NMSE Section 6.1 mentioned, B

Percentage Accurate: 79.2% → 99.6%

Time: 15.9s

Alternatives: 13

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: 79.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -3.6 \cdot 10^{+93}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{a + b}}{a} \cdot \frac{\pi}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -3.6e+93)
   (/ (* 0.5 (/ PI a)) (* a b))
   (* (/ (/ 0.5 (+ a b)) a) (/ PI b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -3.6e+93) {
		tmp = (0.5 * (((double) M_PI) / a)) / (a * b);
	} else {
		tmp = ((0.5 / (a + b)) / a) * (((double) M_PI) / b);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -3.6e+93) {
		tmp = (0.5 * (Math.PI / a)) / (a * b);
	} else {
		tmp = ((0.5 / (a + b)) / a) * (Math.PI / b);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -3.6e+93:
		tmp = (0.5 * (math.pi / a)) / (a * b)
	else:
		tmp = ((0.5 / (a + b)) / a) * (math.pi / b)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -3.6e+93)
		tmp = Float64(Float64(0.5 * Float64(pi / a)) / Float64(a * b));
	else
		tmp = Float64(Float64(Float64(0.5 / Float64(a + b)) / a) * Float64(pi / b));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -3.6e+93)
		tmp = (0.5 * (pi / a)) / (a * b);
	else
		tmp = ((0.5 / (a + b)) / a) * (pi / b);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -3.6e+93], N[(N[(0.5 * N[(Pi / a), $MachinePrecision]), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 / N[(a + b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] * N[(Pi / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.5999999999999999e93

   1. Initial program 74.2%

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

      1. div-inv 74.2%

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

      2. cancel-sign-sub-inv 74.2%

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

      3. metadata-eval 74.2%

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

      4. div-inv 74.2%

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

      5. div-inv 74.3%

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

      6. difference-of-squares 82.0%

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

      7. *-commutative 82.0%

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

      8. frac-add 82.0%

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

      9. associate-*r/ 82.0%

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

   3. Applied egg-rr 99.7%

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

      1. associate-*r/ 99.8%

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

      2. *-commutative 99.8%

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

      3. +-commutative 99.8%

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

      4. *-commutative 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in a around inf 99.8%

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

   if -3.5999999999999999e93 < a

   1. Initial program 82.0%

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

      1. div-inv 82.0%

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

      2. cancel-sign-sub-inv 82.0%

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

      3. metadata-eval 82.0%

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

      4. div-inv 82.0%

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

      5. div-inv 82.0%

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

      6. difference-of-squares 89.3%

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

      7. *-commutative 89.3%

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

      8. frac-add 89.3%

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

      9. associate-*r/ 89.3%

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

   3. Applied egg-rr 99.5%

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

      1. associate-*r/ 99.6%

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

      2. *-commutative 99.6%

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

      3. +-commutative 99.6%

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

      4. *-commutative 99.6%

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

   5. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      2. metadata-eval 99.6%

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

      3. div-inv 99.6%

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

      4. associate-/r* 99.6%

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

      5. +-commutative 99.6%

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

      6. div-inv 99.5%

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

      7. *-commutative 99.5%

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

      8. times-frac 95.2%

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

      9. *-commutative 95.2%

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

      10. +-commutative 95.2%

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

      11. associate-/r* 95.2%

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

      12. metadata-eval 95.2%

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

   7. Applied egg-rr 95.2%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.6 \cdot 10^{+93}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{a + b}}{a} \cdot \frac{\pi}{b}\\ \end{array} \]

Alternative 2: 99.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 8.2 \cdot 10^{+69}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{\frac{0.5}{b}}{a + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{a \cdot 2}}{b}}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 8.2e+69)
   (* (/ PI a) (/ (/ 0.5 b) (+ a b)))
   (/ (/ (/ PI (* a 2.0)) b) b)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 8.2e+69) {
		tmp = (((double) M_PI) / a) * ((0.5 / b) / (a + b));
	} else {
		tmp = ((((double) M_PI) / (a * 2.0)) / b) / b;
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 8.2e+69) {
		tmp = (Math.PI / a) * ((0.5 / b) / (a + b));
	} else {
		tmp = ((Math.PI / (a * 2.0)) / b) / b;
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 8.2e+69:
		tmp = (math.pi / a) * ((0.5 / b) / (a + b))
	else:
		tmp = ((math.pi / (a * 2.0)) / b) / b
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 8.2e+69)
		tmp = Float64(Float64(pi / a) * Float64(Float64(0.5 / b) / Float64(a + b)));
	else
		tmp = Float64(Float64(Float64(pi / Float64(a * 2.0)) / b) / b);
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 8.2e+69)
		tmp = (pi / a) * ((0.5 / b) / (a + b));
	else
		tmp = ((pi / (a * 2.0)) / 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, 8.2e+69], N[(N[(Pi / a), $MachinePrecision] * N[(N[(0.5 / b), $MachinePrecision] / N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi / N[(a * 2.0), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.1999999999999998e69

   1. Initial program 80.4%

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

      1. div-inv 80.4%

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

      2. cancel-sign-sub-inv 80.4%

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

      3. metadata-eval 80.4%

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

      4. div-inv 80.4%

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

      5. div-inv 80.4%

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

      6. difference-of-squares 85.9%

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

      7. *-commutative 85.9%

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

      8. frac-add 85.9%

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

      9. associate-*r/ 85.9%

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

   3. Applied egg-rr 99.5%

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

      1. associate-*r/ 99.6%

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

      2. *-commutative 99.6%

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

      3. +-commutative 99.6%

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

      4. *-commutative 99.6%

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

   5. Simplified 99.6%

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

      1. frac-2neg 99.6%

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

      2. associate-/l/ 98.9%

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

      3. frac-2neg 98.9%

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

      4. distribute-frac-neg 98.9%

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

      5. neg-sub0 98.9%

         \[\leadsto \color{blue}{0 - \frac{-0.5 \cdot \pi}{-\left(a \cdot b\right) \cdot \left(-\left(a + b\right)\right)}} \]

      6. clear-num 98.9%

         \[\leadsto 0 - \color{blue}{\frac{1}{\frac{-\left(a \cdot b\right) \cdot \left(-\left(a + b\right)\right)}{-0.5 \cdot \pi}}} \]

      7. distribute-frac-neg 98.9%

         \[\leadsto 0 - \frac{1}{\color{blue}{-\frac{\left(a \cdot b\right) \cdot \left(-\left(a + b\right)\right)}{-0.5 \cdot \pi}}} \]

      8. associate-/l* 98.8%

         \[\leadsto 0 - \frac{1}{-\color{blue}{\frac{a \cdot b}{\frac{-0.5 \cdot \pi}{-\left(a + b\right)}}}} \]

      9. frac-2neg 98.8%

         \[\leadsto 0 - \frac{1}{-\frac{a \cdot b}{\color{blue}{\frac{0.5 \cdot \pi}{a + b}}}} \]

      10. *-commutative 98.8%

          \[\leadsto 0 - \frac{1}{-\frac{a \cdot b}{\frac{\color{blue}{\pi \cdot 0.5}}{a + b}}} \]

      11. metadata-eval 98.8%

          \[\leadsto 0 - \frac{1}{-\frac{a \cdot b}{\frac{\pi \cdot \color{blue}{\frac{1}{2}}}{a + b}}} \]

      12. div-inv 98.8%

          \[\leadsto 0 - \frac{1}{-\frac{a \cdot b}{\frac{\color{blue}{\frac{\pi}{2}}}{a + b}}} \]

      13. associate-/r* 98.8%

          \[\leadsto 0 - \frac{1}{-\frac{a \cdot b}{\color{blue}{\frac{\pi}{2 \cdot \left(a + b\right)}}}} \]

      14. +-commutative 98.8%

          \[\leadsto 0 - \frac{1}{-\frac{a \cdot b}{\frac{\pi}{2 \cdot \color{blue}{\left(b + a\right)}}}} \]

   7. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{0 - \frac{\pi \cdot \frac{-0.5}{a + b}}{a \cdot b}} \]
   8. Step-by-step derivation

      1. sub0-neg 99.5%

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

      2. times-frac 95.4%

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

      3. distribute-rgt-neg-in 95.4%

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

      4. distribute-frac-neg 95.4%

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

      5. distribute-neg-frac 95.4%

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

      6. metadata-eval 95.4%

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

      7. associate-/l/ 95.1%

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

      8. associate-/r* 95.5%

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

      9. +-commutative 95.5%

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

   9. Simplified 95.5%

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

   if 8.1999999999999998e69 < b

   1. Initial program 80.6%

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

   2. Taylor expanded in b around inf 92.5%

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

      1. unpow2 92.5%

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

   4. Simplified 92.5%

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

      1. associate-*r/ 92.5%

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

      2. *-commutative 92.5%

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

      3. metadata-eval 92.5%

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

      4. div-inv 92.5%

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

      5. associate-*r* 95.6%

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

      6. associate-/r* 98.2%

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

      7. frac-2neg 98.2%

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

      8. distribute-neg-frac 98.2%

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

      9. div-inv 98.2%

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

      10. metadata-eval 98.2%

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

      11. metadata-eval 98.2%

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

      12. distribute-rgt-neg-in 98.2%

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

      13. frac-2neg 98.2%

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

      14. *-commutative 98.2%

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

      15. associate-/l* 98.2%

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

      16. distribute-neg-frac 98.2%

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

      17. metadata-eval 98.2%

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

   6. Applied egg-rr 98.2%

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

      1. clear-num 98.2%

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

      2. associate-/r* 98.2%

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

      3. times-frac 98.2%

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

      4. associate-/r* 98.2%

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

      5. div-inv 98.2%

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

      6. clear-num 98.3%

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

      7. div-inv 98.3%

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

      8. metadata-eval 98.3%

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

      9. times-frac 98.2%

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

   8. Applied egg-rr 98.2%

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

      1. associate-*l/ 98.3%

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

      2. *-lft-identity 98.3%

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

      3. associate-/l/ 98.3%

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

   10. Simplified 98.3%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8.2 \cdot 10^{+69}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{\frac{0.5}{b}}{a + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{a \cdot 2}}{b}}{b}\\ \end{array} \]

Alternative 3: 84.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 5.3 \cdot 10^{-30}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 5.3e-30) (* 0.5 (/ PI (* a (* a b)))) (* 0.5 (/ PI (* a (* b b))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 5.3e-30) {
		tmp = 0.5 * (((double) M_PI) / (a * (a * b)));
	} else {
		tmp = 0.5 * (((double) M_PI) / (a * (b * b)));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 5.3e-30) {
		tmp = 0.5 * (Math.PI / (a * (a * b)));
	} else {
		tmp = 0.5 * (Math.PI / (a * (b * b)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 5.3e-30:
		tmp = 0.5 * (math.pi / (a * (a * b)))
	else:
		tmp = 0.5 * (math.pi / (a * (b * b)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 5.3e-30)
		tmp = Float64(0.5 * Float64(pi / Float64(a * Float64(a * 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 <= 5.3e-30)
		tmp = 0.5 * (pi / (a * (a * b)));
	else
		tmp = 0.5 * (pi / (a * (b * b)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[b, 5.3e-30], N[(0.5 * N[(Pi / N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $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 < 5.29999999999999974e-30

   1. Initial program 77.2%

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

      1. div-inv 77.2%

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

      2. cancel-sign-sub-inv 77.2%

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

      3. metadata-eval 77.2%

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

      4. div-inv 77.2%

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

      5. div-inv 77.2%

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

      6. difference-of-squares 83.6%

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

      7. *-commutative 83.6%

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

      8. frac-add 83.6%

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

      9. associate-*r/ 83.6%

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

   3. Applied egg-rr 99.6%

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

      1. associate-*r/ 99.7%

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

      2. *-commutative 99.7%

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

      3. +-commutative 99.7%

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

      4. *-commutative 99.7%

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

   5. Simplified 99.7%

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

      1. frac-2neg 99.7%

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

      2. associate-/l/ 98.8%

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

      3. distribute-lft-neg-in 98.8%

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

      4. metadata-eval 98.8%

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

      5. *-commutative 98.8%

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

      6. associate-/r* 99.7%

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

      7. frac-2neg 99.7%

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

      8. remove-double-neg 99.7%

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

      9. div-inv 99.6%

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

      10. *-commutative 99.6%

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

      11. associate-/l* 99.6%

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

      12. distribute-neg-frac 99.6%

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

      13. metadata-eval 99.6%

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

   7. Applied egg-rr 99.6%

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

   8. Taylor expanded in a around inf 62.6%

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

      1. unpow2 62.6%

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

      2. associate-*r* 72.9%

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

   10. Simplified 72.9%

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

   if 5.29999999999999974e-30 < b

   1. Initial program 87.0%

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

   2. Taylor expanded in b around inf 85.7%

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

      1. unpow2 85.7%

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

   4. Simplified 85.7%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.3 \cdot 10^{-30}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot b\right)}\\ \end{array} \]

Alternative 4: 90.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 1.8 \cdot 10^{-67}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{\frac{\frac{0.5}{a}}{b}}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 1.8e-67) (* 0.5 (/ PI (* a (* a b)))) (* PI (/ (/ (/ 0.5 a) b) b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 1.8e-67) {
		tmp = 0.5 * (((double) M_PI) / (a * (a * b)));
	} else {
		tmp = ((double) M_PI) * (((0.5 / a) / b) / b);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 1.8e-67) {
		tmp = 0.5 * (Math.PI / (a * (a * b)));
	} else {
		tmp = Math.PI * (((0.5 / a) / b) / b);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 1.8e-67:
		tmp = 0.5 * (math.pi / (a * (a * b)))
	else:
		tmp = math.pi * (((0.5 / a) / b) / b)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 1.8e-67)
		tmp = Float64(0.5 * Float64(pi / Float64(a * Float64(a * b))));
	else
		tmp = Float64(pi * Float64(Float64(Float64(0.5 / a) / b) / b));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.8e-67)
		tmp = 0.5 * (pi / (a * (a * b)));
	else
		tmp = pi * (((0.5 / 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, 1.8e-67], N[(0.5 * N[(Pi / N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(N[(N[(0.5 / a), $MachinePrecision] / b), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.8e-67

   1. Initial program 77.9%

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

      1. div-inv 77.9%

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

      2. cancel-sign-sub-inv 77.9%

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

      3. metadata-eval 77.9%

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

      4. div-inv 77.9%

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

      5. div-inv 77.9%

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

      6. difference-of-squares 84.5%

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

      7. *-commutative 84.5%

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

      8. frac-add 84.5%

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

      9. associate-*r/ 84.5%

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

   3. Applied egg-rr 99.6%

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

      1. associate-*r/ 99.7%

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

      2. *-commutative 99.7%

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

      3. +-commutative 99.7%

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

      4. *-commutative 99.7%

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

   5. Simplified 99.7%

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

      1. frac-2neg 99.7%

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

      2. associate-/l/ 99.0%

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

      3. distribute-lft-neg-in 99.0%

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

      4. metadata-eval 99.0%

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

      5. *-commutative 99.0%

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

      6. associate-/r* 99.7%

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

      7. frac-2neg 99.7%

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

      8. remove-double-neg 99.7%

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

      9. div-inv 99.6%

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

      10. *-commutative 99.6%

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

      11. associate-/l* 99.6%

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

      12. distribute-neg-frac 99.6%

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

      13. metadata-eval 99.6%

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

   7. Applied egg-rr 99.6%

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

   8. Taylor expanded in a around inf 63.6%

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

      1. unpow2 63.6%

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

      2. associate-*r* 73.0%

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

   10. Simplified 73.0%

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

   if 1.8e-67 < b

   1. Initial program 85.2%

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

      1. div-inv 85.2%

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

      2. cancel-sign-sub-inv 85.2%

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

      3. metadata-eval 85.2%

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

      4. div-inv 85.2%

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

      5. div-inv 85.2%

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

      6. difference-of-squares 94.3%

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

      7. *-commutative 94.3%

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

      8. frac-add 94.2%

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

      9. associate-*r/ 94.2%

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

   3. Applied egg-rr 99.5%

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

      1. associate-*r/ 99.6%

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

      2. *-commutative 99.6%

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

      3. +-commutative 99.6%

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

      4. *-commutative 99.6%

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

   5. Simplified 99.6%

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

      1. frac-2neg 99.6%

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

      2. associate-/l/ 97.8%

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

      3. distribute-lft-neg-in 97.8%

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

      4. metadata-eval 97.8%

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

      5. *-commutative 97.8%

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

      6. associate-/r* 99.7%

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

      7. frac-2neg 99.7%

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

      8. remove-double-neg 99.7%

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

      9. div-inv 99.6%

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

      10. *-commutative 99.6%

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

      11. associate-/l* 99.6%

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

      12. distribute-neg-frac 99.6%

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

      13. metadata-eval 99.6%

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

   7. Applied egg-rr 99.6%

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

   8. Taylor expanded in a around 0 83.0%

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

      1. unpow2 83.0%

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

      2. associate-/r* 82.9%

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

      3. associate-/l/ 85.9%

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

      4. associate-/r* 85.8%

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

      5. associate-*r/ 85.8%

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

      6. associate-*r/ 85.8%

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

      7. *-commutative 85.8%

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

      8. associate-*r/ 85.7%

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

      9. *-rgt-identity 85.7%

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

      10. associate-*r/ 85.7%

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

      11. associate-*r* 85.6%

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

      12. associate-*r/ 85.7%

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

      13. *-rgt-identity 85.7%

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

      14. associate-/r* 85.8%

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

   10. Simplified 85.8%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.8 \cdot 10^{-67}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{\frac{\frac{0.5}{a}}{b}}{b}\\ \end{array} \]

Alternative 5: 90.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 1.8 \cdot 10^{-67}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{\frac{\frac{0.5}{a}}{b}}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 1.8e-67) (* (/ PI a) (/ 0.5 (* a b))) (* PI (/ (/ (/ 0.5 a) b) b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 1.8e-67) {
		tmp = (((double) M_PI) / a) * (0.5 / (a * b));
	} else {
		tmp = ((double) M_PI) * (((0.5 / a) / b) / b);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 1.8e-67) {
		tmp = (Math.PI / a) * (0.5 / (a * b));
	} else {
		tmp = Math.PI * (((0.5 / a) / b) / b);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 1.8e-67:
		tmp = (math.pi / a) * (0.5 / (a * b))
	else:
		tmp = math.pi * (((0.5 / a) / b) / b)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 1.8e-67)
		tmp = Float64(Float64(pi / a) * Float64(0.5 / Float64(a * b)));
	else
		tmp = Float64(pi * Float64(Float64(Float64(0.5 / a) / b) / b));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.8e-67)
		tmp = (pi / a) * (0.5 / (a * b));
	else
		tmp = pi * (((0.5 / 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, 1.8e-67], N[(N[(Pi / a), $MachinePrecision] * N[(0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(N[(N[(0.5 / a), $MachinePrecision] / b), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.8e-67

   1. Initial program 77.9%

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

      1. div-inv 77.9%

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

      2. cancel-sign-sub-inv 77.9%

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

      3. metadata-eval 77.9%

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

      4. div-inv 77.9%

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

      5. div-inv 77.9%

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

      6. difference-of-squares 84.5%

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

      7. *-commutative 84.5%

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

      8. frac-add 84.5%

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

      9. associate-*r/ 84.5%

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

   3. Applied egg-rr 99.6%

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

      1. associate-*r/ 99.7%

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

      2. *-commutative 99.7%

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

      3. +-commutative 99.7%

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

      4. *-commutative 99.7%

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

   5. Simplified 99.7%

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

      1. frac-2neg 99.7%

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

      2. associate-/l/ 99.0%

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

      3. frac-2neg 99.0%

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

      4. distribute-frac-neg 99.0%

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

      5. neg-sub0 99.0%

         \[\leadsto \color{blue}{0 - \frac{-0.5 \cdot \pi}{-\left(a \cdot b\right) \cdot \left(-\left(a + b\right)\right)}} \]

      6. clear-num 99.0%

         \[\leadsto 0 - \color{blue}{\frac{1}{\frac{-\left(a \cdot b\right) \cdot \left(-\left(a + b\right)\right)}{-0.5 \cdot \pi}}} \]

      7. distribute-frac-neg 99.0%

         \[\leadsto 0 - \frac{1}{\color{blue}{-\frac{\left(a \cdot b\right) \cdot \left(-\left(a + b\right)\right)}{-0.5 \cdot \pi}}} \]

      8. associate-/l* 98.9%

         \[\leadsto 0 - \frac{1}{-\color{blue}{\frac{a \cdot b}{\frac{-0.5 \cdot \pi}{-\left(a + b\right)}}}} \]

      9. frac-2neg 98.9%

         \[\leadsto 0 - \frac{1}{-\frac{a \cdot b}{\color{blue}{\frac{0.5 \cdot \pi}{a + b}}}} \]

      10. *-commutative 98.9%

          \[\leadsto 0 - \frac{1}{-\frac{a \cdot b}{\frac{\color{blue}{\pi \cdot 0.5}}{a + b}}} \]

      11. metadata-eval 98.9%

          \[\leadsto 0 - \frac{1}{-\frac{a \cdot b}{\frac{\pi \cdot \color{blue}{\frac{1}{2}}}{a + b}}} \]

      12. div-inv 98.9%

          \[\leadsto 0 - \frac{1}{-\frac{a \cdot b}{\frac{\color{blue}{\frac{\pi}{2}}}{a + b}}} \]

      13. associate-/r* 98.9%

          \[\leadsto 0 - \frac{1}{-\frac{a \cdot b}{\color{blue}{\frac{\pi}{2 \cdot \left(a + b\right)}}}} \]

      14. +-commutative 98.9%

          \[\leadsto 0 - \frac{1}{-\frac{a \cdot b}{\frac{\pi}{2 \cdot \color{blue}{\left(b + a\right)}}}} \]

   7. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{0 - \frac{\pi \cdot \frac{-0.5}{a + b}}{a \cdot b}} \]
   8. Step-by-step derivation

      1. sub0-neg 99.6%

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

      2. times-frac 94.6%

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

      3. distribute-rgt-neg-in 94.6%

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

      4. distribute-frac-neg 94.6%

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

      5. distribute-neg-frac 94.6%

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

      6. metadata-eval 94.6%

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

      7. associate-/l/ 94.2%

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

      8. associate-/r* 94.7%

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

      9. +-commutative 94.7%

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

   9. Simplified 94.7%

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

   10. Taylor expanded in b around 0 72.9%

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

   if 1.8e-67 < b

   1. Initial program 85.2%

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

      1. div-inv 85.2%

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

      2. cancel-sign-sub-inv 85.2%

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

      3. metadata-eval 85.2%

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

      4. div-inv 85.2%

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

      5. div-inv 85.2%

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

      6. difference-of-squares 94.3%

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

      7. *-commutative 94.3%

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

      8. frac-add 94.2%

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

      9. associate-*r/ 94.2%

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

   3. Applied egg-rr 99.5%

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

      1. associate-*r/ 99.6%

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

      2. *-commutative 99.6%

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

      3. +-commutative 99.6%

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

      4. *-commutative 99.6%

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

   5. Simplified 99.6%

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

      1. frac-2neg 99.6%

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

      2. associate-/l/ 97.8%

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

      3. distribute-lft-neg-in 97.8%

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

      4. metadata-eval 97.8%

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

      5. *-commutative 97.8%

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

      6. associate-/r* 99.7%

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

      7. frac-2neg 99.7%

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

      8. remove-double-neg 99.7%

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

      9. div-inv 99.6%

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

      10. *-commutative 99.6%

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

      11. associate-/l* 99.6%

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

      12. distribute-neg-frac 99.6%

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

      13. metadata-eval 99.6%

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

   7. Applied egg-rr 99.6%

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

   8. Taylor expanded in a around 0 83.0%

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

      1. unpow2 83.0%

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

      2. associate-/r* 82.9%

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

      3. associate-/l/ 85.9%

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

      4. associate-/r* 85.8%

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

      5. associate-*r/ 85.8%

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

      6. associate-*r/ 85.8%

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

      7. *-commutative 85.8%

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

      8. associate-*r/ 85.7%

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

      9. *-rgt-identity 85.7%

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

      10. associate-*r/ 85.7%

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

      11. associate-*r* 85.6%

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

      12. associate-*r/ 85.7%

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

      13. *-rgt-identity 85.7%

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

      14. associate-/r* 85.8%

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

   10. Simplified 85.8%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.8 \cdot 10^{-67}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{\frac{\frac{0.5}{a}}{b}}{b}\\ \end{array} \]

Alternative 6: 90.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 1.8 \cdot 10^{-67}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\frac{\pi}{b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{\frac{\frac{0.5}{a}}{b}}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 1.8e-67) (* (/ 0.5 a) (/ (/ PI b) a)) (* PI (/ (/ (/ 0.5 a) b) b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 1.8e-67) {
		tmp = (0.5 / a) * ((((double) M_PI) / b) / a);
	} else {
		tmp = ((double) M_PI) * (((0.5 / a) / b) / b);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 1.8e-67) {
		tmp = (0.5 / a) * ((Math.PI / b) / a);
	} else {
		tmp = Math.PI * (((0.5 / a) / b) / b);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 1.8e-67:
		tmp = (0.5 / a) * ((math.pi / b) / a)
	else:
		tmp = math.pi * (((0.5 / a) / b) / b)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 1.8e-67)
		tmp = Float64(Float64(0.5 / a) * Float64(Float64(pi / b) / a));
	else
		tmp = Float64(pi * Float64(Float64(Float64(0.5 / a) / b) / b));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.8e-67)
		tmp = (0.5 / a) * ((pi / b) / a);
	else
		tmp = pi * (((0.5 / 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, 1.8e-67], N[(N[(0.5 / a), $MachinePrecision] * N[(N[(Pi / b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(N[(N[(0.5 / a), $MachinePrecision] / b), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.8e-67

   1. Initial program 77.9%

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

      1. div-inv 77.9%

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

      2. cancel-sign-sub-inv 77.9%

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

      3. metadata-eval 77.9%

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

      4. div-inv 77.9%

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

      5. div-inv 77.9%

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

      6. difference-of-squares 84.5%

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

      7. *-commutative 84.5%

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

      8. frac-add 84.5%

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

      9. associate-*r/ 84.5%

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

   3. Applied egg-rr 99.6%

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

      1. associate-*r/ 99.7%

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

      2. *-commutative 99.7%

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

      3. +-commutative 99.7%

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

      4. *-commutative 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in a around inf 72.9%

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

      1. times-frac 72.9%

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

      2. *-commutative 72.9%

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

      3. associate-/r* 72.9%

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

   8. Applied egg-rr 72.9%

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

      1. frac-2neg 72.9%

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

      2. distribute-frac-neg 72.9%

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

      3. remove-double-neg 72.9%

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

      4. distribute-frac-neg 72.9%

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

      5. frac-2neg 72.9%

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

      6. neg-sub0 72.9%

         \[\leadsto \color{blue}{\left(0 - \left(-\frac{\pi}{a \cdot b}\right)\right)} \cdot \frac{0.5}{a} \]

      7. div-inv 72.9%

         \[\leadsto \left(0 - \left(-\color{blue}{\pi \cdot \frac{1}{a \cdot b}}\right)\right) \cdot \frac{0.5}{a} \]

      8. distribute-rgt-neg-in 72.9%

         \[\leadsto \left(0 - \color{blue}{\pi \cdot \left(-\frac{1}{a \cdot b}\right)}\right) \cdot \frac{0.5}{a} \]

      9. remove-double-neg 72.9%

         \[\leadsto \left(0 - \color{blue}{\left(-\left(-\pi\right)\right)} \cdot \left(-\frac{1}{a \cdot b}\right)\right) \cdot \frac{0.5}{a} \]

      10. cancel-sign-sub 72.9%

          \[\leadsto \color{blue}{\left(0 + \left(-\pi\right) \cdot \left(-\frac{1}{a \cdot b}\right)\right)} \cdot \frac{0.5}{a} \]

      11. associate-/r* 72.8%

          \[\leadsto \left(0 + \left(-\pi\right) \cdot \left(-\color{blue}{\frac{\frac{1}{a}}{b}}\right)\right) \cdot \frac{0.5}{a} \]

   10. Applied egg-rr 72.8%

       \[\leadsto \color{blue}{\left(0 + \left(-\pi\right) \cdot \left(-\frac{\frac{1}{a}}{b}\right)\right)} \cdot \frac{0.5}{a} \]
   11. Step-by-step derivation

       1. +-lft-identity 72.8%

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

       2. distribute-neg-frac 72.8%

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

       3. distribute-neg-frac 72.8%

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

       4. metadata-eval 72.8%

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

       5. distribute-lft-neg-out 72.8%

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

       6. distribute-rgt-neg-out 72.8%

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

       7. distribute-neg-frac 72.8%

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

       8. distribute-neg-frac 72.8%

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

       9. metadata-eval 72.8%

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

       10. associate-*r/ 72.8%

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

       11. associate-*l/ 72.8%

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

       12. associate-*r/ 72.9%

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

       13. *-rgt-identity 72.9%

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

   12. Simplified 72.9%

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

   if 1.8e-67 < b

   1. Initial program 85.2%

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

      1. div-inv 85.2%

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

      2. cancel-sign-sub-inv 85.2%

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

      3. metadata-eval 85.2%

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

      4. div-inv 85.2%

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

      5. div-inv 85.2%

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

      6. difference-of-squares 94.3%

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

      7. *-commutative 94.3%

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

      8. frac-add 94.2%

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

      9. associate-*r/ 94.2%

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

   3. Applied egg-rr 99.5%

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

      1. associate-*r/ 99.6%

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

      2. *-commutative 99.6%

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

      3. +-commutative 99.6%

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

      4. *-commutative 99.6%

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

   5. Simplified 99.6%

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

      1. frac-2neg 99.6%

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

      2. associate-/l/ 97.8%

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

      3. distribute-lft-neg-in 97.8%

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

      4. metadata-eval 97.8%

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

      5. *-commutative 97.8%

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

      6. associate-/r* 99.7%

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

      7. frac-2neg 99.7%

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

      8. remove-double-neg 99.7%

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

      9. div-inv 99.6%

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

      10. *-commutative 99.6%

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

      11. associate-/l* 99.6%

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

      12. distribute-neg-frac 99.6%

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

      13. metadata-eval 99.6%

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

   7. Applied egg-rr 99.6%

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

   8. Taylor expanded in a around 0 83.0%

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

      1. unpow2 83.0%

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

      2. associate-/r* 82.9%

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

      3. associate-/l/ 85.9%

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

      4. associate-/r* 85.8%

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

      5. associate-*r/ 85.8%

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

      6. associate-*r/ 85.8%

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

      7. *-commutative 85.8%

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

      8. associate-*r/ 85.7%

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

      9. *-rgt-identity 85.7%

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

      10. associate-*r/ 85.7%

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

      11. associate-*r* 85.6%

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

      12. associate-*r/ 85.7%

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

      13. *-rgt-identity 85.7%

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

      14. associate-/r* 85.8%

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

   10. Simplified 85.8%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.8 \cdot 10^{-67}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\frac{\pi}{b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{\frac{\frac{0.5}{a}}{b}}{b}\\ \end{array} \]

Alternative 7: 90.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 1.3 \cdot 10^{-67}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{\frac{\frac{0.5}{a}}{b}}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 1.3e-67) (/ (* 0.5 (/ PI a)) (* a b)) (* PI (/ (/ (/ 0.5 a) b) b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 1.3e-67) {
		tmp = (0.5 * (((double) M_PI) / a)) / (a * b);
	} else {
		tmp = ((double) M_PI) * (((0.5 / a) / b) / b);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 1.3e-67) {
		tmp = (0.5 * (Math.PI / a)) / (a * b);
	} else {
		tmp = Math.PI * (((0.5 / a) / b) / b);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 1.3e-67:
		tmp = (0.5 * (math.pi / a)) / (a * b)
	else:
		tmp = math.pi * (((0.5 / a) / b) / b)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 1.3e-67)
		tmp = Float64(Float64(0.5 * Float64(pi / a)) / Float64(a * b));
	else
		tmp = Float64(pi * Float64(Float64(Float64(0.5 / a) / b) / b));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.3e-67)
		tmp = (0.5 * (pi / a)) / (a * b);
	else
		tmp = pi * (((0.5 / 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, 1.3e-67], N[(N[(0.5 * N[(Pi / a), $MachinePrecision]), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(N[(N[(0.5 / a), $MachinePrecision] / b), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.2999999999999999e-67

   1. Initial program 77.9%

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

      1. div-inv 77.9%

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

      2. cancel-sign-sub-inv 77.9%

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

      3. metadata-eval 77.9%

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

      4. div-inv 77.9%

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

      5. div-inv 77.9%

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

      6. difference-of-squares 84.5%

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

      7. *-commutative 84.5%

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

      8. frac-add 84.5%

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

      9. associate-*r/ 84.5%

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

   3. Applied egg-rr 99.6%

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

      1. associate-*r/ 99.7%

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

      2. *-commutative 99.7%

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

      3. +-commutative 99.7%

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

      4. *-commutative 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in a around inf 72.9%

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

   if 1.2999999999999999e-67 < b

   1. Initial program 85.2%

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

      1. div-inv 85.2%

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

      2. cancel-sign-sub-inv 85.2%

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

      3. metadata-eval 85.2%

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

      4. div-inv 85.2%

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

      5. div-inv 85.2%

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

      6. difference-of-squares 94.3%

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

      7. *-commutative 94.3%

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

      8. frac-add 94.2%

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

      9. associate-*r/ 94.2%

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

   3. Applied egg-rr 99.5%

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

      1. associate-*r/ 99.6%

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

      2. *-commutative 99.6%

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

      3. +-commutative 99.6%

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

      4. *-commutative 99.6%

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

   5. Simplified 99.6%

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

      1. frac-2neg 99.6%

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

      2. associate-/l/ 97.8%

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

      3. distribute-lft-neg-in 97.8%

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

      4. metadata-eval 97.8%

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

      5. *-commutative 97.8%

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

      6. associate-/r* 99.7%

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

      7. frac-2neg 99.7%

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

      8. remove-double-neg 99.7%

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

      9. div-inv 99.6%

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

      10. *-commutative 99.6%

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

      11. associate-/l* 99.6%

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

      12. distribute-neg-frac 99.6%

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

      13. metadata-eval 99.6%

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

   7. Applied egg-rr 99.6%

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

   8. Taylor expanded in a around 0 83.0%

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

      1. unpow2 83.0%

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

      2. associate-/r* 82.9%

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

      3. associate-/l/ 85.9%

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

      4. associate-/r* 85.8%

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

      5. associate-*r/ 85.8%

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

      6. associate-*r/ 85.8%

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

      7. *-commutative 85.8%

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

      8. associate-*r/ 85.7%

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

      9. *-rgt-identity 85.7%

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

      10. associate-*r/ 85.7%

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

      11. associate-*r* 85.6%

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

      12. associate-*r/ 85.7%

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

      13. *-rgt-identity 85.7%

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

      14. associate-/r* 85.8%

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

   10. Simplified 85.8%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.3 \cdot 10^{-67}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{\frac{\frac{0.5}{a}}{b}}{b}\\ \end{array} \]

Alternative 8: 90.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 1.8 \cdot 10^{-67}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{a \cdot \frac{b}{\pi}}}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 1.8e-67) (/ (* 0.5 (/ PI a)) (* a b)) (/ (/ 0.5 (* a (/ b PI))) b)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 1.8e-67) {
		tmp = (0.5 * (((double) M_PI) / a)) / (a * b);
	} else {
		tmp = (0.5 / (a * (b / ((double) M_PI)))) / b;
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 1.8e-67) {
		tmp = (0.5 * (Math.PI / a)) / (a * b);
	} else {
		tmp = (0.5 / (a * (b / Math.PI))) / b;
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 1.8e-67:
		tmp = (0.5 * (math.pi / a)) / (a * b)
	else:
		tmp = (0.5 / (a * (b / math.pi))) / b
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 1.8e-67)
		tmp = Float64(Float64(0.5 * Float64(pi / a)) / Float64(a * b));
	else
		tmp = Float64(Float64(0.5 / Float64(a * Float64(b / pi))) / b);
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.8e-67)
		tmp = (0.5 * (pi / a)) / (a * b);
	else
		tmp = (0.5 / (a * (b / pi))) / 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, 1.8e-67], N[(N[(0.5 * N[(Pi / a), $MachinePrecision]), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / N[(a * N[(b / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.8e-67

   1. Initial program 77.9%

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

      1. div-inv 77.9%

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

      2. cancel-sign-sub-inv 77.9%

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

      3. metadata-eval 77.9%

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

      4. div-inv 77.9%

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

      5. div-inv 77.9%

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

      6. difference-of-squares 84.5%

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

      7. *-commutative 84.5%

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

      8. frac-add 84.5%

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

      9. associate-*r/ 84.5%

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

   3. Applied egg-rr 99.6%

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

      1. associate-*r/ 99.7%

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

      2. *-commutative 99.7%

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

      3. +-commutative 99.7%

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

      4. *-commutative 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in a around inf 72.9%

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

   if 1.8e-67 < b

   1. Initial program 85.2%

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

   2. Taylor expanded in b around inf 83.0%

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

      1. unpow2 83.0%

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

   4. Simplified 83.0%

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

      1. associate-*r/ 83.0%

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

      2. *-commutative 83.0%

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

      3. metadata-eval 83.0%

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

      4. div-inv 83.0%

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

      5. associate-*r* 85.0%

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

      6. associate-/r* 85.8%

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

      7. frac-2neg 85.8%

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

      8. distribute-neg-frac 85.8%

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

      9. div-inv 85.8%

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

      10. metadata-eval 85.8%

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

      11. metadata-eval 85.8%

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

      12. distribute-rgt-neg-in 85.8%

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

      13. frac-2neg 85.8%

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

      14. *-commutative 85.8%

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

      15. associate-/l* 85.8%

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

      16. distribute-neg-frac 85.8%

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

      17. metadata-eval 85.8%

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

   6. Applied egg-rr 85.8%

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

      1. associate-/l* 85.7%

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

      2. div-inv 85.8%

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

      3. *-commutative 85.8%

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

      4. clear-num 85.8%

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

      5. remove-double-div 85.9%

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

   8. Applied egg-rr 85.9%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.8 \cdot 10^{-67}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{a \cdot \frac{b}{\pi}}}{b}\\ \end{array} \]

Alternative 9: 90.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 1.8 \cdot 10^{-67}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{a \cdot 2}}{b}}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 1.8e-67) (/ (* 0.5 (/ PI a)) (* a b)) (/ (/ (/ PI (* a 2.0)) b) b)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 1.8e-67) {
		tmp = (0.5 * (((double) M_PI) / a)) / (a * b);
	} else {
		tmp = ((((double) M_PI) / (a * 2.0)) / b) / b;
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 1.8e-67) {
		tmp = (0.5 * (Math.PI / a)) / (a * b);
	} else {
		tmp = ((Math.PI / (a * 2.0)) / b) / b;
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 1.8e-67:
		tmp = (0.5 * (math.pi / a)) / (a * b)
	else:
		tmp = ((math.pi / (a * 2.0)) / b) / b
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 1.8e-67)
		tmp = Float64(Float64(0.5 * Float64(pi / a)) / Float64(a * b));
	else
		tmp = Float64(Float64(Float64(pi / Float64(a * 2.0)) / b) / b);
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.8e-67)
		tmp = (0.5 * (pi / a)) / (a * b);
	else
		tmp = ((pi / (a * 2.0)) / 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, 1.8e-67], N[(N[(0.5 * N[(Pi / a), $MachinePrecision]), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi / N[(a * 2.0), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.8e-67

   1. Initial program 77.9%

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

      1. div-inv 77.9%

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

      2. cancel-sign-sub-inv 77.9%

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

      3. metadata-eval 77.9%

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

      4. div-inv 77.9%

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

      5. div-inv 77.9%

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

      6. difference-of-squares 84.5%

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

      7. *-commutative 84.5%

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

      8. frac-add 84.5%

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

      9. associate-*r/ 84.5%

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

   3. Applied egg-rr 99.6%

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

      1. associate-*r/ 99.7%

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

      2. *-commutative 99.7%

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

      3. +-commutative 99.7%

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

      4. *-commutative 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in a around inf 72.9%

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

   if 1.8e-67 < b

   1. Initial program 85.2%

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

   2. Taylor expanded in b around inf 83.0%

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

      1. unpow2 83.0%

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

   4. Simplified 83.0%

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

      1. associate-*r/ 83.0%

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

      2. *-commutative 83.0%

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

      3. metadata-eval 83.0%

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

      4. div-inv 83.0%

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

      5. associate-*r* 85.0%

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

      6. associate-/r* 85.8%

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

      7. frac-2neg 85.8%

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

      8. distribute-neg-frac 85.8%

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

      9. div-inv 85.8%

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

      10. metadata-eval 85.8%

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

      11. metadata-eval 85.8%

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

      12. distribute-rgt-neg-in 85.8%

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

      13. frac-2neg 85.8%

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

      14. *-commutative 85.8%

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

      15. associate-/l* 85.8%

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

      16. distribute-neg-frac 85.8%

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

      17. metadata-eval 85.8%

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

   6. Applied egg-rr 85.8%

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

      1. clear-num 85.8%

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

      2. associate-/r* 85.8%

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

      3. times-frac 85.8%

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

      4. associate-/r* 85.8%

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

      5. div-inv 85.8%

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

      6. clear-num 85.9%

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

      7. div-inv 85.9%

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

      8. metadata-eval 85.9%

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

      9. times-frac 85.8%

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

   8. Applied egg-rr 85.8%

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

      1. associate-*l/ 85.9%

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

      2. *-lft-identity 85.9%

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

      3. associate-/l/ 85.9%

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

   10. Simplified 85.9%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.8 \cdot 10^{-67}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{a \cdot 2}}{b}}{b}\\ \end{array} \]

Alternative 10: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.4%

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

   1. div-inv 80.4%

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

   2. cancel-sign-sub-inv 80.4%

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

   3. metadata-eval 80.4%

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

   4. div-inv 80.4%

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

   5. div-inv 80.4%

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

   6. difference-of-squares 87.9%

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

   7. *-commutative 87.9%

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

   8. frac-add 87.8%

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

   9. associate-*r/ 87.8%

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

3. Applied egg-rr 99.6%

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

   1. associate-*r/ 99.7%

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

   2. *-commutative 99.7%

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

   3. +-commutative 99.7%

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

   4. *-commutative 99.7%

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

5. Simplified 99.7%

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

6. Final simplification 99.7%

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

Alternative 11: 99.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.4%

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

   1. div-inv 80.4%

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

   2. cancel-sign-sub-inv 80.4%

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

   3. metadata-eval 80.4%

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

   4. div-inv 80.4%

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

   5. div-inv 80.4%

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

   6. difference-of-squares 87.9%

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

   7. *-commutative 87.9%

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

   8. frac-add 87.8%

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

   9. associate-*r/ 87.8%

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

3. Applied egg-rr 99.6%

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

   1. associate-*r/ 99.7%

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

   2. *-commutative 99.7%

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

   3. +-commutative 99.7%

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

   4. *-commutative 99.7%

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

5. Simplified 99.7%

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

   1. frac-2neg 99.7%

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

   2. associate-/l/ 98.6%

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

   3. distribute-lft-neg-in 98.6%

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

   4. metadata-eval 98.6%

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

   5. *-commutative 98.6%

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

   6. associate-/r* 99.7%

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

   7. frac-2neg 99.7%

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

   8. remove-double-neg 99.7%

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

   9. div-inv 99.6%

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

   10. *-commutative 99.6%

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

   11. associate-/l* 99.6%

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

   12. distribute-neg-frac 99.6%

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

   13. metadata-eval 99.6%

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

7. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{\frac{0.5}{\frac{a \cdot b}{\pi}} \cdot \frac{1}{a + b}} \]
8. Step-by-step derivation

   1. clear-num 99.6%

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

   2. frac-times 98.5%

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

   3. metadata-eval 98.5%

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

   4. associate-/l* 98.5%

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

   5. div-inv 98.4%

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

   6. /-rgt-identity 98.4%

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

   7. associate-*l* 94.2%

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

   8. clear-num 94.2%

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

   9. remove-double-div 94.3%

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

9. Applied egg-rr 94.3%

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

    1. associate-*r* 98.5%

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

    2. associate-*r/ 98.5%

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

    3. *-commutative 98.5%

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

    4. associate-*r/ 98.5%

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

11. Simplified 98.5%

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

12. Final simplification 98.5%

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

Alternative 12: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.4%

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

   1. div-inv 80.4%

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

   2. cancel-sign-sub-inv 80.4%

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

   3. metadata-eval 80.4%

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

   4. div-inv 80.4%

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

   5. div-inv 80.4%

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

   6. difference-of-squares 87.9%

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

   7. *-commutative 87.9%

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

   8. frac-add 87.8%

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

   9. associate-*r/ 87.8%

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

3. Applied egg-rr 99.6%

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

4. Final simplification 99.6%

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

Alternative 13: 63.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.4%

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

   1. div-inv 80.4%

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

   2. cancel-sign-sub-inv 80.4%

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

   3. metadata-eval 80.4%

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

   4. div-inv 80.4%

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

   5. div-inv 80.4%

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

   6. difference-of-squares 87.9%

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

   7. *-commutative 87.9%

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

   8. frac-add 87.8%

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

   9. associate-*r/ 87.8%

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

3. Applied egg-rr 99.6%

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

   1. associate-*r/ 99.7%

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

   2. *-commutative 99.7%

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

   3. +-commutative 99.7%

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

   4. *-commutative 99.7%

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

5. Simplified 99.7%

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

   1. frac-2neg 99.7%

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

   2. associate-/l/ 98.6%

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

   3. distribute-lft-neg-in 98.6%

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

   4. metadata-eval 98.6%

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

   5. *-commutative 98.6%

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

   6. associate-/r* 99.7%

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

   7. frac-2neg 99.7%

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

   8. remove-double-neg 99.7%

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

   9. div-inv 99.6%

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

   10. *-commutative 99.6%

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

   11. associate-/l* 99.6%

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

   12. distribute-neg-frac 99.6%

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

   13. metadata-eval 99.6%

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

7. Applied egg-rr 99.6%

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

8. Taylor expanded in a around inf 56.9%

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

   1. unpow2 56.9%

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

   2. associate-*r* 63.8%

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

10. Simplified 63.8%

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

11. Final simplification 63.8%

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

Reproduce

herbie shell --seed 2023297 
(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))))