NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.9% → 93.8%

Time: 8.7s

Alternatives: 11

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: 78.9% 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: 93.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -3.9 \cdot 10^{+133}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -3.9e+133)
   (* 0.5 (/ PI (* a (* a b))))
   (* (* (/ PI 2.0) (/ (/ 1.0 (+ a b)) (- b a))) (+ (/ 1.0 a) (/ -1.0 b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -3.9e+133) {
		tmp = 0.5 * (((double) M_PI) / (a * (a * b)));
	} else {
		tmp = ((((double) M_PI) / 2.0) * ((1.0 / (a + b)) / (b - a))) * ((1.0 / a) + (-1.0 / b));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -3.9e+133) {
		tmp = 0.5 * (Math.PI / (a * (a * b)));
	} else {
		tmp = ((Math.PI / 2.0) * ((1.0 / (a + b)) / (b - a))) * ((1.0 / a) + (-1.0 / b));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -3.9e+133:
		tmp = 0.5 * (math.pi / (a * (a * b)))
	else:
		tmp = ((math.pi / 2.0) * ((1.0 / (a + b)) / (b - a))) * ((1.0 / a) + (-1.0 / b))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -3.9e+133)
		tmp = Float64(0.5 * Float64(pi / Float64(a * Float64(a * b))));
	else
		tmp = Float64(Float64(Float64(pi / 2.0) * Float64(Float64(1.0 / Float64(a + b)) / Float64(b - a))) * Float64(Float64(1.0 / a) + Float64(-1.0 / b)));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -3.9e+133)
		tmp = 0.5 * (pi / (a * (a * b)));
	else
		tmp = ((pi / 2.0) * ((1.0 / (a + b)) / (b - a))) * ((1.0 / a) + (-1.0 / 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, -3.9e+133], N[(0.5 * N[(Pi / N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi / 2.0), $MachinePrecision] * N[(N[(1.0 / N[(a + b), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / a), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.90000000000000014e133

   1. Initial program 61.1%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Step-by-step derivation

      1. inv-pow 61.1%

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

      2. difference-of-squares 73.3%

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

      3. unpow-prod-down 76.5%

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

      4. inv-pow 76.5%

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

      5. inv-pow 76.5%

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

   3. Applied egg-rr 76.5%

      \[\leadsto \left(\frac{\pi}{2} \cdot \color{blue}{\left(\frac{1}{b + a} \cdot \frac{1}{b - a}\right)}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ 76.5%

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

      2. *-rgt-identity 76.5%

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

      3. +-commutative 76.5%

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

   5. Simplified 76.5%

      \[\leadsto \left(\frac{\pi}{2} \cdot \color{blue}{\frac{\frac{1}{a + b}}{b - a}}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   6. Step-by-step derivation

      1. add-cube-cbrt 76.4%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{1}{a} - \frac{1}{b}} \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right)} \]

      2. inv-pow 76.4%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{\color{blue}{{a}^{-1}} - \frac{1}{b}} \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \]

      3. inv-pow 76.4%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - \color{blue}{{b}^{-1}}} \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \]

      4. inv-pow 76.4%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{\color{blue}{{a}^{-1}} - \frac{1}{b}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \]

      5. inv-pow 76.4%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{{a}^{-1} - \color{blue}{{b}^{-1}}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \]

      6. inv-pow 76.4%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{{a}^{-1} - {b}^{-1}}\right) \cdot \sqrt[3]{\color{blue}{{a}^{-1}} - \frac{1}{b}}\right) \]

      7. inv-pow 76.4%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{{a}^{-1} - {b}^{-1}}\right) \cdot \sqrt[3]{{a}^{-1} - \color{blue}{{b}^{-1}}}\right) \]

   7. Applied egg-rr 76.4%

      \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{{a}^{-1} - {b}^{-1}}\right) \cdot \sqrt[3]{{a}^{-1} - {b}^{-1}}\right)} \]

   8. Taylor expanded in a around inf 73.4%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{{a}^{2} \cdot b}} \]
   9. Step-by-step derivation

      1. unpow2 73.4%

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

      2. associate-*l* 98.4%

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

   10. Simplified 98.4%

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

   if -3.90000000000000014e133 < a

   1. Initial program 81.7%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Step-by-step derivation

      1. inv-pow 81.7%

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

      2. difference-of-squares 89.2%

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

      3. unpow-prod-down 89.7%

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

      4. inv-pow 89.7%

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

      5. inv-pow 89.7%

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

   3. Applied egg-rr 89.7%

      \[\leadsto \left(\frac{\pi}{2} \cdot \color{blue}{\left(\frac{1}{b + a} \cdot \frac{1}{b - a}\right)}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ 89.8%

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

      2. *-rgt-identity 89.8%

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

      3. +-commutative 89.8%

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

   5. Simplified 89.8%

      \[\leadsto \left(\frac{\pi}{2} \cdot \color{blue}{\frac{\frac{1}{a + b}}{b - a}}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.9 \cdot 10^{+133}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)\\ \end{array} \]

Alternative 2: 91.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{+147}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(a \cdot b\right)}\\ \mathbf{elif}\;a \leq -6 \cdot 10^{-159}:\\ \;\;\;\;\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{a \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -2e+147)
   (* 0.5 (/ PI (* a (* a b))))
   (if (<= a -6e-159)
     (* (+ (/ 1.0 a) (/ -1.0 b)) (/ (/ PI 2.0) (- (* b b) (* a a))))
     (/ (* 0.5 PI) (* a (* b b))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -2e+147) {
		tmp = 0.5 * (((double) M_PI) / (a * (a * b)));
	} else if (a <= -6e-159) {
		tmp = ((1.0 / a) + (-1.0 / b)) * ((((double) M_PI) / 2.0) / ((b * b) - (a * 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 <= -2e+147) {
		tmp = 0.5 * (Math.PI / (a * (a * b)));
	} else if (a <= -6e-159) {
		tmp = ((1.0 / a) + (-1.0 / b)) * ((Math.PI / 2.0) / ((b * b) - (a * 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 <= -2e+147:
		tmp = 0.5 * (math.pi / (a * (a * b)))
	elif a <= -6e-159:
		tmp = ((1.0 / a) + (-1.0 / b)) * ((math.pi / 2.0) / ((b * b) - (a * 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 <= -2e+147)
		tmp = Float64(0.5 * Float64(pi / Float64(a * Float64(a * b))));
	elseif (a <= -6e-159)
		tmp = Float64(Float64(Float64(1.0 / a) + Float64(-1.0 / b)) * Float64(Float64(pi / 2.0) / Float64(Float64(b * b) - Float64(a * a))));
	else
		tmp = Float64(Float64(0.5 * pi) / Float64(a * Float64(b * b)));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2e+147)
		tmp = 0.5 * (pi / (a * (a * b)));
	elseif (a <= -6e-159)
		tmp = ((1.0 / a) + (-1.0 / b)) * ((pi / 2.0) / ((b * b) - (a * 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, -2e+147], N[(0.5 * N[(Pi / N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -6e-159], N[(N[(N[(1.0 / a), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi / 2.0), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * Pi), $MachinePrecision] / N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2e147

   1. Initial program 58.1%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Step-by-step derivation

      1. inv-pow 58.1%

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

      2. difference-of-squares 71.3%

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

      3. unpow-prod-down 74.7%

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

      4. inv-pow 74.7%

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

      5. inv-pow 74.7%

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

   3. Applied egg-rr 74.7%

      \[\leadsto \left(\frac{\pi}{2} \cdot \color{blue}{\left(\frac{1}{b + a} \cdot \frac{1}{b - a}\right)}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ 74.7%

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

      2. *-rgt-identity 74.7%

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

      3. +-commutative 74.7%

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

   5. Simplified 74.7%

      \[\leadsto \left(\frac{\pi}{2} \cdot \color{blue}{\frac{\frac{1}{a + b}}{b - a}}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   6. Step-by-step derivation

      1. add-cube-cbrt 74.7%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{1}{a} - \frac{1}{b}} \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right)} \]

      2. inv-pow 74.7%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{\color{blue}{{a}^{-1}} - \frac{1}{b}} \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \]

      3. inv-pow 74.7%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - \color{blue}{{b}^{-1}}} \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \]

      4. inv-pow 74.7%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{\color{blue}{{a}^{-1}} - \frac{1}{b}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \]

      5. inv-pow 74.7%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{{a}^{-1} - \color{blue}{{b}^{-1}}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \]

      6. inv-pow 74.7%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{{a}^{-1} - {b}^{-1}}\right) \cdot \sqrt[3]{\color{blue}{{a}^{-1}} - \frac{1}{b}}\right) \]

      7. inv-pow 74.7%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{{a}^{-1} - {b}^{-1}}\right) \cdot \sqrt[3]{{a}^{-1} - \color{blue}{{b}^{-1}}}\right) \]

   7. Applied egg-rr 74.7%

      \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{{a}^{-1} - {b}^{-1}}\right) \cdot \sqrt[3]{{a}^{-1} - {b}^{-1}}\right)} \]

   8. Taylor expanded in a around inf 71.3%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{{a}^{2} \cdot b}} \]
   9. Step-by-step derivation

      1. unpow2 71.3%

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

      2. associate-*l* 98.3%

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

   10. Simplified 98.3%

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

   if -2e147 < a < -6.00000000000000018e-159

   1. Initial program 93.6%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Step-by-step derivation

      1. associate-*r/ 93.8%

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

      2. *-rgt-identity 93.8%

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

      3. sub-neg 93.8%

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

      4. distribute-neg-frac 93.8%

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

      5. metadata-eval 93.8%

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

   3. Simplified 93.8%

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

   if -6.00000000000000018e-159 < a

   1. Initial program 77.1%

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

   2. Taylor expanded in b around inf 62.1%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{a \cdot {b}^{2}}} \]
   3. Step-by-step derivation

      1. associate-*r/ 62.1%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{a \cdot {b}^{2}}} \]

      2. unpow2 62.1%

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

   4. Simplified 62.1%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{a \cdot \left(b \cdot b\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{+147}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(a \cdot b\right)}\\ \mathbf{elif}\;a \leq -6 \cdot 10^{-159}:\\ \;\;\;\;\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{a \cdot \left(b \cdot b\right)}\\ \end{array} \]

Alternative 3: 86.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1.65 \cdot 10^{+79}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(a \cdot b\right)}\\ \mathbf{elif}\;a \leq -3.8 \cdot 10^{-116}:\\ \;\;\;\;\frac{\frac{-1}{b}}{\frac{2}{\frac{\pi}{a + b} \cdot \frac{1}{b - a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{a \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -1.65e+79)
   (* 0.5 (/ PI (* a (* a b))))
   (if (<= a -3.8e-116)
     (/ (/ -1.0 b) (/ 2.0 (* (/ PI (+ a b)) (/ 1.0 (- b a)))))
     (/ (* 0.5 PI) (* a (* b b))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -1.65e+79) {
		tmp = 0.5 * (((double) M_PI) / (a * (a * b)));
	} else if (a <= -3.8e-116) {
		tmp = (-1.0 / b) / (2.0 / ((((double) M_PI) / (a + b)) * (1.0 / (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 <= -1.65e+79) {
		tmp = 0.5 * (Math.PI / (a * (a * b)));
	} else if (a <= -3.8e-116) {
		tmp = (-1.0 / b) / (2.0 / ((Math.PI / (a + b)) * (1.0 / (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 <= -1.65e+79:
		tmp = 0.5 * (math.pi / (a * (a * b)))
	elif a <= -3.8e-116:
		tmp = (-1.0 / b) / (2.0 / ((math.pi / (a + b)) * (1.0 / (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 <= -1.65e+79)
		tmp = Float64(0.5 * Float64(pi / Float64(a * Float64(a * b))));
	elseif (a <= -3.8e-116)
		tmp = Float64(Float64(-1.0 / b) / Float64(2.0 / Float64(Float64(pi / Float64(a + b)) * Float64(1.0 / Float64(b - a)))));
	else
		tmp = Float64(Float64(0.5 * pi) / Float64(a * Float64(b * b)));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.65e+79)
		tmp = 0.5 * (pi / (a * (a * b)));
	elseif (a <= -3.8e-116)
		tmp = (-1.0 / b) / (2.0 / ((pi / (a + b)) * (1.0 / (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, -1.65e+79], N[(0.5 * N[(Pi / N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -3.8e-116], N[(N[(-1.0 / b), $MachinePrecision] / N[(2.0 / N[(N[(Pi / N[(a + b), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * Pi), $MachinePrecision] / N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.6500000000000001e79

   1. Initial program 69.8%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Step-by-step derivation

      1. inv-pow 69.8%

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

      2. difference-of-squares 79.2%

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

      3. unpow-prod-down 81.7%

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

      4. inv-pow 81.7%

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

      5. inv-pow 81.7%

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

   3. Applied egg-rr 81.7%

      \[\leadsto \left(\frac{\pi}{2} \cdot \color{blue}{\left(\frac{1}{b + a} \cdot \frac{1}{b - a}\right)}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ 81.6%

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

      2. *-rgt-identity 81.6%

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

      3. +-commutative 81.6%

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

   5. Simplified 81.6%

      \[\leadsto \left(\frac{\pi}{2} \cdot \color{blue}{\frac{\frac{1}{a + b}}{b - a}}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   6. Step-by-step derivation

      1. add-cube-cbrt 81.6%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{1}{a} - \frac{1}{b}} \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right)} \]

      2. inv-pow 81.6%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{\color{blue}{{a}^{-1}} - \frac{1}{b}} \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \]

      3. inv-pow 81.6%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - \color{blue}{{b}^{-1}}} \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \]

      4. inv-pow 81.6%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{\color{blue}{{a}^{-1}} - \frac{1}{b}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \]

      5. inv-pow 81.6%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{{a}^{-1} - \color{blue}{{b}^{-1}}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \]

      6. inv-pow 81.6%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{{a}^{-1} - {b}^{-1}}\right) \cdot \sqrt[3]{\color{blue}{{a}^{-1}} - \frac{1}{b}}\right) \]

      7. inv-pow 81.6%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{{a}^{-1} - {b}^{-1}}\right) \cdot \sqrt[3]{{a}^{-1} - \color{blue}{{b}^{-1}}}\right) \]

   7. Applied egg-rr 81.6%

      \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{{a}^{-1} - {b}^{-1}}\right) \cdot \sqrt[3]{{a}^{-1} - {b}^{-1}}\right)} \]

   8. Taylor expanded in a around inf 79.1%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{{a}^{2} \cdot b}} \]
   9. Step-by-step derivation

      1. unpow2 79.1%

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

      2. associate-*l* 98.4%

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

   10. Simplified 98.4%

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

   if -1.6500000000000001e79 < a < -3.8000000000000001e-116

   1. Initial program 92.3%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Step-by-step derivation

      1. *-commutative 92.3%

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

      2. associate-*l/ 92.3%

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

      3. associate-*r/ 92.3%

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

      4. associate-/l* 92.4%

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

      5. sub-neg 92.4%

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

      6. distribute-neg-frac 92.4%

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

      7. metadata-eval 92.4%

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

      8. associate-*r/ 92.5%

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

      9. *-rgt-identity 92.5%

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

      10. difference-of-squares 92.5%

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

      11. associate-/r* 92.4%

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

   3. Simplified 92.4%

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

   4. Taylor expanded in a around inf 58.3%

      \[\leadsto \frac{\color{blue}{\frac{-1}{b}}}{\frac{2}{\frac{\frac{\pi}{b + a}}{b - a}}} \]
   5. Step-by-step derivation

      1. div-inv 58.3%

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

   6. Applied egg-rr 58.3%

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

   if -3.8000000000000001e-116 < a

   1. Initial program 77.9%

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

   2. Taylor expanded in b around inf 62.6%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{a \cdot {b}^{2}}} \]
   3. Step-by-step derivation

      1. associate-*r/ 62.6%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{a \cdot {b}^{2}}} \]

      2. unpow2 62.6%

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

   4. Simplified 62.6%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{a \cdot \left(b \cdot b\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.65 \cdot 10^{+79}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(a \cdot b\right)}\\ \mathbf{elif}\;a \leq -3.8 \cdot 10^{-116}:\\ \;\;\;\;\frac{\frac{-1}{b}}{\frac{2}{\frac{\pi}{a + b} \cdot \frac{1}{b - a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{a \cdot \left(b \cdot b\right)}\\ \end{array} \]

Alternative 4: 93.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1.35 \cdot 10^{+152}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a} + \frac{-1}{b}}{\left(b - a\right) \cdot \frac{2}{\frac{\pi}{a + b}}}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -1.35e+152)
   (* 0.5 (/ PI (* a (* a b))))
   (/ (+ (/ 1.0 a) (/ -1.0 b)) (* (- b a) (/ 2.0 (/ PI (+ a b)))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -1.35e+152) {
		tmp = 0.5 * (((double) M_PI) / (a * (a * b)));
	} else {
		tmp = ((1.0 / a) + (-1.0 / b)) / ((b - a) * (2.0 / (((double) M_PI) / (a + b))));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -1.35e+152) {
		tmp = 0.5 * (Math.PI / (a * (a * b)));
	} else {
		tmp = ((1.0 / a) + (-1.0 / b)) / ((b - a) * (2.0 / (Math.PI / (a + b))));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -1.35e+152:
		tmp = 0.5 * (math.pi / (a * (a * b)))
	else:
		tmp = ((1.0 / a) + (-1.0 / b)) / ((b - a) * (2.0 / (math.pi / (a + b))))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -1.35e+152)
		tmp = Float64(0.5 * Float64(pi / Float64(a * Float64(a * b))));
	else
		tmp = Float64(Float64(Float64(1.0 / a) + Float64(-1.0 / b)) / Float64(Float64(b - a) * Float64(2.0 / Float64(pi / Float64(a + b)))));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.35e+152)
		tmp = 0.5 * (pi / (a * (a * b)));
	else
		tmp = ((1.0 / a) + (-1.0 / b)) / ((b - a) * (2.0 / (pi / (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, -1.35e+152], N[(0.5 * N[(Pi / N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / a), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision] / N[(N[(b - a), $MachinePrecision] * N[(2.0 / N[(Pi / N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.35000000000000007e152

   1. Initial program 55.8%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Step-by-step derivation

      1. inv-pow 55.8%

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

      2. difference-of-squares 69.7%

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

      3. unpow-prod-down 73.3%

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

      4. inv-pow 73.3%

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

      5. inv-pow 73.3%

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

   3. Applied egg-rr 73.3%

      \[\leadsto \left(\frac{\pi}{2} \cdot \color{blue}{\left(\frac{1}{b + a} \cdot \frac{1}{b - a}\right)}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ 73.3%

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

      2. *-rgt-identity 73.3%

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

      3. +-commutative 73.3%

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

   5. Simplified 73.3%

      \[\leadsto \left(\frac{\pi}{2} \cdot \color{blue}{\frac{\frac{1}{a + b}}{b - a}}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   6. Step-by-step derivation

      1. add-cube-cbrt 73.3%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{1}{a} - \frac{1}{b}} \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right)} \]

      2. inv-pow 73.3%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{\color{blue}{{a}^{-1}} - \frac{1}{b}} \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \]

      3. inv-pow 73.3%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - \color{blue}{{b}^{-1}}} \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \]

      4. inv-pow 73.3%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{\color{blue}{{a}^{-1}} - \frac{1}{b}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \]

      5. inv-pow 73.3%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{{a}^{-1} - \color{blue}{{b}^{-1}}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \]

      6. inv-pow 73.3%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{{a}^{-1} - {b}^{-1}}\right) \cdot \sqrt[3]{\color{blue}{{a}^{-1}} - \frac{1}{b}}\right) \]

      7. inv-pow 73.3%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{{a}^{-1} - {b}^{-1}}\right) \cdot \sqrt[3]{{a}^{-1} - \color{blue}{{b}^{-1}}}\right) \]

   7. Applied egg-rr 73.3%

      \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{{a}^{-1} - {b}^{-1}}\right) \cdot \sqrt[3]{{a}^{-1} - {b}^{-1}}\right)} \]

   8. Taylor expanded in a around inf 69.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{{a}^{2} \cdot b}} \]
   9. Step-by-step derivation

      1. unpow2 69.7%

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

      2. associate-*l* 98.2%

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

   10. Simplified 98.2%

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

   if -1.35000000000000007e152 < a

   1. Initial program 82.1%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Step-by-step derivation

      1. *-commutative 82.1%

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

      2. associate-*l/ 82.1%

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

      3. associate-*r/ 82.1%

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

      4. associate-/l* 82.1%

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

      5. sub-neg 82.1%

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

      6. distribute-neg-frac 82.1%

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

      7. metadata-eval 82.1%

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

      8. associate-*r/ 82.1%

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

      9. *-rgt-identity 82.1%

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

      10. difference-of-squares 89.4%

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

      11. associate-/r* 89.4%

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

   3. Simplified 89.4%

      \[\leadsto \color{blue}{\frac{\frac{1}{a} + \frac{-1}{b}}{\frac{2}{\frac{\frac{\pi}{b + a}}{b - a}}}} \]
   4. Step-by-step derivation

      1. associate-/r/ 89.4%

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

   5. Applied egg-rr 89.4%

      \[\leadsto \frac{\frac{1}{a} + \frac{-1}{b}}{\color{blue}{\frac{2}{\frac{\pi}{b + a}} \cdot \left(b - a\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.35 \cdot 10^{+152}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a} + \frac{-1}{b}}{\left(b - a\right) \cdot \frac{2}{\frac{\pi}{a + b}}}\\ \end{array} \]

Alternative 5: 93.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a} + \frac{-1}{b}}{\frac{2}{\frac{\frac{\pi}{a + b}}{b - a}}}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -2e+154)
   (* 0.5 (/ PI (* a (* a b))))
   (/ (+ (/ 1.0 a) (/ -1.0 b)) (/ 2.0 (/ (/ PI (+ a b)) (- b a))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -2e+154) {
		tmp = 0.5 * (((double) M_PI) / (a * (a * b)));
	} else {
		tmp = ((1.0 / a) + (-1.0 / b)) / (2.0 / ((((double) M_PI) / (a + b)) / (b - a)));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -2e+154) {
		tmp = 0.5 * (Math.PI / (a * (a * b)));
	} else {
		tmp = ((1.0 / a) + (-1.0 / b)) / (2.0 / ((Math.PI / (a + b)) / (b - a)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -2e+154:
		tmp = 0.5 * (math.pi / (a * (a * b)))
	else:
		tmp = ((1.0 / a) + (-1.0 / b)) / (2.0 / ((math.pi / (a + b)) / (b - a)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -2e+154)
		tmp = Float64(0.5 * Float64(pi / Float64(a * Float64(a * b))));
	else
		tmp = Float64(Float64(Float64(1.0 / a) + Float64(-1.0 / b)) / Float64(2.0 / Float64(Float64(pi / Float64(a + b)) / Float64(b - a))));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2e+154)
		tmp = 0.5 * (pi / (a * (a * b)));
	else
		tmp = ((1.0 / a) + (-1.0 / b)) / (2.0 / ((pi / (a + b)) / (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[a, -2e+154], N[(0.5 * N[(Pi / N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / a), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision] / N[(2.0 / N[(N[(Pi / N[(a + b), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.00000000000000007e154

   1. Initial program 55.8%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Step-by-step derivation

      1. inv-pow 55.8%

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

      2. difference-of-squares 69.7%

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

      3. unpow-prod-down 73.3%

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

      4. inv-pow 73.3%

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

      5. inv-pow 73.3%

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

   3. Applied egg-rr 73.3%

      \[\leadsto \left(\frac{\pi}{2} \cdot \color{blue}{\left(\frac{1}{b + a} \cdot \frac{1}{b - a}\right)}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ 73.3%

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

      2. *-rgt-identity 73.3%

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

      3. +-commutative 73.3%

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

   5. Simplified 73.3%

      \[\leadsto \left(\frac{\pi}{2} \cdot \color{blue}{\frac{\frac{1}{a + b}}{b - a}}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   6. Step-by-step derivation

      1. add-cube-cbrt 73.3%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{1}{a} - \frac{1}{b}} \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right)} \]

      2. inv-pow 73.3%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{\color{blue}{{a}^{-1}} - \frac{1}{b}} \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \]

      3. inv-pow 73.3%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - \color{blue}{{b}^{-1}}} \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \]

      4. inv-pow 73.3%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{\color{blue}{{a}^{-1}} - \frac{1}{b}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \]

      5. inv-pow 73.3%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{{a}^{-1} - \color{blue}{{b}^{-1}}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \]

      6. inv-pow 73.3%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{{a}^{-1} - {b}^{-1}}\right) \cdot \sqrt[3]{\color{blue}{{a}^{-1}} - \frac{1}{b}}\right) \]

      7. inv-pow 73.3%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{{a}^{-1} - {b}^{-1}}\right) \cdot \sqrt[3]{{a}^{-1} - \color{blue}{{b}^{-1}}}\right) \]

   7. Applied egg-rr 73.3%

      \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{{a}^{-1} - {b}^{-1}}\right) \cdot \sqrt[3]{{a}^{-1} - {b}^{-1}}\right)} \]

   8. Taylor expanded in a around inf 69.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{{a}^{2} \cdot b}} \]
   9. Step-by-step derivation

      1. unpow2 69.7%

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

      2. associate-*l* 98.2%

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

   10. Simplified 98.2%

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

   if -2.00000000000000007e154 < a

   1. Initial program 82.1%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Step-by-step derivation

      1. *-commutative 82.1%

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

      2. associate-*l/ 82.1%

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

      3. associate-*r/ 82.1%

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

      4. associate-/l* 82.1%

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

      5. sub-neg 82.1%

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

      6. distribute-neg-frac 82.1%

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

      7. metadata-eval 82.1%

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

      8. associate-*r/ 82.1%

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

      9. *-rgt-identity 82.1%

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

      10. difference-of-squares 89.4%

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

      11. associate-/r* 89.4%

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

   3. Simplified 89.4%

      \[\leadsto \color{blue}{\frac{\frac{1}{a} + \frac{-1}{b}}{\frac{2}{\frac{\frac{\pi}{b + a}}{b - a}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a} + \frac{-1}{b}}{\frac{2}{\frac{\frac{\pi}{a + b}}{b - a}}}\\ \end{array} \]

Alternative 6: 86.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{+153}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(a \cdot b\right)}\\ \mathbf{elif}\;a \leq -3.7 \cdot 10^{-115}:\\ \;\;\;\;\frac{\frac{-1}{b}}{2} \cdot \frac{\frac{\pi}{a + b}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{a \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -1e+153)
   (* 0.5 (/ PI (* a (* a b))))
   (if (<= a -3.7e-115)
     (* (/ (/ -1.0 b) 2.0) (/ (/ PI (+ a b)) (- b a)))
     (/ (* 0.5 PI) (* a (* b b))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -1e+153) {
		tmp = 0.5 * (((double) M_PI) / (a * (a * b)));
	} else if (a <= -3.7e-115) {
		tmp = ((-1.0 / b) / 2.0) * ((((double) M_PI) / (a + b)) / (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 <= -1e+153) {
		tmp = 0.5 * (Math.PI / (a * (a * b)));
	} else if (a <= -3.7e-115) {
		tmp = ((-1.0 / b) / 2.0) * ((Math.PI / (a + b)) / (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 <= -1e+153:
		tmp = 0.5 * (math.pi / (a * (a * b)))
	elif a <= -3.7e-115:
		tmp = ((-1.0 / b) / 2.0) * ((math.pi / (a + b)) / (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 <= -1e+153)
		tmp = Float64(0.5 * Float64(pi / Float64(a * Float64(a * b))));
	elseif (a <= -3.7e-115)
		tmp = Float64(Float64(Float64(-1.0 / b) / 2.0) * Float64(Float64(pi / Float64(a + b)) / Float64(b - a)));
	else
		tmp = Float64(Float64(0.5 * pi) / Float64(a * Float64(b * b)));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1e+153)
		tmp = 0.5 * (pi / (a * (a * b)));
	elseif (a <= -3.7e-115)
		tmp = ((-1.0 / b) / 2.0) * ((pi / (a + b)) / (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, -1e+153], N[(0.5 * N[(Pi / N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -3.7e-115], N[(N[(N[(-1.0 / b), $MachinePrecision] / 2.0), $MachinePrecision] * N[(N[(Pi / N[(a + b), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * Pi), $MachinePrecision] / N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1e153

   1. Initial program 55.8%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Step-by-step derivation

      1. inv-pow 55.8%

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

      2. difference-of-squares 69.7%

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

      3. unpow-prod-down 73.3%

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

      4. inv-pow 73.3%

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

      5. inv-pow 73.3%

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

   3. Applied egg-rr 73.3%

      \[\leadsto \left(\frac{\pi}{2} \cdot \color{blue}{\left(\frac{1}{b + a} \cdot \frac{1}{b - a}\right)}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ 73.3%

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

      2. *-rgt-identity 73.3%

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

      3. +-commutative 73.3%

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

   5. Simplified 73.3%

      \[\leadsto \left(\frac{\pi}{2} \cdot \color{blue}{\frac{\frac{1}{a + b}}{b - a}}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   6. Step-by-step derivation

      1. add-cube-cbrt 73.3%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{1}{a} - \frac{1}{b}} \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right)} \]

      2. inv-pow 73.3%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{\color{blue}{{a}^{-1}} - \frac{1}{b}} \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \]

      3. inv-pow 73.3%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - \color{blue}{{b}^{-1}}} \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \]

      4. inv-pow 73.3%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{\color{blue}{{a}^{-1}} - \frac{1}{b}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \]

      5. inv-pow 73.3%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{{a}^{-1} - \color{blue}{{b}^{-1}}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \]

      6. inv-pow 73.3%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{{a}^{-1} - {b}^{-1}}\right) \cdot \sqrt[3]{\color{blue}{{a}^{-1}} - \frac{1}{b}}\right) \]

      7. inv-pow 73.3%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{{a}^{-1} - {b}^{-1}}\right) \cdot \sqrt[3]{{a}^{-1} - \color{blue}{{b}^{-1}}}\right) \]

   7. Applied egg-rr 73.3%

      \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{{a}^{-1} - {b}^{-1}}\right) \cdot \sqrt[3]{{a}^{-1} - {b}^{-1}}\right)} \]

   8. Taylor expanded in a around inf 69.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{{a}^{2} \cdot b}} \]
   9. Step-by-step derivation

      1. unpow2 69.7%

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

      2. associate-*l* 98.2%

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

   10. Simplified 98.2%

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

   if -1e153 < a < -3.7e-115

   1. Initial program 94.4%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Step-by-step derivation

      1. *-commutative 94.4%

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

      2. associate-*l/ 94.4%

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

      3. associate-*r/ 94.4%

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

      4. associate-/l* 94.5%

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

      5. sub-neg 94.5%

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

      6. distribute-neg-frac 94.5%

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

      7. metadata-eval 94.5%

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

      8. associate-*r/ 94.7%

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

      9. *-rgt-identity 94.7%

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

      10. difference-of-squares 94.7%

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

      11. associate-/r* 94.7%

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

   3. Simplified 94.7%

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

   4. Taylor expanded in a around inf 70.6%

      \[\leadsto \frac{\color{blue}{\frac{-1}{b}}}{\frac{2}{\frac{\frac{\pi}{b + a}}{b - a}}} \]
   5. Step-by-step derivation

      1. associate-/r/ 70.6%

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

      2. +-commutative 70.6%

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

   6. Applied egg-rr 70.6%

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

   if -3.7e-115 < a

   1. Initial program 77.9%

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

   2. Taylor expanded in b around inf 62.6%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{a \cdot {b}^{2}}} \]
   3. Step-by-step derivation

      1. associate-*r/ 62.6%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{a \cdot {b}^{2}}} \]

      2. unpow2 62.6%

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

   4. Simplified 62.6%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{a \cdot \left(b \cdot b\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{+153}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(a \cdot b\right)}\\ \mathbf{elif}\;a \leq -3.7 \cdot 10^{-115}:\\ \;\;\;\;\frac{\frac{-1}{b}}{2} \cdot \frac{\frac{\pi}{a + b}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{a \cdot \left(b \cdot b\right)}\\ \end{array} \]

Alternative 7: 86.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(a \cdot b\right)}\\ \mathbf{elif}\;a \leq -2.9 \cdot 10^{-115}:\\ \;\;\;\;\frac{\frac{-1}{b}}{\frac{2}{\frac{\frac{\pi}{a + b}}{b - a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{a \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -2e+154)
   (* 0.5 (/ PI (* a (* a b))))
   (if (<= a -2.9e-115)
     (/ (/ -1.0 b) (/ 2.0 (/ (/ PI (+ a b)) (- b a))))
     (/ (* 0.5 PI) (* a (* b b))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -2e+154) {
		tmp = 0.5 * (((double) M_PI) / (a * (a * b)));
	} else if (a <= -2.9e-115) {
		tmp = (-1.0 / b) / (2.0 / ((((double) M_PI) / (a + b)) / (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 <= -2e+154) {
		tmp = 0.5 * (Math.PI / (a * (a * b)));
	} else if (a <= -2.9e-115) {
		tmp = (-1.0 / b) / (2.0 / ((Math.PI / (a + b)) / (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 <= -2e+154:
		tmp = 0.5 * (math.pi / (a * (a * b)))
	elif a <= -2.9e-115:
		tmp = (-1.0 / b) / (2.0 / ((math.pi / (a + b)) / (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 <= -2e+154)
		tmp = Float64(0.5 * Float64(pi / Float64(a * Float64(a * b))));
	elseif (a <= -2.9e-115)
		tmp = Float64(Float64(-1.0 / b) / Float64(2.0 / Float64(Float64(pi / Float64(a + b)) / Float64(b - a))));
	else
		tmp = Float64(Float64(0.5 * pi) / Float64(a * Float64(b * b)));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2e+154)
		tmp = 0.5 * (pi / (a * (a * b)));
	elseif (a <= -2.9e-115)
		tmp = (-1.0 / b) / (2.0 / ((pi / (a + b)) / (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, -2e+154], N[(0.5 * N[(Pi / N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -2.9e-115], N[(N[(-1.0 / b), $MachinePrecision] / N[(2.0 / N[(N[(Pi / N[(a + b), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * Pi), $MachinePrecision] / N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.00000000000000007e154

   1. Initial program 55.8%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Step-by-step derivation

      1. inv-pow 55.8%

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

      2. difference-of-squares 69.7%

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

      3. unpow-prod-down 73.3%

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

      4. inv-pow 73.3%

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

      5. inv-pow 73.3%

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

   3. Applied egg-rr 73.3%

      \[\leadsto \left(\frac{\pi}{2} \cdot \color{blue}{\left(\frac{1}{b + a} \cdot \frac{1}{b - a}\right)}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ 73.3%

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

      2. *-rgt-identity 73.3%

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

      3. +-commutative 73.3%

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

   5. Simplified 73.3%

      \[\leadsto \left(\frac{\pi}{2} \cdot \color{blue}{\frac{\frac{1}{a + b}}{b - a}}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   6. Step-by-step derivation

      1. add-cube-cbrt 73.3%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{1}{a} - \frac{1}{b}} \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right)} \]

      2. inv-pow 73.3%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{\color{blue}{{a}^{-1}} - \frac{1}{b}} \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \]

      3. inv-pow 73.3%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - \color{blue}{{b}^{-1}}} \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \]

      4. inv-pow 73.3%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{\color{blue}{{a}^{-1}} - \frac{1}{b}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \]

      5. inv-pow 73.3%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{{a}^{-1} - \color{blue}{{b}^{-1}}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \]

      6. inv-pow 73.3%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{{a}^{-1} - {b}^{-1}}\right) \cdot \sqrt[3]{\color{blue}{{a}^{-1}} - \frac{1}{b}}\right) \]

      7. inv-pow 73.3%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{{a}^{-1} - {b}^{-1}}\right) \cdot \sqrt[3]{{a}^{-1} - \color{blue}{{b}^{-1}}}\right) \]

   7. Applied egg-rr 73.3%

      \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{{a}^{-1} - {b}^{-1}}\right) \cdot \sqrt[3]{{a}^{-1} - {b}^{-1}}\right)} \]

   8. Taylor expanded in a around inf 69.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{{a}^{2} \cdot b}} \]
   9. Step-by-step derivation

      1. unpow2 69.7%

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

      2. associate-*l* 98.2%

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

   10. Simplified 98.2%

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

   if -2.00000000000000007e154 < a < -2.8999999999999998e-115

   1. Initial program 94.4%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Step-by-step derivation

      1. *-commutative 94.4%

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

      2. associate-*l/ 94.4%

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

      3. associate-*r/ 94.4%

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

      4. associate-/l* 94.5%

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

      5. sub-neg 94.5%

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

      6. distribute-neg-frac 94.5%

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

      7. metadata-eval 94.5%

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

      8. associate-*r/ 94.7%

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

      9. *-rgt-identity 94.7%

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

      10. difference-of-squares 94.7%

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

      11. associate-/r* 94.7%

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

   3. Simplified 94.7%

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

   4. Taylor expanded in a around inf 70.6%

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

   if -2.8999999999999998e-115 < a

   1. Initial program 77.9%

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

   2. Taylor expanded in b around inf 62.6%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{a \cdot {b}^{2}}} \]
   3. Step-by-step derivation

      1. associate-*r/ 62.6%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{a \cdot {b}^{2}}} \]

      2. unpow2 62.6%

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

   4. Simplified 62.6%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{a \cdot \left(b \cdot b\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(a \cdot b\right)}\\ \mathbf{elif}\;a \leq -2.9 \cdot 10^{-115}:\\ \;\;\;\;\frac{\frac{-1}{b}}{\frac{2}{\frac{\frac{\pi}{a + b}}{b - a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{a \cdot \left(b \cdot b\right)}\\ \end{array} \]

Alternative 8: 84.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1.48 \cdot 10^{-89}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a}}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -1.48e-89)
   (* 0.5 (/ PI (* a (* a b))))
   (* 0.5 (/ (/ PI a) (* b b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -1.48e-89) {
		tmp = 0.5 * (((double) M_PI) / (a * (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 <= -1.48e-89) {
		tmp = 0.5 * (Math.PI / (a * (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 <= -1.48e-89:
		tmp = 0.5 * (math.pi / (a * (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 <= -1.48e-89)
		tmp = Float64(0.5 * Float64(pi / Float64(a * Float64(a * b))));
	else
		tmp = Float64(0.5 * Float64(Float64(pi / a) / Float64(b * b)));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.48e-89)
		tmp = 0.5 * (pi / (a * (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, -1.48e-89], N[(0.5 * N[(Pi / N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(Pi / a), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.48000000000000007e-89

   1. Initial program 77.8%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Step-by-step derivation

      1. inv-pow 77.8%

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

      2. difference-of-squares 84.3%

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

      3. unpow-prod-down 86.9%

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

      4. inv-pow 86.9%

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

      5. inv-pow 86.9%

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

   3. Applied egg-rr 86.9%

      \[\leadsto \left(\frac{\pi}{2} \cdot \color{blue}{\left(\frac{1}{b + a} \cdot \frac{1}{b - a}\right)}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ 86.9%

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

      2. *-rgt-identity 86.9%

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

      3. +-commutative 86.9%

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

   5. Simplified 86.9%

      \[\leadsto \left(\frac{\pi}{2} \cdot \color{blue}{\frac{\frac{1}{a + b}}{b - a}}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   6. Step-by-step derivation

      1. add-cube-cbrt 86.5%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{1}{a} - \frac{1}{b}} \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right)} \]

      2. inv-pow 86.5%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{\color{blue}{{a}^{-1}} - \frac{1}{b}} \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \]

      3. inv-pow 86.5%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - \color{blue}{{b}^{-1}}} \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \]

      4. inv-pow 86.5%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{\color{blue}{{a}^{-1}} - \frac{1}{b}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \]

      5. inv-pow 86.5%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{{a}^{-1} - \color{blue}{{b}^{-1}}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \]

      6. inv-pow 86.5%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{{a}^{-1} - {b}^{-1}}\right) \cdot \sqrt[3]{\color{blue}{{a}^{-1}} - \frac{1}{b}}\right) \]

      7. inv-pow 86.5%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{{a}^{-1} - {b}^{-1}}\right) \cdot \sqrt[3]{{a}^{-1} - \color{blue}{{b}^{-1}}}\right) \]

   7. Applied egg-rr 86.5%

      \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{{a}^{-1} - {b}^{-1}}\right) \cdot \sqrt[3]{{a}^{-1} - {b}^{-1}}\right)} \]

   8. Taylor expanded in a around inf 69.6%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{{a}^{2} \cdot b}} \]
   9. Step-by-step derivation

      1. unpow2 69.6%

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

      2. associate-*l* 82.8%

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

   10. Simplified 82.8%

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

   if -1.48000000000000007e-89 < a

   1. Initial program 78.7%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Step-by-step derivation

      1. inv-pow 78.7%

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

      2. difference-of-squares 87.6%

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

      3. unpow-prod-down 87.9%

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

      4. inv-pow 87.9%

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

      5. inv-pow 87.9%

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

   3. Applied egg-rr 87.9%

      \[\leadsto \left(\frac{\pi}{2} \cdot \color{blue}{\left(\frac{1}{b + a} \cdot \frac{1}{b - a}\right)}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ 88.0%

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

      2. *-rgt-identity 88.0%

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

      3. +-commutative 88.0%

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

   5. Simplified 88.0%

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

   6. Taylor expanded in a around 0 64.1%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{a \cdot {b}^{2}}} \]
   7. Step-by-step derivation

      1. associate-/r* 64.1%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{\frac{\pi}{a}}{{b}^{2}}} \]

      2. unpow2 64.1%

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

   8. Simplified 64.1%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.48 \cdot 10^{-89}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a}}{b \cdot b}\\ \end{array} \]

Alternative 9: 84.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1.55 \cdot 10^{-89}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -1.55e-89)
   (* 0.5 (/ PI (* a (* a b))))
   (* (/ PI a) (/ 0.5 (* b b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -1.55e-89) {
		tmp = 0.5 * (((double) M_PI) / (a * (a * b)));
	} else {
		tmp = (((double) M_PI) / a) * (0.5 / (b * b));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -1.55e-89) {
		tmp = 0.5 * (Math.PI / (a * (a * b)));
	} else {
		tmp = (Math.PI / a) * (0.5 / (b * b));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -1.55e-89:
		tmp = 0.5 * (math.pi / (a * (a * b)))
	else:
		tmp = (math.pi / a) * (0.5 / (b * b))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -1.55e-89)
		tmp = Float64(0.5 * Float64(pi / Float64(a * Float64(a * b))));
	else
		tmp = Float64(Float64(pi / a) * Float64(0.5 / Float64(b * b)));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.55e-89)
		tmp = 0.5 * (pi / (a * (a * b)));
	else
		tmp = (pi / a) * (0.5 / (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, -1.55e-89], N[(0.5 * N[(Pi / N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / a), $MachinePrecision] * N[(0.5 / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.54999999999999998e-89

   1. Initial program 77.8%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Step-by-step derivation

      1. inv-pow 77.8%

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

      2. difference-of-squares 84.3%

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

      3. unpow-prod-down 86.9%

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

      4. inv-pow 86.9%

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

      5. inv-pow 86.9%

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

   3. Applied egg-rr 86.9%

      \[\leadsto \left(\frac{\pi}{2} \cdot \color{blue}{\left(\frac{1}{b + a} \cdot \frac{1}{b - a}\right)}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ 86.9%

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

      2. *-rgt-identity 86.9%

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

      3. +-commutative 86.9%

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

   5. Simplified 86.9%

      \[\leadsto \left(\frac{\pi}{2} \cdot \color{blue}{\frac{\frac{1}{a + b}}{b - a}}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   6. Step-by-step derivation

      1. add-cube-cbrt 86.5%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{1}{a} - \frac{1}{b}} \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right)} \]

      2. inv-pow 86.5%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{\color{blue}{{a}^{-1}} - \frac{1}{b}} \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \]

      3. inv-pow 86.5%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - \color{blue}{{b}^{-1}}} \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \]

      4. inv-pow 86.5%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{\color{blue}{{a}^{-1}} - \frac{1}{b}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \]

      5. inv-pow 86.5%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{{a}^{-1} - \color{blue}{{b}^{-1}}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \]

      6. inv-pow 86.5%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{{a}^{-1} - {b}^{-1}}\right) \cdot \sqrt[3]{\color{blue}{{a}^{-1}} - \frac{1}{b}}\right) \]

      7. inv-pow 86.5%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{{a}^{-1} - {b}^{-1}}\right) \cdot \sqrt[3]{{a}^{-1} - \color{blue}{{b}^{-1}}}\right) \]

   7. Applied egg-rr 86.5%

      \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{{a}^{-1} - {b}^{-1}}\right) \cdot \sqrt[3]{{a}^{-1} - {b}^{-1}}\right)} \]

   8. Taylor expanded in a around inf 69.6%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{{a}^{2} \cdot b}} \]
   9. Step-by-step derivation

      1. unpow2 69.6%

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

      2. associate-*l* 82.8%

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

   10. Simplified 82.8%

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

   if -1.54999999999999998e-89 < a

   1. Initial program 78.7%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Step-by-step derivation

      1. inv-pow 78.7%

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

      2. difference-of-squares 87.6%

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

      3. unpow-prod-down 87.9%

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

      4. inv-pow 87.9%

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

      5. inv-pow 87.9%

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

   3. Applied egg-rr 87.9%

      \[\leadsto \left(\frac{\pi}{2} \cdot \color{blue}{\left(\frac{1}{b + a} \cdot \frac{1}{b - a}\right)}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ 88.0%

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

      2. *-rgt-identity 88.0%

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

      3. +-commutative 88.0%

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

   5. Simplified 88.0%

      \[\leadsto \left(\frac{\pi}{2} \cdot \color{blue}{\frac{\frac{1}{a + b}}{b - a}}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   6. Step-by-step derivation

      1. add-cube-cbrt 87.5%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{1}{a} - \frac{1}{b}} \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right)} \]

      2. inv-pow 87.5%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{\color{blue}{{a}^{-1}} - \frac{1}{b}} \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \]

      3. inv-pow 87.5%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - \color{blue}{{b}^{-1}}} \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \]

      4. inv-pow 87.5%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{\color{blue}{{a}^{-1}} - \frac{1}{b}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \]

      5. inv-pow 87.5%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{{a}^{-1} - \color{blue}{{b}^{-1}}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \]

      6. inv-pow 87.5%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{{a}^{-1} - {b}^{-1}}\right) \cdot \sqrt[3]{\color{blue}{{a}^{-1}} - \frac{1}{b}}\right) \]

      7. inv-pow 87.5%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{{a}^{-1} - {b}^{-1}}\right) \cdot \sqrt[3]{{a}^{-1} - \color{blue}{{b}^{-1}}}\right) \]

   7. Applied egg-rr 87.5%

      \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{{a}^{-1} - {b}^{-1}}\right) \cdot \sqrt[3]{{a}^{-1} - {b}^{-1}}\right)} \]

   8. Taylor expanded in a around 0 64.1%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{a \cdot {b}^{2}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 64.1%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{a \cdot {b}^{2}}} \]

      2. *-commutative 64.1%

         \[\leadsto \frac{\color{blue}{\pi \cdot 0.5}}{a \cdot {b}^{2}} \]

      3. times-frac 64.1%

         \[\leadsto \color{blue}{\frac{\pi}{a} \cdot \frac{0.5}{{b}^{2}}} \]

      4. unpow2 64.1%

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

   10. Simplified 64.1%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.55 \cdot 10^{-89}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{b \cdot b}\\ \end{array} \]

Alternative 10: 84.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1.55 \cdot 10^{-89}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{a \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -1.55e-89)
   (* 0.5 (/ PI (* a (* a b))))
   (/ (* 0.5 PI) (* a (* b b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -1.55e-89) {
		tmp = 0.5 * (((double) M_PI) / (a * (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 <= -1.55e-89) {
		tmp = 0.5 * (Math.PI / (a * (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 <= -1.55e-89:
		tmp = 0.5 * (math.pi / (a * (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 <= -1.55e-89)
		tmp = Float64(0.5 * Float64(pi / Float64(a * Float64(a * b))));
	else
		tmp = Float64(Float64(0.5 * pi) / Float64(a * Float64(b * b)));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.55e-89)
		tmp = 0.5 * (pi / (a * (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, -1.55e-89], N[(0.5 * N[(Pi / N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * Pi), $MachinePrecision] / N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.54999999999999998e-89

   1. Initial program 77.8%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Step-by-step derivation

      1. inv-pow 77.8%

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

      2. difference-of-squares 84.3%

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

      3. unpow-prod-down 86.9%

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

      4. inv-pow 86.9%

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

      5. inv-pow 86.9%

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

   3. Applied egg-rr 86.9%

      \[\leadsto \left(\frac{\pi}{2} \cdot \color{blue}{\left(\frac{1}{b + a} \cdot \frac{1}{b - a}\right)}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ 86.9%

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

      2. *-rgt-identity 86.9%

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

      3. +-commutative 86.9%

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

   5. Simplified 86.9%

      \[\leadsto \left(\frac{\pi}{2} \cdot \color{blue}{\frac{\frac{1}{a + b}}{b - a}}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   6. Step-by-step derivation

      1. add-cube-cbrt 86.5%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{1}{a} - \frac{1}{b}} \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right)} \]

      2. inv-pow 86.5%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{\color{blue}{{a}^{-1}} - \frac{1}{b}} \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \]

      3. inv-pow 86.5%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - \color{blue}{{b}^{-1}}} \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \]

      4. inv-pow 86.5%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{\color{blue}{{a}^{-1}} - \frac{1}{b}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \]

      5. inv-pow 86.5%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{{a}^{-1} - \color{blue}{{b}^{-1}}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \]

      6. inv-pow 86.5%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{{a}^{-1} - {b}^{-1}}\right) \cdot \sqrt[3]{\color{blue}{{a}^{-1}} - \frac{1}{b}}\right) \]

      7. inv-pow 86.5%

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{{a}^{-1} - {b}^{-1}}\right) \cdot \sqrt[3]{{a}^{-1} - \color{blue}{{b}^{-1}}}\right) \]

   7. Applied egg-rr 86.5%

      \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{{a}^{-1} - {b}^{-1}}\right) \cdot \sqrt[3]{{a}^{-1} - {b}^{-1}}\right)} \]

   8. Taylor expanded in a around inf 69.6%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{{a}^{2} \cdot b}} \]
   9. Step-by-step derivation

      1. unpow2 69.6%

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

      2. associate-*l* 82.8%

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

   10. Simplified 82.8%

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

   if -1.54999999999999998e-89 < a

   1. Initial program 78.7%

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

   2. Taylor expanded in b around inf 64.1%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{a \cdot {b}^{2}}} \]
   3. Step-by-step derivation

      1. associate-*r/ 64.1%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{a \cdot {b}^{2}}} \]

      2. unpow2 64.1%

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

   4. Simplified 64.1%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{a \cdot \left(b \cdot b\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.55 \cdot 10^{-89}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{a \cdot \left(b \cdot b\right)}\\ \end{array} \]

Alternative 11: 63.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.4%

   \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
2. Step-by-step derivation

   1. inv-pow 78.4%

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

   2. difference-of-squares 86.6%

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

   3. unpow-prod-down 87.6%

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

   4. inv-pow 87.6%

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

   5. inv-pow 87.6%

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

3. Applied egg-rr 87.6%

   \[\leadsto \left(\frac{\pi}{2} \cdot \color{blue}{\left(\frac{1}{b + a} \cdot \frac{1}{b - a}\right)}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
4. Step-by-step derivation

   1. associate-*r/ 87.6%

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

   2. *-rgt-identity 87.6%

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

   3. +-commutative 87.6%

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

5. Simplified 87.6%

   \[\leadsto \left(\frac{\pi}{2} \cdot \color{blue}{\frac{\frac{1}{a + b}}{b - a}}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
6. Step-by-step derivation

   1. add-cube-cbrt 87.2%

      \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{1}{a} - \frac{1}{b}} \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right)} \]

   2. inv-pow 87.2%

      \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{\color{blue}{{a}^{-1}} - \frac{1}{b}} \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \]

   3. inv-pow 87.2%

      \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - \color{blue}{{b}^{-1}}} \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \]

   4. inv-pow 87.2%

      \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{\color{blue}{{a}^{-1}} - \frac{1}{b}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \]

   5. inv-pow 87.2%

      \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{{a}^{-1} - \color{blue}{{b}^{-1}}}\right) \cdot \sqrt[3]{\frac{1}{a} - \frac{1}{b}}\right) \]

   6. inv-pow 87.2%

      \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{{a}^{-1} - {b}^{-1}}\right) \cdot \sqrt[3]{\color{blue}{{a}^{-1}} - \frac{1}{b}}\right) \]

   7. inv-pow 87.2%

      \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{{a}^{-1} - {b}^{-1}}\right) \cdot \sqrt[3]{{a}^{-1} - \color{blue}{{b}^{-1}}}\right) \]

7. Applied egg-rr 87.2%

   \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\frac{1}{a + b}}{b - a}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{{a}^{-1} - {b}^{-1}} \cdot \sqrt[3]{{a}^{-1} - {b}^{-1}}\right) \cdot \sqrt[3]{{a}^{-1} - {b}^{-1}}\right)} \]

8. Taylor expanded in a around inf 55.7%

   \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{{a}^{2} \cdot b}} \]
9. Step-by-step derivation

   1. unpow2 55.7%

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

   2. associate-*l* 61.9%

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

10. Simplified 61.9%

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

11. Final simplification 61.9%

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

Reproduce

herbie shell --seed 2023293 
(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))))