2isqrt (example 3.6)

Average Error: 19.9 → 0.4

Time: 10.5s

Precision: binary64

Cost: 13892

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq 1.5631665013639515 \cdot 10^{+82}:\\ \;\;\;\;\frac{1}{\left({x}^{-0.5} + {\left(x + 1\right)}^{-0.5}\right) \cdot \left(x \cdot \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}{x}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x 1.5631665013639515e+82)
   (/ 1.0 (* (+ (pow x -0.5) (pow (+ x 1.0) -0.5)) (* x (+ x 1.0))))
   (/ (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt 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 (x <= 1.5631665013639515e+82) {
		tmp = 1.0 / ((pow(x, -0.5) + pow((x + 1.0), -0.5)) * (x * (x + 1.0)));
	} else {
		tmp = (1.0 / (sqrt((x + 1.0)) + sqrt(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 (x <= 1.5631665013639515d+82) then
        tmp = 1.0d0 / (((x ** (-0.5d0)) + ((x + 1.0d0) ** (-0.5d0))) * (x * (x + 1.0d0)))
    else
        tmp = (1.0d0 / (sqrt((x + 1.0d0)) + sqrt(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 (x <= 1.5631665013639515e+82) {
		tmp = 1.0 / ((Math.pow(x, -0.5) + Math.pow((x + 1.0), -0.5)) * (x * (x + 1.0)));
	} else {
		tmp = (1.0 / (Math.sqrt((x + 1.0)) + Math.sqrt(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 x <= 1.5631665013639515e+82:
		tmp = 1.0 / ((math.pow(x, -0.5) + math.pow((x + 1.0), -0.5)) * (x * (x + 1.0)))
	else:
		tmp = (1.0 / (math.sqrt((x + 1.0)) + math.sqrt(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 (x <= 1.5631665013639515e+82)
		tmp = Float64(1.0 / Float64(Float64((x ^ -0.5) + (Float64(x + 1.0) ^ -0.5)) * Float64(x * Float64(x + 1.0))));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(Float64(x + 1.0)) + sqrt(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 (x <= 1.5631665013639515e+82)
		tmp = 1.0 / (((x ^ -0.5) + ((x + 1.0) ^ -0.5)) * (x * (x + 1.0)));
	else
		tmp = (1.0 / (sqrt((x + 1.0)) + sqrt(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[x, 1.5631665013639515e+82], N[(1.0 / N[(N[(N[Power[x, -0.5], $MachinePrecision] + N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] * N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Target

Original	19.9
Target	0.6
Herbie	0.4

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

Derivation

1. Split input into 2 regimes

2. if x < 1.56316650136395147e82

   1. Initial program 11.2

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

   2. Applied egg-rr 11.4

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

   3. Applied egg-rr 0.5

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

   4. Applied egg-rr 0.5

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

   if 1.56316650136395147e82 < x

   1. Initial program 34.7

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

   2. Applied egg-rr 34.7

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

   3. Applied egg-rr 34.7

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

   4. Taylor expanded in x around inf 34.7

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

   5. Taylor expanded in x around 0 0.2

      \[\leadsto \frac{\frac{\color{blue}{1}}{\sqrt{x + 1} + \sqrt{x}}}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.5631665013639515 \cdot 10^{+82}:\\ \;\;\;\;\frac{1}{\left({x}^{-0.5} + {\left(x + 1\right)}^{-0.5}\right) \cdot \left(x \cdot \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}{x}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	26820

\[\begin{array}{l} t_0 := \sqrt{x + 1}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{t_0} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{1}{t_0 + \sqrt{x}}}{x}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\ \end{array} \]

Alternative 2
Error	0.4
Cost	20224

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

Alternative 3
Error	0.5
Cost	14016

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

Alternative 4
Error	0.8
Cost	13380

\[\begin{array}{l} \mathbf{if}\;x \leq 1462373.4357213434:\\ \;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot {x}^{1.5}}\\ \end{array} \]

Alternative 5
Error	1.3
Cost	7044

\[\begin{array}{l} \mathbf{if}\;x \leq 0.029275776111882246:\\ \;\;\;\;-1 + \left({x}^{-0.5} + x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot {x}^{1.5}}\\ \end{array} \]

Alternative 6
Error	1.5
Cost	6916

\[\begin{array}{l} \mathbf{if}\;x \leq 0.029275776111882246:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot {x}^{1.5}}\\ \end{array} \]

Alternative 7
Error	20.8
Cost	6852

\[\begin{array}{l} \mathbf{if}\;x \leq 9.724842711893388 \cdot 10^{+151}:\\ \;\;\;\;\frac{1}{x + \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(1 - x\right)}{x}\\ \end{array} \]

Alternative 8
Error	20.9
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 0.029275776111882246:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(1 - x\right)}{x}\\ \end{array} \]

Alternative 9
Error	50.5
Cost	448

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

Alternative 10
Error	59.3
Cost	192

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

Alternative 11
Error	62.8
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2022294 
(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)))))