2isqrt (example 3.6)

Percentage Accurate: 69.0% → 99.7%

Time: 10.3s

Alternatives: 8

Speedup: 2.0×

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: 69.0% 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: 99.7% accurate, 0.4× speedup

Math

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

FPCore

(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))
     (* (/ 1.0 (+ (sqrt x) t_0)) (/ 1.0 (sqrt (* x (+ 1.0 x))))))))

C

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 = (1.0 / (sqrt(x) + t_0)) * (1.0 / sqrt((x * (1.0 + x))));
	}
	return tmp;
}

Fortran

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 = (1.0d0 / (sqrt(x) + t_0)) * (1.0d0 / sqrt((x * (1.0d0 + x))))
    end if
    code = tmp
end function

Java

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 = (1.0 / (Math.sqrt(x) + t_0)) * (1.0 / Math.sqrt((x * (1.0 + x))));
	}
	return tmp;
}

Python

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 = (1.0 / (math.sqrt(x) + t_0)) * (1.0 / math.sqrt((x * (1.0 + x))))
	return tmp

Julia

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(1.0 / Float64(sqrt(x) + t_0)) * Float64(1.0 / sqrt(Float64(x * Float64(1.0 + x)))));
	end
	return tmp
end

MATLAB

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 = (1.0 / (sqrt(x) + t_0)) * (1.0 / sqrt((x * (1.0 + x))));
	end
	tmp_2 = tmp;
end

Wolfram

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[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(x * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

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1. Add Preprocessing

Alternative 2: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{t_0} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_0} \cdot \frac{1}{\left(x + 0.5\right) - \frac{0.125}{x}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ 1.0 x))))
   (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 t_0)) 5e-8)
     (* (/ 1.0 (+ (sqrt x) t_0)) (/ 1.0 (- (+ x 0.5) (/ 0.125 x))))
     (- (pow x -0.5) (pow (+ 1.0 x) -0.5)))))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x));
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / t_0)) <= 5e-8) {
		tmp = (1.0 / (sqrt(x) + t_0)) * (1.0 / ((x + 0.5) - (0.125 / x)));
	} else {
		tmp = pow(x, -0.5) - pow((1.0 + x), -0.5);
	}
	return tmp;
}

Fortran

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)) <= 5d-8) then
        tmp = (1.0d0 / (sqrt(x) + t_0)) * (1.0d0 / ((x + 0.5d0) - (0.125d0 / x)))
    else
        tmp = (x ** (-0.5d0)) - ((1.0d0 + x) ** (-0.5d0))
    end if
    code = tmp
end function

Java

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)) <= 5e-8) {
		tmp = (1.0 / (Math.sqrt(x) + t_0)) * (1.0 / ((x + 0.5) - (0.125 / x)));
	} else {
		tmp = Math.pow(x, -0.5) - Math.pow((1.0 + x), -0.5);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 + x))
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / t_0)) <= 5e-8:
		tmp = (1.0 / (math.sqrt(x) + t_0)) * (1.0 / ((x + 0.5) - (0.125 / x)))
	else:
		tmp = math.pow(x, -0.5) - math.pow((1.0 + x), -0.5)
	return tmp

Julia

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)) <= 5e-8)
		tmp = Float64(Float64(1.0 / Float64(sqrt(x) + t_0)) * Float64(1.0 / Float64(Float64(x + 0.5) - Float64(0.125 / x))));
	else
		tmp = Float64((x ^ -0.5) - (Float64(1.0 + x) ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((1.0 + x));
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + (-1.0 / t_0)) <= 5e-8)
		tmp = (1.0 / (sqrt(x) + t_0)) * (1.0 / ((x + 0.5) - (0.125 / x)));
	else
		tmp = (x ^ -0.5) - ((1.0 + x) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

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], 5e-8], N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(x + 0.5), $MachinePrecision] - N[(0.125 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

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1. Add Preprocessing

Alternative 3: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{t_0} \leq 4 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_0} \cdot \frac{1}{x + 0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ 1.0 x))))
   (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 t_0)) 4e-9)
     (* (/ 1.0 (+ (sqrt x) t_0)) (/ 1.0 (+ x 0.5)))
     (- (pow x -0.5) (pow (+ 1.0 x) -0.5)))))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x));
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / t_0)) <= 4e-9) {
		tmp = (1.0 / (sqrt(x) + t_0)) * (1.0 / (x + 0.5));
	} else {
		tmp = pow(x, -0.5) - pow((1.0 + x), -0.5);
	}
	return tmp;
}

Fortran

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)) <= 4d-9) then
        tmp = (1.0d0 / (sqrt(x) + t_0)) * (1.0d0 / (x + 0.5d0))
    else
        tmp = (x ** (-0.5d0)) - ((1.0d0 + x) ** (-0.5d0))
    end if
    code = tmp
end function

Java

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)) <= 4e-9) {
		tmp = (1.0 / (Math.sqrt(x) + t_0)) * (1.0 / (x + 0.5));
	} else {
		tmp = Math.pow(x, -0.5) - Math.pow((1.0 + x), -0.5);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 + x))
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / t_0)) <= 4e-9:
		tmp = (1.0 / (math.sqrt(x) + t_0)) * (1.0 / (x + 0.5))
	else:
		tmp = math.pow(x, -0.5) - math.pow((1.0 + x), -0.5)
	return tmp

Julia

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)) <= 4e-9)
		tmp = Float64(Float64(1.0 / Float64(sqrt(x) + t_0)) * Float64(1.0 / Float64(x + 0.5)));
	else
		tmp = Float64((x ^ -0.5) - (Float64(1.0 + x) ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((1.0 + x));
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + (-1.0 / t_0)) <= 4e-9)
		tmp = (1.0 / (sqrt(x) + t_0)) * (1.0 / (x + 0.5));
	else
		tmp = (x ^ -0.5) - ((1.0 + x) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

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], 4e-9], N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(x + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

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1. Add Preprocessing

Alternative 4: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 2 \cdot 10^{-15}:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 (sqrt (+ 1.0 x)))) 2e-15)
   (* 0.5 (pow x -1.5))
   (- (pow x -0.5) (pow (+ 1.0 x) -0.5))))

C

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 2e-15) {
		tmp = 0.5 * pow(x, -1.5);
	} else {
		tmp = pow(x, -0.5) - pow((1.0 + x), -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((1.0d0 / sqrt(x)) + ((-1.0d0) / sqrt((1.0d0 + x)))) <= 2d-15) then
        tmp = 0.5d0 * (x ** (-1.5d0))
    else
        tmp = (x ** (-0.5d0)) - ((1.0d0 + x) ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((1.0 / Math.sqrt(x)) + (-1.0 / Math.sqrt((1.0 + x)))) <= 2e-15) {
		tmp = 0.5 * Math.pow(x, -1.5);
	} else {
		tmp = Math.pow(x, -0.5) - Math.pow((1.0 + x), -0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / math.sqrt((1.0 + x)))) <= 2e-15:
		tmp = 0.5 * math.pow(x, -1.5)
	else:
		tmp = math.pow(x, -0.5) - math.pow((1.0 + x), -0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / sqrt(Float64(1.0 + x)))) <= 2e-15)
		tmp = Float64(0.5 * (x ^ -1.5));
	else
		tmp = Float64((x ^ -0.5) - (Float64(1.0 + x) ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 2e-15)
		tmp = 0.5 * (x ^ -1.5);
	else
		tmp = (x ^ -0.5) - ((1.0 + x) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-15], N[(0.5 * N[Power[x, -1.5], $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

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Alternative 5: 99.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(1.0 / N[Sqrt[N[Sqrt[x], $MachinePrecision] ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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Alternative 6: 98.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.68:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.68) (+ (pow x -0.5) -1.0) (* 0.5 (pow x -1.5))))

C

double code(double x) {
	double tmp;
	if (x <= 0.68) {
		tmp = pow(x, -0.5) + -1.0;
	} else {
		tmp = 0.5 * pow(x, -1.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.68d0) then
        tmp = (x ** (-0.5d0)) + (-1.0d0)
    else
        tmp = 0.5d0 * (x ** (-1.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.68) {
		tmp = Math.pow(x, -0.5) + -1.0;
	} else {
		tmp = 0.5 * Math.pow(x, -1.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.68:
		tmp = math.pow(x, -0.5) + -1.0
	else:
		tmp = 0.5 * math.pow(x, -1.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.68)
		tmp = Float64((x ^ -0.5) + -1.0);
	else
		tmp = Float64(0.5 * (x ^ -1.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.68)
		tmp = (x ^ -0.5) + -1.0;
	else
		tmp = 0.5 * (x ^ -1.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.68], N[(N[Power[x, -0.5], $MachinePrecision] + -1.0), $MachinePrecision], N[(0.5 * N[Power[x, -1.5], $MachinePrecision]), $MachinePrecision]]

Derivation

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Alternative 7: 51.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot {x}^{-1.5} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 0.5 (pow x -1.5)))

C

double code(double x) {
	return 0.5 * pow(x, -1.5);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0 * (x ** (-1.5d0))
end function

Java

public static double code(double x) {
	return 0.5 * Math.pow(x, -1.5);
}

Python

def code(x):
	return 0.5 * math.pow(x, -1.5)

Julia

function code(x)
	return Float64(0.5 * (x ^ -1.5))
end

MATLAB

function tmp = code(x)
	tmp = 0.5 * (x ^ -1.5);
end

Wolfram

code[x_] := N[(0.5 * N[Power[x, -1.5], $MachinePrecision]), $MachinePrecision]

Derivation

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

Alternative 8: 2.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	return -pow(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 -Math.pow(x, -0.5);
}

Python

def code(x):
	return -math.pow(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[Power[x, -0.5], $MachinePrecision])

Derivation

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

Developer target: 98.9% 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 2023341 
(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)))))