NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.5% → 99.7%

Time: 10.6s

Alternatives: 9

Speedup: 1.1×

Specification

Math

\[\begin{array}{l} \\ \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (* (* (/ PI 2.0) (/ 1.0 (- (* b b) (* a a)))) (- (/ 1.0 a) (/ 1.0 b))))

C

double code(double a, double b) {
	return ((((double) M_PI) / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
}

Java

public static double code(double a, double b) {
	return ((Math.PI / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
}

Python

def code(a, b):
	return ((math.pi / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b))

Julia

function code(a, b)
	return Float64(Float64(Float64(pi / 2.0) * Float64(1.0 / Float64(Float64(b * b) - Float64(a * a)))) * Float64(Float64(1.0 / a) - Float64(1.0 / b)))
end

MATLAB

function tmp = code(a, b)
	tmp = ((pi / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
end

Wolfram

code[a_, b_] := N[(N[(N[(Pi / 2.0), $MachinePrecision] * N[(1.0 / N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / a), $MachinePrecision] - N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 78.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (* (* (/ PI 2.0) (/ 1.0 (- (* b b) (* a a)))) (- (/ 1.0 a) (/ 1.0 b))))

C

double code(double a, double b) {
	return ((((double) M_PI) / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
}

Java

public static double code(double a, double b) {
	return ((Math.PI / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
}

Python

def code(a, b):
	return ((math.pi / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b))

Julia

function code(a, b)
	return Float64(Float64(Float64(pi / 2.0) * Float64(1.0 / Float64(Float64(b * b) - Float64(a * a)))) * Float64(Float64(1.0 / a) - Float64(1.0 / b)))
end

MATLAB

function tmp = code(a, b)
	tmp = ((pi / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
end

Wolfram

code[a_, b_] := N[(N[(N[(Pi / 2.0), $MachinePrecision] * N[(1.0 / N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / a), $MachinePrecision] - N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \frac{\frac{\frac{\pi}{b}}{a \cdot 2}}{b + a} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (/ (/ (/ PI b) (* a 2.0)) (+ b a)))

C

assert(a < b);
double code(double a, double b) {
	return ((((double) M_PI) / b) / (a * 2.0)) / (b + a);
}

Java

assert a < b;
public static double code(double a, double b) {
	return ((Math.PI / b) / (a * 2.0)) / (b + a);
}

Python

[a, b] = sort([a, b])
def code(a, b):
	return ((math.pi / b) / (a * 2.0)) / (b + a)

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(Float64(Float64(pi / b) / Float64(a * 2.0)) / Float64(b + a))
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = ((pi / b) / (a * 2.0)) / (b + a);
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[(N[(N[(Pi / b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision] / N[(b + a), $MachinePrecision]), $MachinePrecision]

Derivation

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

Alternative 2: 69.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 7 \cdot 10^{-98}:\\ \;\;\;\;\frac{\frac{0.5}{\frac{a}{\pi}}}{b \cdot a}\\ \mathbf{elif}\;b \leq 2.65 \cdot 10^{-55}:\\ \;\;\;\;\frac{-0.5}{a \cdot \frac{b \cdot a}{\pi}}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{+112}:\\ \;\;\;\;\frac{\frac{0.5}{a \cdot \frac{b}{\pi}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a \cdot \left(-b\right)} \cdot \frac{0.5}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 7e-98)
   (/ (/ 0.5 (/ a PI)) (* b a))
   (if (<= b 2.65e-55)
     (/ -0.5 (* a (/ (* b a) PI)))
     (if (<= b 5.2e+112)
       (/ (/ 0.5 (* a (/ b PI))) a)
       (* (/ PI (* a (- b))) (/ 0.5 b))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 7e-98) {
		tmp = (0.5 / (a / ((double) M_PI))) / (b * a);
	} else if (b <= 2.65e-55) {
		tmp = -0.5 / (a * ((b * a) / ((double) M_PI)));
	} else if (b <= 5.2e+112) {
		tmp = (0.5 / (a * (b / ((double) M_PI)))) / a;
	} else {
		tmp = (((double) M_PI) / (a * -b)) * (0.5 / b);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 7e-98) {
		tmp = (0.5 / (a / Math.PI)) / (b * a);
	} else if (b <= 2.65e-55) {
		tmp = -0.5 / (a * ((b * a) / Math.PI));
	} else if (b <= 5.2e+112) {
		tmp = (0.5 / (a * (b / Math.PI))) / a;
	} else {
		tmp = (Math.PI / (a * -b)) * (0.5 / b);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 7e-98:
		tmp = (0.5 / (a / math.pi)) / (b * a)
	elif b <= 2.65e-55:
		tmp = -0.5 / (a * ((b * a) / math.pi))
	elif b <= 5.2e+112:
		tmp = (0.5 / (a * (b / math.pi))) / a
	else:
		tmp = (math.pi / (a * -b)) * (0.5 / b)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 7e-98)
		tmp = Float64(Float64(0.5 / Float64(a / pi)) / Float64(b * a));
	elseif (b <= 2.65e-55)
		tmp = Float64(-0.5 / Float64(a * Float64(Float64(b * a) / pi)));
	elseif (b <= 5.2e+112)
		tmp = Float64(Float64(0.5 / Float64(a * Float64(b / pi))) / a);
	else
		tmp = Float64(Float64(pi / Float64(a * Float64(-b))) * Float64(0.5 / b));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 7e-98)
		tmp = (0.5 / (a / pi)) / (b * a);
	elseif (b <= 2.65e-55)
		tmp = -0.5 / (a * ((b * a) / pi));
	elseif (b <= 5.2e+112)
		tmp = (0.5 / (a * (b / pi))) / a;
	else
		tmp = (pi / (a * -b)) * (0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[b, 7e-98], N[(N[(0.5 / N[(a / Pi), $MachinePrecision]), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.65e-55], N[(-0.5 / N[(a * N[(N[(b * a), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.2e+112], N[(N[(0.5 / N[(a * N[(b / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(Pi / N[(a * (-b)), $MachinePrecision]), $MachinePrecision] * N[(0.5 / b), $MachinePrecision]), $MachinePrecision]]]]

Derivation

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

Alternative 3: 64.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -6.8 \cdot 10^{-176}:\\ \;\;\;\;\frac{\pi}{b \cdot a} \cdot \frac{0.5}{a}\\ \mathbf{elif}\;a \leq 1.45 \cdot 10^{-307}:\\ \;\;\;\;\frac{-0.5}{a \cdot \frac{b \cdot a}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{\frac{0.5}{a}}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -6.8e-176)
   (* (/ PI (* b a)) (/ 0.5 a))
   (if (<= a 1.45e-307)
     (/ -0.5 (* a (/ (* b a) PI)))
     (* (/ PI a) (/ (/ 0.5 a) b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -6.8e-176) {
		tmp = (((double) M_PI) / (b * a)) * (0.5 / a);
	} else if (a <= 1.45e-307) {
		tmp = -0.5 / (a * ((b * a) / ((double) M_PI)));
	} else {
		tmp = (((double) M_PI) / a) * ((0.5 / a) / b);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -6.8e-176) {
		tmp = (Math.PI / (b * a)) * (0.5 / a);
	} else if (a <= 1.45e-307) {
		tmp = -0.5 / (a * ((b * a) / Math.PI));
	} else {
		tmp = (Math.PI / a) * ((0.5 / a) / b);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -6.8e-176:
		tmp = (math.pi / (b * a)) * (0.5 / a)
	elif a <= 1.45e-307:
		tmp = -0.5 / (a * ((b * a) / math.pi))
	else:
		tmp = (math.pi / a) * ((0.5 / a) / b)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -6.8e-176)
		tmp = Float64(Float64(pi / Float64(b * a)) * Float64(0.5 / a));
	elseif (a <= 1.45e-307)
		tmp = Float64(-0.5 / Float64(a * Float64(Float64(b * a) / pi)));
	else
		tmp = Float64(Float64(pi / a) * Float64(Float64(0.5 / a) / b));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -6.8e-176)
		tmp = (pi / (b * a)) * (0.5 / a);
	elseif (a <= 1.45e-307)
		tmp = -0.5 / (a * ((b * a) / pi));
	else
		tmp = (pi / a) * ((0.5 / a) / b);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -6.8e-176], N[(N[(Pi / N[(b * a), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.45e-307], N[(-0.5 / N[(a * N[(N[(b * a), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / a), $MachinePrecision] * N[(N[(0.5 / a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

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

Alternative 4: 64.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -5.5 \cdot 10^{-174}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{b \cdot a}}{a}\\ \mathbf{elif}\;a \leq -1.4 \cdot 10^{-308}:\\ \;\;\;\;\frac{-0.5}{a \cdot \frac{b \cdot a}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{\frac{0.5}{a}}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -5.5e-174)
   (/ (* 0.5 (/ PI (* b a))) a)
   (if (<= a -1.4e-308)
     (/ -0.5 (* a (/ (* b a) PI)))
     (* (/ PI a) (/ (/ 0.5 a) b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -5.5e-174) {
		tmp = (0.5 * (((double) M_PI) / (b * a))) / a;
	} else if (a <= -1.4e-308) {
		tmp = -0.5 / (a * ((b * a) / ((double) M_PI)));
	} else {
		tmp = (((double) M_PI) / a) * ((0.5 / a) / b);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -5.5e-174) {
		tmp = (0.5 * (Math.PI / (b * a))) / a;
	} else if (a <= -1.4e-308) {
		tmp = -0.5 / (a * ((b * a) / Math.PI));
	} else {
		tmp = (Math.PI / a) * ((0.5 / a) / b);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -5.5e-174:
		tmp = (0.5 * (math.pi / (b * a))) / a
	elif a <= -1.4e-308:
		tmp = -0.5 / (a * ((b * a) / math.pi))
	else:
		tmp = (math.pi / a) * ((0.5 / a) / b)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -5.5e-174)
		tmp = Float64(Float64(0.5 * Float64(pi / Float64(b * a))) / a);
	elseif (a <= -1.4e-308)
		tmp = Float64(-0.5 / Float64(a * Float64(Float64(b * a) / pi)));
	else
		tmp = Float64(Float64(pi / a) * Float64(Float64(0.5 / a) / b));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -5.5e-174)
		tmp = (0.5 * (pi / (b * a))) / a;
	elseif (a <= -1.4e-308)
		tmp = -0.5 / (a * ((b * a) / pi));
	else
		tmp = (pi / a) * ((0.5 / a) / b);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -5.5e-174], N[(N[(0.5 * N[(Pi / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[a, -1.4e-308], N[(-0.5 / N[(a * N[(N[(b * a), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / a), $MachinePrecision] * N[(N[(0.5 / a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

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

Alternative 5: 64.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -2.7 \cdot 10^{-176}:\\ \;\;\;\;\frac{\frac{0.5}{\frac{a}{\pi}}}{b \cdot a}\\ \mathbf{elif}\;a \leq 1.4 \cdot 10^{-307}:\\ \;\;\;\;\frac{-0.5}{a \cdot \frac{b \cdot a}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{\frac{0.5}{a}}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -2.7e-176)
   (/ (/ 0.5 (/ a PI)) (* b a))
   (if (<= a 1.4e-307)
     (/ -0.5 (* a (/ (* b a) PI)))
     (* (/ PI a) (/ (/ 0.5 a) b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -2.7e-176) {
		tmp = (0.5 / (a / ((double) M_PI))) / (b * a);
	} else if (a <= 1.4e-307) {
		tmp = -0.5 / (a * ((b * a) / ((double) M_PI)));
	} else {
		tmp = (((double) M_PI) / a) * ((0.5 / a) / b);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -2.7e-176) {
		tmp = (0.5 / (a / Math.PI)) / (b * a);
	} else if (a <= 1.4e-307) {
		tmp = -0.5 / (a * ((b * a) / Math.PI));
	} else {
		tmp = (Math.PI / a) * ((0.5 / a) / b);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -2.7e-176:
		tmp = (0.5 / (a / math.pi)) / (b * a)
	elif a <= 1.4e-307:
		tmp = -0.5 / (a * ((b * a) / math.pi))
	else:
		tmp = (math.pi / a) * ((0.5 / a) / b)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -2.7e-176)
		tmp = Float64(Float64(0.5 / Float64(a / pi)) / Float64(b * a));
	elseif (a <= 1.4e-307)
		tmp = Float64(-0.5 / Float64(a * Float64(Float64(b * a) / pi)));
	else
		tmp = Float64(Float64(pi / a) * Float64(Float64(0.5 / a) / b));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.7e-176)
		tmp = (0.5 / (a / pi)) / (b * a);
	elseif (a <= 1.4e-307)
		tmp = -0.5 / (a * ((b * a) / pi));
	else
		tmp = (pi / a) * ((0.5 / a) / b);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -2.7e-176], N[(N[(0.5 / N[(a / Pi), $MachinePrecision]), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.4e-307], N[(-0.5 / N[(a * N[(N[(b * a), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / a), $MachinePrecision] * N[(N[(0.5 / a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

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

Alternative 6: 91.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 9.6 \cdot 10^{-143}:\\ \;\;\;\;\frac{\frac{0.5}{\frac{a}{\pi}}}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b \cdot a} \cdot \frac{0.5}{b - a}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 9.6e-143)
   (/ (/ 0.5 (/ a PI)) (* b a))
   (* (/ PI (* b a)) (/ 0.5 (- b a)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 9.6e-143) {
		tmp = (0.5 / (a / ((double) M_PI))) / (b * a);
	} else {
		tmp = (((double) M_PI) / (b * a)) * (0.5 / (b - a));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 9.6e-143) {
		tmp = (0.5 / (a / Math.PI)) / (b * a);
	} else {
		tmp = (Math.PI / (b * a)) * (0.5 / (b - a));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 9.6e-143:
		tmp = (0.5 / (a / math.pi)) / (b * a)
	else:
		tmp = (math.pi / (b * a)) * (0.5 / (b - a))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 9.6e-143)
		tmp = Float64(Float64(0.5 / Float64(a / pi)) / Float64(b * a));
	else
		tmp = Float64(Float64(pi / Float64(b * a)) * Float64(0.5 / Float64(b - a)));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 9.6e-143)
		tmp = (0.5 / (a / pi)) / (b * a);
	else
		tmp = (pi / (b * a)) * (0.5 / (b - a));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[b, 9.6e-143], N[(N[(0.5 / N[(a / Pi), $MachinePrecision]), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / N[(b * a), $MachinePrecision]), $MachinePrecision] * N[(0.5 / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

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Alternative 7: 99.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \frac{0.5 \cdot \frac{\pi}{b \cdot a}}{b + a} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (/ (* 0.5 (/ PI (* b a))) (+ b a)))

C

assert(a < b);
double code(double a, double b) {
	return (0.5 * (((double) M_PI) / (b * a))) / (b + a);
}

Java

assert a < b;
public static double code(double a, double b) {
	return (0.5 * (Math.PI / (b * a))) / (b + a);
}

Python

[a, b] = sort([a, b])
def code(a, b):
	return (0.5 * (math.pi / (b * a))) / (b + a)

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(Float64(0.5 * Float64(pi / Float64(b * a))) / Float64(b + a))
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = (0.5 * (pi / (b * a))) / (b + a);
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[(N[(0.5 * N[(Pi / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b + a), $MachinePrecision]), $MachinePrecision]

Derivation

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Alternative 8: 63.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \frac{\pi}{a} \cdot \frac{\frac{0.5}{a}}{b} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (* (/ PI a) (/ (/ 0.5 a) b)))

C

assert(a < b);
double code(double a, double b) {
	return (((double) M_PI) / a) * ((0.5 / a) / b);
}

Java

assert a < b;
public static double code(double a, double b) {
	return (Math.PI / a) * ((0.5 / a) / b);
}

Python

[a, b] = sort([a, b])
def code(a, b):
	return (math.pi / a) * ((0.5 / a) / b)

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(Float64(pi / a) * Float64(Float64(0.5 / a) / b))
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = (pi / a) * ((0.5 / a) / b);
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[(N[(Pi / a), $MachinePrecision] * N[(N[(0.5 / a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]

Derivation

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1. Add Preprocessing

Alternative 9: 63.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \frac{\pi}{b \cdot a} \cdot \frac{0.5}{a} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (* (/ PI (* b a)) (/ 0.5 a)))

C

assert(a < b);
double code(double a, double b) {
	return (((double) M_PI) / (b * a)) * (0.5 / a);
}

Java

assert a < b;
public static double code(double a, double b) {
	return (Math.PI / (b * a)) * (0.5 / a);
}

Python

[a, b] = sort([a, b])
def code(a, b):
	return (math.pi / (b * a)) * (0.5 / a)

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(Float64(pi / Float64(b * a)) * Float64(0.5 / a))
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = (pi / (b * a)) * (0.5 / a);
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[(N[(Pi / N[(b * a), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision]

Derivation

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Reproduce

herbie shell --seed 2023350 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2.0) (/ 1.0 (- (* b b) (* a a)))) (- (/ 1.0 a) (/ 1.0 b))))