2isqrt (example 3.6)

Percentage Accurate: 69.0% → 99.7%

Time: 7.4s

Precision: binary64

Cost: 33732

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{t_0} \leq 0:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x + \left(1 - x\right)}{\sqrt{x} + t_0}}{\sqrt{x \cdot \left(1 + x\right)}}\\ \end{array} \]

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)) 0.0)
     (* 0.5 (pow x -1.5))
     (/ (/ (+ x (- 1.0 x)) (+ (sqrt x) t_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)) <= 0.0) {
		tmp = 0.5 * pow(x, -1.5);
	} else {
		tmp = ((x + (1.0 - x)) / (sqrt(x) + t_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)) <= 0.0d0) then
        tmp = 0.5d0 * (x ** (-1.5d0))
    else
        tmp = ((x + (1.0d0 - x)) / (sqrt(x) + t_0)) / 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)) <= 0.0) {
		tmp = 0.5 * Math.pow(x, -1.5);
	} else {
		tmp = ((x + (1.0 - x)) / (Math.sqrt(x) + t_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)) <= 0.0:
		tmp = 0.5 * math.pow(x, -1.5)
	else:
		tmp = ((x + (1.0 - x)) / (math.sqrt(x) + t_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)) <= 0.0)
		tmp = Float64(0.5 * (x ^ -1.5));
	else
		tmp = Float64(Float64(Float64(x + Float64(1.0 - x)) / Float64(sqrt(x) + t_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)) <= 0.0)
		tmp = 0.5 * (x ^ -1.5);
	else
		tmp = ((x + (1.0 - x)) / (sqrt(x) + t_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], 0.0], N[(0.5 * N[Power[x, -1.5], $MachinePrecision]), $MachinePrecision], N[(N[(N[(x + N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(x * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Target

Original	69.0%
Target	98.9%
Herbie	99.7%

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

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 36.5%

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

   2. Applied egg-rr 36.5%

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

      [Start] 36.5%	\[ \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
      flip-- [=>] 36.5%	\[ \color{blue}{\frac{\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \cdot \frac{1}{\sqrt{x + 1}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}} \]
      frac-times [=>] 16.4%	\[ \frac{\color{blue}{\frac{1 \cdot 1}{\sqrt{x} \cdot \sqrt{x}}} - \frac{1}{\sqrt{x + 1}} \cdot \frac{1}{\sqrt{x + 1}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \]
      metadata-eval [=>] 16.4%	\[ \frac{\frac{\color{blue}{1}}{\sqrt{x} \cdot \sqrt{x}} - \frac{1}{\sqrt{x + 1}} \cdot \frac{1}{\sqrt{x + 1}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \]
      add-sqr-sqrt [<=] 17.7%	\[ \frac{\frac{1}{\color{blue}{x}} - \frac{1}{\sqrt{x + 1}} \cdot \frac{1}{\sqrt{x + 1}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \]
      frac-times [=>] 24.7%	\[ \frac{\frac{1}{x} - \color{blue}{\frac{1 \cdot 1}{\sqrt{x + 1} \cdot \sqrt{x + 1}}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \]
      metadata-eval [=>] 24.7%	\[ \frac{\frac{1}{x} - \frac{\color{blue}{1}}{\sqrt{x + 1} \cdot \sqrt{x + 1}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \]
      add-sqr-sqrt [<=] 36.5%	\[ \frac{\frac{1}{x} - \frac{1}{\color{blue}{x + 1}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \]
      +-commutative [=>] 36.5%	\[ \frac{\frac{1}{x} - \frac{1}{\color{blue}{1 + x}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \]
      pow1/2 [=>] 36.5%	\[ \frac{\frac{1}{x} - \frac{1}{1 + x}}{\frac{1}{\color{blue}{{x}^{0.5}}} + \frac{1}{\sqrt{x + 1}}} \]
      pow-flip [=>] 36.5%	\[ \frac{\frac{1}{x} - \frac{1}{1 + x}}{\color{blue}{{x}^{\left(-0.5\right)}} + \frac{1}{\sqrt{x + 1}}} \]
      metadata-eval [=>] 36.5%	\[ \frac{\frac{1}{x} - \frac{1}{1 + x}}{{x}^{\color{blue}{-0.5}} + \frac{1}{\sqrt{x + 1}}} \]
      inv-pow [=>] 36.5%	\[ \frac{\frac{1}{x} - \frac{1}{1 + x}}{{x}^{-0.5} + \color{blue}{{\left(\sqrt{x + 1}\right)}^{-1}}} \]
      sqrt-pow2 [=>] 36.5%	\[ \frac{\frac{1}{x} - \frac{1}{1 + x}}{{x}^{-0.5} + \color{blue}{{\left(x + 1\right)}^{\left(\frac{-1}{2}\right)}}} \]
      +-commutative [=>] 36.5%	\[ \frac{\frac{1}{x} - \frac{1}{1 + x}}{{x}^{-0.5} + {\color{blue}{\left(1 + x\right)}}^{\left(\frac{-1}{2}\right)}} \]
      metadata-eval [=>] 36.5%	\[ \frac{\frac{1}{x} - \frac{1}{1 + x}}{{x}^{-0.5} + {\left(1 + x\right)}^{\color{blue}{-0.5}}} \]

   3. Taylor expanded in x around inf 67.0%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}} \]

   4. Simplified 100.0%

      \[\leadsto \color{blue}{0.5 \cdot {x}^{-1.5}} \]
      Step-by-step derivation

      [Start] 67.0%	\[ 0.5 \cdot \sqrt{\frac{1}{{x}^{3}}} \]
      exp-to-pow [<=] 64.1%	\[ 0.5 \cdot \sqrt{\frac{1}{\color{blue}{e^{\log x \cdot 3}}}} \]
      *-commutative [<=] 64.1%	\[ 0.5 \cdot \sqrt{\frac{1}{e^{\color{blue}{3 \cdot \log x}}}} \]
      exp-neg [<=] 64.1%	\[ 0.5 \cdot \sqrt{\color{blue}{e^{-3 \cdot \log x}}} \]
      distribute-lft-neg-in [=>] 64.1%	\[ 0.5 \cdot \sqrt{e^{\color{blue}{\left(-3\right) \cdot \log x}}} \]
      metadata-eval [=>] 64.1%	\[ 0.5 \cdot \sqrt{e^{\color{blue}{-3} \cdot \log x}} \]
      *-commutative [=>] 64.1%	\[ 0.5 \cdot \sqrt{e^{\color{blue}{\log x \cdot -3}}} \]
      exp-to-pow [=>] 67.0%	\[ 0.5 \cdot \sqrt{\color{blue}{{x}^{-3}}} \]
      metadata-eval [<=] 67.0%	\[ 0.5 \cdot \sqrt{{x}^{\color{blue}{\left(2 \cdot -1.5\right)}}} \]
      pow-sqr [<=] 67.1%	\[ 0.5 \cdot \sqrt{\color{blue}{{x}^{-1.5} \cdot {x}^{-1.5}}} \]
      rem-sqrt-square [=>] 100.0%	\[ 0.5 \cdot \color{blue}{\left|{x}^{-1.5}\right|} \]
      rem-square-sqrt [<=] 99.4%	\[ 0.5 \cdot \left|\color{blue}{\sqrt{{x}^{-1.5}} \cdot \sqrt{{x}^{-1.5}}}\right| \]
      fabs-sqr [=>] 99.4%	\[ 0.5 \cdot \color{blue}{\left(\sqrt{{x}^{-1.5}} \cdot \sqrt{{x}^{-1.5}}\right)} \]
      rem-square-sqrt [=>] 100.0%	\[ 0.5 \cdot \color{blue}{{x}^{-1.5}} \]

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

   1. Initial program 98.2%

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

   2. Applied egg-rr 98.2%

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

      [Start] 98.2%	\[ \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
      frac-sub [=>] 98.2%	\[ \color{blue}{\frac{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt{x + 1}}} \]
      *-un-lft-identity [<=] 98.2%	\[ \frac{\color{blue}{\sqrt{x + 1}} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt{x + 1}} \]
      +-commutative [=>] 98.2%	\[ \frac{\sqrt{\color{blue}{1 + x}} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt{x + 1}} \]
      *-rgt-identity [=>] 98.2%	\[ \frac{\sqrt{1 + x} - \color{blue}{\sqrt{x}}}{\sqrt{x} \cdot \sqrt{x + 1}} \]
      sqrt-unprod [=>] 98.2%	\[ \frac{\sqrt{1 + x} - \sqrt{x}}{\color{blue}{\sqrt{x \cdot \left(x + 1\right)}}} \]
      +-commutative [=>] 98.2%	\[ \frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{x \cdot \color{blue}{\left(1 + x\right)}}} \]

   3. Applied egg-rr 99.6%

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

      [Start] 98.2%	\[ \frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{x \cdot \left(1 + x\right)}} \]
      add-sqr-sqrt [=>] 98.2%	\[ \frac{\sqrt{1 + \color{blue}{\sqrt{x} \cdot \sqrt{x}}} - \sqrt{x}}{\sqrt{x \cdot \left(1 + x\right)}} \]
      hypot-1-def [=>] 98.2%	\[ \frac{\color{blue}{\mathsf{hypot}\left(1, \sqrt{x}\right)} - \sqrt{x}}{\sqrt{x \cdot \left(1 + x\right)}} \]
      flip-- [=>] 98.4%	\[ \frac{\color{blue}{\frac{\mathsf{hypot}\left(1, \sqrt{x}\right) \cdot \mathsf{hypot}\left(1, \sqrt{x}\right) - \sqrt{x} \cdot \sqrt{x}}{\mathsf{hypot}\left(1, \sqrt{x}\right) + \sqrt{x}}}}{\sqrt{x \cdot \left(1 + x\right)}} \]
      div-inv [=>] 98.4%	\[ \frac{\color{blue}{\left(\mathsf{hypot}\left(1, \sqrt{x}\right) \cdot \mathsf{hypot}\left(1, \sqrt{x}\right) - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\mathsf{hypot}\left(1, \sqrt{x}\right) + \sqrt{x}}}}{\sqrt{x \cdot \left(1 + x\right)}} \]
      hypot-1-def [<=] 98.6%	\[ \frac{\left(\color{blue}{\sqrt{1 + \sqrt{x} \cdot \sqrt{x}}} \cdot \mathsf{hypot}\left(1, \sqrt{x}\right) - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\mathsf{hypot}\left(1, \sqrt{x}\right) + \sqrt{x}}}{\sqrt{x \cdot \left(1 + x\right)}} \]
      add-sqr-sqrt [<=] 98.4%	\[ \frac{\left(\sqrt{1 + \color{blue}{x}} \cdot \mathsf{hypot}\left(1, \sqrt{x}\right) - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\mathsf{hypot}\left(1, \sqrt{x}\right) + \sqrt{x}}}{\sqrt{x \cdot \left(1 + x\right)}} \]
      hypot-1-def [<=] 98.6%	\[ \frac{\left(\sqrt{1 + x} \cdot \color{blue}{\sqrt{1 + \sqrt{x} \cdot \sqrt{x}}} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\mathsf{hypot}\left(1, \sqrt{x}\right) + \sqrt{x}}}{\sqrt{x \cdot \left(1 + x\right)}} \]
      add-sqr-sqrt [<=] 98.4%	\[ \frac{\left(\sqrt{1 + x} \cdot \sqrt{1 + \color{blue}{x}} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\mathsf{hypot}\left(1, \sqrt{x}\right) + \sqrt{x}}}{\sqrt{x \cdot \left(1 + x\right)}} \]
      add-sqr-sqrt [<=] 99.1%	\[ \frac{\left(\color{blue}{\left(1 + x\right)} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\mathsf{hypot}\left(1, \sqrt{x}\right) + \sqrt{x}}}{\sqrt{x \cdot \left(1 + x\right)}} \]
      +-commutative [=>] 99.1%	\[ \frac{\left(\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\mathsf{hypot}\left(1, \sqrt{x}\right) + \sqrt{x}}}{\sqrt{x \cdot \left(1 + x\right)}} \]
      add-sqr-sqrt [<=] 99.6%	\[ \frac{\left(\left(x + 1\right) - \color{blue}{x}\right) \cdot \frac{1}{\mathsf{hypot}\left(1, \sqrt{x}\right) + \sqrt{x}}}{\sqrt{x \cdot \left(1 + x\right)}} \]
      associate--l+ [=>] 99.6%	\[ \frac{\color{blue}{\left(x + \left(1 - x\right)\right)} \cdot \frac{1}{\mathsf{hypot}\left(1, \sqrt{x}\right) + \sqrt{x}}}{\sqrt{x \cdot \left(1 + x\right)}} \]
      +-commutative [=>] 99.6%	\[ \frac{\left(x + \left(1 - x\right)\right) \cdot \frac{1}{\color{blue}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)}}}{\sqrt{x \cdot \left(1 + x\right)}} \]
      hypot-1-def [<=] 99.6%	\[ \frac{\left(x + \left(1 - x\right)\right) \cdot \frac{1}{\sqrt{x} + \color{blue}{\sqrt{1 + \sqrt{x} \cdot \sqrt{x}}}}}{\sqrt{x \cdot \left(1 + x\right)}} \]
      add-sqr-sqrt [<=] 99.6%	\[ \frac{\left(x + \left(1 - x\right)\right) \cdot \frac{1}{\sqrt{x} + \sqrt{1 + \color{blue}{x}}}}{\sqrt{x \cdot \left(1 + x\right)}} \]

   4. Simplified 99.6%

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

      [Start] 99.6%	\[ \frac{\left(x + \left(1 - x\right)\right) \cdot \frac{1}{\sqrt{x} + \sqrt{x + 1}}}{\sqrt{x \cdot \left(1 + x\right)}} \]
      associate-*r/ [=>] 99.6%	\[ \frac{\color{blue}{\frac{\left(x + \left(1 - x\right)\right) \cdot 1}{\sqrt{x} + \sqrt{x + 1}}}}{\sqrt{x \cdot \left(1 + x\right)}} \]
      *-rgt-identity [=>] 99.6%	\[ \frac{\frac{\color{blue}{x + \left(1 - x\right)}}{\sqrt{x} + \sqrt{x + 1}}}{\sqrt{x \cdot \left(1 + x\right)}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 0:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x + \left(1 - x\right)}{\sqrt{x} + \sqrt{1 + x}}}{\sqrt{x \cdot \left(1 + x\right)}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.7%
Cost	33732

\[\begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{t_0} \leq 0:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x + \left(1 - x\right)}{\sqrt{x} + t_0}}{\sqrt{x \cdot \left(1 + x\right)}}\\ \end{array} \]

Alternative 2
Accuracy	99.5%
Cost	26692

\[\begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 10^{-13}:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

Alternative 3
Accuracy	98.5%
Cost	7044

\[\begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\left({x}^{-0.5} + x \cdot 0.5\right) + -1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \end{array} \]

Alternative 4
Accuracy	98.2%
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 0.66:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \end{array} \]

Alternative 5
Accuracy	51.7%
Cost	6656

\[0.5 \cdot {x}^{-1.5} \]

Alternative 6
Accuracy	7.4%
Cost	320

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

Alternative 7
Accuracy	7.4%
Cost	192

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

Reproduce

herbie shell --seed 2023178 
(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)))))