NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.2% → 99.5%

Time: 4.0s

Alternatives: 10

Speedup: 1.8×

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: 78.2% 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: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{+62}:\\ \;\;\;\;\frac{\frac{\frac{-1}{b} \cdot \pi}{2 \cdot a}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{a}}{\left(b + a\right) \cdot 2}}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -5e+62)
   (/ (/ (* (/ -1.0 b) PI) (* 2.0 a)) (- b a))
   (/ (/ (/ PI a) (* (+ b a) 2.0)) b)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -5e+62) {
		tmp = (((-1.0 / b) * ((double) M_PI)) / (2.0 * a)) / (b - a);
	} else {
		tmp = ((((double) M_PI) / a) / ((b + a) * 2.0)) / b;
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -5e+62) {
		tmp = (((-1.0 / b) * Math.PI) / (2.0 * a)) / (b - a);
	} else {
		tmp = ((Math.PI / a) / ((b + a) * 2.0)) / b;
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -5e+62:
		tmp = (((-1.0 / b) * math.pi) / (2.0 * a)) / (b - a)
	else:
		tmp = ((math.pi / a) / ((b + a) * 2.0)) / b
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -5e+62)
		tmp = Float64(Float64(Float64(Float64(-1.0 / b) * pi) / Float64(2.0 * a)) / Float64(b - a));
	else
		tmp = Float64(Float64(Float64(pi / a) / Float64(Float64(b + a) * 2.0)) / b);
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -5e+62)
		tmp = (((-1.0 / b) * pi) / (2.0 * a)) / (b - a);
	else
		tmp = ((pi / a) / ((b + a) * 2.0)) / 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, -5e+62], N[(N[(N[(N[(-1.0 / b), $MachinePrecision] * Pi), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi / a), $MachinePrecision] / N[(N[(b + a), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.00000000000000029e62

   1. Initial program 78.2%

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

   3. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\frac{\frac{\frac{b - a}{b \cdot a} \cdot \pi}{\left(b + a\right) \cdot 2}}{b - a}} \]

   4. Taylor expanded in a around inf

      \[\leadsto \frac{\frac{\color{blue}{\frac{-1}{b}} \cdot \pi}{\left(b + a\right) \cdot 2}}{b - a} \]

   6. Applied rewrites 70.1%

      \[\leadsto \frac{\frac{\color{blue}{\frac{-1}{b}} \cdot \pi}{\left(b + a\right) \cdot 2}}{b - a} \]

   7. Taylor expanded in a around inf

      \[\leadsto \frac{\frac{\frac{-1}{b} \cdot \pi}{\color{blue}{2 \cdot a}}}{b - a} \]

   9. Applied rewrites 66.6%

      \[\leadsto \frac{\frac{\frac{-1}{b} \cdot \pi}{\color{blue}{2 \cdot a}}}{b - a} \]

   if -5.00000000000000029e62 < a

   1. Initial program 78.2%

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

   3. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\frac{\frac{\frac{b - a}{b \cdot a} \cdot \pi}{\left(b + a\right) \cdot 2}}{b - a}} \]

   4. Taylor expanded in a around 0

      \[\leadsto \frac{\frac{\frac{\color{blue}{b}}{b \cdot a} \cdot \pi}{\left(b + a\right) \cdot 2}}{b - a} \]

   6. Applied rewrites 69.8%

      \[\leadsto \frac{\frac{\frac{\color{blue}{b}}{b \cdot a} \cdot \pi}{\left(b + a\right) \cdot 2}}{b - a} \]

   7. Taylor expanded in a around 0

      \[\leadsto \frac{\frac{\frac{b}{b \cdot a} \cdot \pi}{\left(b + a\right) \cdot 2}}{\color{blue}{b}} \]

   9. Applied rewrites 93.8%

      \[\leadsto \frac{\frac{\frac{b}{b \cdot a} \cdot \pi}{\left(b + a\right) \cdot 2}}{\color{blue}{b}} \]

   10. Taylor expanded in a around 0

       \[\leadsto \frac{\frac{\color{blue}{\frac{\mathsf{PI}\left(\right)}{a}}}{\left(b + a\right) \cdot 2}}{b} \]

   12. Applied rewrites 93.9%

       \[\leadsto \frac{\frac{\color{blue}{\frac{\pi}{a}}}{\left(b + a\right) \cdot 2}}{b} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -3.75 \cdot 10^{+67}:\\ \;\;\;\;\frac{-0.5 \cdot \left(\pi \cdot \frac{1}{a \cdot b}\right)}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{a}}{\left(b + a\right) \cdot 2}}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -3.75e+67)
   (/ (* -0.5 (* PI (/ 1.0 (* a b)))) (- b a))
   (/ (/ (/ PI a) (* (+ b a) 2.0)) b)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -3.75e+67) {
		tmp = (-0.5 * (((double) M_PI) * (1.0 / (a * b)))) / (b - a);
	} else {
		tmp = ((((double) M_PI) / a) / ((b + a) * 2.0)) / b;
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -3.75e+67) {
		tmp = (-0.5 * (Math.PI * (1.0 / (a * b)))) / (b - a);
	} else {
		tmp = ((Math.PI / a) / ((b + a) * 2.0)) / b;
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -3.75e+67:
		tmp = (-0.5 * (math.pi * (1.0 / (a * b)))) / (b - a)
	else:
		tmp = ((math.pi / a) / ((b + a) * 2.0)) / b
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -3.75e+67)
		tmp = Float64(Float64(-0.5 * Float64(pi * Float64(1.0 / Float64(a * b)))) / Float64(b - a));
	else
		tmp = Float64(Float64(Float64(pi / a) / Float64(Float64(b + a) * 2.0)) / b);
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -3.75e+67)
		tmp = (-0.5 * (pi * (1.0 / (a * b)))) / (b - a);
	else
		tmp = ((pi / a) / ((b + a) * 2.0)) / 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, -3.75e+67], N[(N[(-0.5 * N[(Pi * N[(1.0 / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi / a), $MachinePrecision] / N[(N[(b + a), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.7500000000000003e67

   1. Initial program 78.2%

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

   3. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\frac{\frac{\frac{b - a}{b \cdot a} \cdot \pi}{\left(b + a\right) \cdot 2}}{b - a}} \]

   4. Taylor expanded in a around inf

      \[\leadsto \frac{\color{blue}{\frac{-1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a \cdot b}}}{b - a} \]

   6. Applied rewrites 66.6%

      \[\leadsto \frac{\color{blue}{-0.5 \cdot \frac{\pi}{a \cdot b}}}{b - a} \]

   8. Applied rewrites 66.6%

      \[\leadsto \frac{-0.5 \cdot \left(\pi \cdot \color{blue}{\frac{1}{a \cdot b}}\right)}{b - a} \]

   if -3.7500000000000003e67 < a

   1. Initial program 78.2%

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

   3. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\frac{\frac{\frac{b - a}{b \cdot a} \cdot \pi}{\left(b + a\right) \cdot 2}}{b - a}} \]

   4. Taylor expanded in a around 0

      \[\leadsto \frac{\frac{\frac{\color{blue}{b}}{b \cdot a} \cdot \pi}{\left(b + a\right) \cdot 2}}{b - a} \]

   6. Applied rewrites 69.8%

      \[\leadsto \frac{\frac{\frac{\color{blue}{b}}{b \cdot a} \cdot \pi}{\left(b + a\right) \cdot 2}}{b - a} \]

   7. Taylor expanded in a around 0

      \[\leadsto \frac{\frac{\frac{b}{b \cdot a} \cdot \pi}{\left(b + a\right) \cdot 2}}{\color{blue}{b}} \]

   9. Applied rewrites 93.8%

      \[\leadsto \frac{\frac{\frac{b}{b \cdot a} \cdot \pi}{\left(b + a\right) \cdot 2}}{\color{blue}{b}} \]

   10. Taylor expanded in a around 0

       \[\leadsto \frac{\frac{\color{blue}{\frac{\mathsf{PI}\left(\right)}{a}}}{\left(b + a\right) \cdot 2}}{b} \]

   12. Applied rewrites 93.9%

       \[\leadsto \frac{\frac{\color{blue}{\frac{\pi}{a}}}{\left(b + a\right) \cdot 2}}{b} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -3.95 \cdot 10^{+118}:\\ \;\;\;\;\frac{-0.5 \cdot \pi}{\left(b - a\right) \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{a}}{\left(b + a\right) \cdot 2}}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -3.95e+118)
   (/ (* -0.5 PI) (* (- b a) (* b a)))
   (/ (/ (/ PI a) (* (+ b a) 2.0)) b)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -3.95e+118) {
		tmp = (-0.5 * ((double) M_PI)) / ((b - a) * (b * a));
	} else {
		tmp = ((((double) M_PI) / a) / ((b + a) * 2.0)) / b;
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -3.95e+118) {
		tmp = (-0.5 * Math.PI) / ((b - a) * (b * a));
	} else {
		tmp = ((Math.PI / a) / ((b + a) * 2.0)) / b;
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -3.95e+118:
		tmp = (-0.5 * math.pi) / ((b - a) * (b * a))
	else:
		tmp = ((math.pi / a) / ((b + a) * 2.0)) / b
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -3.95e+118)
		tmp = Float64(Float64(-0.5 * pi) / Float64(Float64(b - a) * Float64(b * a)));
	else
		tmp = Float64(Float64(Float64(pi / a) / Float64(Float64(b + a) * 2.0)) / b);
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -3.95e+118)
		tmp = (-0.5 * pi) / ((b - a) * (b * a));
	else
		tmp = ((pi / a) / ((b + a) * 2.0)) / 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, -3.95e+118], N[(N[(-0.5 * Pi), $MachinePrecision] / N[(N[(b - a), $MachinePrecision] * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi / a), $MachinePrecision] / N[(N[(b + a), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.9500000000000002e118

   1. Initial program 78.2%

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

   3. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\frac{\frac{0.5 \cdot \pi}{b + a} \cdot \left(b - a\right)}{\left(b - a\right) \cdot \left(b \cdot a\right)}} \]

   4. Taylor expanded in a around inf

      \[\leadsto \frac{\color{blue}{\frac{-1}{2} \cdot \mathsf{PI}\left(\right)}}{\left(b - a\right) \cdot \left(b \cdot a\right)} \]

   6. Applied rewrites 66.3%

      \[\leadsto \frac{\color{blue}{-0.5 \cdot \pi}}{\left(b - a\right) \cdot \left(b \cdot a\right)} \]

   if -3.9500000000000002e118 < a

   1. Initial program 78.2%

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

   3. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\frac{\frac{\frac{b - a}{b \cdot a} \cdot \pi}{\left(b + a\right) \cdot 2}}{b - a}} \]

   4. Taylor expanded in a around 0

      \[\leadsto \frac{\frac{\frac{\color{blue}{b}}{b \cdot a} \cdot \pi}{\left(b + a\right) \cdot 2}}{b - a} \]

   6. Applied rewrites 69.8%

      \[\leadsto \frac{\frac{\frac{\color{blue}{b}}{b \cdot a} \cdot \pi}{\left(b + a\right) \cdot 2}}{b - a} \]

   7. Taylor expanded in a around 0

      \[\leadsto \frac{\frac{\frac{b}{b \cdot a} \cdot \pi}{\left(b + a\right) \cdot 2}}{\color{blue}{b}} \]

   9. Applied rewrites 93.8%

      \[\leadsto \frac{\frac{\frac{b}{b \cdot a} \cdot \pi}{\left(b + a\right) \cdot 2}}{\color{blue}{b}} \]

   10. Taylor expanded in a around 0

       \[\leadsto \frac{\frac{\color{blue}{\frac{\mathsf{PI}\left(\right)}{a}}}{\left(b + a\right) \cdot 2}}{b} \]

   12. Applied rewrites 93.9%

       \[\leadsto \frac{\frac{\color{blue}{\frac{\pi}{a}}}{\left(b + a\right) \cdot 2}}{b} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = ((((b - a) / (b * a)) * 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[(N[(N[(b - a), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision] / N[(N[(b + a), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.2%

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

3. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{\frac{\frac{b - a}{b \cdot a} \cdot \pi}{\left(b + a\right) \cdot 2}}{b - a}} \]
4. Add Preprocessing

Alternative 5: 99.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = ((((b - a) * pi) / ((b + b) * a)) / (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[(N[(N[(N[(b - a), $MachinePrecision] * Pi), $MachinePrecision] / N[(N[(b + b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] / N[(b + a), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.2%

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

3. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\frac{\frac{\frac{\left(b - a\right) \cdot \pi}{\left(b + b\right) \cdot a}}{b + a}}{b - a}} \]
4. Add Preprocessing

Alternative 6: 99.0% accurate, 1.1× speedup

Math

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

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)) (* (- b a) (* b a))))

C

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

Python

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

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(Float64(Float64(Float64(0.5 * pi) / Float64(b + a)) * Float64(b - a)) / Float64(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)) / ((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[(N[(N[(0.5 * Pi), $MachinePrecision] / N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] / N[(N[(b - a), $MachinePrecision] * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.2%

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

3. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\frac{\frac{0.5 \cdot \pi}{b + a} \cdot \left(b - a\right)}{\left(b - a\right) \cdot \left(b \cdot a\right)}} \]
4. Add Preprocessing

Alternative 7: 91.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -3.1 \cdot 10^{-56}:\\ \;\;\;\;\frac{-0.5 \cdot \pi}{\left(b - a\right) \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a \cdot b}}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -3.1e-56)
   (/ (* -0.5 PI) (* (- b a) (* b a)))
   (/ (* 0.5 (/ PI (* a b))) b)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -3.1e-56) {
		tmp = (-0.5 * ((double) M_PI)) / ((b - a) * (b * a));
	} else {
		tmp = (0.5 * (((double) M_PI) / (a * b))) / b;
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -3.1e-56) {
		tmp = (-0.5 * Math.PI) / ((b - a) * (b * a));
	} else {
		tmp = (0.5 * (Math.PI / (a * b))) / b;
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -3.1e-56:
		tmp = (-0.5 * math.pi) / ((b - a) * (b * a))
	else:
		tmp = (0.5 * (math.pi / (a * b))) / b
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -3.1e-56)
		tmp = Float64(Float64(-0.5 * pi) / Float64(Float64(b - a) * Float64(b * a)));
	else
		tmp = Float64(Float64(0.5 * Float64(pi / Float64(a * b))) / b);
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -3.1e-56)
		tmp = (-0.5 * pi) / ((b - a) * (b * a));
	else
		tmp = (0.5 * (pi / (a * b))) / 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, -3.1e-56], N[(N[(-0.5 * Pi), $MachinePrecision] / N[(N[(b - a), $MachinePrecision] * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.09999999999999987e-56

   1. Initial program 78.2%

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

   3. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\frac{\frac{0.5 \cdot \pi}{b + a} \cdot \left(b - a\right)}{\left(b - a\right) \cdot \left(b \cdot a\right)}} \]

   4. Taylor expanded in a around inf

      \[\leadsto \frac{\color{blue}{\frac{-1}{2} \cdot \mathsf{PI}\left(\right)}}{\left(b - a\right) \cdot \left(b \cdot a\right)} \]

   6. Applied rewrites 66.3%

      \[\leadsto \frac{\color{blue}{-0.5 \cdot \pi}}{\left(b - a\right) \cdot \left(b \cdot a\right)} \]

   if -3.09999999999999987e-56 < a

   1. Initial program 78.2%

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

   3. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\frac{\frac{\frac{b - a}{b \cdot a} \cdot \pi}{\left(b + a\right) \cdot 2}}{b - a}} \]

   4. Taylor expanded in a around 0

      \[\leadsto \frac{\frac{\frac{\color{blue}{b}}{b \cdot a} \cdot \pi}{\left(b + a\right) \cdot 2}}{b - a} \]

   6. Applied rewrites 69.8%

      \[\leadsto \frac{\frac{\frac{\color{blue}{b}}{b \cdot a} \cdot \pi}{\left(b + a\right) \cdot 2}}{b - a} \]

   7. Taylor expanded in a around 0

      \[\leadsto \frac{\frac{\frac{b}{b \cdot a} \cdot \pi}{\left(b + a\right) \cdot 2}}{\color{blue}{b}} \]

   9. Applied rewrites 93.8%

      \[\leadsto \frac{\frac{\frac{b}{b \cdot a} \cdot \pi}{\left(b + a\right) \cdot 2}}{\color{blue}{b}} \]

   10. Taylor expanded in a around 0

       \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a \cdot b}}}{b} \]

   12. Applied rewrites 63.0%

       \[\leadsto \frac{\color{blue}{0.5 \cdot \frac{\pi}{a \cdot b}}}{b} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 84.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -5.8 \cdot 10^{-56}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{a}}{2 \cdot a}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a \cdot b}}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -5.8e-56)
   (/ (/ (/ PI a) (* 2.0 a)) b)
   (/ (* 0.5 (/ PI (* a b))) b)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -5.8e-56) {
		tmp = ((((double) M_PI) / a) / (2.0 * a)) / b;
	} else {
		tmp = (0.5 * (((double) M_PI) / (a * b))) / b;
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -5.8e-56) {
		tmp = ((Math.PI / a) / (2.0 * a)) / b;
	} else {
		tmp = (0.5 * (Math.PI / (a * b))) / b;
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -5.8e-56:
		tmp = ((math.pi / a) / (2.0 * a)) / b
	else:
		tmp = (0.5 * (math.pi / (a * b))) / b
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -5.8e-56)
		tmp = Float64(Float64(Float64(pi / a) / Float64(2.0 * a)) / b);
	else
		tmp = Float64(Float64(0.5 * Float64(pi / Float64(a * b))) / b);
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -5.8e-56)
		tmp = ((pi / a) / (2.0 * a)) / b;
	else
		tmp = (0.5 * (pi / (a * b))) / 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, -5.8e-56], N[(N[(N[(Pi / a), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], N[(N[(0.5 * N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.79999999999999982e-56

   1. Initial program 78.2%

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

   3. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\frac{\frac{\frac{b - a}{b \cdot a} \cdot \pi}{\left(b + a\right) \cdot 2}}{b - a}} \]

   4. Taylor expanded in a around 0

      \[\leadsto \frac{\frac{\frac{\color{blue}{b}}{b \cdot a} \cdot \pi}{\left(b + a\right) \cdot 2}}{b - a} \]

   6. Applied rewrites 69.8%

      \[\leadsto \frac{\frac{\frac{\color{blue}{b}}{b \cdot a} \cdot \pi}{\left(b + a\right) \cdot 2}}{b - a} \]

   7. Taylor expanded in a around 0

      \[\leadsto \frac{\frac{\frac{b}{b \cdot a} \cdot \pi}{\left(b + a\right) \cdot 2}}{\color{blue}{b}} \]

   9. Applied rewrites 93.8%

      \[\leadsto \frac{\frac{\frac{b}{b \cdot a} \cdot \pi}{\left(b + a\right) \cdot 2}}{\color{blue}{b}} \]

   10. Taylor expanded in a around 0

       \[\leadsto \frac{\frac{\color{blue}{\frac{\mathsf{PI}\left(\right)}{a}}}{\left(b + a\right) \cdot 2}}{b} \]

   12. Applied rewrites 93.9%

       \[\leadsto \frac{\frac{\color{blue}{\frac{\pi}{a}}}{\left(b + a\right) \cdot 2}}{b} \]

   13. Taylor expanded in a around inf

       \[\leadsto \frac{\frac{\frac{\pi}{a}}{\color{blue}{2 \cdot a}}}{b} \]

   15. Applied rewrites 57.3%

       \[\leadsto \frac{\frac{\frac{\pi}{a}}{\color{blue}{2 \cdot a}}}{b} \]

   if -5.79999999999999982e-56 < a

   1. Initial program 78.2%

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

   3. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\frac{\frac{\frac{b - a}{b \cdot a} \cdot \pi}{\left(b + a\right) \cdot 2}}{b - a}} \]

   4. Taylor expanded in a around 0

      \[\leadsto \frac{\frac{\frac{\color{blue}{b}}{b \cdot a} \cdot \pi}{\left(b + a\right) \cdot 2}}{b - a} \]

   6. Applied rewrites 69.8%

      \[\leadsto \frac{\frac{\frac{\color{blue}{b}}{b \cdot a} \cdot \pi}{\left(b + a\right) \cdot 2}}{b - a} \]

   7. Taylor expanded in a around 0

      \[\leadsto \frac{\frac{\frac{b}{b \cdot a} \cdot \pi}{\left(b + a\right) \cdot 2}}{\color{blue}{b}} \]

   9. Applied rewrites 93.8%

      \[\leadsto \frac{\frac{\frac{b}{b \cdot a} \cdot \pi}{\left(b + a\right) \cdot 2}}{\color{blue}{b}} \]

   10. Taylor expanded in a around 0

       \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a \cdot b}}}{b} \]

   12. Applied rewrites 63.0%

       \[\leadsto \frac{\color{blue}{0.5 \cdot \frac{\pi}{a \cdot b}}}{b} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 63.0% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 78.2%

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

3. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{\frac{\frac{b - a}{b \cdot a} \cdot \pi}{\left(b + a\right) \cdot 2}}{b - a}} \]

4. Taylor expanded in a around 0

   \[\leadsto \frac{\frac{\frac{\color{blue}{b}}{b \cdot a} \cdot \pi}{\left(b + a\right) \cdot 2}}{b - a} \]

6. Applied rewrites 69.8%

   \[\leadsto \frac{\frac{\frac{\color{blue}{b}}{b \cdot a} \cdot \pi}{\left(b + a\right) \cdot 2}}{b - a} \]

7. Taylor expanded in a around 0

   \[\leadsto \frac{\frac{\frac{b}{b \cdot a} \cdot \pi}{\left(b + a\right) \cdot 2}}{\color{blue}{b}} \]

9. Applied rewrites 93.8%

   \[\leadsto \frac{\frac{\frac{b}{b \cdot a} \cdot \pi}{\left(b + a\right) \cdot 2}}{\color{blue}{b}} \]

10. Taylor expanded in a around 0

    \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a \cdot b}}}{b} \]

12. Applied rewrites 63.0%

    \[\leadsto \frac{\color{blue}{0.5 \cdot \frac{\pi}{a \cdot b}}}{b} \]
13. Add Preprocessing

Alternative 10: 30.8% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 78.2%

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

3. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{\frac{\frac{b - a}{b \cdot a} \cdot \pi}{\left(b + a\right) \cdot 2}}{b - a}} \]

4. Taylor expanded in a around 0

   \[\leadsto \frac{\frac{\frac{\color{blue}{b}}{b \cdot a} \cdot \pi}{\left(b + a\right) \cdot 2}}{b - a} \]

6. Applied rewrites 69.8%

   \[\leadsto \frac{\frac{\frac{\color{blue}{b}}{b \cdot a} \cdot \pi}{\left(b + a\right) \cdot 2}}{b - a} \]

7. Taylor expanded in a around 0

   \[\leadsto \frac{\frac{\frac{b}{b \cdot a} \cdot \pi}{\left(b + a\right) \cdot 2}}{\color{blue}{b}} \]

9. Applied rewrites 93.8%

   \[\leadsto \frac{\frac{\frac{b}{b \cdot a} \cdot \pi}{\left(b + a\right) \cdot 2}}{\color{blue}{b}} \]

10. Taylor expanded in a around inf

    \[\leadsto \frac{\color{blue}{\frac{-1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a \cdot b}}}{b} \]

12. Applied rewrites 30.8%

    \[\leadsto \frac{\color{blue}{-0.5 \cdot \frac{\pi}{a \cdot b}}}{b} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025161 
(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))))