2isqrt (example 3.6)

Percentage Accurate: 69.7% → 99.8%

Time: 12.1s

Alternatives: 12

Speedup: 1.0×

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: 69.7% 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: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x}\\ t_1 := {t_0}^{-0.5}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{t_0} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left({x}^{-1.5}, 0.5, {x}^{-2.5} \cdot -0.375\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_1, -t_1, {x}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ 1.0 x))) (t_1 (pow t_0 -0.5)))
   (if (<= (- (/ 1.0 (sqrt x)) (/ 1.0 t_0)) 2e-11)
     (fma (pow x -1.5) 0.5 (* (pow x -2.5) -0.375))
     (fma t_1 (- t_1) (pow x -0.5)))))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x));
	double t_1 = pow(t_0, -0.5);
	double tmp;
	if (((1.0 / sqrt(x)) - (1.0 / t_0)) <= 2e-11) {
		tmp = fma(pow(x, -1.5), 0.5, (pow(x, -2.5) * -0.375));
	} else {
		tmp = fma(t_1, -t_1, pow(x, -0.5));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 + x))
	t_1 = t_0 ^ -0.5
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) - Float64(1.0 / t_0)) <= 2e-11)
		tmp = fma((x ^ -1.5), 0.5, Float64((x ^ -2.5) * -0.375));
	else
		tmp = fma(t_1, Float64(-t_1), (x ^ -0.5));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[t$95$0, -0.5], $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], 2e-11], N[(N[Power[x, -1.5], $MachinePrecision] * 0.5 + N[(N[Power[x, -2.5], $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * (-t$95$1) + N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]]]]

Derivation

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

Alternative 2: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{1 + x}} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left({x}^{-1.5}, 0.5, {x}^{-2.5} \cdot -0.375\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ 1.0 x)))) 2e-11)
   (fma (pow x -1.5) 0.5 (* (pow x -2.5) -0.375))
   (- (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-11) {
		tmp = fma(pow(x, -1.5), 0.5, (pow(x, -2.5) * -0.375));
	} else {
		tmp = pow(x, -0.5) - 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-11)
		tmp = fma((x ^ -1.5), 0.5, Float64((x ^ -2.5) * -0.375));
	else
		tmp = Float64((x ^ -0.5) - (Float64(1.0 + x) ^ -0.5));
	end
	return 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-11], N[(N[Power[x, -1.5], $MachinePrecision] * 0.5 + N[(N[Power[x, -2.5], $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

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

Alternative 3: 91.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 95000000:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{{x}^{2}}}{2 \cdot \sqrt{\frac{1}{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 95000000.0)
   (- (pow x -0.5) (pow (+ 1.0 x) -0.5))
   (/ (/ 1.0 (pow x 2.0)) (* 2.0 (sqrt (/ 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= 95000000.0) {
		tmp = pow(x, -0.5) - pow((1.0 + x), -0.5);
	} else {
		tmp = (1.0 / pow(x, 2.0)) / (2.0 * sqrt((1.0 / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 95000000.0d0) then
        tmp = (x ** (-0.5d0)) - ((1.0d0 + x) ** (-0.5d0))
    else
        tmp = (1.0d0 / (x ** 2.0d0)) / (2.0d0 * sqrt((1.0d0 / x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 95000000.0) {
		tmp = Math.pow(x, -0.5) - Math.pow((1.0 + x), -0.5);
	} else {
		tmp = (1.0 / Math.pow(x, 2.0)) / (2.0 * Math.sqrt((1.0 / x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 95000000.0:
		tmp = math.pow(x, -0.5) - math.pow((1.0 + x), -0.5)
	else:
		tmp = (1.0 / math.pow(x, 2.0)) / (2.0 * math.sqrt((1.0 / x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 95000000.0)
		tmp = Float64((x ^ -0.5) - (Float64(1.0 + x) ^ -0.5));
	else
		tmp = Float64(Float64(1.0 / (x ^ 2.0)) / Float64(2.0 * sqrt(Float64(1.0 / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 95000000.0)
		tmp = (x ^ -0.5) - ((1.0 + x) ^ -0.5);
	else
		tmp = (1.0 / (x ^ 2.0)) / (2.0 * sqrt((1.0 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 95000000.0], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

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

Alternative 4: 83.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 95000000:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 95000000.0)
   (- (pow x -0.5) (pow (+ 1.0 x) -0.5))
   (* 0.5 (sqrt (/ 1.0 (pow x 3.0))))))

C

double code(double x) {
	double tmp;
	if (x <= 95000000.0) {
		tmp = pow(x, -0.5) - pow((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 <= 95000000.0d0) then
        tmp = (x ** (-0.5d0)) - ((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 <= 95000000.0) {
		tmp = Math.pow(x, -0.5) - Math.pow((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 <= 95000000.0:
		tmp = math.pow(x, -0.5) - math.pow((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 <= 95000000.0)
		tmp = Float64((x ^ -0.5) - (Float64(1.0 + 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 <= 95000000.0)
		tmp = (x ^ -0.5) - ((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, 95000000.0], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(1.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

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

Alternative 5: 82.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = pow(x, -0.5) + (-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 <= 1.0d0) then
        tmp = (x ** (-0.5d0)) + ((-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 <= 1.0) {
		tmp = Math.pow(x, -0.5) + (-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 <= 1.0:
		tmp = math.pow(x, -0.5) + (-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 <= 1.0)
		tmp = Float64((x ^ -0.5) + 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 <= 1.0)
		tmp = (x ^ -0.5) + (-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, 1.0], N[(N[Power[x, -0.5], $MachinePrecision] + N[(-1.0 - N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(1.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

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

Alternative 6: 68.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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

Alternative 7: 69.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;{x}^{-0.5} + \left(-1 - x \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x \cdot \left(1 + x\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.4)
   (+ (pow x -0.5) (- -1.0 (* x -0.5)))
   (/ 1.0 (sqrt (* x (+ 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.4) {
		tmp = pow(x, -0.5) + (-1.0 - (x * -0.5));
	} else {
		tmp = 1.0 / sqrt((x * (1.0 + x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.4d0) then
        tmp = (x ** (-0.5d0)) + ((-1.0d0) - (x * (-0.5d0)))
    else
        tmp = 1.0d0 / sqrt((x * (1.0d0 + x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.4) {
		tmp = Math.pow(x, -0.5) + (-1.0 - (x * -0.5));
	} else {
		tmp = 1.0 / Math.sqrt((x * (1.0 + x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.4:
		tmp = math.pow(x, -0.5) + (-1.0 - (x * -0.5))
	else:
		tmp = 1.0 / math.sqrt((x * (1.0 + x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.4)
		tmp = Float64((x ^ -0.5) + Float64(-1.0 - Float64(x * -0.5)));
	else
		tmp = Float64(1.0 / sqrt(Float64(x * Float64(1.0 + x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.4)
		tmp = (x ^ -0.5) + (-1.0 - (x * -0.5));
	else
		tmp = 1.0 / sqrt((x * (1.0 + x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.4], N[(N[Power[x, -0.5], $MachinePrecision] + N[(-1.0 - N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sqrt[N[(x * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

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

Alternative 8: 53.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.58:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.58) (+ (pow x -0.5) -1.0) (/ 1.0 (+ x 0.5))))

C

double code(double x) {
	double tmp;
	if (x <= 0.58) {
		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 <= 0.58d0) 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 <= 0.58) {
		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 <= 0.58:
		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 <= 0.58)
		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 <= 0.58)
		tmp = (x ^ -0.5) + -1.0;
	else
		tmp = 1.0 / (x + 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.58], 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 9: 52.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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

Alternative 10: 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 11: 7.4% accurate, 69.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return 1.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

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

Alternative 12: 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

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 2023364 
(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)))))