NMSE Section 6.1 mentioned, B

Percentage Accurate: 77.6% → 99.7%

Time: 11.3s

Alternatives: 9

Speedup: 1.9×

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: 77.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.9× speedup

Math

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

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 * Float64(pi / 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 * N[(Pi / a), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] / N[(a + b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.1%

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

   1. un-div-inv N/A

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

   2. associate-/r* N/A

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

   3. associate-*l/ N/A

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

   4. distribute-lft-out-- N/A

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

   5. div-inv N/A

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

   6. div-inv N/A

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

   7. *-commutative N/A

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

   8. associate-/l/ N/A

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

   9. difference-of-squares N/A

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

   10. associate-/r* N/A

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

   11. *-rgt-identity N/A

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

   12. *-lft-identity N/A

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

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\frac{\mathsf{PI}\left(\right)}{a} - \frac{\mathsf{PI}\left(\right)}{b}}{2}}{b + a}\right), \color{blue}{\left(1 \cdot b - a \cdot 1\right)}\right) \]

4. Applied egg-rr 99.7%

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

   1. associate-/l/ N/A

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

   2. associate-/r* N/A

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

   3. div-inv N/A

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

   4. metadata-eval N/A

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

   5. associate-*r/ N/A

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

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{\mathsf{PI}\left(\right)}{a} - \frac{\mathsf{PI}\left(\right)}{b}\right) \cdot \frac{\frac{1}{2}}{b - a}\right), \color{blue}{\left(b + a\right)}\right) \]

6. Applied egg-rr 99.7%

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

7. Taylor expanded in a around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a \cdot b}\right)}, \mathsf{+.f64}\left(a, b\right)\right) \]
8. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \frac{\frac{\mathsf{PI}\left(\right)}{a}}{b}\right), \mathsf{+.f64}\left(a, b\right)\right) \]

   2. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a}}{b}\right), \mathsf{+.f64}\left(\color{blue}{a}, b\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a}\right), b\right), \mathsf{+.f64}\left(\color{blue}{a}, b\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{\mathsf{PI}\left(\right)}{a}\right)\right), b\right), \mathsf{+.f64}\left(a, b\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{PI}\left(\right), a\right)\right), b\right), \mathsf{+.f64}\left(a, b\right)\right) \]

   6. PI-lowering-PI.f64 99.7%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), a\right)\right), b\right), \mathsf{+.f64}\left(a, b\right)\right) \]

9. Simplified 99.7%

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

Alternative 2: 91.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -6.5 \cdot 10^{-27}:\\ \;\;\;\;\frac{\frac{-0.5 \cdot \frac{\pi}{b}}{a}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{0.5}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -6.5e-27)
   (/ (/ (* -0.5 (/ PI b)) a) (- b a))
   (* (/ PI b) (/ 0.5 (* a b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -6.5e-27) {
		tmp = ((-0.5 * (((double) M_PI) / b)) / a) / (b - a);
	} else {
		tmp = (((double) M_PI) / b) * (0.5 / (a * b));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -6.5e-27) {
		tmp = ((-0.5 * (Math.PI / b)) / a) / (b - a);
	} else {
		tmp = (Math.PI / b) * (0.5 / (a * b));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -6.5e-27:
		tmp = ((-0.5 * (math.pi / b)) / a) / (b - a)
	else:
		tmp = (math.pi / b) * (0.5 / (a * b))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -6.5e-27)
		tmp = Float64(Float64(Float64(-0.5 * Float64(pi / b)) / a) / Float64(b - a));
	else
		tmp = Float64(Float64(pi / b) * Float64(0.5 / Float64(a * b)));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -6.5e-27)
		tmp = ((-0.5 * (pi / b)) / a) / (b - a);
	else
		tmp = (pi / b) * (0.5 / (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -6.5e-27], N[(N[(N[(-0.5 * N[(Pi / b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / b), $MachinePrecision] * N[(0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.50000000000000025e-27

   1. Initial program 76.3%

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

      1. un-div-inv N/A

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

      2. associate-/r* N/A

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

      3. associate-*l/ N/A

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

      4. distribute-lft-out-- N/A

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

      5. div-inv N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

      8. associate-/l/ N/A

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

      9. difference-of-squares N/A

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

      10. associate-/r* N/A

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

      11. *-rgt-identity N/A

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

      12. *-lft-identity N/A

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

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\frac{\mathsf{PI}\left(\right)}{a} - \frac{\mathsf{PI}\left(\right)}{b}}{2}}{b + a}\right), \color{blue}{\left(1 \cdot b - a \cdot 1\right)}\right) \]

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in a around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a \cdot b}\right)}, \mathsf{\_.f64}\left(b, a\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{a \cdot b}\right), \mathsf{\_.f64}\left(\color{blue}{b}, a\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{b \cdot a}\right), \mathsf{\_.f64}\left(b, a\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{b}}{a}\right), \mathsf{\_.f64}\left(\color{blue}{b}, a\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{-1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{b}}{a}\right), \mathsf{\_.f64}\left(b, a\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{b}\right), a\right), \mathsf{\_.f64}\left(\color{blue}{b}, a\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \left(\frac{\mathsf{PI}\left(\right)}{b}\right)\right), a\right), \mathsf{\_.f64}\left(b, a\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{PI}\left(\right), b\right)\right), a\right), \mathsf{\_.f64}\left(b, a\right)\right) \]

      8. PI-lowering-PI.f64 94.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), b\right)\right), a\right), \mathsf{\_.f64}\left(b, a\right)\right) \]

   7. Simplified 94.1%

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

   if -6.50000000000000025e-27 < a

   1. Initial program 78.6%

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

      1. un-div-inv N/A

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

      2. associate-/r* N/A

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

      3. associate-*l/ N/A

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

      4. distribute-lft-out-- N/A

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

      5. div-inv N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

      8. associate-/l/ N/A

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

      9. difference-of-squares N/A

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

      10. associate-/r* N/A

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

      11. *-rgt-identity N/A

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

      12. *-lft-identity N/A

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

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\frac{\mathsf{PI}\left(\right)}{a} - \frac{\mathsf{PI}\left(\right)}{b}}{2}}{b + a}\right), \color{blue}{\left(1 \cdot b - a \cdot 1\right)}\right) \]

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{{b}^{2}}\right), \color{blue}{a}\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{2} \cdot \mathsf{PI}\left(\right)}{{b}^{2}}\right), a\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}{{b}^{2}}\right), a\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{\frac{1}{2}}{{b}^{2}}\right), a\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(\frac{\frac{1}{2}}{{b}^{2}}\right)\right), a\right) \]

      10. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\frac{\frac{1}{2}}{{b}^{2}}\right)\right), a\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\frac{1}{2}, \left({b}^{2}\right)\right)\right), a\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\frac{1}{2}, \left(b \cdot b\right)\right)\right), a\right) \]

      13. *-lowering-*.f64 68.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, b\right)\right)\right), a\right) \]

   7. Simplified 68.7%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{b \cdot b}\right), \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{a}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(b \cdot b\right)\right), \left(\frac{\color{blue}{\mathsf{PI}\left(\right)}}{a}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, b\right)\right), \left(\frac{\mathsf{PI}\left(\right)}{a}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{/.f64}\left(\mathsf{PI}\left(\right), \color{blue}{a}\right)\right) \]

      7. PI-lowering-PI.f64 68.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), a\right)\right) \]

   9. Applied egg-rr 68.7%

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

       1. frac-times N/A

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

       2. *-commutative N/A

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

       3. times-frac N/A

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

       4. associate-/l/ N/A

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

       5. times-frac N/A

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

       6. *-commutative N/A

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

       7. associate-/l* N/A

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

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{b}\right), \color{blue}{\left(\frac{\frac{1}{2}}{a \cdot b}\right)}\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), b\right), \left(\frac{\color{blue}{\frac{1}{2}}}{a \cdot b}\right)\right) \]

       10. PI-lowering-PI.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), b\right), \left(\frac{\frac{1}{2}}{a \cdot b}\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), b\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(a \cdot b\right)}\right)\right) \]

       12. *-lowering-*.f64 76.7%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), b\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{b}\right)\right)\right) \]

   11. Applied egg-rr 76.7%

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

Alternative 3: 89.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -4.6 \cdot 10^{-26}:\\ \;\;\;\;\frac{\frac{\frac{\frac{\pi}{a}}{2}}{b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{0.5}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -4.6e-26)
   (/ (/ (/ (/ PI a) 2.0) b) a)
   (* (/ PI b) (/ 0.5 (* a b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -4.6e-26) {
		tmp = (((((double) M_PI) / a) / 2.0) / b) / a;
	} else {
		tmp = (((double) M_PI) / b) * (0.5 / (a * b));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -4.6e-26) {
		tmp = (((Math.PI / a) / 2.0) / b) / a;
	} else {
		tmp = (Math.PI / b) * (0.5 / (a * b));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -4.6e-26:
		tmp = (((math.pi / a) / 2.0) / b) / a
	else:
		tmp = (math.pi / b) * (0.5 / (a * b))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -4.6e-26)
		tmp = Float64(Float64(Float64(Float64(pi / a) / 2.0) / b) / a);
	else
		tmp = Float64(Float64(pi / b) * Float64(0.5 / Float64(a * b)));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -4.6e-26)
		tmp = (((pi / a) / 2.0) / b) / a;
	else
		tmp = (pi / b) * (0.5 / (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -4.6e-26], N[(N[(N[(N[(Pi / a), $MachinePrecision] / 2.0), $MachinePrecision] / b), $MachinePrecision] / a), $MachinePrecision], N[(N[(Pi / b), $MachinePrecision] * N[(0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.60000000000000018e-26

   1. Initial program 76.3%

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

   3. Taylor expanded in b around 0

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

      1. associate-/r* N/A

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

      2. associate-*r/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{{a}^{2}}\right), \color{blue}{b}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{\mathsf{PI}\left(\right)}{{a}^{2}}\right)\right), b\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{PI}\left(\right), \left({a}^{2}\right)\right)\right), b\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left({a}^{2}\right)\right)\right), b\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left(a \cdot a\right)\right)\right), b\right) \]

      8. *-lowering-*.f64 71.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(a, a\right)\right)\right), b\right) \]

   5. Simplified 71.1%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\mathsf{PI}\left(\right)}{a \cdot a}}{b}\right), \color{blue}{\frac{1}{2}}\right) \]

      4. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{\mathsf{PI}\left(\right)}{a}}{a}}{b}\right), \frac{1}{2}\right) \]

      5. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\mathsf{PI}\left(\right)}{a}}{b \cdot a}\right), \frac{1}{2}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\mathsf{PI}\left(\right)}{a}}{a \cdot b}\right), \frac{1}{2}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{a}\right), \left(a \cdot b\right)\right), \frac{1}{2}\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), a\right), \left(a \cdot b\right)\right), \frac{1}{2}\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), a\right), \left(a \cdot b\right)\right), \frac{1}{2}\right) \]

      10. *-lowering-*.f64 83.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), a\right), \mathsf{*.f64}\left(a, b\right)\right), \frac{1}{2}\right) \]

   7. Applied egg-rr 83.0%

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

      1. associate-*l/ N/A

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

      2. metadata-eval N/A

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

      3. div-inv N/A

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

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

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\frac{\mathsf{PI}\left(\right)}{a}}{2}}{b}\right), \color{blue}{a}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{\mathsf{PI}\left(\right)}{a}}{2}\right), b\right), a\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{a}\right), 2\right), b\right), a\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), a\right), 2\right), b\right), a\right) \]

      10. PI-lowering-PI.f64 83.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), a\right), 2\right), b\right), a\right) \]

   9. Applied egg-rr 83.1%

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

   if -4.60000000000000018e-26 < a

   1. Initial program 78.6%

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

      1. un-div-inv N/A

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

      2. associate-/r* N/A

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

      3. associate-*l/ N/A

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

      4. distribute-lft-out-- N/A

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

      5. div-inv N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

      8. associate-/l/ N/A

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

      9. difference-of-squares N/A

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

      10. associate-/r* N/A

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

      11. *-rgt-identity N/A

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

      12. *-lft-identity N/A

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

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\frac{\mathsf{PI}\left(\right)}{a} - \frac{\mathsf{PI}\left(\right)}{b}}{2}}{b + a}\right), \color{blue}{\left(1 \cdot b - a \cdot 1\right)}\right) \]

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{{b}^{2}}\right), \color{blue}{a}\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{2} \cdot \mathsf{PI}\left(\right)}{{b}^{2}}\right), a\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}{{b}^{2}}\right), a\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{\frac{1}{2}}{{b}^{2}}\right), a\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(\frac{\frac{1}{2}}{{b}^{2}}\right)\right), a\right) \]

      10. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\frac{\frac{1}{2}}{{b}^{2}}\right)\right), a\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\frac{1}{2}, \left({b}^{2}\right)\right)\right), a\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\frac{1}{2}, \left(b \cdot b\right)\right)\right), a\right) \]

      13. *-lowering-*.f64 68.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, b\right)\right)\right), a\right) \]

   7. Simplified 68.7%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{b \cdot b}\right), \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{a}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(b \cdot b\right)\right), \left(\frac{\color{blue}{\mathsf{PI}\left(\right)}}{a}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, b\right)\right), \left(\frac{\mathsf{PI}\left(\right)}{a}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{/.f64}\left(\mathsf{PI}\left(\right), \color{blue}{a}\right)\right) \]

      7. PI-lowering-PI.f64 68.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), a\right)\right) \]

   9. Applied egg-rr 68.7%

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

       1. frac-times N/A

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

       2. *-commutative N/A

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

       3. times-frac N/A

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

       4. associate-/l/ N/A

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

       5. times-frac N/A

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

       6. *-commutative N/A

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

       7. associate-/l* N/A

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

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{b}\right), \color{blue}{\left(\frac{\frac{1}{2}}{a \cdot b}\right)}\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), b\right), \left(\frac{\color{blue}{\frac{1}{2}}}{a \cdot b}\right)\right) \]

       10. PI-lowering-PI.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), b\right), \left(\frac{\frac{1}{2}}{a \cdot b}\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), b\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(a \cdot b\right)}\right)\right) \]

       12. *-lowering-*.f64 76.7%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), b\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{b}\right)\right)\right) \]

   11. Applied egg-rr 76.7%

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

Alternative 4: 89.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -2.5 \cdot 10^{-25}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{0.5}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -2.5e-25)
   (* 0.5 (/ (/ PI a) (* a b)))
   (* (/ PI b) (/ 0.5 (* a b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -2.5e-25) {
		tmp = 0.5 * ((((double) M_PI) / a) / (a * b));
	} else {
		tmp = (((double) M_PI) / b) * (0.5 / (a * b));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -2.5e-25) {
		tmp = 0.5 * ((Math.PI / a) / (a * b));
	} else {
		tmp = (Math.PI / b) * (0.5 / (a * b));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -2.5e-25:
		tmp = 0.5 * ((math.pi / a) / (a * b))
	else:
		tmp = (math.pi / b) * (0.5 / (a * b))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -2.5e-25)
		tmp = Float64(0.5 * Float64(Float64(pi / a) / Float64(a * b)));
	else
		tmp = Float64(Float64(pi / b) * Float64(0.5 / Float64(a * b)));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.5e-25)
		tmp = 0.5 * ((pi / a) / (a * b));
	else
		tmp = (pi / b) * (0.5 / (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -2.5e-25], N[(0.5 * N[(N[(Pi / a), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / b), $MachinePrecision] * N[(0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.49999999999999981e-25

   1. Initial program 76.3%

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

   3. Taylor expanded in b around 0

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

      1. associate-/r* N/A

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

      2. associate-*r/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{{a}^{2}}\right), \color{blue}{b}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{\mathsf{PI}\left(\right)}{{a}^{2}}\right)\right), b\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{PI}\left(\right), \left({a}^{2}\right)\right)\right), b\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left({a}^{2}\right)\right)\right), b\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left(a \cdot a\right)\right)\right), b\right) \]

      8. *-lowering-*.f64 71.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(a, a\right)\right)\right), b\right) \]

   5. Simplified 71.1%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\mathsf{PI}\left(\right)}{a \cdot a}}{b}\right), \color{blue}{\frac{1}{2}}\right) \]

      4. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{\mathsf{PI}\left(\right)}{a}}{a}}{b}\right), \frac{1}{2}\right) \]

      5. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\mathsf{PI}\left(\right)}{a}}{b \cdot a}\right), \frac{1}{2}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\mathsf{PI}\left(\right)}{a}}{a \cdot b}\right), \frac{1}{2}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{a}\right), \left(a \cdot b\right)\right), \frac{1}{2}\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), a\right), \left(a \cdot b\right)\right), \frac{1}{2}\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), a\right), \left(a \cdot b\right)\right), \frac{1}{2}\right) \]

      10. *-lowering-*.f64 83.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), a\right), \mathsf{*.f64}\left(a, b\right)\right), \frac{1}{2}\right) \]

   7. Applied egg-rr 83.0%

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

   if -2.49999999999999981e-25 < a

   1. Initial program 78.6%

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

      1. un-div-inv N/A

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

      2. associate-/r* N/A

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

      3. associate-*l/ N/A

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

      4. distribute-lft-out-- N/A

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

      5. div-inv N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

      8. associate-/l/ N/A

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

      9. difference-of-squares N/A

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

      10. associate-/r* N/A

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

      11. *-rgt-identity N/A

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

      12. *-lft-identity N/A

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

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\frac{\mathsf{PI}\left(\right)}{a} - \frac{\mathsf{PI}\left(\right)}{b}}{2}}{b + a}\right), \color{blue}{\left(1 \cdot b - a \cdot 1\right)}\right) \]

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{{b}^{2}}\right), \color{blue}{a}\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{2} \cdot \mathsf{PI}\left(\right)}{{b}^{2}}\right), a\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}{{b}^{2}}\right), a\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{\frac{1}{2}}{{b}^{2}}\right), a\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(\frac{\frac{1}{2}}{{b}^{2}}\right)\right), a\right) \]

      10. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\frac{\frac{1}{2}}{{b}^{2}}\right)\right), a\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\frac{1}{2}, \left({b}^{2}\right)\right)\right), a\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\frac{1}{2}, \left(b \cdot b\right)\right)\right), a\right) \]

      13. *-lowering-*.f64 68.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, b\right)\right)\right), a\right) \]

   7. Simplified 68.7%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{b \cdot b}\right), \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{a}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(b \cdot b\right)\right), \left(\frac{\color{blue}{\mathsf{PI}\left(\right)}}{a}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, b\right)\right), \left(\frac{\mathsf{PI}\left(\right)}{a}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{/.f64}\left(\mathsf{PI}\left(\right), \color{blue}{a}\right)\right) \]

      7. PI-lowering-PI.f64 68.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), a\right)\right) \]

   9. Applied egg-rr 68.7%

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

       1. frac-times N/A

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

       2. *-commutative N/A

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

       3. times-frac N/A

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

       4. associate-/l/ N/A

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

       5. times-frac N/A

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

       6. *-commutative N/A

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

       7. associate-/l* N/A

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

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{b}\right), \color{blue}{\left(\frac{\frac{1}{2}}{a \cdot b}\right)}\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), b\right), \left(\frac{\color{blue}{\frac{1}{2}}}{a \cdot b}\right)\right) \]

       10. PI-lowering-PI.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), b\right), \left(\frac{\frac{1}{2}}{a \cdot b}\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), b\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(a \cdot b\right)}\right)\right) \]

       12. *-lowering-*.f64 76.7%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), b\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{b}\right)\right)\right) \]

   11. Applied egg-rr 76.7%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.5 \cdot 10^{-25}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{0.5}{a \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -4.6 \cdot 10^{-26}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{\frac{0.5}{b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{0.5}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -4.6e-26)
   (* (/ PI a) (/ (/ 0.5 b) a))
   (* (/ PI b) (/ 0.5 (* a b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -4.6e-26) {
		tmp = (((double) M_PI) / a) * ((0.5 / b) / a);
	} else {
		tmp = (((double) M_PI) / b) * (0.5 / (a * b));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -4.6e-26) {
		tmp = (Math.PI / a) * ((0.5 / b) / a);
	} else {
		tmp = (Math.PI / b) * (0.5 / (a * b));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -4.6e-26:
		tmp = (math.pi / a) * ((0.5 / b) / a)
	else:
		tmp = (math.pi / b) * (0.5 / (a * b))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -4.6e-26)
		tmp = Float64(Float64(pi / a) * Float64(Float64(0.5 / b) / a));
	else
		tmp = Float64(Float64(pi / b) * Float64(0.5 / Float64(a * b)));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -4.6e-26)
		tmp = (pi / a) * ((0.5 / b) / a);
	else
		tmp = (pi / b) * (0.5 / (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -4.6e-26], N[(N[(Pi / a), $MachinePrecision] * N[(N[(0.5 / b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / b), $MachinePrecision] * N[(0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.60000000000000018e-26

   1. Initial program 76.3%

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

   3. Taylor expanded in a around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a} + \frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{b}\right), \color{blue}{\left({a}^{2}\right)}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{-1}{2} \cdot \mathsf{PI}\left(\right)}{a} + \frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{b}\right), \left({a}^{2}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot \frac{-1}{2}}{a} + \frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{b}\right), \left({a}^{2}\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{\frac{-1}{2}}{a} + \frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{b}\right), \left({a}^{2}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{\frac{-1}{2}}{a} + \frac{\frac{1}{2} \cdot \mathsf{PI}\left(\right)}{b}\right), \left({a}^{2}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{\frac{-1}{2}}{a} + \frac{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}{b}\right), \left({a}^{2}\right)\right) \]

      7. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{\frac{-1}{2}}{a} + \mathsf{PI}\left(\right) \cdot \frac{\frac{1}{2}}{b}\right), \left({a}^{2}\right)\right) \]

      8. distribute-lft-out N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left(\frac{\frac{-1}{2}}{a} + \frac{\frac{1}{2}}{b}\right)\right), \left({\color{blue}{a}}^{2}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(\frac{\frac{-1}{2}}{a} + \frac{\frac{1}{2}}{b}\right)\right), \left({\color{blue}{a}}^{2}\right)\right) \]

      10. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\frac{\frac{-1}{2}}{a} + \frac{\frac{1}{2}}{b}\right)\right), \left({a}^{2}\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{+.f64}\left(\left(\frac{\frac{-1}{2}}{a}\right), \left(\frac{\frac{1}{2}}{b}\right)\right)\right), \left({a}^{2}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \left(\frac{\frac{1}{2}}{b}\right)\right)\right), \left({a}^{2}\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{/.f64}\left(\frac{1}{2}, b\right)\right)\right), \left({a}^{2}\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{/.f64}\left(\frac{1}{2}, b\right)\right)\right), \left(a \cdot \color{blue}{a}\right)\right) \]

      15. *-lowering-*.f64 67.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{/.f64}\left(\frac{1}{2}, b\right)\right)\right), \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right) \]

   5. Simplified 67.7%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{-1}{2}}{a} + \frac{\frac{1}{2}}{b}}{a}\right), \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{a}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{-1}{2}}{a} + \frac{\frac{1}{2}}{b}\right), a\right), \left(\frac{\color{blue}{\mathsf{PI}\left(\right)}}{a}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{-1}{2}}{a}\right), \left(\frac{\frac{1}{2}}{b}\right)\right), a\right), \left(\frac{\mathsf{PI}\left(\right)}{a}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \left(\frac{\frac{1}{2}}{b}\right)\right), a\right), \left(\frac{\mathsf{PI}\left(\right)}{a}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{/.f64}\left(\frac{1}{2}, b\right)\right), a\right), \left(\frac{\mathsf{PI}\left(\right)}{a}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{/.f64}\left(\frac{1}{2}, b\right)\right), a\right), \mathsf{/.f64}\left(\mathsf{PI}\left(\right), \color{blue}{a}\right)\right) \]

      9. PI-lowering-PI.f64 79.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{/.f64}\left(\frac{1}{2}, b\right)\right), a\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), a\right)\right) \]

   7. Applied egg-rr 79.6%

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

   8. Taylor expanded in a around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{\frac{1}{2}}{a \cdot b}\right)}, \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), a\right)\right) \]
   9. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{1}{2}}{b}}{a}\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{PI.f64}\left(\right)}, a\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{1}{2} \cdot 1}{b}}{a}\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), a\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{2} \cdot \frac{1}{b}}{a}\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), a\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \frac{1}{b}\right), a\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{PI.f64}\left(\right)}, a\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{1}{2} \cdot 1}{b}\right), a\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), a\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{1}{2}}{b}\right), a\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), a\right)\right) \]

      7. /-lowering-/.f64 83.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, b\right), a\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), a\right)\right) \]

   10. Simplified 83.0%

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

   if -4.60000000000000018e-26 < a

   1. Initial program 78.6%

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

      1. un-div-inv N/A

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

      2. associate-/r* N/A

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

      3. associate-*l/ N/A

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

      4. distribute-lft-out-- N/A

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

      5. div-inv N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

      8. associate-/l/ N/A

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

      9. difference-of-squares N/A

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

      10. associate-/r* N/A

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

      11. *-rgt-identity N/A

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

      12. *-lft-identity N/A

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

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\frac{\mathsf{PI}\left(\right)}{a} - \frac{\mathsf{PI}\left(\right)}{b}}{2}}{b + a}\right), \color{blue}{\left(1 \cdot b - a \cdot 1\right)}\right) \]

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{{b}^{2}}\right), \color{blue}{a}\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{2} \cdot \mathsf{PI}\left(\right)}{{b}^{2}}\right), a\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}{{b}^{2}}\right), a\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{\frac{1}{2}}{{b}^{2}}\right), a\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(\frac{\frac{1}{2}}{{b}^{2}}\right)\right), a\right) \]

      10. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\frac{\frac{1}{2}}{{b}^{2}}\right)\right), a\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\frac{1}{2}, \left({b}^{2}\right)\right)\right), a\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\frac{1}{2}, \left(b \cdot b\right)\right)\right), a\right) \]

      13. *-lowering-*.f64 68.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, b\right)\right)\right), a\right) \]

   7. Simplified 68.7%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{b \cdot b}\right), \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{a}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(b \cdot b\right)\right), \left(\frac{\color{blue}{\mathsf{PI}\left(\right)}}{a}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, b\right)\right), \left(\frac{\mathsf{PI}\left(\right)}{a}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{/.f64}\left(\mathsf{PI}\left(\right), \color{blue}{a}\right)\right) \]

      7. PI-lowering-PI.f64 68.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), a\right)\right) \]

   9. Applied egg-rr 68.7%

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

       1. frac-times N/A

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

       2. *-commutative N/A

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

       3. times-frac N/A

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

       4. associate-/l/ N/A

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

       5. times-frac N/A

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

       6. *-commutative N/A

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

       7. associate-/l* N/A

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

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{b}\right), \color{blue}{\left(\frac{\frac{1}{2}}{a \cdot b}\right)}\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), b\right), \left(\frac{\color{blue}{\frac{1}{2}}}{a \cdot b}\right)\right) \]

       10. PI-lowering-PI.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), b\right), \left(\frac{\frac{1}{2}}{a \cdot b}\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), b\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(a \cdot b\right)}\right)\right) \]

       12. *-lowering-*.f64 76.7%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), b\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{b}\right)\right)\right) \]

   11. Applied egg-rr 76.7%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.6 \cdot 10^{-26}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{\frac{0.5}{b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{0.5}{a \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} t_0 := \frac{0.5}{a \cdot b}\\ \mathbf{if}\;a \leq -7.2 \cdot 10^{-26}:\\ \;\;\;\;\frac{\pi}{a} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ 0.5 (* a b))))
   (if (<= a -7.2e-26) (* (/ PI a) t_0) (* (/ PI b) t_0))))

C

assert(a < b);
double code(double a, double b) {
	double t_0 = 0.5 / (a * b);
	double tmp;
	if (a <= -7.2e-26) {
		tmp = (((double) M_PI) / a) * t_0;
	} else {
		tmp = (((double) M_PI) / b) * t_0;
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double t_0 = 0.5 / (a * b);
	double tmp;
	if (a <= -7.2e-26) {
		tmp = (Math.PI / a) * t_0;
	} else {
		tmp = (Math.PI / b) * t_0;
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	t_0 = 0.5 / (a * b)
	tmp = 0
	if a <= -7.2e-26:
		tmp = (math.pi / a) * t_0
	else:
		tmp = (math.pi / b) * t_0
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	t_0 = Float64(0.5 / Float64(a * b))
	tmp = 0.0
	if (a <= -7.2e-26)
		tmp = Float64(Float64(pi / a) * t_0);
	else
		tmp = Float64(Float64(pi / b) * t_0);
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	t_0 = 0.5 / (a * b);
	tmp = 0.0;
	if (a <= -7.2e-26)
		tmp = (pi / a) * t_0;
	else
		tmp = (pi / b) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := Block[{t$95$0 = N[(0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -7.2e-26], N[(N[(Pi / a), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(Pi / b), $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -7.2000000000000003e-26

   1. Initial program 76.3%

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

   3. Taylor expanded in b around 0

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

      1. associate-/r* N/A

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

      2. associate-*r/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{{a}^{2}}\right), \color{blue}{b}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{\mathsf{PI}\left(\right)}{{a}^{2}}\right)\right), b\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{PI}\left(\right), \left({a}^{2}\right)\right)\right), b\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left({a}^{2}\right)\right)\right), b\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left(a \cdot a\right)\right)\right), b\right) \]

      8. *-lowering-*.f64 71.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(a, a\right)\right)\right), b\right) \]

   5. Simplified 71.1%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\mathsf{PI}\left(\right)}{a \cdot a}}{b}\right), \color{blue}{\frac{1}{2}}\right) \]

      4. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{\mathsf{PI}\left(\right)}{a}}{a}}{b}\right), \frac{1}{2}\right) \]

      5. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\mathsf{PI}\left(\right)}{a}}{b \cdot a}\right), \frac{1}{2}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\mathsf{PI}\left(\right)}{a}}{a \cdot b}\right), \frac{1}{2}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{a}\right), \left(a \cdot b\right)\right), \frac{1}{2}\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), a\right), \left(a \cdot b\right)\right), \frac{1}{2}\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), a\right), \left(a \cdot b\right)\right), \frac{1}{2}\right) \]

      10. *-lowering-*.f64 83.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), a\right), \mathsf{*.f64}\left(a, b\right)\right), \frac{1}{2}\right) \]

   7. Applied egg-rr 83.0%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{a}\right), \color{blue}{\left(\frac{\frac{1}{2}}{a \cdot b}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), a\right), \left(\frac{\color{blue}{\frac{1}{2}}}{a \cdot b}\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), a\right), \left(\frac{\frac{1}{2}}{a \cdot b}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), a\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(a \cdot b\right)}\right)\right) \]

      7. *-lowering-*.f64 83.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), a\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{b}\right)\right)\right) \]

   9. Applied egg-rr 83.1%

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

   if -7.2000000000000003e-26 < a

   1. Initial program 78.6%

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

      1. un-div-inv N/A

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

      2. associate-/r* N/A

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

      3. associate-*l/ N/A

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

      4. distribute-lft-out-- N/A

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

      5. div-inv N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

      8. associate-/l/ N/A

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

      9. difference-of-squares N/A

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

      10. associate-/r* N/A

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

      11. *-rgt-identity N/A

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

      12. *-lft-identity N/A

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

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\frac{\mathsf{PI}\left(\right)}{a} - \frac{\mathsf{PI}\left(\right)}{b}}{2}}{b + a}\right), \color{blue}{\left(1 \cdot b - a \cdot 1\right)}\right) \]

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{{b}^{2}}\right), \color{blue}{a}\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{2} \cdot \mathsf{PI}\left(\right)}{{b}^{2}}\right), a\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}{{b}^{2}}\right), a\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{\frac{1}{2}}{{b}^{2}}\right), a\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(\frac{\frac{1}{2}}{{b}^{2}}\right)\right), a\right) \]

      10. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\frac{\frac{1}{2}}{{b}^{2}}\right)\right), a\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\frac{1}{2}, \left({b}^{2}\right)\right)\right), a\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\frac{1}{2}, \left(b \cdot b\right)\right)\right), a\right) \]

      13. *-lowering-*.f64 68.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, b\right)\right)\right), a\right) \]

   7. Simplified 68.7%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{b \cdot b}\right), \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{a}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(b \cdot b\right)\right), \left(\frac{\color{blue}{\mathsf{PI}\left(\right)}}{a}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, b\right)\right), \left(\frac{\mathsf{PI}\left(\right)}{a}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{/.f64}\left(\mathsf{PI}\left(\right), \color{blue}{a}\right)\right) \]

      7. PI-lowering-PI.f64 68.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), a\right)\right) \]

   9. Applied egg-rr 68.7%

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

       1. frac-times N/A

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

       2. *-commutative N/A

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

       3. times-frac N/A

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

       4. associate-/l/ N/A

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

       5. times-frac N/A

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

       6. *-commutative N/A

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

       7. associate-/l* N/A

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

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{b}\right), \color{blue}{\left(\frac{\frac{1}{2}}{a \cdot b}\right)}\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), b\right), \left(\frac{\color{blue}{\frac{1}{2}}}{a \cdot b}\right)\right) \]

       10. PI-lowering-PI.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), b\right), \left(\frac{\frac{1}{2}}{a \cdot b}\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), b\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(a \cdot b\right)}\right)\right) \]

       12. *-lowering-*.f64 76.7%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), b\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{b}\right)\right)\right) \]

   11. Applied egg-rr 76.7%

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

Alternative 7: 89.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1.6 \cdot 10^{-25}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{b}}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -1.6e-25)
   (* (/ PI a) (/ 0.5 (* a b)))
   (* 0.5 (/ (/ PI b) (* a b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -1.6e-25) {
		tmp = (((double) M_PI) / a) * (0.5 / (a * b));
	} else {
		tmp = 0.5 * ((((double) M_PI) / b) / (a * b));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -1.6e-25) {
		tmp = (Math.PI / a) * (0.5 / (a * b));
	} else {
		tmp = 0.5 * ((Math.PI / b) / (a * b));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -1.6e-25:
		tmp = (math.pi / a) * (0.5 / (a * b))
	else:
		tmp = 0.5 * ((math.pi / b) / (a * b))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -1.6e-25)
		tmp = Float64(Float64(pi / a) * Float64(0.5 / Float64(a * b)));
	else
		tmp = Float64(0.5 * Float64(Float64(pi / b) / Float64(a * b)));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.6e-25)
		tmp = (pi / a) * (0.5 / (a * b));
	else
		tmp = 0.5 * ((pi / b) / (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -1.6e-25], N[(N[(Pi / a), $MachinePrecision] * N[(0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(Pi / b), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.6000000000000001e-25

   1. Initial program 76.3%

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

   3. Taylor expanded in b around 0

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

      1. associate-/r* N/A

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

      2. associate-*r/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{{a}^{2}}\right), \color{blue}{b}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{\mathsf{PI}\left(\right)}{{a}^{2}}\right)\right), b\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{PI}\left(\right), \left({a}^{2}\right)\right)\right), b\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left({a}^{2}\right)\right)\right), b\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left(a \cdot a\right)\right)\right), b\right) \]

      8. *-lowering-*.f64 71.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(a, a\right)\right)\right), b\right) \]

   5. Simplified 71.1%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\mathsf{PI}\left(\right)}{a \cdot a}}{b}\right), \color{blue}{\frac{1}{2}}\right) \]

      4. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{\mathsf{PI}\left(\right)}{a}}{a}}{b}\right), \frac{1}{2}\right) \]

      5. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\mathsf{PI}\left(\right)}{a}}{b \cdot a}\right), \frac{1}{2}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\mathsf{PI}\left(\right)}{a}}{a \cdot b}\right), \frac{1}{2}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{a}\right), \left(a \cdot b\right)\right), \frac{1}{2}\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), a\right), \left(a \cdot b\right)\right), \frac{1}{2}\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), a\right), \left(a \cdot b\right)\right), \frac{1}{2}\right) \]

      10. *-lowering-*.f64 83.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), a\right), \mathsf{*.f64}\left(a, b\right)\right), \frac{1}{2}\right) \]

   7. Applied egg-rr 83.0%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{a}\right), \color{blue}{\left(\frac{\frac{1}{2}}{a \cdot b}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), a\right), \left(\frac{\color{blue}{\frac{1}{2}}}{a \cdot b}\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), a\right), \left(\frac{\frac{1}{2}}{a \cdot b}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), a\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(a \cdot b\right)}\right)\right) \]

      7. *-lowering-*.f64 83.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), a\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{b}\right)\right)\right) \]

   9. Applied egg-rr 83.1%

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

   if -1.6000000000000001e-25 < a

   1. Initial program 78.6%

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

      1. un-div-inv N/A

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

      2. associate-/r* N/A

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

      3. associate-*l/ N/A

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

      4. distribute-lft-out-- N/A

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

      5. div-inv N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

      8. associate-/l/ N/A

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

      9. difference-of-squares N/A

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

      10. associate-/r* N/A

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

      11. *-rgt-identity N/A

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

      12. *-lft-identity N/A

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

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\frac{\mathsf{PI}\left(\right)}{a} - \frac{\mathsf{PI}\left(\right)}{b}}{2}}{b + a}\right), \color{blue}{\left(1 \cdot b - a \cdot 1\right)}\right) \]

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{{b}^{2}}\right), \color{blue}{a}\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{2} \cdot \mathsf{PI}\left(\right)}{{b}^{2}}\right), a\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}{{b}^{2}}\right), a\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{\frac{1}{2}}{{b}^{2}}\right), a\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(\frac{\frac{1}{2}}{{b}^{2}}\right)\right), a\right) \]

      10. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\frac{\frac{1}{2}}{{b}^{2}}\right)\right), a\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\frac{1}{2}, \left({b}^{2}\right)\right)\right), a\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\frac{1}{2}, \left(b \cdot b\right)\right)\right), a\right) \]

      13. *-lowering-*.f64 68.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, b\right)\right)\right), a\right) \]

   7. Simplified 68.7%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{b \cdot b}\right), \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{a}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(b \cdot b\right)\right), \left(\frac{\color{blue}{\mathsf{PI}\left(\right)}}{a}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, b\right)\right), \left(\frac{\mathsf{PI}\left(\right)}{a}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{/.f64}\left(\mathsf{PI}\left(\right), \color{blue}{a}\right)\right) \]

      7. PI-lowering-PI.f64 68.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), a\right)\right) \]

   9. Applied egg-rr 68.7%

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

       1. *-commutative N/A

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

       2. associate-/r* N/A

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

       3. frac-times N/A

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

       4. associate-/l* N/A

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

       5. *-commutative N/A

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

       6. associate-*r/ N/A

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

       7. associate-/l* N/A

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

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{\frac{\mathsf{PI}\left(\right)}{b}}{a \cdot b}\right)}\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{b}\right), \color{blue}{\left(a \cdot b\right)}\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), b\right), \left(\color{blue}{a} \cdot b\right)\right)\right) \]

       11. PI-lowering-PI.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), b\right), \left(a \cdot b\right)\right)\right) \]

       12. *-lowering-*.f64 76.7%

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), b\right), \mathsf{*.f64}\left(a, \color{blue}{b}\right)\right)\right) \]

   11. Applied egg-rr 76.7%

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

Alternative 8: 89.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -2.3 \cdot 10^{-25}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{b}}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -2.3e-25)
   (* PI (/ 0.5 (* a (* a b))))
   (* 0.5 (/ (/ PI b) (* a b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -2.3e-25) {
		tmp = ((double) M_PI) * (0.5 / (a * (a * b)));
	} else {
		tmp = 0.5 * ((((double) M_PI) / b) / (a * b));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -2.3e-25) {
		tmp = Math.PI * (0.5 / (a * (a * b)));
	} else {
		tmp = 0.5 * ((Math.PI / b) / (a * b));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -2.3e-25:
		tmp = math.pi * (0.5 / (a * (a * b)))
	else:
		tmp = 0.5 * ((math.pi / b) / (a * b))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -2.3e-25)
		tmp = Float64(pi * Float64(0.5 / Float64(a * Float64(a * b))));
	else
		tmp = Float64(0.5 * Float64(Float64(pi / b) / Float64(a * b)));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.3e-25)
		tmp = pi * (0.5 / (a * (a * b)));
	else
		tmp = 0.5 * ((pi / b) / (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -2.3e-25], N[(Pi * N[(0.5 / N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(Pi / b), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.2999999999999999e-25

   1. Initial program 76.3%

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

   3. Taylor expanded in b around 0

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

      1. associate-/r* N/A

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

      2. associate-*r/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{{a}^{2}}\right), \color{blue}{b}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{\mathsf{PI}\left(\right)}{{a}^{2}}\right)\right), b\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{PI}\left(\right), \left({a}^{2}\right)\right)\right), b\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left({a}^{2}\right)\right)\right), b\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left(a \cdot a\right)\right)\right), b\right) \]

      8. *-lowering-*.f64 71.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(a, a\right)\right)\right), b\right) \]

   5. Simplified 71.1%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\mathsf{PI}\left(\right)}{a \cdot a}}{b}\right), \color{blue}{\frac{1}{2}}\right) \]

      4. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{\mathsf{PI}\left(\right)}{a}}{a}}{b}\right), \frac{1}{2}\right) \]

      5. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\mathsf{PI}\left(\right)}{a}}{b \cdot a}\right), \frac{1}{2}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\mathsf{PI}\left(\right)}{a}}{a \cdot b}\right), \frac{1}{2}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{a}\right), \left(a \cdot b\right)\right), \frac{1}{2}\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), a\right), \left(a \cdot b\right)\right), \frac{1}{2}\right) \]

      9. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), a\right), \left(a \cdot b\right)\right), \frac{1}{2}\right) \]

      10. *-lowering-*.f64 83.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), a\right), \mathsf{*.f64}\left(a, b\right)\right), \frac{1}{2}\right) \]

   7. Applied egg-rr 83.0%

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

      1. associate-/l/ N/A

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

      2. associate-*l/ N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right), \color{blue}{\left(\left(a \cdot b\right) \cdot a\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \frac{1}{2}\right), \left(\color{blue}{\left(a \cdot b\right)} \cdot a\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{2}\right), \left(\left(\color{blue}{a} \cdot b\right) \cdot a\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{2}\right), \left(a \cdot \color{blue}{\left(a \cdot b\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{2}\right), \mathsf{*.f64}\left(a, \color{blue}{\left(a \cdot b\right)}\right)\right) \]

      8. *-lowering-*.f64 84.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{2}\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{b}\right)\right)\right) \]

   9. Applied egg-rr 84.3%

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

       1. associate-/l* N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{a \cdot \left(a \cdot b\right)}\right), \color{blue}{\mathsf{PI}\left(\right)}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(a \cdot \left(a \cdot b\right)\right)\right), \mathsf{PI}\left(\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \left(a \cdot b\right)\right)\right), \mathsf{PI}\left(\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, b\right)\right)\right), \mathsf{PI}\left(\right)\right) \]

       7. PI-lowering-PI.f64 84.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, b\right)\right)\right), \mathsf{PI.f64}\left(\right)\right) \]

   11. Applied egg-rr 84.2%

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

   if -2.2999999999999999e-25 < a

   1. Initial program 78.6%

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

      1. un-div-inv N/A

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

      2. associate-/r* N/A

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

      3. associate-*l/ N/A

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

      4. distribute-lft-out-- N/A

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

      5. div-inv N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

      8. associate-/l/ N/A

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

      9. difference-of-squares N/A

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

      10. associate-/r* N/A

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

      11. *-rgt-identity N/A

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

      12. *-lft-identity N/A

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

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\frac{\mathsf{PI}\left(\right)}{a} - \frac{\mathsf{PI}\left(\right)}{b}}{2}}{b + a}\right), \color{blue}{\left(1 \cdot b - a \cdot 1\right)}\right) \]

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{{b}^{2}}\right), \color{blue}{a}\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{2} \cdot \mathsf{PI}\left(\right)}{{b}^{2}}\right), a\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}{{b}^{2}}\right), a\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{\frac{1}{2}}{{b}^{2}}\right), a\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(\frac{\frac{1}{2}}{{b}^{2}}\right)\right), a\right) \]

      10. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\frac{\frac{1}{2}}{{b}^{2}}\right)\right), a\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\frac{1}{2}, \left({b}^{2}\right)\right)\right), a\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\frac{1}{2}, \left(b \cdot b\right)\right)\right), a\right) \]

      13. *-lowering-*.f64 68.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, b\right)\right)\right), a\right) \]

   7. Simplified 68.7%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{b \cdot b}\right), \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{a}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(b \cdot b\right)\right), \left(\frac{\color{blue}{\mathsf{PI}\left(\right)}}{a}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, b\right)\right), \left(\frac{\mathsf{PI}\left(\right)}{a}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{/.f64}\left(\mathsf{PI}\left(\right), \color{blue}{a}\right)\right) \]

      7. PI-lowering-PI.f64 68.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), a\right)\right) \]

   9. Applied egg-rr 68.7%

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

       1. *-commutative N/A

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

       2. associate-/r* N/A

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

       3. frac-times N/A

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

       4. associate-/l* N/A

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

       5. *-commutative N/A

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

       6. associate-*r/ N/A

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

       7. associate-/l* N/A

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

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{\frac{\mathsf{PI}\left(\right)}{b}}{a \cdot b}\right)}\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{b}\right), \color{blue}{\left(a \cdot b\right)}\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), b\right), \left(\color{blue}{a} \cdot b\right)\right)\right) \]

       11. PI-lowering-PI.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), b\right), \left(a \cdot b\right)\right)\right) \]

       12. *-lowering-*.f64 76.7%

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), b\right), \mathsf{*.f64}\left(a, \color{blue}{b}\right)\right)\right) \]

   11. Applied egg-rr 76.7%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.3 \cdot 10^{-25}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{b}}{a \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.1% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 78.1%

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

   1. un-div-inv N/A

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

   2. associate-/r* N/A

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

   3. associate-*l/ N/A

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

   4. distribute-lft-out-- N/A

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

   5. div-inv N/A

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

   6. div-inv N/A

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

   7. *-commutative N/A

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

   8. associate-/l/ N/A

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

   9. difference-of-squares N/A

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

   10. associate-/r* N/A

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

   11. *-rgt-identity N/A

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

   12. *-lft-identity N/A

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

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\frac{\mathsf{PI}\left(\right)}{a} - \frac{\mathsf{PI}\left(\right)}{b}}{2}}{b + a}\right), \color{blue}{\left(1 \cdot b - a \cdot 1\right)}\right) \]

4. Applied egg-rr 99.7%

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

5. Taylor expanded in a around 0

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. associate-/r* N/A

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

   4. associate-*r/ N/A

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

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{{b}^{2}}\right), \color{blue}{a}\right) \]

   6. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{2} \cdot \mathsf{PI}\left(\right)}{{b}^{2}}\right), a\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}{{b}^{2}}\right), a\right) \]

   8. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{\frac{1}{2}}{{b}^{2}}\right), a\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(\frac{\frac{1}{2}}{{b}^{2}}\right)\right), a\right) \]

   10. PI-lowering-PI.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\frac{\frac{1}{2}}{{b}^{2}}\right)\right), a\right) \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\frac{1}{2}, \left({b}^{2}\right)\right)\right), a\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\frac{1}{2}, \left(b \cdot b\right)\right)\right), a\right) \]

   13. *-lowering-*.f64 63.1%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, b\right)\right)\right), a\right) \]

7. Simplified 63.1%

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{b \cdot b}\right), \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{a}\right)}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(b \cdot b\right)\right), \left(\frac{\color{blue}{\mathsf{PI}\left(\right)}}{a}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, b\right)\right), \left(\frac{\mathsf{PI}\left(\right)}{a}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{/.f64}\left(\mathsf{PI}\left(\right), \color{blue}{a}\right)\right) \]

   7. PI-lowering-PI.f64 63.1%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), a\right)\right) \]

9. Applied egg-rr 63.1%

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

    1. *-commutative N/A

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

    2. associate-/r* N/A

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

    3. frac-times N/A

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

    4. associate-/l* N/A

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

    5. *-commutative N/A

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

    6. associate-*r/ N/A

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

    7. associate-/l* N/A

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

    8. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{\frac{\mathsf{PI}\left(\right)}{b}}{a \cdot b}\right)}\right) \]

    9. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{b}\right), \color{blue}{\left(a \cdot b\right)}\right)\right) \]

    10. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), b\right), \left(\color{blue}{a} \cdot b\right)\right)\right) \]

    11. PI-lowering-PI.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), b\right), \left(a \cdot b\right)\right)\right) \]

    12. *-lowering-*.f64 69.2%

        \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), b\right), \mathsf{*.f64}\left(a, \color{blue}{b}\right)\right)\right) \]

11. Applied egg-rr 69.2%

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

Reproduce

herbie shell --seed 2024144 
(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))))