2isqrt (example 3.6)

Percentage Accurate: 68.2% → 91.7%

Time: 13.7s

Alternatives: 14

Speedup: 1.8×

Specification

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / sqrt(x)) - (1.0d0 / sqrt((x + 1.0d0)))
end function

Java

public static double code(double x) {
	return (1.0 / Math.sqrt(x)) - (1.0 / Math.sqrt((x + 1.0)));
}

Python

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

Julia

function code(x)
	return Float64(Float64(1.0 / sqrt(x)) - Float64(1.0 / sqrt(Float64(x + 1.0))))
end

MATLAB

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

Initial Program: 68.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / sqrt(x)) - (1.0d0 / sqrt((x + 1.0d0)))
end function

Java

public static double code(double x) {
	return (1.0 / Math.sqrt(x)) - (1.0 / Math.sqrt((x + 1.0)));
}

Python

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

Julia

function code(x)
	return Float64(Float64(1.0 / sqrt(x)) - Float64(1.0 / sqrt(Float64(x + 1.0))))
end

MATLAB

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

Alternative 1: 91.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{-1}{x}}{-1 - x}}{{x}^{-0.5} + \frac{1}{\sqrt{x + 1}}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (/ (/ -1.0 x) (- -1.0 x)) (+ (pow x -0.5) (/ 1.0 (sqrt (+ x 1.0))))))

C

double code(double x) {
	return ((-1.0 / x) / (-1.0 - x)) / (pow(x, -0.5) + (1.0 / sqrt((x + 1.0))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (((-1.0d0) / x) / ((-1.0d0) - x)) / ((x ** (-0.5d0)) + (1.0d0 / sqrt((x + 1.0d0))))
end function

Java

public static double code(double x) {
	return ((-1.0 / x) / (-1.0 - x)) / (Math.pow(x, -0.5) + (1.0 / Math.sqrt((x + 1.0))));
}

Python

def code(x):
	return ((-1.0 / x) / (-1.0 - x)) / (math.pow(x, -0.5) + (1.0 / math.sqrt((x + 1.0))))

Julia

function code(x)
	return Float64(Float64(Float64(-1.0 / x) / Float64(-1.0 - x)) / Float64((x ^ -0.5) + Float64(1.0 / sqrt(Float64(x + 1.0)))))
end

MATLAB

function tmp = code(x)
	tmp = ((-1.0 / x) / (-1.0 - x)) / ((x ^ -0.5) + (1.0 / sqrt((x + 1.0))));
end

Wolfram

code[x_] := N[(N[(N[(-1.0 / x), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(N[Power[x, -0.5], $MachinePrecision] + N[(1.0 / N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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Alternative 2: 82.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x + 1}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{t_0} \leq 5 \cdot 10^{-16}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ x 1.0))))
   (if (<= (- (/ 1.0 (sqrt x)) (/ 1.0 t_0)) 5e-16)
     (* 0.5 (sqrt (/ 1.0 (pow x 3.0))))
     (+ (pow x -0.5) (/ -1.0 t_0)))))

C

double code(double x) {
	double t_0 = sqrt((x + 1.0));
	double tmp;
	if (((1.0 / sqrt(x)) - (1.0 / t_0)) <= 5e-16) {
		tmp = 0.5 * sqrt((1.0 / pow(x, 3.0)));
	} else {
		tmp = pow(x, -0.5) + (-1.0 / t_0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((x + 1.0d0))
    if (((1.0d0 / sqrt(x)) - (1.0d0 / t_0)) <= 5d-16) then
        tmp = 0.5d0 * sqrt((1.0d0 / (x ** 3.0d0)))
    else
        tmp = (x ** (-0.5d0)) + ((-1.0d0) / t_0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((x + 1.0));
	double tmp;
	if (((1.0 / Math.sqrt(x)) - (1.0 / t_0)) <= 5e-16) {
		tmp = 0.5 * Math.sqrt((1.0 / Math.pow(x, 3.0)));
	} else {
		tmp = Math.pow(x, -0.5) + (-1.0 / t_0);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((x + 1.0))
	tmp = 0
	if ((1.0 / math.sqrt(x)) - (1.0 / t_0)) <= 5e-16:
		tmp = 0.5 * math.sqrt((1.0 / math.pow(x, 3.0)))
	else:
		tmp = math.pow(x, -0.5) + (-1.0 / t_0)
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(x + 1.0))
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) - Float64(1.0 / t_0)) <= 5e-16)
		tmp = Float64(0.5 * sqrt(Float64(1.0 / (x ^ 3.0))));
	else
		tmp = Float64((x ^ -0.5) + Float64(-1.0 / t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((x + 1.0));
	tmp = 0.0;
	if (((1.0 / sqrt(x)) - (1.0 / t_0)) <= 5e-16)
		tmp = 0.5 * sqrt((1.0 / (x ^ 3.0)));
	else
		tmp = (x ^ -0.5) + (-1.0 / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], 5e-16], N[(0.5 * N[Sqrt[N[(1.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] + N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

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Alternative 3: 91.1% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (/ -1.0 (* (+ (pow x -0.5) (pow (+ x 1.0) -0.5)) (* x (- -1.0 x)))))

C

double code(double x) {
	return -1.0 / ((pow(x, -0.5) + pow((x + 1.0), -0.5)) * (x * (-1.0 - x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-1.0d0) / (((x ** (-0.5d0)) + ((x + 1.0d0) ** (-0.5d0))) * (x * ((-1.0d0) - x)))
end function

Java

public static double code(double x) {
	return -1.0 / ((Math.pow(x, -0.5) + Math.pow((x + 1.0), -0.5)) * (x * (-1.0 - x)));
}

Python

def code(x):
	return -1.0 / ((math.pow(x, -0.5) + math.pow((x + 1.0), -0.5)) * (x * (-1.0 - x)))

Julia

function code(x)
	return Float64(-1.0 / Float64(Float64((x ^ -0.5) + (Float64(x + 1.0) ^ -0.5)) * Float64(x * Float64(-1.0 - x))))
end

MATLAB

function tmp = code(x)
	tmp = -1.0 / (((x ^ -0.5) + ((x + 1.0) ^ -0.5)) * (x * (-1.0 - x)));
end

Wolfram

code[x_] := 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[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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Alternative 4: 91.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{-1}{x}}{-1 - x}}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (/ (/ -1.0 x) (- -1.0 x)) (+ (pow x -0.5) (pow (+ x 1.0) -0.5))))

C

double code(double x) {
	return ((-1.0 / x) / (-1.0 - x)) / (pow(x, -0.5) + pow((x + 1.0), -0.5));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (((-1.0d0) / x) / ((-1.0d0) - x)) / ((x ** (-0.5d0)) + ((x + 1.0d0) ** (-0.5d0)))
end function

Java

public static double code(double x) {
	return ((-1.0 / x) / (-1.0 - x)) / (Math.pow(x, -0.5) + Math.pow((x + 1.0), -0.5));
}

Python

def code(x):
	return ((-1.0 / x) / (-1.0 - x)) / (math.pow(x, -0.5) + math.pow((x + 1.0), -0.5))

Julia

function code(x)
	return Float64(Float64(Float64(-1.0 / x) / Float64(-1.0 - x)) / Float64((x ^ -0.5) + (Float64(x + 1.0) ^ -0.5)))
end

MATLAB

function tmp = code(x)
	tmp = ((-1.0 / x) / (-1.0 - x)) / ((x ^ -0.5) + ((x + 1.0) ^ -0.5));
end

Wolfram

code[x_] := N[(N[(N[(-1.0 / x), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(N[Power[x, -0.5], $MachinePrecision] + N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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Alternative 5: 82.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 90000000:\\ \;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 90000000.0)
   (- (pow x -0.5) (pow (+ x 1.0) -0.5))
   (* 0.5 (sqrt (/ 1.0 (pow x 3.0))))))

C

double code(double x) {
	double tmp;
	if (x <= 90000000.0) {
		tmp = pow(x, -0.5) - pow((x + 1.0), -0.5);
	} else {
		tmp = 0.5 * sqrt((1.0 / pow(x, 3.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 90000000.0d0) then
        tmp = (x ** (-0.5d0)) - ((x + 1.0d0) ** (-0.5d0))
    else
        tmp = 0.5d0 * sqrt((1.0d0 / (x ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 90000000.0) {
		tmp = Math.pow(x, -0.5) - Math.pow((x + 1.0), -0.5);
	} else {
		tmp = 0.5 * Math.sqrt((1.0 / Math.pow(x, 3.0)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 90000000.0:
		tmp = math.pow(x, -0.5) - math.pow((x + 1.0), -0.5)
	else:
		tmp = 0.5 * math.sqrt((1.0 / math.pow(x, 3.0)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 90000000.0)
		tmp = Float64((x ^ -0.5) - (Float64(x + 1.0) ^ -0.5));
	else
		tmp = Float64(0.5 * sqrt(Float64(1.0 / (x ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 90000000.0)
		tmp = (x ^ -0.5) - ((x + 1.0) ^ -0.5);
	else
		tmp = 0.5 * sqrt((1.0 / (x ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 90000000.0], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(1.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

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Alternative 6: 82.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.7:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.7)
   (+ (pow x -0.5) (/ -1.0 (+ 1.0 (* x 0.5))))
   (* 0.5 (sqrt (/ 1.0 (pow x 3.0))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.7) {
		tmp = pow(x, -0.5) + (-1.0 / (1.0 + (x * 0.5)));
	} else {
		tmp = 0.5 * sqrt((1.0 / pow(x, 3.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.7d0) then
        tmp = (x ** (-0.5d0)) + ((-1.0d0) / (1.0d0 + (x * 0.5d0)))
    else
        tmp = 0.5d0 * sqrt((1.0d0 / (x ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.7) {
		tmp = Math.pow(x, -0.5) + (-1.0 / (1.0 + (x * 0.5)));
	} else {
		tmp = 0.5 * Math.sqrt((1.0 / Math.pow(x, 3.0)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.7:
		tmp = math.pow(x, -0.5) + (-1.0 / (1.0 + (x * 0.5)))
	else:
		tmp = 0.5 * math.sqrt((1.0 / math.pow(x, 3.0)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.7)
		tmp = Float64((x ^ -0.5) + Float64(-1.0 / Float64(1.0 + Float64(x * 0.5))));
	else
		tmp = Float64(0.5 * sqrt(Float64(1.0 / (x ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.7)
		tmp = (x ^ -0.5) + (-1.0 / (1.0 + (x * 0.5)));
	else
		tmp = 0.5 * sqrt((1.0 / (x ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.7], N[(N[Power[x, -0.5], $MachinePrecision] + N[(-1.0 / N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(1.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

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Alternative 7: 67.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.2:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{1}{x + 1}}{\sqrt{\frac{1}{x}} \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.2)
   (+ (pow x -0.5) (/ -1.0 (+ 1.0 (* x 0.5))))
   (/ (- (/ 1.0 x) (/ 1.0 (+ x 1.0))) (* (sqrt (/ 1.0 x)) 2.0))))

C

double code(double x) {
	double tmp;
	if (x <= 1.2) {
		tmp = pow(x, -0.5) + (-1.0 / (1.0 + (x * 0.5)));
	} else {
		tmp = ((1.0 / x) - (1.0 / (x + 1.0))) / (sqrt((1.0 / x)) * 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.2d0) then
        tmp = (x ** (-0.5d0)) + ((-1.0d0) / (1.0d0 + (x * 0.5d0)))
    else
        tmp = ((1.0d0 / x) - (1.0d0 / (x + 1.0d0))) / (sqrt((1.0d0 / x)) * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.2) {
		tmp = Math.pow(x, -0.5) + (-1.0 / (1.0 + (x * 0.5)));
	} else {
		tmp = ((1.0 / x) - (1.0 / (x + 1.0))) / (Math.sqrt((1.0 / x)) * 2.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.2:
		tmp = math.pow(x, -0.5) + (-1.0 / (1.0 + (x * 0.5)))
	else:
		tmp = ((1.0 / x) - (1.0 / (x + 1.0))) / (math.sqrt((1.0 / x)) * 2.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.2)
		tmp = Float64((x ^ -0.5) + Float64(-1.0 / Float64(1.0 + Float64(x * 0.5))));
	else
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(1.0 / Float64(x + 1.0))) / Float64(sqrt(Float64(1.0 / x)) * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.2)
		tmp = (x ^ -0.5) + (-1.0 / (1.0 + (x * 0.5)));
	else
		tmp = ((1.0 / x) - (1.0 / (x + 1.0))) / (sqrt((1.0 / x)) * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.2], N[(N[Power[x, -0.5], $MachinePrecision] + N[(-1.0 / N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

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Alternative 8: 70.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.35:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 + x \cdot 0}{x \cdot \left(-1 - x\right)}}{\sqrt{\frac{1}{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.35)
   (+ (pow x -0.5) (/ -1.0 (+ 1.0 (* x 0.5))))
   (/ (/ (+ -1.0 (* x 0.0)) (* x (- -1.0 x))) (sqrt (/ 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= 2.35) {
		tmp = pow(x, -0.5) + (-1.0 / (1.0 + (x * 0.5)));
	} else {
		tmp = ((-1.0 + (x * 0.0)) / (x * (-1.0 - x))) / sqrt((1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.35d0) then
        tmp = (x ** (-0.5d0)) + ((-1.0d0) / (1.0d0 + (x * 0.5d0)))
    else
        tmp = (((-1.0d0) + (x * 0.0d0)) / (x * ((-1.0d0) - x))) / sqrt((1.0d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.35) {
		tmp = Math.pow(x, -0.5) + (-1.0 / (1.0 + (x * 0.5)));
	} else {
		tmp = ((-1.0 + (x * 0.0)) / (x * (-1.0 - x))) / Math.sqrt((1.0 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.35:
		tmp = math.pow(x, -0.5) + (-1.0 / (1.0 + (x * 0.5)))
	else:
		tmp = ((-1.0 + (x * 0.0)) / (x * (-1.0 - x))) / math.sqrt((1.0 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.35)
		tmp = Float64((x ^ -0.5) + Float64(-1.0 / Float64(1.0 + Float64(x * 0.5))));
	else
		tmp = Float64(Float64(Float64(-1.0 + Float64(x * 0.0)) / Float64(x * Float64(-1.0 - x))) / sqrt(Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.35)
		tmp = (x ^ -0.5) + (-1.0 / (1.0 + (x * 0.5)));
	else
		tmp = ((-1.0 + (x * 0.0)) / (x * (-1.0 - x))) / sqrt((1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.35], N[(N[Power[x, -0.5], $MachinePrecision] + N[(-1.0 / N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-1.0 + N[(x * 0.0), $MachinePrecision]), $MachinePrecision] / N[(x * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 9: 67.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{+122}:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{x} + \frac{1}{x}}{\sqrt{\frac{1}{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 8.5e+122)
   (+ (pow x -0.5) (/ -1.0 (+ 1.0 (* x 0.5))))
   (/ (+ (/ -1.0 x) (/ 1.0 x)) (sqrt (/ 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= 8.5e+122) {
		tmp = pow(x, -0.5) + (-1.0 / (1.0 + (x * 0.5)));
	} else {
		tmp = ((-1.0 / x) + (1.0 / x)) / sqrt((1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 8.5d+122) then
        tmp = (x ** (-0.5d0)) + ((-1.0d0) / (1.0d0 + (x * 0.5d0)))
    else
        tmp = (((-1.0d0) / x) + (1.0d0 / x)) / sqrt((1.0d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 8.5e+122) {
		tmp = Math.pow(x, -0.5) + (-1.0 / (1.0 + (x * 0.5)));
	} else {
		tmp = ((-1.0 / x) + (1.0 / x)) / Math.sqrt((1.0 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 8.5e+122:
		tmp = math.pow(x, -0.5) + (-1.0 / (1.0 + (x * 0.5)))
	else:
		tmp = ((-1.0 / x) + (1.0 / x)) / math.sqrt((1.0 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 8.5e+122)
		tmp = Float64((x ^ -0.5) + Float64(-1.0 / Float64(1.0 + Float64(x * 0.5))));
	else
		tmp = Float64(Float64(Float64(-1.0 / x) + Float64(1.0 / x)) / sqrt(Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 8.5e+122)
		tmp = (x ^ -0.5) + (-1.0 / (1.0 + (x * 0.5)));
	else
		tmp = ((-1.0 / x) + (1.0 / x)) / sqrt((1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 8.5e+122], N[(N[Power[x, -0.5], $MachinePrecision] + N[(-1.0 / N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-1.0 / x), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 10: 51.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ {x}^{-0.5} + \frac{-1}{1 + x \cdot 0.5} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ (pow x -0.5) (/ -1.0 (+ 1.0 (* x 0.5)))))

C

double code(double x) {
	return pow(x, -0.5) + (-1.0 / (1.0 + (x * 0.5)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x ** (-0.5d0)) + ((-1.0d0) / (1.0d0 + (x * 0.5d0)))
end function

Java

public static double code(double x) {
	return Math.pow(x, -0.5) + (-1.0 / (1.0 + (x * 0.5)));
}

Python

def code(x):
	return math.pow(x, -0.5) + (-1.0 / (1.0 + (x * 0.5)))

Julia

function code(x)
	return Float64((x ^ -0.5) + Float64(-1.0 / Float64(1.0 + Float64(x * 0.5))))
end

MATLAB

function tmp = code(x)
	tmp = (x ^ -0.5) + (-1.0 / (1.0 + (x * 0.5)));
end

Wolfram

code[x_] := N[(N[Power[x, -0.5], $MachinePrecision] + N[(-1.0 / N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 11: 51.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.44:\\ \;\;\;\;-1 + {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{x}} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.44)
   (+ -1.0 (pow x -0.5))
   (* (sqrt (/ 1.0 x)) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 0.44) {
		tmp = -1.0 + pow(x, -0.5);
	} else {
		tmp = sqrt((1.0 / x)) * 0.3333333333333333;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.44d0) then
        tmp = (-1.0d0) + (x ** (-0.5d0))
    else
        tmp = sqrt((1.0d0 / x)) * 0.3333333333333333d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.44) {
		tmp = -1.0 + Math.pow(x, -0.5);
	} else {
		tmp = Math.sqrt((1.0 / x)) * 0.3333333333333333;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.44:
		tmp = -1.0 + math.pow(x, -0.5)
	else:
		tmp = math.sqrt((1.0 / x)) * 0.3333333333333333
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.44)
		tmp = Float64(-1.0 + (x ^ -0.5));
	else
		tmp = Float64(sqrt(Float64(1.0 / x)) * 0.3333333333333333);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.44)
		tmp = -1.0 + (x ^ -0.5);
	else
		tmp = sqrt((1.0 / x)) * 0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.44], N[(-1.0 + N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 12: 52.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x + {x}^{0.5}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (+ x (pow x 0.5))))

C

double code(double x) {
	return 1.0 / (x + pow(x, 0.5));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (x + (x ** 0.5d0))
end function

Java

public static double code(double x) {
	return 1.0 / (x + Math.pow(x, 0.5));
}

Python

def code(x):
	return 1.0 / (x + math.pow(x, 0.5))

Julia

function code(x)
	return Float64(1.0 / Float64(x + (x ^ 0.5)))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (x + (x ^ 0.5));
end

Wolfram

code[x_] := N[(1.0 / N[(x + N[Power[x, 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 13: 50.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ -1 + {x}^{-0.5} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ -1.0 (pow x -0.5)))

C

double code(double x) {
	return -1.0 + pow(x, -0.5);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-1.0d0) + (x ** (-0.5d0))
end function

Java

public static double code(double x) {
	return -1.0 + Math.pow(x, -0.5);
}

Python

def code(x):
	return -1.0 + math.pow(x, -0.5)

Julia

function code(x)
	return Float64(-1.0 + (x ^ -0.5))
end

MATLAB

function tmp = code(x)
	tmp = -1.0 + (x ^ -0.5);
end

Wolfram

code[x_] := N[(-1.0 + N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 14: 3.9% accurate, 69.7× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x 0.5))

C

double code(double x) {
	return x * 0.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * 0.5d0
end function

Java

public static double code(double x) {
	return x * 0.5;
}

Python

def code(x):
	return x * 0.5

Julia

function code(x)
	return Float64(x * 0.5)
end

MATLAB

function tmp = code(x)
	tmp = x * 0.5;
end

Wolfram

code[x_] := N[(x * 0.5), $MachinePrecision]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Developer target: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (((x + 1.0d0) * sqrt(x)) + (x * sqrt((x + 1.0d0))))
end function

Java

public static double code(double x) {
	return 1.0 / (((x + 1.0) * Math.sqrt(x)) + (x * Math.sqrt((x + 1.0))));
}

Python

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

Julia

function code(x)
	return Float64(1.0 / Float64(Float64(Float64(x + 1.0) * sqrt(x)) + Float64(x * sqrt(Float64(x + 1.0)))))
end

MATLAB

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

Wolfram

code[x_] := N[(1.0 / N[(N[(N[(x + 1.0), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(x * N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023347 
(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)))))