frac-times [<=] 96.3%	\[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \color{blue}{\left(\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x + 1}}\right)} \]
      un-div-inv [=>] 96.3%	\[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \color{blue}{\frac{\frac{1}{\sqrt{x}}}{\sqrt{x + 1}}} \]
      pow1/2 [=>] 96.3%	\[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\frac{1}{\color{blue}{{x}^{0.5}}}}{\sqrt{x + 1}} \]
      pow-flip [=>] 96.7%	\[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\color{blue}{{x}^{\left(-0.5\right)}}}{\sqrt{x + 1}} \]
      metadata-eval [=>] 96.7%	\[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{x}^{\color{blue}{-0.5}}}{\sqrt{x + 1}} \]
      +-commutative [=>] 96.7%	\[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{x}^{-0.5}}{\sqrt{\color{blue}{1 + x}}} \]

   3. Simplified 96.7%

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

      [Start] 96.7%	\[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{x}^{-0.5}}{\sqrt{1 + x}} \]
      associate-*r/ [=>] 96.7%	\[ \color{blue}{\frac{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot {x}^{-0.5}}{\sqrt{1 + x}}} \]

   4. Applied egg-rr 99.8%

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

      [Start] 96.7%	\[ \frac{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot {x}^{-0.5}}{\sqrt{1 + x}} \]
      flip-- [=>] 97.7%	\[ \frac{\color{blue}{\frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}}} \cdot {x}^{-0.5}}{\sqrt{1 + x}} \]
      add-sqr-sqrt [<=] 97.1%	\[ \frac{\frac{\color{blue}{\left(1 + x\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{-0.5}}{\sqrt{1 + x}} \]
      add-sqr-sqrt [<=] 99.8%	\[ \frac{\frac{\left(1 + x\right) - \color{blue}{x}}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{-0.5}}{\sqrt{1 + x}} \]

   5. Simplified 99.8%

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

      [Start] 99.8%	\[ \frac{\frac{\left(1 + x\right) - x}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{-0.5}}{\sqrt{1 + x}} \]
      associate--l+ [=>] 99.8%	\[ \frac{\frac{\color{blue}{1 + \left(x - x\right)}}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{-0.5}}{\sqrt{1 + x}} \]
      +-inverses [=>] 99.8%	\[ \frac{\frac{1 + \color{blue}{0}}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{-0.5}}{\sqrt{1 + x}} \]
      metadata-eval [=>] 99.8%	\[ \frac{\frac{\color{blue}{1}}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{-0.5}}{\sqrt{1 + x}} \]

   6. Applied egg-rr 87.9%

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

      [Start] 99.8%	\[ \frac{\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{-0.5}}{\sqrt{1 + x}} \]
      expm1-log1p-u [=>] 92.9%	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{-0.5}}{\sqrt{1 + x}}\right)\right)} \]
      expm1-udef [=>] 87.9%	\[ \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{-0.5}}{\sqrt{1 + x}}\right)} - 1} \]
      associate-*l/ [=>] 87.9%	\[ e^{\mathsf{log1p}\left(\frac{\color{blue}{\frac{1 \cdot {x}^{-0.5}}{\sqrt{1 + x} + \sqrt{x}}}}{\sqrt{1 + x}}\right)} - 1 \]
      *-un-lft-identity [<=] 87.9%	\[ e^{\mathsf{log1p}\left(\frac{\frac{\color{blue}{{x}^{-0.5}}}{\sqrt{1 + x} + \sqrt{x}}}{\sqrt{1 + x}}\right)} - 1 \]

   7. Simplified 99.9%

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

      [Start] 87.9%	\[ e^{\mathsf{log1p}\left(\frac{\frac{{x}^{-0.5}}{\sqrt{1 + x} + \sqrt{x}}}{\sqrt{1 + x}}\right)} - 1 \]
      expm1-def [=>] 92.9%	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{{x}^{-0.5}}{\sqrt{1 + x} + \sqrt{x}}}{\sqrt{1 + x}}\right)\right)} \]
      expm1-log1p [=>] 99.8%	\[ \color{blue}{\frac{\frac{{x}^{-0.5}}{\sqrt{1 + x} + \sqrt{x}}}{\sqrt{1 + x}}} \]
      associate-/l/ [=>] 99.8%	\[ \color{blue}{\frac{{x}^{-0.5}}{\sqrt{1 + x} \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)}} \]
      distribute-lft-in [=>] 99.8%	\[ \frac{{x}^{-0.5}}{\color{blue}{\sqrt{1 + x} \cdot \sqrt{1 + x} + \sqrt{1 + x} \cdot \sqrt{x}}} \]
      rem-square-sqrt [=>] 99.9%	\[ \frac{{x}^{-0.5}}{\color{blue}{\left(1 + x\right)} + \sqrt{1 + x} \cdot \sqrt{x}} \]
      +-commutative [=>] 99.9%	\[ \frac{{x}^{-0.5}}{\color{blue}{\left(x + 1\right)} + \sqrt{1 + x} \cdot \sqrt{x}} \]
      +-commutative [=>] 99.9%	\[ \frac{{x}^{-0.5}}{\left(x + 1\right) + \sqrt{\color{blue}{x + 1}} \cdot \sqrt{x}} \]

   8. Applied egg-rr 99.9%

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

      [Start] 99.9%	\[ \frac{{x}^{-0.5}}{\left(x + 1\right) + \sqrt{x + 1} \cdot \sqrt{x}} \]
      pow1 [=>] 99.9%	\[ \frac{{x}^{-0.5}}{\left(x + 1\right) + \color{blue}{{\left(\sqrt{x + 1} \cdot \sqrt{x}\right)}^{1}}} \]
      sqrt-unprod [=>] 99.9%	\[ \frac{{x}^{-0.5}}{\left(x + 1\right) + {\color{blue}{\left(\sqrt{\left(x + 1\right) \cdot x}\right)}}^{1}} \]

   9. Simplified 99.9%

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

      [Start] 99.9%	\[ \frac{{x}^{-0.5}}{\left(x + 1\right) + {\left(\sqrt{\left(x + 1\right) \cdot x}\right)}^{1}} \]
      unpow1 [=>] 99.9%	\[ \frac{{x}^{-0.5}}{\left(x + 1\right) + \color{blue}{\sqrt{\left(x + 1\right) \cdot x}}} \]
      *-commutative [=>] 99.9%	\[ \frac{{x}^{-0.5}}{\left(x + 1\right) + \sqrt{\color{blue}{x \cdot \left(x + 1\right)}}} \]
      distribute-lft-in [=>] 99.9%	\[ \frac{{x}^{-0.5}}{\left(x + 1\right) + \sqrt{\color{blue}{x \cdot x + x \cdot 1}}} \]
      *-rgt-identity [=>] 99.9%	\[ \frac{{x}^{-0.5}}{\left(x + 1\right) + \sqrt{x \cdot x + \color{blue}{x}}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{x + 1}} \leq 0:\\ \;\;\;\;\frac{{x}^{-0.5}}{x + \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.5}}{\left(x + 1\right) + \sqrt{x + x \cdot x}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.8%
Cost	27012

\[\begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{x + 1}} \leq 0:\\ \;\;\;\;\frac{{x}^{-0.5}}{x + \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.5}}{\left(x + 1\right) + \sqrt{x + x \cdot x}}\\ \end{array} \]

Alternative 2
Accuracy	99.8%
Cost	26756

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

Alternative 3
Accuracy	99.8%
Cost	19968

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

Alternative 4
Accuracy	99.8%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;x \leq 650:\\ \;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.5}}{\left(x + 1\right) + \left(\frac{0.0625}{x \cdot x} + \left(\left(x + 0.5\right) + \frac{-0.125}{x}\right)\right)}\\ \end{array} \]

Alternative 5
Accuracy	99.1%
Cost	7812

\[\begin{array}{l} \mathbf{if}\;x \leq 0.6:\\ \;\;\;\;\frac{1}{\sqrt{x}} + \frac{-1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.5}}{\left(x + 1\right) + \left(\frac{0.0625}{x \cdot x} + \left(\left(x + 0.5\right) + \frac{-0.125}{x}\right)\right)}\\ \end{array} \]

Alternative 6
Accuracy	99.0%
Cost	7428

\[\begin{array}{l} \mathbf{if}\;x \leq 0.58:\\ \;\;\;\;\frac{1}{\sqrt{x}} + \frac{-1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.5}}{\left(x + 1\right) + \left(\left(x + 0.5\right) + \frac{-0.125}{x}\right)}\\ \end{array} \]

Alternative 7
Accuracy	98.8%
Cost	7236

\[\begin{array}{l} \mathbf{if}\;x \leq 0.62:\\ \;\;\;\;\frac{1}{\sqrt{x}} + \frac{-1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.5}}{\left(x + 1\right) + \left(x + 0.5\right)}\\ \end{array} \]

Alternative 8
Accuracy	98.8%
Cost	7172

\[\begin{array}{l} \mathbf{if}\;x \leq 0.41:\\ \;\;\;\;\frac{1}{\sqrt{x}} + \left(-1 - x \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.5}}{\left(x + 1\right) + \left(x + 0.5\right)}\\ \end{array} \]

Alternative 9
Accuracy	98.4%
Cost	7108

\[\begin{array}{l} \mathbf{if}\;x \leq 0.6:\\ \;\;\;\;\frac{1}{\sqrt{x}} + \left(-1 - x \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.5}}{x + \left(x + 1\right)}\\ \end{array} \]

Alternative 10
Accuracy	98.3%
Cost	7044

\[\begin{array}{l} \mathbf{if}\;x \leq 0.27:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.5}}{x + \left(x + 1\right)}\\ \end{array} \]

Alternative 11
Accuracy	98.2%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;x \leq 0.48:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{x}}{\sqrt{x + 1}}\\ \end{array} \]

Alternative 12
Accuracy	98.2%
Cost	6916

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

Alternative 13
Accuracy	52.3%
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 0.8:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5}\\ \end{array} \]

Alternative 14
Accuracy	12.2%
Cost	6656

\[{x}^{-0.5} \cdot 2 \]

Alternative 15
Accuracy	51.0%
Cost	6528

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

Alternative 16
Accuracy	1.9%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023263 
(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)))))