NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.6% → 99.6%

Time: 10.1s

Alternatives: 10

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{+147}:\\ \;\;\;\;\frac{\pi}{b \cdot a} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{\frac{\frac{0.5}{a}}{a + b}}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -5e+147)
   (* (/ PI (* b a)) (/ 0.5 a))
   (* PI (/ (/ (/ 0.5 a) (+ a b)) b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -5e+147) {
		tmp = (((double) M_PI) / (b * a)) * (0.5 / a);
	} else {
		tmp = ((double) M_PI) * (((0.5 / a) / (a + b)) / b);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -5e+147) {
		tmp = (Math.PI / (b * a)) * (0.5 / a);
	} else {
		tmp = Math.PI * (((0.5 / a) / (a + b)) / b);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -5e+147:
		tmp = (math.pi / (b * a)) * (0.5 / a)
	else:
		tmp = math.pi * (((0.5 / a) / (a + b)) / b)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -5e+147)
		tmp = Float64(Float64(pi / Float64(b * a)) * Float64(0.5 / a));
	else
		tmp = Float64(pi * Float64(Float64(Float64(0.5 / a) / Float64(a + b)) / b));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -5e+147)
		tmp = (pi / (b * a)) * (0.5 / a);
	else
		tmp = pi * (((0.5 / a) / (a + b)) / b);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -5e+147], N[(N[(Pi / N[(b * a), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(N[(N[(0.5 / a), $MachinePrecision] / N[(a + b), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.0000000000000002e147

   1. Initial program 56.8%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 76.0

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

   4. Applied rewrites 76.0%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. *-commutative N/A

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

      10. associate-/l/ N/A

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

      11. associate-*r/ N/A

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

      12. *-commutative N/A

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

      13. pow2 N/A

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

      14. times-frac N/A

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

      15. associate-/l/ N/A

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

      16. *-commutative N/A

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

      17. associate-/l/ N/A

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

      18. lower-*.f64 N/A

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

      19. associate-/l/ N/A

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

      20. lower-/.f64 N/A

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

      21. lift-PI.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 N/A

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

      24. lower-/.f64 99.8

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

   6. Applied rewrites 99.8%

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

   if -5.0000000000000002e147 < a

   1. Initial program 85.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-rgt-identity N/A

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

      8. difference-of-squares N/A

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

      9. associate-/r* N/A

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

      10. *-lft-identity N/A

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

      11. *-rgt-identity N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. *-lft-identity N/A

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

      17. *-rgt-identity N/A

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

      18. lower--.f64 92.1

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

   3. Applied rewrites 92.1%

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

   5. Applied rewrites 99.0%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-*l/ N/A

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

      9. associate-/l* N/A

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

      10. frac-2neg-rev N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. metadata-eval N/A

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

      15. frac-2neg-rev N/A

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

      16. lower-/.f64 N/A

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

      17. *-commutative N/A

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

      18. associate-*l* N/A

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

   7. Applied rewrites 91.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. count-2-rev N/A

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

      7. associate-/r* N/A

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

      8. associate-/r* N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. *-commutative N/A

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

      13. associate-/r* N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. associate-*r/ N/A

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

      17. metadata-eval N/A

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

      18. lift-/.f64 N/A

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

      19. lift-+.f64 99.6

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

   9. Applied rewrites 99.6%

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

Alternative 2: 99.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1.3 \cdot 10^{+157}:\\ \;\;\;\;\frac{\pi}{b \cdot a} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{b}}{\left(a + a\right) \cdot \left(a + b\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -1.3e+157)
   (* (/ PI (* b a)) (/ 0.5 a))
   (/ (/ PI b) (* (+ a a) (+ a b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -1.3e+157) {
		tmp = (((double) M_PI) / (b * a)) * (0.5 / a);
	} else {
		tmp = (((double) M_PI) / b) / ((a + a) * (a + b));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -1.3e+157) {
		tmp = (Math.PI / (b * a)) * (0.5 / a);
	} else {
		tmp = (Math.PI / b) / ((a + a) * (a + b));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -1.3e+157:
		tmp = (math.pi / (b * a)) * (0.5 / a)
	else:
		tmp = (math.pi / b) / ((a + a) * (a + b))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -1.3e+157)
		tmp = Float64(Float64(pi / Float64(b * a)) * Float64(0.5 / a));
	else
		tmp = Float64(Float64(pi / b) / Float64(Float64(a + a) * 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 <= -1.3e+157)
		tmp = (pi / (b * a)) * (0.5 / a);
	else
		tmp = (pi / b) / ((a + a) * (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, -1.3e+157], N[(N[(Pi / N[(b * a), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / b), $MachinePrecision] / N[(N[(a + a), $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.30000000000000005e157

   1. Initial program 55.7%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 75.8

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

   4. Applied rewrites 75.8%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. *-commutative N/A

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

      10. associate-/l/ N/A

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

      11. associate-*r/ N/A

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

      12. *-commutative N/A

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

      13. pow2 N/A

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

      14. times-frac N/A

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

      15. associate-/l/ N/A

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

      16. *-commutative N/A

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

      17. associate-/l/ N/A

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

      18. lower-*.f64 N/A

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

      19. associate-/l/ N/A

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

      20. lower-/.f64 N/A

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

      21. lift-PI.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 N/A

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

      24. lower-/.f64 99.8

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

   6. Applied rewrites 99.8%

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

   if -1.30000000000000005e157 < a

   1. Initial program 85.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-rgt-identity N/A

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

      8. difference-of-squares N/A

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

      9. associate-/r* N/A

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

      10. *-lft-identity N/A

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

      11. *-rgt-identity N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. *-lft-identity N/A

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

      17. *-rgt-identity N/A

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

      18. lower--.f64 92.2

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

   3. Applied rewrites 92.2%

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

   5. Applied rewrites 98.9%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. *-rgt-identity N/A

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

      9. associate-*l* N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. count-2-rev N/A

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

      19. lower-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 99.3

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

   7. Applied rewrites 99.3%

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

Alternative 3: 99.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1.7 \cdot 10^{+89}:\\ \;\;\;\;\frac{\frac{\pi}{b} \cdot \frac{0.5}{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{\left(a \cdot \left(a + b\right)\right) \cdot \left(b + b\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -1.7e+89)
   (/ (* (/ PI b) (/ 0.5 a)) a)
   (/ PI (* (* a (+ a b)) (+ b b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -1.7e+89) {
		tmp = ((((double) M_PI) / b) * (0.5 / a)) / a;
	} else {
		tmp = ((double) M_PI) / ((a * (a + b)) * (b + b));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -1.7e+89) {
		tmp = ((Math.PI / b) * (0.5 / a)) / a;
	} else {
		tmp = Math.PI / ((a * (a + b)) * (b + b));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -1.7e+89:
		tmp = ((math.pi / b) * (0.5 / a)) / a
	else:
		tmp = math.pi / ((a * (a + b)) * (b + b))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -1.7e+89)
		tmp = Float64(Float64(Float64(pi / b) * Float64(0.5 / a)) / a);
	else
		tmp = Float64(pi / Float64(Float64(a * Float64(a + b)) * Float64(b + b)));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.7e+89)
		tmp = ((pi / b) * (0.5 / a)) / a;
	else
		tmp = pi / ((a * (a + b)) * (b + b));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -1.7e+89], N[(N[(N[(Pi / b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(Pi / N[(N[(a * N[(a + b), $MachinePrecision]), $MachinePrecision] * N[(b + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.7000000000000001e89

   1. Initial program 67.5%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 81.3

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

   4. Applied rewrites 81.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 98.5

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

   6. Applied rewrites 98.5%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. frac-times N/A

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

      8. lift-/.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/r* N/A

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

      15. lift-/.f64 N/A

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

      16. associate-*l/ N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lift-/.f64 N/A

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

      20. lift-PI.f64 N/A

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

      21. lift-/.f64 99.4

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

   8. Applied rewrites 99.4%

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

   if -1.7000000000000001e89 < a

   1. Initial program 83.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-rgt-identity N/A

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

      8. difference-of-squares N/A

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

      9. associate-/r* N/A

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

      10. *-lft-identity N/A

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

      11. *-rgt-identity N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. *-lft-identity N/A

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

      17. *-rgt-identity N/A

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

      18. lower--.f64 91.3

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

   3. Applied rewrites 91.3%

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

   5. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. *-rgt-identity N/A

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

      9. associate-*l* N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. count-2-rev N/A

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

      19. lower-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 99.6

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

   7. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-/l/ N/A

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

      8. count-2-rev N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. count-2-rev N/A

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

      13. lower-/.f64 N/A

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

      14. lift-PI.f64 N/A

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

      15. count-2-rev N/A

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

      16. *-commutative N/A

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

      17. associate-*l* N/A

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

      18. *-commutative N/A

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

      19. associate-*l* N/A

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

      20. distribute-lft-out N/A

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

      21. pow2 N/A

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

      22. +-commutative N/A

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

   9. Applied rewrites 99.1%

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

Alternative 4: 98.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \frac{\pi}{\left(b \cdot a\right) \cdot \left(\left(b + a\right) \cdot 2\right)} \cdot 1 \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (* (/ PI (* (* b a) (* (+ b a) 2.0))) 1.0))

C

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

Java

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

Python

[a, b] = sort([a, b])
def code(a, b):
	return (math.pi / ((b * a) * ((b + a) * 2.0))) * 1.0

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(Float64(pi / Float64(Float64(b * a) * Float64(Float64(b + a) * 2.0))) * 1.0)
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = (pi / ((b * a) * ((b + a) * 2.0))) * 1.0;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[(N[(Pi / N[(N[(b * a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]

Derivation

1. Initial program 78.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift--.f64 N/A

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

   6. associate-/l* N/A

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

   7. *-rgt-identity N/A

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

   8. difference-of-squares N/A

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

   9. associate-/r* N/A

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

   10. *-lft-identity N/A

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

   11. *-rgt-identity N/A

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

   12. lower-/.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. *-lft-identity N/A

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

   17. *-rgt-identity N/A

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

   18. lower--.f64 88.6

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

3. Applied rewrites 88.6%

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

5. Applied rewrites 98.9%

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

Alternative 5: 90.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 1.08 \cdot 10^{-72}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{b \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{b}}{\left(a + a\right) \cdot b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 1.08e-72) (* (/ (/ PI a) (* b a)) 0.5) (/ (/ PI b) (* (+ a a) b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 1.08e-72) {
		tmp = ((((double) M_PI) / a) / (b * a)) * 0.5;
	} else {
		tmp = (((double) M_PI) / b) / ((a + a) * b);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 1.08e-72) {
		tmp = ((Math.PI / a) / (b * a)) * 0.5;
	} else {
		tmp = (Math.PI / b) / ((a + a) * b);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 1.08e-72:
		tmp = ((math.pi / a) / (b * a)) * 0.5
	else:
		tmp = (math.pi / b) / ((a + a) * b)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 1.08e-72)
		tmp = Float64(Float64(Float64(pi / a) / Float64(b * a)) * 0.5);
	else
		tmp = Float64(Float64(pi / b) / Float64(Float64(a + a) * b));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.08e-72)
		tmp = ((pi / a) / (b * a)) * 0.5;
	else
		tmp = (pi / b) / ((a + 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, 1.08e-72], N[(N[(N[(Pi / a), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(Pi / b), $MachinePrecision] / N[(N[(a + a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.07999999999999998e-72

   1. Initial program 78.5%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 79.7

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

   4. Applied rewrites 79.7%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. frac-2neg N/A

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

      8. associate-/r* N/A

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

      9. pow2 N/A

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

      10. associate-/r* N/A

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

      11. associate-/l/ N/A

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

      12. lower-/.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-PI.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 91.5

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

   6. Applied rewrites 91.5%

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

   if 1.07999999999999998e-72 < b

   1. Initial program 78.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-rgt-identity N/A

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

      8. difference-of-squares N/A

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

      9. associate-/r* N/A

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

      10. *-lft-identity N/A

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

      11. *-rgt-identity N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. *-lft-identity N/A

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

      17. *-rgt-identity N/A

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

      18. lower--.f64 88.6

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

   3. Applied rewrites 88.6%

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

   5. Applied rewrites 98.7%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. *-rgt-identity N/A

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

      9. associate-*l* N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. count-2-rev N/A

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

      19. lower-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 99.4

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 89.2%

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

Alternative 6: 90.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 1.08 \cdot 10^{-72}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{b}}{\left(a + a\right) \cdot b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 1.08e-72) (* (/ PI a) (/ 0.5 (* b a))) (/ (/ PI b) (* (+ a a) b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 1.08e-72) {
		tmp = (((double) M_PI) / a) * (0.5 / (b * a));
	} else {
		tmp = (((double) M_PI) / b) / ((a + a) * b);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 1.08e-72) {
		tmp = (Math.PI / a) * (0.5 / (b * a));
	} else {
		tmp = (Math.PI / b) / ((a + a) * b);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 1.08e-72:
		tmp = (math.pi / a) * (0.5 / (b * a))
	else:
		tmp = (math.pi / b) / ((a + a) * b)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 1.08e-72)
		tmp = Float64(Float64(pi / a) * Float64(0.5 / Float64(b * a)));
	else
		tmp = Float64(Float64(pi / b) / Float64(Float64(a + a) * b));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.08e-72)
		tmp = (pi / a) * (0.5 / (b * a));
	else
		tmp = (pi / b) / ((a + 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, 1.08e-72], N[(N[(Pi / a), $MachinePrecision] * N[(0.5 / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / b), $MachinePrecision] / N[(N[(a + a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.07999999999999998e-72

   1. Initial program 78.5%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 79.7

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

   4. Applied rewrites 79.7%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*l* N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. lift-PI.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 91.5

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

   6. Applied rewrites 91.5%

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

   if 1.07999999999999998e-72 < b

   1. Initial program 78.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-rgt-identity N/A

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

      8. difference-of-squares N/A

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

      9. associate-/r* N/A

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

      10. *-lft-identity N/A

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

      11. *-rgt-identity N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. *-lft-identity N/A

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

      17. *-rgt-identity N/A

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

      18. lower--.f64 88.6

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

   3. Applied rewrites 88.6%

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

   5. Applied rewrites 98.7%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. *-rgt-identity N/A

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

      9. associate-*l* N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. count-2-rev N/A

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

      19. lower-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 99.4

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 89.2%

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

Alternative 7: 90.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 1.08 \cdot 10^{-72}:\\ \;\;\;\;\pi \cdot \frac{\frac{0.5}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{b}}{\left(a + a\right) \cdot b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 1.08e-72) (* PI (/ (/ 0.5 a) (* a b))) (/ (/ PI b) (* (+ a a) b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 1.08e-72) {
		tmp = ((double) M_PI) * ((0.5 / a) / (a * b));
	} else {
		tmp = (((double) M_PI) / b) / ((a + a) * b);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 1.08e-72) {
		tmp = Math.PI * ((0.5 / a) / (a * b));
	} else {
		tmp = (Math.PI / b) / ((a + a) * b);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 1.08e-72:
		tmp = math.pi * ((0.5 / a) / (a * b))
	else:
		tmp = (math.pi / b) / ((a + a) * b)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 1.08e-72)
		tmp = Float64(pi * Float64(Float64(0.5 / a) / Float64(a * b)));
	else
		tmp = Float64(Float64(pi / b) / Float64(Float64(a + a) * b));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.08e-72)
		tmp = pi * ((0.5 / a) / (a * b));
	else
		tmp = (pi / b) / ((a + 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, 1.08e-72], N[(Pi * N[(N[(0.5 / a), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / b), $MachinePrecision] / N[(N[(a + a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.07999999999999998e-72

   1. Initial program 78.5%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 79.7

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

   4. Applied rewrites 79.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 91.0

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

   6. Applied rewrites 91.0%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. frac-times N/A

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

      8. associate-*l/ N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-/.f64 91.4

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 91.4

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

   8. Applied rewrites 91.4%

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

   if 1.07999999999999998e-72 < b

   1. Initial program 78.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-rgt-identity N/A

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

      8. difference-of-squares N/A

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

      9. associate-/r* N/A

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

      10. *-lft-identity N/A

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

      11. *-rgt-identity N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. *-lft-identity N/A

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

      17. *-rgt-identity N/A

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

      18. lower--.f64 88.6

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

   3. Applied rewrites 88.6%

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

   5. Applied rewrites 98.7%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. *-rgt-identity N/A

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

      9. associate-*l* N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. count-2-rev N/A

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

      19. lower-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 99.4

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 89.2%

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

Alternative 8: 90.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 1.08 \cdot 10^{-72}:\\ \;\;\;\;\frac{\pi}{a \cdot \left(b \cdot a\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{b}}{\left(a + a\right) \cdot b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 1.08e-72) (* (/ PI (* a (* b a))) 0.5) (/ (/ PI b) (* (+ a a) b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 1.08e-72) {
		tmp = (((double) M_PI) / (a * (b * a))) * 0.5;
	} else {
		tmp = (((double) M_PI) / b) / ((a + a) * b);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 1.08e-72) {
		tmp = (Math.PI / (a * (b * a))) * 0.5;
	} else {
		tmp = (Math.PI / b) / ((a + a) * b);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 1.08e-72:
		tmp = (math.pi / (a * (b * a))) * 0.5
	else:
		tmp = (math.pi / b) / ((a + a) * b)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 1.08e-72)
		tmp = Float64(Float64(pi / Float64(a * Float64(b * a))) * 0.5);
	else
		tmp = Float64(Float64(pi / b) / Float64(Float64(a + a) * b));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.08e-72)
		tmp = (pi / (a * (b * a))) * 0.5;
	else
		tmp = (pi / b) / ((a + 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, 1.08e-72], N[(N[(Pi / N[(a * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(Pi / b), $MachinePrecision] / N[(N[(a + a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.07999999999999998e-72

   1. Initial program 78.5%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 79.7

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

   4. Applied rewrites 79.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 91.0

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

   6. Applied rewrites 91.0%

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

   if 1.07999999999999998e-72 < b

   1. Initial program 78.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-rgt-identity N/A

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

      8. difference-of-squares N/A

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

      9. associate-/r* N/A

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

      10. *-lft-identity N/A

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

      11. *-rgt-identity N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. *-lft-identity N/A

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

      17. *-rgt-identity N/A

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

      18. lower--.f64 88.6

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

   3. Applied rewrites 88.6%

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

   5. Applied rewrites 98.7%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. *-rgt-identity N/A

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

      9. associate-*l* N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. count-2-rev N/A

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

      19. lower-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 99.4

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 89.2%

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

Alternative 9: 84.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 1.08 \cdot 10^{-72}:\\ \;\;\;\;\frac{\pi}{a \cdot \left(b \cdot a\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 1.08e-72)
   (* (/ PI (* a (* b a))) 0.5)
   (* (/ PI (* (* b b) a)) 0.5)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 1.08e-72) {
		tmp = (((double) M_PI) / (a * (b * a))) * 0.5;
	} else {
		tmp = (((double) M_PI) / ((b * b) * a)) * 0.5;
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 1.08e-72) {
		tmp = (Math.PI / (a * (b * a))) * 0.5;
	} else {
		tmp = (Math.PI / ((b * b) * a)) * 0.5;
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 1.08e-72:
		tmp = (math.pi / (a * (b * a))) * 0.5
	else:
		tmp = (math.pi / ((b * b) * a)) * 0.5
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 1.08e-72)
		tmp = Float64(Float64(pi / Float64(a * Float64(b * a))) * 0.5);
	else
		tmp = Float64(Float64(pi / Float64(Float64(b * b) * a)) * 0.5);
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.08e-72)
		tmp = (pi / (a * (b * a))) * 0.5;
	else
		tmp = (pi / ((b * b) * a)) * 0.5;
	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, 1.08e-72], N[(N[(Pi / N[(a * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(Pi / N[(N[(b * b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.07999999999999998e-72

   1. Initial program 78.5%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 79.7

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

   4. Applied rewrites 79.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 91.0

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

   6. Applied rewrites 91.0%

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

   if 1.07999999999999998e-72 < b

   1. Initial program 78.7%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 77.7

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

   4. Applied rewrites 77.7%

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

Alternative 10: 62.8% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.6%

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

2. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 57.2

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

4. Applied rewrites 57.2%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 62.8

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

6. Applied rewrites 62.8%

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

Reproduce

herbie shell --seed 2025130 
(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))))