NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.8% → 96.9%

Time: 9.0s

Alternatives: 10

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: 78.8% 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: 96.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -2.05 \cdot 10^{+96}:\\ \;\;\;\;\frac{\pi}{\frac{a \cdot \left(a \cdot b\right)}{0.5}}\\ \mathbf{elif}\;a \leq -1.76 \cdot 10^{-264}:\\ \;\;\;\;\left(0.5 \cdot \left(\frac{\pi}{a + b} \cdot \frac{1}{b - a}\right)\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{b} \cdot \frac{\frac{1}{a}}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -2.05e+96)
   (/ PI (/ (* a (* a b)) 0.5))
   (if (<= a -1.76e-264)
     (* (* 0.5 (* (/ PI (+ a b)) (/ 1.0 (- b a)))) (+ (/ 1.0 a) (/ -1.0 b)))
     (* (/ (* PI 0.5) b) (/ (/ 1.0 a) b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -2.05e+96) {
		tmp = ((double) M_PI) / ((a * (a * b)) / 0.5);
	} else if (a <= -1.76e-264) {
		tmp = (0.5 * ((((double) M_PI) / (a + b)) * (1.0 / (b - a)))) * ((1.0 / a) + (-1.0 / b));
	} else {
		tmp = ((((double) M_PI) * 0.5) / b) * ((1.0 / a) / b);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -2.05e+96) {
		tmp = Math.PI / ((a * (a * b)) / 0.5);
	} else if (a <= -1.76e-264) {
		tmp = (0.5 * ((Math.PI / (a + b)) * (1.0 / (b - a)))) * ((1.0 / a) + (-1.0 / b));
	} else {
		tmp = ((Math.PI * 0.5) / b) * ((1.0 / a) / b);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -2.05e+96:
		tmp = math.pi / ((a * (a * b)) / 0.5)
	elif a <= -1.76e-264:
		tmp = (0.5 * ((math.pi / (a + b)) * (1.0 / (b - a)))) * ((1.0 / a) + (-1.0 / b))
	else:
		tmp = ((math.pi * 0.5) / b) * ((1.0 / a) / b)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -2.05e+96)
		tmp = Float64(pi / Float64(Float64(a * Float64(a * b)) / 0.5));
	elseif (a <= -1.76e-264)
		tmp = Float64(Float64(0.5 * Float64(Float64(pi / Float64(a + b)) * Float64(1.0 / Float64(b - a)))) * Float64(Float64(1.0 / a) + Float64(-1.0 / b)));
	else
		tmp = Float64(Float64(Float64(pi * 0.5) / b) * Float64(Float64(1.0 / a) / b));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.05e+96)
		tmp = pi / ((a * (a * b)) / 0.5);
	elseif (a <= -1.76e-264)
		tmp = (0.5 * ((pi / (a + b)) * (1.0 / (b - a)))) * ((1.0 / a) + (-1.0 / b));
	else
		tmp = ((pi * 0.5) / b) * ((1.0 / a) / b);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -2.05e+96], N[(Pi / N[(N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision] / 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -1.76e-264], N[(N[(0.5 * N[(N[(Pi / N[(a + b), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / a), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi * 0.5), $MachinePrecision] / b), $MachinePrecision] * N[(N[(1.0 / a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.04999999999999999e96

   1. Initial program 68.6%

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

      1. times-frac 68.6%

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

      2. *-commutative 68.6%

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

      3. times-frac 68.6%

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

      4. difference-of-squares 85.0%

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

      5. associate-/r* 86.2%

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

      6. metadata-eval 86.2%

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

      7. sub-neg 86.2%

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

      8. distribute-neg-frac 86.2%

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

      9. metadata-eval 86.2%

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

   3. Simplified 86.2%

      \[\leadsto \color{blue}{\left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)} \]
   4. Step-by-step derivation

      1. clear-num 86.3%

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

      2. inv-pow 86.3%

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

   5. Applied egg-rr 86.3%

      \[\leadsto \left(\frac{\color{blue}{{\left(\frac{b + a}{\pi}\right)}^{-1}}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right) \]
   6. Step-by-step derivation

      1. unpow-1 86.3%

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

      2. +-commutative 86.3%

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

   7. Simplified 86.3%

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

   8. Taylor expanded in a around inf 85.1%

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

      1. *-commutative 85.1%

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

      2. associate-/r/ 85.1%

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

      3. unpow2 85.1%

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

      4. associate-*l* 98.3%

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

   10. Simplified 98.3%

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

   if -2.04999999999999999e96 < a < -1.76000000000000008e-264

   1. Initial program 82.5%

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

      1. times-frac 82.5%

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

      2. *-commutative 82.5%

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

      3. times-frac 82.5%

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

      4. difference-of-squares 88.8%

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

      5. associate-/r* 90.4%

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

      6. metadata-eval 90.4%

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

      7. sub-neg 90.4%

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

      8. distribute-neg-frac 90.4%

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

      9. metadata-eval 90.4%

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

   3. Simplified 90.4%

      \[\leadsto \color{blue}{\left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)} \]
   4. Step-by-step derivation

      1. div-inv 90.4%

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

   5. Applied egg-rr 90.4%

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

   if -1.76000000000000008e-264 < a

   1. Initial program 79.0%

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

      1. associate-*r/ 79.1%

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

      2. *-rgt-identity 79.1%

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

      3. sub-neg 79.1%

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

      4. distribute-neg-frac 79.1%

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

      5. metadata-eval 79.1%

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

   3. Simplified 79.1%

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

      1. associate-*l/ 79.1%

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

      2. div-inv 79.1%

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

      3. metadata-eval 79.1%

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

   5. Applied egg-rr 79.1%

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

   6. Taylor expanded in b around inf 55.2%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)}{\color{blue}{{b}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 55.2%

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

   8. Simplified 55.2%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)}{\color{blue}{b \cdot b}} \]
   9. Step-by-step derivation

      1. times-frac 59.8%

         \[\leadsto \color{blue}{\frac{\pi \cdot 0.5}{b} \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b}} \]

   10. Applied egg-rr 59.8%

       \[\leadsto \color{blue}{\frac{\pi \cdot 0.5}{b} \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b}} \]

   11. Taylor expanded in a around 0 69.2%

       \[\leadsto \frac{\pi \cdot 0.5}{b} \cdot \frac{\color{blue}{\frac{1}{a}}}{b} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.05 \cdot 10^{+96}:\\ \;\;\;\;\frac{\pi}{\frac{a \cdot \left(a \cdot b\right)}{0.5}}\\ \mathbf{elif}\;a \leq -1.76 \cdot 10^{-264}:\\ \;\;\;\;\left(0.5 \cdot \left(\frac{\pi}{a + b} \cdot \frac{1}{b - a}\right)\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{b} \cdot \frac{\frac{1}{a}}{b}\\ \end{array} \]

Alternative 2: 96.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -6.5 \cdot 10^{+95}:\\ \;\;\;\;\frac{\pi}{\frac{a \cdot \left(a \cdot b\right)}{0.5}}\\ \mathbf{elif}\;a \leq -1.76 \cdot 10^{-264}:\\ \;\;\;\;\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \left(0.5 \cdot \frac{\frac{\pi}{a + b}}{b - a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{b} \cdot \frac{\frac{1}{a}}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -6.5e+95)
   (/ PI (/ (* a (* a b)) 0.5))
   (if (<= a -1.76e-264)
     (* (+ (/ 1.0 a) (/ -1.0 b)) (* 0.5 (/ (/ PI (+ a b)) (- b a))))
     (* (/ (* PI 0.5) b) (/ (/ 1.0 a) b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -6.5e+95) {
		tmp = ((double) M_PI) / ((a * (a * b)) / 0.5);
	} else if (a <= -1.76e-264) {
		tmp = ((1.0 / a) + (-1.0 / b)) * (0.5 * ((((double) M_PI) / (a + b)) / (b - a)));
	} else {
		tmp = ((((double) M_PI) * 0.5) / b) * ((1.0 / a) / b);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -6.5e+95) {
		tmp = Math.PI / ((a * (a * b)) / 0.5);
	} else if (a <= -1.76e-264) {
		tmp = ((1.0 / a) + (-1.0 / b)) * (0.5 * ((Math.PI / (a + b)) / (b - a)));
	} else {
		tmp = ((Math.PI * 0.5) / b) * ((1.0 / a) / b);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -6.5e+95:
		tmp = math.pi / ((a * (a * b)) / 0.5)
	elif a <= -1.76e-264:
		tmp = ((1.0 / a) + (-1.0 / b)) * (0.5 * ((math.pi / (a + b)) / (b - a)))
	else:
		tmp = ((math.pi * 0.5) / b) * ((1.0 / a) / b)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -6.5e+95)
		tmp = Float64(pi / Float64(Float64(a * Float64(a * b)) / 0.5));
	elseif (a <= -1.76e-264)
		tmp = Float64(Float64(Float64(1.0 / a) + Float64(-1.0 / b)) * Float64(0.5 * Float64(Float64(pi / Float64(a + b)) / Float64(b - a))));
	else
		tmp = Float64(Float64(Float64(pi * 0.5) / b) * Float64(Float64(1.0 / a) / b));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -6.5e+95)
		tmp = pi / ((a * (a * b)) / 0.5);
	elseif (a <= -1.76e-264)
		tmp = ((1.0 / a) + (-1.0 / b)) * (0.5 * ((pi / (a + b)) / (b - a)));
	else
		tmp = ((pi * 0.5) / b) * ((1.0 / a) / b);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -6.5e+95], N[(Pi / N[(N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision] / 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -1.76e-264], N[(N[(N[(1.0 / a), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[(N[(Pi / N[(a + b), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi * 0.5), $MachinePrecision] / b), $MachinePrecision] * N[(N[(1.0 / a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -6.5e95

   1. Initial program 69.2%

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

      1. times-frac 69.2%

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

      2. *-commutative 69.2%

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

      3. times-frac 69.2%

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

      4. difference-of-squares 85.2%

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

      5. associate-/r* 86.5%

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

      6. metadata-eval 86.5%

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

      7. sub-neg 86.5%

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

      8. distribute-neg-frac 86.5%

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

      9. metadata-eval 86.5%

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

   3. Simplified 86.5%

      \[\leadsto \color{blue}{\left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)} \]
   4. Step-by-step derivation

      1. clear-num 86.6%

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

      2. inv-pow 86.6%

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

   5. Applied egg-rr 86.6%

      \[\leadsto \left(\frac{\color{blue}{{\left(\frac{b + a}{\pi}\right)}^{-1}}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right) \]
   6. Step-by-step derivation

      1. unpow-1 86.6%

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

      2. +-commutative 86.6%

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

   7. Simplified 86.6%

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

   8. Taylor expanded in a around inf 85.4%

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

      1. *-commutative 85.4%

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

      2. associate-/r/ 85.4%

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

      3. unpow2 85.4%

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

      4. associate-*l* 98.3%

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

   10. Simplified 98.3%

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

   if -6.5e95 < a < -1.76000000000000008e-264

   1. Initial program 82.3%

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

      1. times-frac 82.3%

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

      2. *-commutative 82.3%

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

      3. times-frac 82.3%

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

      4. difference-of-squares 88.7%

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

      5. associate-/r* 90.3%

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

      6. metadata-eval 90.3%

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

      7. sub-neg 90.3%

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

      8. distribute-neg-frac 90.3%

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

      9. metadata-eval 90.3%

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

   3. Simplified 90.3%

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

   if -1.76000000000000008e-264 < a

   1. Initial program 79.0%

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

      1. associate-*r/ 79.1%

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

      2. *-rgt-identity 79.1%

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

      3. sub-neg 79.1%

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

      4. distribute-neg-frac 79.1%

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

      5. metadata-eval 79.1%

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

   3. Simplified 79.1%

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

      1. associate-*l/ 79.1%

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

      2. div-inv 79.1%

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

      3. metadata-eval 79.1%

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

   5. Applied egg-rr 79.1%

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

   6. Taylor expanded in b around inf 55.2%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)}{\color{blue}{{b}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 55.2%

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

   8. Simplified 55.2%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)}{\color{blue}{b \cdot b}} \]
   9. Step-by-step derivation

      1. times-frac 59.8%

         \[\leadsto \color{blue}{\frac{\pi \cdot 0.5}{b} \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b}} \]

   10. Applied egg-rr 59.8%

       \[\leadsto \color{blue}{\frac{\pi \cdot 0.5}{b} \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b}} \]

   11. Taylor expanded in a around 0 69.2%

       \[\leadsto \frac{\pi \cdot 0.5}{b} \cdot \frac{\color{blue}{\frac{1}{a}}}{b} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.5 \cdot 10^{+95}:\\ \;\;\;\;\frac{\pi}{\frac{a \cdot \left(a \cdot b\right)}{0.5}}\\ \mathbf{elif}\;a \leq -1.76 \cdot 10^{-264}:\\ \;\;\;\;\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \left(0.5 \cdot \frac{\frac{\pi}{a + b}}{b - a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{b} \cdot \frac{\frac{1}{a}}{b}\\ \end{array} \]

Alternative 3: 96.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 2.8 \cdot 10^{-104}:\\ \;\;\;\;\frac{\pi}{\frac{a \cdot \left(a \cdot b\right)}{0.5}}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+153}:\\ \;\;\;\;\frac{\pi}{b \cdot b - a \cdot a} \cdot \left(\frac{0.5}{a} + \frac{-0.5}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{b} \cdot \frac{\frac{1}{a}}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 2.8e-104)
   (/ PI (/ (* a (* a b)) 0.5))
   (if (<= b 2e+153)
     (* (/ PI (- (* b b) (* a a))) (+ (/ 0.5 a) (/ -0.5 b)))
     (* (/ (* PI 0.5) b) (/ (/ 1.0 a) b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 2.8e-104) {
		tmp = ((double) M_PI) / ((a * (a * b)) / 0.5);
	} else if (b <= 2e+153) {
		tmp = (((double) M_PI) / ((b * b) - (a * a))) * ((0.5 / a) + (-0.5 / b));
	} else {
		tmp = ((((double) M_PI) * 0.5) / b) * ((1.0 / a) / b);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 2.8e-104) {
		tmp = Math.PI / ((a * (a * b)) / 0.5);
	} else if (b <= 2e+153) {
		tmp = (Math.PI / ((b * b) - (a * a))) * ((0.5 / a) + (-0.5 / b));
	} else {
		tmp = ((Math.PI * 0.5) / b) * ((1.0 / a) / b);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 2.8e-104:
		tmp = math.pi / ((a * (a * b)) / 0.5)
	elif b <= 2e+153:
		tmp = (math.pi / ((b * b) - (a * a))) * ((0.5 / a) + (-0.5 / b))
	else:
		tmp = ((math.pi * 0.5) / b) * ((1.0 / a) / b)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 2.8e-104)
		tmp = Float64(pi / Float64(Float64(a * Float64(a * b)) / 0.5));
	elseif (b <= 2e+153)
		tmp = Float64(Float64(pi / Float64(Float64(b * b) - Float64(a * a))) * Float64(Float64(0.5 / a) + Float64(-0.5 / b)));
	else
		tmp = Float64(Float64(Float64(pi * 0.5) / b) * Float64(Float64(1.0 / a) / b));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2.8e-104)
		tmp = pi / ((a * (a * b)) / 0.5);
	elseif (b <= 2e+153)
		tmp = (pi / ((b * b) - (a * a))) * ((0.5 / a) + (-0.5 / b));
	else
		tmp = ((pi * 0.5) / b) * ((1.0 / 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[b, 2.8e-104], N[(Pi / N[(N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision] / 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2e+153], N[(N[(Pi / N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 / a), $MachinePrecision] + N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi * 0.5), $MachinePrecision] / b), $MachinePrecision] * N[(N[(1.0 / a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 2.8e-104

   1. Initial program 75.6%

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

      1. times-frac 75.6%

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

      2. *-commutative 75.6%

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

      3. times-frac 75.6%

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

      4. difference-of-squares 84.0%

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

      5. associate-/r* 85.6%

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

      6. metadata-eval 85.6%

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

      7. sub-neg 85.6%

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

      8. distribute-neg-frac 85.6%

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

      9. metadata-eval 85.6%

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

   3. Simplified 85.6%

      \[\leadsto \color{blue}{\left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)} \]
   4. Step-by-step derivation

      1. clear-num 85.6%

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

      2. inv-pow 85.6%

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

   5. Applied egg-rr 85.6%

      \[\leadsto \left(\frac{\color{blue}{{\left(\frac{b + a}{\pi}\right)}^{-1}}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right) \]
   6. Step-by-step derivation

      1. unpow-1 85.6%

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

      2. +-commutative 85.6%

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

   7. Simplified 85.6%

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

   8. Taylor expanded in a around inf 60.9%

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

      1. *-commutative 60.9%

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

      2. associate-/r/ 60.9%

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

      3. unpow2 60.9%

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

      4. associate-*l* 69.3%

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

   10. Simplified 69.3%

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

   if 2.8e-104 < b < 2e153

   1. Initial program 99.3%

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

      1. times-frac 99.5%

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

      2. *-commutative 99.5%

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

      3. times-frac 99.5%

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

      4. difference-of-squares 99.5%

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

      5. associate-/r* 99.4%

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

      6. metadata-eval 99.4%

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

      7. sub-neg 99.4%

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

      8. distribute-neg-frac 99.4%

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

      9. metadata-eval 99.4%

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

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)} \]
   4. Step-by-step derivation

      1. distribute-lft-in 99.4%

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

      2. associate-/l/ 99.5%

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

      3. associate-/l/ 99.5%

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

   5. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\left(\frac{\pi}{\left(b - a\right) \cdot \left(b + a\right)} \cdot 0.5\right) \cdot \frac{1}{a} + \left(\frac{\pi}{\left(b - a\right) \cdot \left(b + a\right)} \cdot 0.5\right) \cdot \frac{-1}{b}} \]
   6. Step-by-step derivation

      1. distribute-lft-out 99.5%

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

      2. associate-*r* 99.5%

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

      3. associate-*l/ 99.5%

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

      4. *-commutative 99.5%

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

      5. difference-of-squares 99.5%

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

      6. associate-*l/ 99.5%

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

      7. distribute-lft-in 99.5%

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

      8. associate-*r/ 99.5%

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

      9. metadata-eval 99.5%

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

      10. associate-*r/ 99.5%

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

      11. metadata-eval 99.5%

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

   7. Simplified 99.5%

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

   if 2e153 < b

   1. Initial program 47.8%

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

      1. associate-*r/ 47.8%

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

      2. *-rgt-identity 47.8%

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

      3. sub-neg 47.8%

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

      4. distribute-neg-frac 47.8%

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

      5. metadata-eval 47.8%

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

   3. Simplified 47.8%

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

      1. associate-*l/ 47.8%

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

      2. div-inv 47.8%

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

      3. metadata-eval 47.8%

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

   5. Applied egg-rr 47.8%

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

   6. Taylor expanded in b around inf 89.2%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)}{\color{blue}{{b}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 89.2%

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

   8. Simplified 89.2%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)}{\color{blue}{b \cdot b}} \]
   9. Step-by-step derivation

      1. times-frac 99.9%

         \[\leadsto \color{blue}{\frac{\pi \cdot 0.5}{b} \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b}} \]

   10. Applied egg-rr 99.9%

       \[\leadsto \color{blue}{\frac{\pi \cdot 0.5}{b} \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b}} \]

   11. Taylor expanded in a around 0 99.9%

       \[\leadsto \frac{\pi \cdot 0.5}{b} \cdot \frac{\color{blue}{\frac{1}{a}}}{b} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.8 \cdot 10^{-104}:\\ \;\;\;\;\frac{\pi}{\frac{a \cdot \left(a \cdot b\right)}{0.5}}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+153}:\\ \;\;\;\;\frac{\pi}{b \cdot b - a \cdot a} \cdot \left(\frac{0.5}{a} + \frac{-0.5}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{b} \cdot \frac{\frac{1}{a}}{b}\\ \end{array} \]

Alternative 4: 92.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -3 \cdot 10^{+94}:\\ \;\;\;\;\frac{\pi}{\frac{a \cdot \left(a \cdot b\right)}{0.5}}\\ \mathbf{elif}\;a \leq -2.8 \cdot 10^{-116}:\\ \;\;\;\;\frac{\frac{-0.5}{a + b}}{b - a} \cdot \frac{\pi}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{b} \cdot \frac{\frac{1}{a}}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -3e+94)
   (/ PI (/ (* a (* a b)) 0.5))
   (if (<= a -2.8e-116)
     (* (/ (/ -0.5 (+ a b)) (- b a)) (/ PI b))
     (* (/ (* PI 0.5) b) (/ (/ 1.0 a) b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -3e+94) {
		tmp = ((double) M_PI) / ((a * (a * b)) / 0.5);
	} else if (a <= -2.8e-116) {
		tmp = ((-0.5 / (a + b)) / (b - a)) * (((double) M_PI) / b);
	} else {
		tmp = ((((double) M_PI) * 0.5) / b) * ((1.0 / a) / b);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -3e+94) {
		tmp = Math.PI / ((a * (a * b)) / 0.5);
	} else if (a <= -2.8e-116) {
		tmp = ((-0.5 / (a + b)) / (b - a)) * (Math.PI / b);
	} else {
		tmp = ((Math.PI * 0.5) / b) * ((1.0 / a) / b);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -3e+94:
		tmp = math.pi / ((a * (a * b)) / 0.5)
	elif a <= -2.8e-116:
		tmp = ((-0.5 / (a + b)) / (b - a)) * (math.pi / b)
	else:
		tmp = ((math.pi * 0.5) / b) * ((1.0 / a) / b)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -3e+94)
		tmp = Float64(pi / Float64(Float64(a * Float64(a * b)) / 0.5));
	elseif (a <= -2.8e-116)
		tmp = Float64(Float64(Float64(-0.5 / Float64(a + b)) / Float64(b - a)) * Float64(pi / b));
	else
		tmp = Float64(Float64(Float64(pi * 0.5) / b) * Float64(Float64(1.0 / a) / b));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -3e+94)
		tmp = pi / ((a * (a * b)) / 0.5);
	elseif (a <= -2.8e-116)
		tmp = ((-0.5 / (a + b)) / (b - a)) * (pi / b);
	else
		tmp = ((pi * 0.5) / b) * ((1.0 / 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, -3e+94], N[(Pi / N[(N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision] / 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -2.8e-116], N[(N[(N[(-0.5 / N[(a + b), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(Pi / b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi * 0.5), $MachinePrecision] / b), $MachinePrecision] * N[(N[(1.0 / a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.0000000000000001e94

   1. Initial program 70.4%

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

      1. times-frac 70.4%

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

      2. *-commutative 70.4%

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

      3. times-frac 70.4%

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

      4. difference-of-squares 85.8%

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

      5. associate-/r* 87.0%

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

      6. metadata-eval 87.0%

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

      7. sub-neg 87.0%

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

      8. distribute-neg-frac 87.0%

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

      9. metadata-eval 87.0%

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

   3. Simplified 87.0%

      \[\leadsto \color{blue}{\left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)} \]
   4. Step-by-step derivation

      1. clear-num 87.1%

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

      2. inv-pow 87.1%

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

   5. Applied egg-rr 87.1%

      \[\leadsto \left(\frac{\color{blue}{{\left(\frac{b + a}{\pi}\right)}^{-1}}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right) \]
   6. Step-by-step derivation

      1. unpow-1 87.1%

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

      2. +-commutative 87.1%

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

   7. Simplified 87.1%

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

   8. Taylor expanded in a around inf 85.9%

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

      1. *-commutative 85.9%

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

      2. associate-/r/ 85.9%

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

      3. unpow2 85.9%

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

      4. associate-*l* 98.4%

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

   10. Simplified 98.4%

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

   if -3.0000000000000001e94 < a < -2.7999999999999999e-116

   1. Initial program 97.2%

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

      1. associate-*r/ 97.2%

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

      2. *-rgt-identity 97.2%

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

      3. sub-neg 97.2%

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

      4. distribute-neg-frac 97.2%

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

      5. metadata-eval 97.2%

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

   3. Simplified 97.2%

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

      1. associate-*l/ 97.2%

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

      2. div-inv 97.2%

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

      3. metadata-eval 97.2%

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

   5. Applied egg-rr 97.2%

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

   6. Taylor expanded in a around inf 72.9%

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

      1. expm1-log1p-u 52.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.5 \cdot \frac{\pi}{b}}{b \cdot b - a \cdot a}\right)\right)} \]

      2. expm1-udef 44.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-0.5 \cdot \frac{\pi}{b}}{b \cdot b - a \cdot a}\right)} - 1} \]

      3. associate-/l* 44.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{-0.5}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{b}}}}\right)} - 1 \]

   8. Applied egg-rr 44.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-0.5}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{b}}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 52.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.5}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{b}}}\right)\right)} \]

      2. expm1-log1p 73.0%

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

      3. associate-/r/ 73.0%

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

      4. difference-of-squares 73.0%

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

      5. associate-/r* 73.0%

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

   10. Simplified 73.0%

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

   if -2.7999999999999999e-116 < a

   1. Initial program 75.3%

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

      1. associate-*r/ 75.4%

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

      2. *-rgt-identity 75.4%

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

      3. sub-neg 75.4%

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

      4. distribute-neg-frac 75.4%

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

      5. metadata-eval 75.4%

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

   3. Simplified 75.4%

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

      1. associate-*l/ 75.4%

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

      2. div-inv 75.4%

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

      3. metadata-eval 75.4%

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

   5. Applied egg-rr 75.4%

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

   6. Taylor expanded in b around inf 55.7%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)}{\color{blue}{{b}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 55.7%

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

   8. Simplified 55.7%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)}{\color{blue}{b \cdot b}} \]
   9. Step-by-step derivation

      1. times-frac 64.0%

         \[\leadsto \color{blue}{\frac{\pi \cdot 0.5}{b} \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b}} \]

   10. Applied egg-rr 64.0%

       \[\leadsto \color{blue}{\frac{\pi \cdot 0.5}{b} \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b}} \]

   11. Taylor expanded in a around 0 72.7%

       \[\leadsto \frac{\pi \cdot 0.5}{b} \cdot \frac{\color{blue}{\frac{1}{a}}}{b} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3 \cdot 10^{+94}:\\ \;\;\;\;\frac{\pi}{\frac{a \cdot \left(a \cdot b\right)}{0.5}}\\ \mathbf{elif}\;a \leq -2.8 \cdot 10^{-116}:\\ \;\;\;\;\frac{\frac{-0.5}{a + b}}{b - a} \cdot \frac{\pi}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{b} \cdot \frac{\frac{1}{a}}{b}\\ \end{array} \]

Alternative 5: 91.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 7.5 \cdot 10^{-67}:\\ \;\;\;\;\frac{\pi}{\frac{a \cdot \left(a \cdot b\right)}{0.5}}\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{+133}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{b \cdot b - a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{b} \cdot \frac{\frac{1}{a}}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 7.5e-67)
   (/ PI (/ (* a (* a b)) 0.5))
   (if (<= b 7.2e+133)
     (/ (* 0.5 (/ PI a)) (- (* b b) (* a a)))
     (* (/ (* PI 0.5) b) (/ (/ 1.0 a) b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 7.5e-67) {
		tmp = ((double) M_PI) / ((a * (a * b)) / 0.5);
	} else if (b <= 7.2e+133) {
		tmp = (0.5 * (((double) M_PI) / a)) / ((b * b) - (a * a));
	} else {
		tmp = ((((double) M_PI) * 0.5) / b) * ((1.0 / a) / b);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 7.5e-67) {
		tmp = Math.PI / ((a * (a * b)) / 0.5);
	} else if (b <= 7.2e+133) {
		tmp = (0.5 * (Math.PI / a)) / ((b * b) - (a * a));
	} else {
		tmp = ((Math.PI * 0.5) / b) * ((1.0 / a) / b);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 7.5e-67:
		tmp = math.pi / ((a * (a * b)) / 0.5)
	elif b <= 7.2e+133:
		tmp = (0.5 * (math.pi / a)) / ((b * b) - (a * a))
	else:
		tmp = ((math.pi * 0.5) / b) * ((1.0 / a) / b)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 7.5e-67)
		tmp = Float64(pi / Float64(Float64(a * Float64(a * b)) / 0.5));
	elseif (b <= 7.2e+133)
		tmp = Float64(Float64(0.5 * Float64(pi / a)) / Float64(Float64(b * b) - Float64(a * a)));
	else
		tmp = Float64(Float64(Float64(pi * 0.5) / b) * Float64(Float64(1.0 / a) / b));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 7.5e-67)
		tmp = pi / ((a * (a * b)) / 0.5);
	elseif (b <= 7.2e+133)
		tmp = (0.5 * (pi / a)) / ((b * b) - (a * a));
	else
		tmp = ((pi * 0.5) / b) * ((1.0 / 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[b, 7.5e-67], N[(Pi / N[(N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision] / 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7.2e+133], N[(N[(0.5 * N[(Pi / a), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi * 0.5), $MachinePrecision] / b), $MachinePrecision] * N[(N[(1.0 / a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 7.5000000000000005e-67

   1. Initial program 76.4%

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

      1. times-frac 76.4%

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

      2. *-commutative 76.4%

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

      3. times-frac 76.4%

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

      4. difference-of-squares 84.6%

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

      5. associate-/r* 86.1%

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

      6. metadata-eval 86.1%

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

      7. sub-neg 86.1%

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

      8. distribute-neg-frac 86.1%

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

      9. metadata-eval 86.1%

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

   3. Simplified 86.1%

      \[\leadsto \color{blue}{\left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)} \]
   4. Step-by-step derivation

      1. clear-num 86.0%

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

      2. inv-pow 86.0%

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

   5. Applied egg-rr 86.0%

      \[\leadsto \left(\frac{\color{blue}{{\left(\frac{b + a}{\pi}\right)}^{-1}}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right) \]
   6. Step-by-step derivation

      1. unpow-1 86.0%

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

      2. +-commutative 86.0%

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

   7. Simplified 86.0%

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

   8. Taylor expanded in a around inf 61.1%

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

      1. *-commutative 61.1%

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

      2. associate-/r/ 61.1%

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

      3. unpow2 61.1%

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

      4. associate-*l* 69.3%

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

   10. Simplified 69.3%

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

   if 7.5000000000000005e-67 < b < 7.19999999999999956e133

   1. Initial program 99.3%

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

      1. associate-*r/ 99.5%

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

      2. *-rgt-identity 99.5%

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

      3. sub-neg 99.5%

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

      4. distribute-neg-frac 99.5%

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

      5. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

      1. associate-*l/ 99.6%

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

      2. div-inv 99.6%

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

      3. metadata-eval 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in a around 0 88.4%

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

   if 7.19999999999999956e133 < b

   1. Initial program 55.4%

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

      1. associate-*r/ 55.5%

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

      2. *-rgt-identity 55.5%

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

      3. sub-neg 55.5%

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

      4. distribute-neg-frac 55.5%

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

      5. metadata-eval 55.5%

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

   3. Simplified 55.5%

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

      1. associate-*l/ 55.3%

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

      2. div-inv 55.3%

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

      3. metadata-eval 55.3%

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

   5. Applied egg-rr 55.3%

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

   6. Taylor expanded in b around inf 90.6%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)}{\color{blue}{{b}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 90.6%

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

   8. Simplified 90.6%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)}{\color{blue}{b \cdot b}} \]
   9. Step-by-step derivation

      1. times-frac 99.8%

         \[\leadsto \color{blue}{\frac{\pi \cdot 0.5}{b} \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b}} \]

   10. Applied egg-rr 99.8%

       \[\leadsto \color{blue}{\frac{\pi \cdot 0.5}{b} \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b}} \]

   11. Taylor expanded in a around 0 99.8%

       \[\leadsto \frac{\pi \cdot 0.5}{b} \cdot \frac{\color{blue}{\frac{1}{a}}}{b} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.5 \cdot 10^{-67}:\\ \;\;\;\;\frac{\pi}{\frac{a \cdot \left(a \cdot b\right)}{0.5}}\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{+133}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{b \cdot b - a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{b} \cdot \frac{\frac{1}{a}}{b}\\ \end{array} \]

Alternative 6: 89.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 1.15 \cdot 10^{-45}:\\ \;\;\;\;\frac{\pi}{\frac{a \cdot \left(a \cdot b\right)}{0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{b} \cdot \frac{\frac{1}{a}}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 1.15e-45)
   (/ PI (/ (* a (* a b)) 0.5))
   (* (/ (* PI 0.5) b) (/ (/ 1.0 a) b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 1.15e-45) {
		tmp = ((double) M_PI) / ((a * (a * b)) / 0.5);
	} else {
		tmp = ((((double) M_PI) * 0.5) / b) * ((1.0 / a) / b);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 1.15e-45) {
		tmp = Math.PI / ((a * (a * b)) / 0.5);
	} else {
		tmp = ((Math.PI * 0.5) / b) * ((1.0 / a) / b);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 1.15e-45:
		tmp = math.pi / ((a * (a * b)) / 0.5)
	else:
		tmp = ((math.pi * 0.5) / b) * ((1.0 / a) / b)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 1.15e-45)
		tmp = Float64(pi / Float64(Float64(a * Float64(a * b)) / 0.5));
	else
		tmp = Float64(Float64(Float64(pi * 0.5) / b) * Float64(Float64(1.0 / a) / b));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.15e-45)
		tmp = pi / ((a * (a * b)) / 0.5);
	else
		tmp = ((pi * 0.5) / b) * ((1.0 / 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[b, 1.15e-45], N[(Pi / N[(N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision] / 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi * 0.5), $MachinePrecision] / b), $MachinePrecision] * N[(N[(1.0 / a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.14999999999999996e-45

   1. Initial program 76.9%

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

      1. times-frac 76.9%

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

      2. *-commutative 76.9%

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

      3. times-frac 76.9%

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

      4. difference-of-squares 84.9%

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

      5. associate-/r* 86.3%

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

      6. metadata-eval 86.3%

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

      7. sub-neg 86.3%

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

      8. distribute-neg-frac 86.3%

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

      9. metadata-eval 86.3%

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

   3. Simplified 86.3%

      \[\leadsto \color{blue}{\left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)} \]
   4. Step-by-step derivation

      1. clear-num 86.3%

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

      2. inv-pow 86.3%

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

   5. Applied egg-rr 86.3%

      \[\leadsto \left(\frac{\color{blue}{{\left(\frac{b + a}{\pi}\right)}^{-1}}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right) \]
   6. Step-by-step derivation

      1. unpow-1 86.3%

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

      2. +-commutative 86.3%

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

   7. Simplified 86.3%

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

   8. Taylor expanded in a around inf 61.5%

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

      1. *-commutative 61.5%

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

      2. associate-/r/ 61.5%

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

      3. unpow2 61.5%

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

      4. associate-*l* 69.4%

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

   10. Simplified 69.4%

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

   if 1.14999999999999996e-45 < b

   1. Initial program 80.7%

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

      1. associate-*r/ 80.8%

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

      2. *-rgt-identity 80.8%

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

      3. sub-neg 80.8%

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

      4. distribute-neg-frac 80.8%

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

      5. metadata-eval 80.8%

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

   3. Simplified 80.8%

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

      1. associate-*l/ 80.8%

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

      2. div-inv 80.8%

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

      3. metadata-eval 80.8%

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

   5. Applied egg-rr 80.8%

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

   6. Taylor expanded in b around inf 72.3%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)}{\color{blue}{{b}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 72.3%

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

   8. Simplified 72.3%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)}{\color{blue}{b \cdot b}} \]
   9. Step-by-step derivation

      1. times-frac 76.2%

         \[\leadsto \color{blue}{\frac{\pi \cdot 0.5}{b} \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b}} \]

   10. Applied egg-rr 76.2%

       \[\leadsto \color{blue}{\frac{\pi \cdot 0.5}{b} \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b}} \]

   11. Taylor expanded in a around 0 82.5%

       \[\leadsto \frac{\pi \cdot 0.5}{b} \cdot \frac{\color{blue}{\frac{1}{a}}}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.15 \cdot 10^{-45}:\\ \;\;\;\;\frac{\pi}{\frac{a \cdot \left(a \cdot b\right)}{0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{b} \cdot \frac{\frac{1}{a}}{b}\\ \end{array} \]

Alternative 7: 78.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 1.15 \cdot 10^{-45}:\\ \;\;\;\;\frac{\pi}{a \cdot a} \cdot \frac{0.5}{b}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 1.15e-45)
   (* (/ PI (* a a)) (/ 0.5 b))
   (* 0.5 (/ PI (* a (* b b))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 1.15e-45) {
		tmp = (((double) M_PI) / (a * a)) * (0.5 / 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 (b <= 1.15e-45) {
		tmp = (Math.PI / (a * a)) * (0.5 / 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 b <= 1.15e-45:
		tmp = (math.pi / (a * a)) * (0.5 / 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 (b <= 1.15e-45)
		tmp = Float64(Float64(pi / Float64(a * a)) * Float64(0.5 / b));
	else
		tmp = Float64(0.5 * Float64(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 (b <= 1.15e-45)
		tmp = (pi / (a * a)) * (0.5 / 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[b, 1.15e-45], N[(N[(Pi / N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(0.5 / b), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(Pi / N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.14999999999999996e-45

   1. Initial program 76.9%

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

      1. times-frac 76.9%

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

      2. *-commutative 76.9%

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

      3. times-frac 76.9%

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

      4. difference-of-squares 84.9%

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

      5. associate-/r* 86.3%

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

      6. metadata-eval 86.3%

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

      7. sub-neg 86.3%

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

      8. distribute-neg-frac 86.3%

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

      9. metadata-eval 86.3%

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

   3. Simplified 86.3%

      \[\leadsto \color{blue}{\left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)} \]
   4. Step-by-step derivation

      1. div-inv 86.4%

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

   5. Applied egg-rr 86.4%

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

   6. Taylor expanded in b around 0 61.5%

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

      1. associate-*r/ 61.5%

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

      2. *-commutative 61.5%

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

      3. times-frac 61.4%

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

      4. unpow2 61.4%

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

   8. Simplified 61.4%

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

   if 1.14999999999999996e-45 < b

   1. Initial program 80.7%

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

      1. *-commutative 80.7%

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

      2. associate-/r/ 80.7%

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

      3. associate-*l/ 80.7%

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

      4. *-commutative 80.7%

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

      5. associate-/r/ 80.7%

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

      6. times-frac 80.7%

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

   3. Simplified 80.8%

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

   4. Taylor expanded in b around inf 78.4%

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

      1. unpow2 78.4%

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

   6. Simplified 78.4%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.15 \cdot 10^{-45}:\\ \;\;\;\;\frac{\pi}{a \cdot a} \cdot \frac{0.5}{b}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot b\right)}\\ \end{array} \]

Alternative 8: 84.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 1.15 \cdot 10^{-45}:\\ \;\;\;\;\frac{\pi}{a \cdot a} \cdot \frac{0.5}{b}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 1.15e-45)
   (* (/ PI (* a a)) (/ 0.5 b))
   (* 0.5 (/ PI (* b (* a b))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 1.15e-45) {
		tmp = (((double) M_PI) / (a * a)) * (0.5 / b);
	} else {
		tmp = 0.5 * (((double) M_PI) / (b * (a * b)));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 1.15e-45) {
		tmp = (Math.PI / (a * a)) * (0.5 / b);
	} else {
		tmp = 0.5 * (Math.PI / (b * (a * b)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 1.15e-45:
		tmp = (math.pi / (a * a)) * (0.5 / b)
	else:
		tmp = 0.5 * (math.pi / (b * (a * b)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 1.15e-45)
		tmp = Float64(Float64(pi / Float64(a * a)) * Float64(0.5 / b));
	else
		tmp = Float64(0.5 * Float64(pi / Float64(b * Float64(a * b))));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.15e-45)
		tmp = (pi / (a * a)) * (0.5 / b);
	else
		tmp = 0.5 * (pi / (b * (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[b, 1.15e-45], N[(N[(Pi / N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(0.5 / b), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(Pi / N[(b * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.14999999999999996e-45

   1. Initial program 76.9%

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

      1. times-frac 76.9%

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

      2. *-commutative 76.9%

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

      3. times-frac 76.9%

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

      4. difference-of-squares 84.9%

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

      5. associate-/r* 86.3%

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

      6. metadata-eval 86.3%

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

      7. sub-neg 86.3%

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

      8. distribute-neg-frac 86.3%

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

      9. metadata-eval 86.3%

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

   3. Simplified 86.3%

      \[\leadsto \color{blue}{\left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)} \]
   4. Step-by-step derivation

      1. div-inv 86.4%

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

   5. Applied egg-rr 86.4%

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

   6. Taylor expanded in b around 0 61.5%

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

      1. associate-*r/ 61.5%

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

      2. *-commutative 61.5%

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

      3. times-frac 61.4%

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

      4. unpow2 61.4%

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

   8. Simplified 61.4%

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

   if 1.14999999999999996e-45 < b

   1. Initial program 80.7%

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

      1. *-commutative 80.7%

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

      2. associate-/r/ 80.7%

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

      3. associate-*l/ 80.7%

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

      4. *-commutative 80.7%

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

      5. associate-/r/ 80.7%

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

      6. times-frac 80.7%

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

   3. Simplified 80.8%

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

   4. Taylor expanded in b around inf 78.4%

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

      1. unpow2 78.4%

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

   6. Simplified 78.4%

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

   7. Taylor expanded in a around 0 78.4%

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

      1. *-commutative 78.4%

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

      2. unpow2 78.4%

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

      3. associate-*l* 81.8%

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

      4. *-commutative 81.8%

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

   9. Simplified 81.8%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.15 \cdot 10^{-45}:\\ \;\;\;\;\frac{\pi}{a \cdot a} \cdot \frac{0.5}{b}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot b\right)}\\ \end{array} \]

Alternative 9: 89.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 1.15 \cdot 10^{-45}:\\ \;\;\;\;\frac{\pi}{\frac{a \cdot \left(a \cdot b\right)}{0.5}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 1.15e-45)
   (/ PI (/ (* a (* a b)) 0.5))
   (* 0.5 (/ PI (* b (* a b))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 1.15e-45) {
		tmp = ((double) M_PI) / ((a * (a * b)) / 0.5);
	} else {
		tmp = 0.5 * (((double) M_PI) / (b * (a * b)));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 1.15e-45) {
		tmp = Math.PI / ((a * (a * b)) / 0.5);
	} else {
		tmp = 0.5 * (Math.PI / (b * (a * b)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 1.15e-45:
		tmp = math.pi / ((a * (a * b)) / 0.5)
	else:
		tmp = 0.5 * (math.pi / (b * (a * b)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 1.15e-45)
		tmp = Float64(pi / Float64(Float64(a * Float64(a * b)) / 0.5));
	else
		tmp = Float64(0.5 * Float64(pi / Float64(b * Float64(a * b))));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.15e-45)
		tmp = pi / ((a * (a * b)) / 0.5);
	else
		tmp = 0.5 * (pi / (b * (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[b, 1.15e-45], N[(Pi / N[(N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision] / 0.5), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(Pi / N[(b * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.14999999999999996e-45

   1. Initial program 76.9%

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

      1. times-frac 76.9%

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

      2. *-commutative 76.9%

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

      3. times-frac 76.9%

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

      4. difference-of-squares 84.9%

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

      5. associate-/r* 86.3%

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

      6. metadata-eval 86.3%

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

      7. sub-neg 86.3%

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

      8. distribute-neg-frac 86.3%

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

      9. metadata-eval 86.3%

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

   3. Simplified 86.3%

      \[\leadsto \color{blue}{\left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)} \]
   4. Step-by-step derivation

      1. clear-num 86.3%

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

      2. inv-pow 86.3%

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

   5. Applied egg-rr 86.3%

      \[\leadsto \left(\frac{\color{blue}{{\left(\frac{b + a}{\pi}\right)}^{-1}}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right) \]
   6. Step-by-step derivation

      1. unpow-1 86.3%

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

      2. +-commutative 86.3%

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

   7. Simplified 86.3%

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

   8. Taylor expanded in a around inf 61.5%

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

      1. *-commutative 61.5%

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

      2. associate-/r/ 61.5%

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

      3. unpow2 61.5%

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

      4. associate-*l* 69.4%

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

   10. Simplified 69.4%

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

   if 1.14999999999999996e-45 < b

   1. Initial program 80.7%

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

      1. *-commutative 80.7%

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

      2. associate-/r/ 80.7%

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

      3. associate-*l/ 80.7%

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

      4. *-commutative 80.7%

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

      5. associate-/r/ 80.7%

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

      6. times-frac 80.7%

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

   3. Simplified 80.8%

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

   4. Taylor expanded in b around inf 78.4%

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

      1. unpow2 78.4%

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

   6. Simplified 78.4%

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

   7. Taylor expanded in a around 0 78.4%

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

      1. *-commutative 78.4%

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

      2. unpow2 78.4%

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

      3. associate-*l* 81.8%

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

      4. *-commutative 81.8%

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

   9. Simplified 81.8%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.15 \cdot 10^{-45}:\\ \;\;\;\;\frac{\pi}{\frac{a \cdot \left(a \cdot b\right)}{0.5}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot b\right)}\\ \end{array} \]

Alternative 10: 57.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.1%

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

   1. times-frac 78.1%

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

   2. *-commutative 78.1%

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

   3. times-frac 78.1%

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

   4. difference-of-squares 88.3%

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

   5. associate-/r* 89.3%

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

   6. metadata-eval 89.3%

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

   7. sub-neg 89.3%

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

   8. distribute-neg-frac 89.3%

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

   9. metadata-eval 89.3%

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

3. Simplified 89.3%

   \[\leadsto \color{blue}{\left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)} \]
4. Step-by-step derivation

   1. div-inv 89.3%

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

5. Applied egg-rr 89.3%

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

6. Taylor expanded in b around 0 59.1%

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

   1. associate-*r/ 59.1%

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

   2. *-commutative 59.1%

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

   3. times-frac 59.0%

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

   4. unpow2 59.0%

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

8. Simplified 59.0%

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

9. Final simplification 59.0%

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

Reproduce

herbie shell --seed 2023200 
(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))))