NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.4% → 99.7%

Time: 11.9s

Alternatives: 11

Speedup: 1.6×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.4% 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, 1.3× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{+128}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{\left(a + b\right) \cdot \left(b \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{a \cdot b}}{2}}{b - a}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 5e+128)
   (/ (/ PI a) (* (+ a b) (* b 2.0)))
   (/ (/ (/ PI (* a b)) 2.0) (- b a))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 5e+128) {
		tmp = (((double) M_PI) / a) / ((a + b) * (b * 2.0));
	} else {
		tmp = ((((double) M_PI) / (a * b)) / 2.0) / (b - a);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 5e+128) {
		tmp = (Math.PI / a) / ((a + b) * (b * 2.0));
	} else {
		tmp = ((Math.PI / (a * b)) / 2.0) / (b - a);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 5e+128:
		tmp = (math.pi / a) / ((a + b) * (b * 2.0))
	else:
		tmp = ((math.pi / (a * b)) / 2.0) / (b - a)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 5e+128)
		tmp = Float64(Float64(pi / a) / Float64(Float64(a + b) * Float64(b * 2.0)));
	else
		tmp = Float64(Float64(Float64(pi / Float64(a * b)) / 2.0) / Float64(b - a));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 5e+128)
		tmp = (pi / a) / ((a + b) * (b * 2.0));
	else
		tmp = ((pi / (a * b)) / 2.0) / (b - a);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[b, 5e+128], N[(N[(Pi / a), $MachinePrecision] / N[(N[(a + b), $MachinePrecision] * N[(b * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5e128

   1. Initial program 79.7%

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

      1. associate-*r/ N/A

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

      2. *-rgt-identity N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. associate-/r* N/A

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

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

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

   3. Simplified 84.8%

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

   5. Taylor expanded in a around 0

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

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

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

      2. PI-lowering-PI.f64 59.0%

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

   7. Simplified 59.0%

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

   8. Taylor expanded in b around inf

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

   10. Simplified 95.2%

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

   if 5e128 < b

   1. Initial program 49.2%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. 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. difference-of-squares N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. associate-/r* N/A

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

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

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in a around 0

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

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

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

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

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

      3. *-lowering-*.f64 99.9%

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

   7. Simplified 99.9%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{+128}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{\left(a + b\right) \cdot \left(b \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{a \cdot b}}{2}}{b - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = ((((pi / a) - (pi / b)) / (a + b)) / 2.0) / (b - a);
end

Wolfram

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

Derivation

1. Initial program 75.5%

   \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
2. 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. difference-of-squares N/A

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

   11. associate-/r* N/A

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

   12. associate-/r* N/A

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

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

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

4. Applied egg-rr 99.6%

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

5. Final simplification 99.6%

   \[\leadsto \frac{\frac{\frac{\frac{\pi}{a} - \frac{\pi}{b}}{a + b}}{2}}{b - a} \]
6. Add Preprocessing

Alternative 3: 99.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.5%

   \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
2. 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. difference-of-squares N/A

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

   11. associate-/r* N/A

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

   12. associate-/r* N/A

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

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

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

4. Applied egg-rr 99.6%

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

   1. associate-/r* N/A

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

   2. clear-num N/A

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

   3. associate-/l/ N/A

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

   4. associate-/r* N/A

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

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

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

   6. associate-/r* N/A

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

   7. metadata-eval N/A

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

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

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

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

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

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

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

   11. +-commutative N/A

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

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

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

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

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

   14. /-lowering-/.f64 N/A

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

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

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

   16. /-lowering-/.f64 N/A

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

   17. PI-lowering-PI.f64 99.6%

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

6. Applied egg-rr 99.6%

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

Alternative 4: 99.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{+122}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{\left(a + b\right) \cdot \left(b \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{\frac{b}{\frac{\pi}{a}}}}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 5e+122)
   (/ (/ PI a) (* (+ a b) (* b 2.0)))
   (/ (/ 0.5 (/ b (/ PI a))) b)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 5e+122) {
		tmp = (((double) M_PI) / a) / ((a + b) * (b * 2.0));
	} else {
		tmp = (0.5 / (b / (((double) M_PI) / a))) / b;
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 5e+122) {
		tmp = (Math.PI / a) / ((a + b) * (b * 2.0));
	} else {
		tmp = (0.5 / (b / (Math.PI / a))) / b;
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 5e+122:
		tmp = (math.pi / a) / ((a + b) * (b * 2.0))
	else:
		tmp = (0.5 / (b / (math.pi / a))) / b
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 5e+122)
		tmp = Float64(Float64(pi / a) / Float64(Float64(a + b) * Float64(b * 2.0)));
	else
		tmp = Float64(Float64(0.5 / Float64(b / Float64(pi / a))) / b);
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 5e+122)
		tmp = (pi / a) / ((a + b) * (b * 2.0));
	else
		tmp = (0.5 / (b / (pi / a))) / b;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[b, 5e+122], N[(N[(Pi / a), $MachinePrecision] / N[(N[(a + b), $MachinePrecision] * N[(b * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / N[(b / N[(Pi / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.99999999999999989e122

   1. Initial program 79.5%

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

      1. associate-*r/ N/A

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

      2. *-rgt-identity N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. associate-/r* N/A

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

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

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

   3. Simplified 84.7%

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

   5. Taylor expanded in a around 0

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

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

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

      2. PI-lowering-PI.f64 58.6%

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

   7. Simplified 58.6%

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

   8. Taylor expanded in b around inf

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

   10. Simplified 95.2%

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

   if 4.99999999999999989e122 < b

   1. Initial program 51.9%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a \cdot {b}^{2}}} \]
   4. 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. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. associate-*r/ N/A

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

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

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

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

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

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

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

      13. PI-lowering-PI.f64 73.6%

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

   5. Simplified 73.6%

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

      1. associate-/l/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 99.7%

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

   7. Applied egg-rr 99.7%

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

      1. associate-*l/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r/ N/A

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

      5. associate-*r/ N/A

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

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

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

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

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

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

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

      11. PI-lowering-PI.f64 99.8%

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

   9. Applied egg-rr 99.8%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{+122}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{\left(a + b\right) \cdot \left(b \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{\frac{b}{\frac{\pi}{a}}}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 1.7 \cdot 10^{-64}:\\ \;\;\;\;\frac{\frac{0.5}{\frac{b}{\frac{\pi}{a}}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{b - a}}{a \cdot \frac{b}{\pi}}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 1.7e-64)
   (/ (/ 0.5 (/ b (/ PI a))) a)
   (/ (/ 0.5 (- b a)) (* a (/ b PI)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 1.7e-64) {
		tmp = (0.5 / (b / (((double) M_PI) / a))) / a;
	} else {
		tmp = (0.5 / (b - a)) / (a * (b / ((double) M_PI)));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 1.7e-64) {
		tmp = (0.5 / (b / (Math.PI / a))) / a;
	} else {
		tmp = (0.5 / (b - a)) / (a * (b / Math.PI));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 1.7e-64:
		tmp = (0.5 / (b / (math.pi / a))) / a
	else:
		tmp = (0.5 / (b - a)) / (a * (b / math.pi))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 1.7e-64)
		tmp = Float64(Float64(0.5 / Float64(b / Float64(pi / a))) / a);
	else
		tmp = Float64(Float64(0.5 / Float64(b - a)) / Float64(a * Float64(b / pi)));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.7e-64)
		tmp = (0.5 / (b / (pi / a))) / a;
	else
		tmp = (0.5 / (b - a)) / (a * (b / pi));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[b, 1.7e-64], N[(N[(0.5 / N[(b / N[(Pi / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(0.5 / N[(b - a), $MachinePrecision]), $MachinePrecision] / N[(a * N[(b / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.70000000000000006e-64

   1. Initial program 75.0%

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

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

      2. /-lowering-/.f64 N/A

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

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

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

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

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 57.1%

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

   5. Simplified 57.1%

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

      1. associate-*l* N/A

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

      2. times-frac N/A

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

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

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

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

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

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

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

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

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

      7. *-lowering-*.f64 69.9%

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

   7. Applied egg-rr 69.9%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. frac-times N/A

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

      4. metadata-eval N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. PI-lowering-PI.f64 69.7%

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

   9. Applied egg-rr 69.7%

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

       1. associate-/r* N/A

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

       2. associate-*r/ N/A

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

       3. associate-/r/ N/A

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

       4. *-commutative N/A

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

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

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

       6. *-commutative N/A

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

       7. associate-/r/ N/A

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

       8. associate-*r/ N/A

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

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

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

       10. clear-num N/A

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

       11. un-div-inv N/A

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

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

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

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

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

       14. PI-lowering-PI.f64 69.9%

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

   11. Applied egg-rr 69.9%

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

   if 1.70000000000000006e-64 < b

   1. Initial program 76.6%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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. difference-of-squares N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. associate-/r* N/A

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

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

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

   4. Applied egg-rr 99.6%

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

      1. associate-/r* N/A

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

      2. clear-num N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

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

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

      6. associate-/r* N/A

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

      7. metadata-eval N/A

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

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

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

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

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

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

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

      11. +-commutative N/A

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

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

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

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

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

      14. /-lowering-/.f64 N/A

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

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

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

      16. /-lowering-/.f64 N/A

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

      17. PI-lowering-PI.f64 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in a around 0

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

      1. associate-/l* N/A

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

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

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

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

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

      4. PI-lowering-PI.f64 90.7%

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

   9. Simplified 90.7%

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

Alternative 6: 91.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1.8 \cdot 10^{-87}:\\ \;\;\;\;\frac{\frac{\pi}{a \cdot b} \cdot -0.5}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{\frac{b}{0.5}}}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -1.8e-87)
   (/ (* (/ PI (* a b)) -0.5) (- b a))
   (/ (/ PI (/ b 0.5)) (* a b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -1.8e-87) {
		tmp = ((((double) M_PI) / (a * b)) * -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 <= -1.8e-87) {
		tmp = ((Math.PI / (a * b)) * -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 <= -1.8e-87:
		tmp = ((math.pi / (a * b)) * -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 <= -1.8e-87)
		tmp = Float64(Float64(Float64(pi / Float64(a * b)) * -0.5) / Float64(b - a));
	else
		tmp = Float64(Float64(pi / Float64(b / 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 <= -1.8e-87)
		tmp = ((pi / (a * b)) * -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, -1.8e-87], N[(N[(N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / N[(b / 0.5), $MachinePrecision]), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.79999999999999996e-87

   1. Initial program 74.8%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. 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. difference-of-squares N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. associate-/r* N/A

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

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

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

   4. Applied egg-rr 99.6%

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

   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. *-lowering-*.f64 N/A

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

      2. /-lowering-/.f64 N/A

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

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

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

      4. *-lowering-*.f64 82.9%

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

   7. Simplified 82.9%

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

   if -1.79999999999999996e-87 < a

   1. Initial program 75.9%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a \cdot {b}^{2}}} \]
   4. 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. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. associate-*r/ N/A

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

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

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

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

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

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

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

      13. PI-lowering-PI.f64 55.1%

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

   5. Simplified 55.1%

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

      1. associate-/l/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 65.9%

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

   7. Applied egg-rr 65.9%

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. clear-num N/A

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

      6. div-inv N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. div-inv N/A

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

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

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

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

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. div-inv N/A

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

      15. /-lowering-/.f64 N/A

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 66.0%

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

   9. Applied egg-rr 66.0%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.8 \cdot 10^{-87}:\\ \;\;\;\;\frac{\frac{\pi}{a \cdot b} \cdot -0.5}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{\frac{b}{0.5}}}{a \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 3.7 \cdot 10^{-63}:\\ \;\;\;\;\frac{\frac{0.5}{\frac{b}{\frac{\pi}{a}}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{\frac{b}{0.5}}}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 3.7e-63) (/ (/ 0.5 (/ b (/ PI a))) a) (/ (/ PI (/ b 0.5)) (* a b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 3.7e-63) {
		tmp = (0.5 / (b / (((double) M_PI) / a))) / 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 (b <= 3.7e-63) {
		tmp = (0.5 / (b / (Math.PI / a))) / 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 b <= 3.7e-63:
		tmp = (0.5 / (b / (math.pi / a))) / 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 (b <= 3.7e-63)
		tmp = Float64(Float64(0.5 / Float64(b / Float64(pi / a))) / a);
	else
		tmp = Float64(Float64(pi / Float64(b / 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 (b <= 3.7e-63)
		tmp = (0.5 / (b / (pi / a))) / 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[b, 3.7e-63], N[(N[(0.5 / N[(b / N[(Pi / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(Pi / N[(b / 0.5), $MachinePrecision]), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.70000000000000012e-63

   1. Initial program 75.0%

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

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

      2. /-lowering-/.f64 N/A

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

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

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

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

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 57.1%

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

   5. Simplified 57.1%

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

      1. associate-*l* N/A

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

      2. times-frac N/A

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

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

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

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

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

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

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

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

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

      7. *-lowering-*.f64 69.9%

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

   7. Applied egg-rr 69.9%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. frac-times N/A

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

      4. metadata-eval N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. PI-lowering-PI.f64 69.7%

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

   9. Applied egg-rr 69.7%

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

       1. associate-/r* N/A

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

       2. associate-*r/ N/A

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

       3. associate-/r/ N/A

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

       4. *-commutative N/A

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

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

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

       6. *-commutative N/A

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

       7. associate-/r/ N/A

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

       8. associate-*r/ N/A

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

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

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

       10. clear-num N/A

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

       11. un-div-inv N/A

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

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

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

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

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

       14. PI-lowering-PI.f64 69.9%

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

   11. Applied egg-rr 69.9%

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

   if 3.70000000000000012e-63 < b

   1. Initial program 76.6%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a \cdot {b}^{2}}} \]
   4. 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. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. associate-*r/ N/A

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

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

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

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

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

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

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

      13. PI-lowering-PI.f64 71.5%

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

   5. Simplified 71.5%

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

      1. associate-/l/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 83.9%

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

   7. Applied egg-rr 83.9%

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. clear-num N/A

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

      6. div-inv N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. div-inv N/A

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

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

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

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

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. div-inv N/A

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

      15. /-lowering-/.f64 N/A

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 84.0%

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

   9. Applied egg-rr 84.0%

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

Alternative 8: 89.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 1.26 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{0.5}{\frac{b}{\frac{\pi}{a}}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{0.5}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 1.26e-62)
   (/ (/ 0.5 (/ b (/ PI a))) a)
   (* (/ PI b) (/ 0.5 (* a b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 1.26e-62) {
		tmp = (0.5 / (b / (((double) M_PI) / a))) / 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 (b <= 1.26e-62) {
		tmp = (0.5 / (b / (Math.PI / a))) / 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 b <= 1.26e-62:
		tmp = (0.5 / (b / (math.pi / a))) / 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 (b <= 1.26e-62)
		tmp = Float64(Float64(0.5 / Float64(b / Float64(pi / a))) / 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 (b <= 1.26e-62)
		tmp = (0.5 / (b / (pi / a))) / 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[b, 1.26e-62], N[(N[(0.5 / N[(b / N[(Pi / a), $MachinePrecision]), $MachinePrecision]), $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 b < 1.26e-62

   1. Initial program 75.0%

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

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

      2. /-lowering-/.f64 N/A

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

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

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

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

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 57.1%

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

   5. Simplified 57.1%

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

      1. associate-*l* N/A

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

      2. times-frac N/A

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

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

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

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

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

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

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

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

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

      7. *-lowering-*.f64 69.9%

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

   7. Applied egg-rr 69.9%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. frac-times N/A

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

      4. metadata-eval N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. PI-lowering-PI.f64 69.7%

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

   9. Applied egg-rr 69.7%

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

       1. associate-/r* N/A

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

       2. associate-*r/ N/A

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

       3. associate-/r/ N/A

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

       4. *-commutative N/A

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

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

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

       6. *-commutative N/A

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

       7. associate-/r/ N/A

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

       8. associate-*r/ N/A

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

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

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

       10. clear-num N/A

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

       11. un-div-inv N/A

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

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

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

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

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

       14. PI-lowering-PI.f64 69.9%

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

   11. Applied egg-rr 69.9%

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

   if 1.26e-62 < b

   1. Initial program 76.6%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a \cdot {b}^{2}}} \]
   4. 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. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. associate-*r/ N/A

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

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

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

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

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

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

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

      13. PI-lowering-PI.f64 71.5%

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

   5. Simplified 71.5%

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

      1. associate-/l/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 83.9%

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

   7. Applied egg-rr 83.9%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.26 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{0.5}{\frac{b}{\frac{\pi}{a}}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{0.5}{a \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 4.3 \cdot 10^{-64}:\\ \;\;\;\;\frac{\pi}{a \cdot b} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{0.5}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 4.3e-64) (* (/ PI (* a b)) (/ 0.5 a)) (* (/ PI b) (/ 0.5 (* a b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 4.3e-64) {
		tmp = (((double) M_PI) / (a * b)) * (0.5 / 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 (b <= 4.3e-64) {
		tmp = (Math.PI / (a * b)) * (0.5 / 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 b <= 4.3e-64:
		tmp = (math.pi / (a * b)) * (0.5 / 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 (b <= 4.3e-64)
		tmp = Float64(Float64(pi / Float64(a * b)) * Float64(0.5 / 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 (b <= 4.3e-64)
		tmp = (pi / (a * b)) * (0.5 / 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[b, 4.3e-64], N[(N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(0.5 / 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 b < 4.29999999999999973e-64

   1. Initial program 75.0%

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

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

      2. /-lowering-/.f64 N/A

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

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

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

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

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 57.1%

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

   5. Simplified 57.1%

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

      1. associate-*l* N/A

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

      2. times-frac N/A

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

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

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

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

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

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

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

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

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

      7. *-lowering-*.f64 69.9%

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

   7. Applied egg-rr 69.9%

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

   if 4.29999999999999973e-64 < b

   1. Initial program 76.6%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a \cdot {b}^{2}}} \]
   4. 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. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. associate-*r/ N/A

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

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

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

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

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

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

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

      13. PI-lowering-PI.f64 71.5%

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

   5. Simplified 71.5%

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

      1. associate-/l/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 83.9%

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

   7. Applied egg-rr 83.9%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.3 \cdot 10^{-64}:\\ \;\;\;\;\frac{\pi}{a \cdot b} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{0.5}{a \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \frac{1}{a + b} \cdot \frac{\pi \cdot 0.5}{a \cdot b} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (* (/ 1.0 (+ a b)) (/ (* PI 0.5) (* a b))))

C

assert(a < b);
double code(double a, double b) {
	return (1.0 / (a + b)) * ((((double) M_PI) * 0.5) / (a * b));
}

Java

assert a < b;
public static double code(double a, double b) {
	return (1.0 / (a + b)) * ((Math.PI * 0.5) / (a * b));
}

Python

[a, b] = sort([a, b])
def code(a, b):
	return (1.0 / (a + b)) * ((math.pi * 0.5) / (a * b))

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(Float64(1.0 / Float64(a + b)) * Float64(Float64(pi * 0.5) / Float64(a * b)))
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = (1.0 / (a + b)) * ((pi * 0.5) / (a * b));
end

Wolfram

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

Derivation

1. Initial program 75.5%

   \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
2. 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. difference-of-squares N/A

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

   11. clear-num N/A

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

   12. inv-pow N/A

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

   13. associate-/l* N/A

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

   14. unpow-prod-down N/A

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

4. Applied egg-rr 99.5%

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

5. Taylor expanded in b around 0

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f64 N/A

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

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

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

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

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

   5. *-lowering-*.f64 99.6%

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

7. Simplified 99.6%

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

8. Final simplification 99.6%

   \[\leadsto \frac{1}{a + b} \cdot \frac{\pi \cdot 0.5}{a \cdot b} \]
9. Add Preprocessing

Alternative 11: 63.9% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.5%

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

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

   2. /-lowering-/.f64 N/A

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

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

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

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

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

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

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

   6. unpow2 N/A

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

   7. *-lowering-*.f64 54.2%

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

5. Simplified 54.2%

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

   1. associate-*l* N/A

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

   2. times-frac N/A

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

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

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

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

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

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

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

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

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

   7. *-lowering-*.f64 63.2%

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

7. Applied egg-rr 63.2%

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

8. Final simplification 63.2%

   \[\leadsto \frac{\pi}{a \cdot b} \cdot \frac{0.5}{a} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024152 
(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))))