2isqrt (example 3.6)

Percentage Accurate: 68.8% → 99.7%

Time: 12.7s

Alternatives: 14

Speedup: 1.9×

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: 68.8% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sqrt{1 + x}}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} - t_0 \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{0.5}{x} + \left(\frac{0.3125}{{x}^{3}} + \frac{-0.375}{{x}^{2}}\right)}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sqrt (+ 1.0 x)))))
   (if (<= (- (/ 1.0 (sqrt x)) t_0) 2e-9)
     (/
      (+ (/ 0.5 x) (+ (/ 0.3125 (pow x 3.0)) (/ -0.375 (pow x 2.0))))
      (sqrt x))
     (- (pow x -0.5) t_0))))

C

double code(double x) {
	double t_0 = 1.0 / sqrt((1.0 + x));
	double tmp;
	if (((1.0 / sqrt(x)) - t_0) <= 2e-9) {
		tmp = ((0.5 / x) + ((0.3125 / pow(x, 3.0)) + (-0.375 / pow(x, 2.0)))) / sqrt(x);
	} else {
		tmp = pow(x, -0.5) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 / sqrt((1.0d0 + x))
    if (((1.0d0 / sqrt(x)) - t_0) <= 2d-9) then
        tmp = ((0.5d0 / x) + ((0.3125d0 / (x ** 3.0d0)) + ((-0.375d0) / (x ** 2.0d0)))) / sqrt(x)
    else
        tmp = (x ** (-0.5d0)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 1.0 / Math.sqrt((1.0 + x));
	double tmp;
	if (((1.0 / Math.sqrt(x)) - t_0) <= 2e-9) {
		tmp = ((0.5 / x) + ((0.3125 / Math.pow(x, 3.0)) + (-0.375 / Math.pow(x, 2.0)))) / Math.sqrt(x);
	} else {
		tmp = Math.pow(x, -0.5) - t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = 1.0 / math.sqrt((1.0 + x))
	tmp = 0
	if ((1.0 / math.sqrt(x)) - t_0) <= 2e-9:
		tmp = ((0.5 / x) + ((0.3125 / math.pow(x, 3.0)) + (-0.375 / math.pow(x, 2.0)))) / math.sqrt(x)
	else:
		tmp = math.pow(x, -0.5) - t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(1.0 / sqrt(Float64(1.0 + x)))
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) - t_0) <= 2e-9)
		tmp = Float64(Float64(Float64(0.5 / x) + Float64(Float64(0.3125 / (x ^ 3.0)) + Float64(-0.375 / (x ^ 2.0)))) / sqrt(x));
	else
		tmp = Float64((x ^ -0.5) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 1.0 / sqrt((1.0 + x));
	tmp = 0.0;
	if (((1.0 / sqrt(x)) - t_0) <= 2e-9)
		tmp = ((0.5 / x) + ((0.3125 / (x ^ 3.0)) + (-0.375 / (x ^ 2.0)))) / sqrt(x);
	else
		tmp = (x ^ -0.5) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], 2e-9], N[(N[(N[(0.5 / x), $MachinePrecision] + N[(N[(0.3125 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-0.375 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - t$95$0), $MachinePrecision]]]

Derivation

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1. Add Preprocessing

Alternative 2: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x}\\ t_1 := \frac{1}{t_0}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} - t_1 \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_0} \cdot \frac{1}{x + \left(0.5 + \frac{-0.125}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ 1.0 x))) (t_1 (/ 1.0 t_0)))
   (if (<= (- (/ 1.0 (sqrt x)) t_1) 2e-9)
     (* (/ 1.0 (+ (sqrt x) t_0)) (/ 1.0 (+ x (+ 0.5 (/ -0.125 x)))))
     (- (pow x -0.5) t_1))))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x));
	double t_1 = 1.0 / t_0;
	double tmp;
	if (((1.0 / sqrt(x)) - t_1) <= 2e-9) {
		tmp = (1.0 / (sqrt(x) + t_0)) * (1.0 / (x + (0.5 + (-0.125 / x))));
	} else {
		tmp = pow(x, -0.5) - t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((1.0d0 + x))
    t_1 = 1.0d0 / t_0
    if (((1.0d0 / sqrt(x)) - t_1) <= 2d-9) then
        tmp = (1.0d0 / (sqrt(x) + t_0)) * (1.0d0 / (x + (0.5d0 + ((-0.125d0) / x))))
    else
        tmp = (x ** (-0.5d0)) - t_1
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 + x));
	double t_1 = 1.0 / t_0;
	double tmp;
	if (((1.0 / Math.sqrt(x)) - t_1) <= 2e-9) {
		tmp = (1.0 / (Math.sqrt(x) + t_0)) * (1.0 / (x + (0.5 + (-0.125 / x))));
	} else {
		tmp = Math.pow(x, -0.5) - t_1;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 + x))
	t_1 = 1.0 / t_0
	tmp = 0
	if ((1.0 / math.sqrt(x)) - t_1) <= 2e-9:
		tmp = (1.0 / (math.sqrt(x) + t_0)) * (1.0 / (x + (0.5 + (-0.125 / x))))
	else:
		tmp = math.pow(x, -0.5) - t_1
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 + x))
	t_1 = Float64(1.0 / t_0)
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) - t_1) <= 2e-9)
		tmp = Float64(Float64(1.0 / Float64(sqrt(x) + t_0)) * Float64(1.0 / Float64(x + Float64(0.5 + Float64(-0.125 / x)))));
	else
		tmp = Float64((x ^ -0.5) - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((1.0 + x));
	t_1 = 1.0 / t_0;
	tmp = 0.0;
	if (((1.0 / sqrt(x)) - t_1) <= 2e-9)
		tmp = (1.0 / (sqrt(x) + t_0)) * (1.0 / (x + (0.5 + (-0.125 / x))));
	else
		tmp = (x ^ -0.5) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / t$95$0), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], 2e-9], N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(x + N[(0.5 + N[(-0.125 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - t$95$1), $MachinePrecision]]]]

Derivation

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1. Add Preprocessing

Alternative 3: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x}\\ t_1 := \frac{1}{t_0}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} - t_1 \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_0} \cdot \frac{1}{x + 0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ 1.0 x))) (t_1 (/ 1.0 t_0)))
   (if (<= (- (/ 1.0 (sqrt x)) t_1) 2e-9)
     (* (/ 1.0 (+ (sqrt x) t_0)) (/ 1.0 (+ x 0.5)))
     (- (pow x -0.5) t_1))))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x));
	double t_1 = 1.0 / t_0;
	double tmp;
	if (((1.0 / sqrt(x)) - t_1) <= 2e-9) {
		tmp = (1.0 / (sqrt(x) + t_0)) * (1.0 / (x + 0.5));
	} else {
		tmp = pow(x, -0.5) - t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((1.0d0 + x))
    t_1 = 1.0d0 / t_0
    if (((1.0d0 / sqrt(x)) - t_1) <= 2d-9) then
        tmp = (1.0d0 / (sqrt(x) + t_0)) * (1.0d0 / (x + 0.5d0))
    else
        tmp = (x ** (-0.5d0)) - t_1
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 + x));
	double t_1 = 1.0 / t_0;
	double tmp;
	if (((1.0 / Math.sqrt(x)) - t_1) <= 2e-9) {
		tmp = (1.0 / (Math.sqrt(x) + t_0)) * (1.0 / (x + 0.5));
	} else {
		tmp = Math.pow(x, -0.5) - t_1;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 + x))
	t_1 = 1.0 / t_0
	tmp = 0
	if ((1.0 / math.sqrt(x)) - t_1) <= 2e-9:
		tmp = (1.0 / (math.sqrt(x) + t_0)) * (1.0 / (x + 0.5))
	else:
		tmp = math.pow(x, -0.5) - t_1
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 + x))
	t_1 = Float64(1.0 / t_0)
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) - t_1) <= 2e-9)
		tmp = Float64(Float64(1.0 / Float64(sqrt(x) + t_0)) * Float64(1.0 / Float64(x + 0.5)));
	else
		tmp = Float64((x ^ -0.5) - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((1.0 + x));
	t_1 = 1.0 / t_0;
	tmp = 0.0;
	if (((1.0 / sqrt(x)) - t_1) <= 2e-9)
		tmp = (1.0 / (sqrt(x) + t_0)) * (1.0 / (x + 0.5));
	else
		tmp = (x ^ -0.5) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / t$95$0), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], 2e-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] - t$95$1), $MachinePrecision]]]]

Derivation

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1. Add Preprocessing

Alternative 4: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sqrt{1 + x}}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} - t_0 \leq 1.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{0.5}{x} - \frac{0.375}{{x}^{2}}}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sqrt (+ 1.0 x)))))
   (if (<= (- (/ 1.0 (sqrt x)) t_0) 1.5e-9)
     (/ (- (/ 0.5 x) (/ 0.375 (pow x 2.0))) (sqrt x))
     (- (pow x -0.5) t_0))))

C

double code(double x) {
	double t_0 = 1.0 / sqrt((1.0 + x));
	double tmp;
	if (((1.0 / sqrt(x)) - t_0) <= 1.5e-9) {
		tmp = ((0.5 / x) - (0.375 / pow(x, 2.0))) / sqrt(x);
	} else {
		tmp = pow(x, -0.5) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 / sqrt((1.0d0 + x))
    if (((1.0d0 / sqrt(x)) - t_0) <= 1.5d-9) then
        tmp = ((0.5d0 / x) - (0.375d0 / (x ** 2.0d0))) / sqrt(x)
    else
        tmp = (x ** (-0.5d0)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 1.0 / Math.sqrt((1.0 + x));
	double tmp;
	if (((1.0 / Math.sqrt(x)) - t_0) <= 1.5e-9) {
		tmp = ((0.5 / x) - (0.375 / Math.pow(x, 2.0))) / Math.sqrt(x);
	} else {
		tmp = Math.pow(x, -0.5) - t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = 1.0 / math.sqrt((1.0 + x))
	tmp = 0
	if ((1.0 / math.sqrt(x)) - t_0) <= 1.5e-9:
		tmp = ((0.5 / x) - (0.375 / math.pow(x, 2.0))) / math.sqrt(x)
	else:
		tmp = math.pow(x, -0.5) - t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(1.0 / sqrt(Float64(1.0 + x)))
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) - t_0) <= 1.5e-9)
		tmp = Float64(Float64(Float64(0.5 / x) - Float64(0.375 / (x ^ 2.0))) / sqrt(x));
	else
		tmp = Float64((x ^ -0.5) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 1.0 / sqrt((1.0 + x));
	tmp = 0.0;
	if (((1.0 / sqrt(x)) - t_0) <= 1.5e-9)
		tmp = ((0.5 / x) - (0.375 / (x ^ 2.0))) / sqrt(x);
	else
		tmp = (x ^ -0.5) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], 1.5e-9], N[(N[(N[(0.5 / x), $MachinePrecision] - N[(0.375 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - t$95$0), $MachinePrecision]]]

Derivation

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1. Add Preprocessing

Alternative 5: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sqrt{1 + x}}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} - t_0 \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{0.5}{x}}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sqrt (+ 1.0 x)))))
   (if (<= (- (/ 1.0 (sqrt x)) t_0) 5e-14)
     (/ (/ 0.5 x) (sqrt x))
     (- (pow x -0.5) t_0))))

C

double code(double x) {
	double t_0 = 1.0 / sqrt((1.0 + x));
	double tmp;
	if (((1.0 / sqrt(x)) - t_0) <= 5e-14) {
		tmp = (0.5 / x) / sqrt(x);
	} else {
		tmp = pow(x, -0.5) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 / sqrt((1.0d0 + x))
    if (((1.0d0 / sqrt(x)) - t_0) <= 5d-14) then
        tmp = (0.5d0 / x) / sqrt(x)
    else
        tmp = (x ** (-0.5d0)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 1.0 / Math.sqrt((1.0 + x));
	double tmp;
	if (((1.0 / Math.sqrt(x)) - t_0) <= 5e-14) {
		tmp = (0.5 / x) / Math.sqrt(x);
	} else {
		tmp = Math.pow(x, -0.5) - t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = 1.0 / math.sqrt((1.0 + x))
	tmp = 0
	if ((1.0 / math.sqrt(x)) - t_0) <= 5e-14:
		tmp = (0.5 / x) / math.sqrt(x)
	else:
		tmp = math.pow(x, -0.5) - t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(1.0 / sqrt(Float64(1.0 + x)))
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) - t_0) <= 5e-14)
		tmp = Float64(Float64(0.5 / x) / sqrt(x));
	else
		tmp = Float64((x ^ -0.5) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 1.0 / sqrt((1.0 + x));
	tmp = 0.0;
	if (((1.0 / sqrt(x)) - t_0) <= 5e-14)
		tmp = (0.5 / x) / sqrt(x);
	else
		tmp = (x ^ -0.5) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], 5e-14], N[(N[(0.5 / x), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - t$95$0), $MachinePrecision]]]

Derivation

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1. Add Preprocessing

Alternative 6: 99.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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1. Add Preprocessing

Alternative 7: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 122000000:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{x}}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 122000000.0)
   (- (pow x -0.5) (pow (+ 1.0 x) -0.5))
   (/ (/ 0.5 x) (sqrt x))))

C

double code(double x) {
	double tmp;
	if (x <= 122000000.0) {
		tmp = pow(x, -0.5) - pow((1.0 + x), -0.5);
	} else {
		tmp = (0.5 / x) / sqrt(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 122000000.0d0) then
        tmp = (x ** (-0.5d0)) - ((1.0d0 + x) ** (-0.5d0))
    else
        tmp = (0.5d0 / x) / sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 122000000.0) {
		tmp = Math.pow(x, -0.5) - Math.pow((1.0 + x), -0.5);
	} else {
		tmp = (0.5 / x) / Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 122000000.0:
		tmp = math.pow(x, -0.5) - math.pow((1.0 + x), -0.5)
	else:
		tmp = (0.5 / x) / math.sqrt(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 122000000.0)
		tmp = Float64((x ^ -0.5) - (Float64(1.0 + x) ^ -0.5));
	else
		tmp = Float64(Float64(0.5 / x) / sqrt(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 122000000.0)
		tmp = (x ^ -0.5) - ((1.0 + x) ^ -0.5);
	else
		tmp = (0.5 / x) / sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 122000000.0], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / x), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

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1. Add Preprocessing

Alternative 8: 98.5% accurate, 1.8× speedup

Math

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

FPCore

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

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 / x) / sqrt(x);
	}
	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 / x) / sqrt(x)
    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 / x) / Math.sqrt(x);
	}
	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 / x) / math.sqrt(x)
	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(Float64(0.5 / x) / sqrt(x));
	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 / x) / sqrt(x);
	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[(N[(0.5 / x), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

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1. Add Preprocessing

Alternative 9: 98.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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1. Add Preprocessing

Alternative 10: 98.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 0.68) {
		tmp = pow(x, -0.5) + -1.0;
	} else {
		tmp = (0.5 / x) / sqrt(x);
	}
	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) / sqrt(x)
    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 / x) / Math.sqrt(x);
	}
	return tmp;
}

Python

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

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.68)
		tmp = Float64((x ^ -0.5) + -1.0);
	else
		tmp = Float64(Float64(0.5 / x) / sqrt(x));
	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) / sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

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

Alternative 11: 53.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.6:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.6) (+ (pow x -0.5) -1.0) (/ 1.0 (+ x 0.5))))

C

double code(double x) {
	double tmp;
	if (x <= 0.6) {
		tmp = pow(x, -0.5) + -1.0;
	} else {
		tmp = 1.0 / (x + 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.6d0) then
        tmp = (x ** (-0.5d0)) + (-1.0d0)
    else
        tmp = 1.0d0 / (x + 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.6) {
		tmp = Math.pow(x, -0.5) + -1.0;
	} else {
		tmp = 1.0 / (x + 0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.6:
		tmp = math.pow(x, -0.5) + -1.0
	else:
		tmp = 1.0 / (x + 0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.6)
		tmp = Float64((x ^ -0.5) + -1.0);
	else
		tmp = Float64(1.0 / Float64(x + 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.6)
		tmp = (x ^ -0.5) + -1.0;
	else
		tmp = 1.0 / (x + 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.6], N[(N[Power[x, -0.5], $MachinePrecision] + -1.0), $MachinePrecision], N[(1.0 / N[(x + 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

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

Alternative 12: 7.4% accurate, 41.8× speedup

Math

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

FPCore

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

C

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

Python

def code(x):
	return 1.0 / (x + 0.5)

Julia

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

MATLAB

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

Wolfram

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

Derivation

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

Alternative 13: 1.9% accurate, 209.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -1.0)

C

double code(double x) {
	return -1.0;
}

Fortran

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

Java

public static double code(double x) {
	return -1.0;
}

Python

def code(x):
	return -1.0

Julia

function code(x)
	return -1.0
end

MATLAB

function tmp = code(x)
	tmp = -1.0;
end

Wolfram

code[x_] := -1.0

Derivation

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

Alternative 14: 5.8% accurate, 209.0× speedup

Math

\[\begin{array}{l} \\ 2 \end{array} \]

FPCore

(FPCore (x) :precision binary64 2.0)

C

double code(double x) {
	return 2.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0
end function

Java

public static double code(double x) {
	return 2.0;
}

Python

def code(x):
	return 2.0

Julia

function code(x)
	return 2.0
end

MATLAB

function tmp = code(x)
	tmp = 2.0;
end

Wolfram

code[x_] := 2.0

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 2024006 
(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)))))