NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.8% → 99.7%

Time: 3.4s

Alternatives: 10

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -8.8 \cdot 10^{+85}:\\ \;\;\;\;\frac{\pi}{a \cdot b} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{b \cdot \left(a \cdot \left(a + b\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -8.8e+85)
   (* (/ PI (* a b)) (/ 0.5 a))
   (/ (* 0.5 PI) (* b (* a (+ a b))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -8.8e+85) {
		tmp = (((double) M_PI) / (a * b)) * (0.5 / a);
	} else {
		tmp = (0.5 * ((double) M_PI)) / (b * (a * (a + b)));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -8.8e+85) {
		tmp = (Math.PI / (a * b)) * (0.5 / a);
	} else {
		tmp = (0.5 * Math.PI) / (b * (a * (a + b)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -8.8e+85:
		tmp = (math.pi / (a * b)) * (0.5 / a)
	else:
		tmp = (0.5 * math.pi) / (b * (a * (a + b)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -8.8e+85)
		tmp = Float64(Float64(pi / Float64(a * b)) * Float64(0.5 / a));
	else
		tmp = Float64(Float64(0.5 * pi) / Float64(b * Float64(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 <= -8.8e+85)
		tmp = (pi / (a * b)) * (0.5 / a);
	else
		tmp = (0.5 * 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[a, -8.8e+85], N[(N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * Pi), $MachinePrecision] / N[(b * N[(a * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -8.8000000000000007e85

   1. Initial program 78.8%

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

   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. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-PI.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 56.7

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

   4. Applied rewrites 56.7%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 62.3

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 62.3

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

   6. Applied rewrites 62.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 62.4

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

   8. Applied rewrites 62.4%

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

   if -8.8000000000000007e85 < a

   1. Initial program 78.8%

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

      1. 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. mult-flip-rev N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. difference-of-squares N/A

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

      8. associate-/r* N/A

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

      9. *-lft-identity N/A

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

      10. /-rgt-identity N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. div-flip N/A

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

      17. associate-/r/ N/A

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

      18. lower-*.f64 N/A

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

      19. metadata-eval N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. *-lft-identity N/A

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

      23. *-rgt-identity N/A

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

   3. Applied rewrites 88.6%

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

   5. Applied rewrites 99.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. div-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. distribute-rgt-out N/A

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

      15. lower-*.f64 N/A

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

      16. distribute-rgt-out N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 N/A

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

      19. lower-*.f64 93.5

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-+.f64 93.5

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

   7. Applied rewrites 93.5%

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

Alternative 2: 99.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.8%

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

   1. 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. mult-flip-rev N/A

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

   4. lift--.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. difference-of-squares N/A

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

   8. associate-/r* N/A

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

   9. *-lft-identity N/A

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

   10. /-rgt-identity N/A

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

   11. mult-flip N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lift-/.f64 N/A

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

   16. div-flip N/A

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

   17. associate-/r/ N/A

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

   18. lower-*.f64 N/A

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

   19. metadata-eval N/A

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

   20. +-commutative N/A

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

   21. lower-+.f64 N/A

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

   22. *-lft-identity N/A

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

   23. *-rgt-identity N/A

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

3. Applied rewrites 88.6%

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

5. Applied rewrites 99.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lift-/.f64 N/A

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

   5. div-flip-rev N/A

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

   6. lower-/.f64 N/A

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

   7. div-flip-rev N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/r* N/A

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

   10. associate-/r/ N/A

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

   11. div-flip N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 99.7

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 99.7

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

   17. lift-+.f64 N/A

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

   18. +-commutative N/A

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

   19. lower-+.f64 99.7

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

7. Applied rewrites 99.7%

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

Alternative 3: 99.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.8%

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

   1. 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. mult-flip-rev N/A

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

   4. lift--.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. difference-of-squares N/A

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

   8. associate-/r* N/A

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

   9. *-lft-identity N/A

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

   10. /-rgt-identity N/A

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

   11. mult-flip N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lift-/.f64 N/A

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

   16. div-flip N/A

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

   17. associate-/r/ N/A

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

   18. lower-*.f64 N/A

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

   19. metadata-eval N/A

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

   20. +-commutative N/A

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

   21. lower-+.f64 N/A

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

   22. *-lft-identity N/A

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

   23. *-rgt-identity N/A

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

3. Applied rewrites 88.6%

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

5. Applied rewrites 99.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lift-/.f64 N/A

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

   5. div-flip-rev N/A

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

   6. lower-/.f64 N/A

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

   7. div-flip-rev N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/r* N/A

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

   10. associate-/r/ N/A

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

   11. div-flip N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 99.7

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 99.7

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

   17. lift-+.f64 N/A

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

   18. +-commutative N/A

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

   19. lower-+.f64 99.7

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

7. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lift-/.f64 N/A

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

   5. frac-times N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 99.1

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 99.1

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 99.1

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

9. Applied rewrites 99.1%

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

Alternative 4: 89.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -7 \cdot 10^{-94}:\\ \;\;\;\;\frac{\pi}{a \cdot b} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\frac{\pi}{b}}{a}}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -7e-94) (* (/ PI (* a b)) (/ 0.5 a)) (* 0.5 (/ (/ (/ PI b) a) b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -7e-94) {
		tmp = (((double) M_PI) / (a * b)) * (0.5 / a);
	} else {
		tmp = 0.5 * (((((double) M_PI) / b) / a) / b);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -7e-94) {
		tmp = (Math.PI / (a * b)) * (0.5 / a);
	} else {
		tmp = 0.5 * (((Math.PI / b) / a) / b);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -7e-94:
		tmp = (math.pi / (a * b)) * (0.5 / a)
	else:
		tmp = 0.5 * (((math.pi / b) / a) / b)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -7e-94)
		tmp = Float64(Float64(pi / Float64(a * b)) * Float64(0.5 / a));
	else
		tmp = Float64(0.5 * Float64(Float64(Float64(pi / b) / a) / b));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -7e-94)
		tmp = (pi / (a * b)) * (0.5 / a);
	else
		tmp = 0.5 * (((pi / b) / a) / b);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -7e-94], N[(N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(N[(Pi / b), $MachinePrecision] / a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.99999999999999996e-94

   1. Initial program 78.8%

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

   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. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-PI.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 56.7

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

   4. Applied rewrites 56.7%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 62.3

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 62.3

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

   6. Applied rewrites 62.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 62.4

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

   8. Applied rewrites 62.4%

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

   if -6.99999999999999996e-94 < a

   1. Initial program 78.8%

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

      1. 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. mult-flip-rev N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. difference-of-squares N/A

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

      8. associate-/r* N/A

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

      9. *-lft-identity N/A

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

      10. /-rgt-identity N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. div-flip N/A

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

      17. associate-/r/ N/A

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

      18. lower-*.f64 N/A

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

      19. metadata-eval N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. *-lft-identity N/A

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

      23. *-rgt-identity N/A

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

   3. Applied rewrites 88.6%

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

   4. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-PI.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 57.2

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

   6. Applied rewrites 57.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. associate-/r* N/A

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

      7. associate-/r* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 63.0

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 63.0

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

   8. Applied rewrites 63.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 63.0

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

   10. Applied rewrites 63.0%

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

Alternative 5: 89.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -7 \cdot 10^{-94}:\\ \;\;\;\;\frac{\pi}{a \cdot b} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{0.5}{b \cdot a}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -7e-94) (* (/ PI (* a b)) (/ 0.5 a)) (* (/ PI b) (/ 0.5 (* b a)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -7e-94) {
		tmp = (((double) M_PI) / (a * b)) * (0.5 / a);
	} else {
		tmp = (((double) M_PI) / b) * (0.5 / (b * a));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -7e-94) {
		tmp = (Math.PI / (a * b)) * (0.5 / a);
	} else {
		tmp = (Math.PI / b) * (0.5 / (b * a));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -7e-94:
		tmp = (math.pi / (a * b)) * (0.5 / a)
	else:
		tmp = (math.pi / b) * (0.5 / (b * a))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -7e-94)
		tmp = Float64(Float64(pi / Float64(a * b)) * Float64(0.5 / a));
	else
		tmp = Float64(Float64(pi / b) * Float64(0.5 / Float64(b * a)));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -7e-94)
		tmp = (pi / (a * b)) * (0.5 / a);
	else
		tmp = (pi / b) * (0.5 / (b * a));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -7e-94], N[(N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / b), $MachinePrecision] * N[(0.5 / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.99999999999999996e-94

   1. Initial program 78.8%

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

   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. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-PI.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 56.7

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

   4. Applied rewrites 56.7%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 62.3

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 62.3

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

   6. Applied rewrites 62.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 62.4

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

   8. Applied rewrites 62.4%

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

   if -6.99999999999999996e-94 < a

   1. Initial program 78.8%

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

      1. 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. mult-flip-rev N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. difference-of-squares N/A

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

      8. associate-/r* N/A

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

      9. *-lft-identity N/A

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

      10. /-rgt-identity N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. div-flip N/A

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

      17. associate-/r/ N/A

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

      18. lower-*.f64 N/A

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

      19. metadata-eval N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. *-lft-identity N/A

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

      23. *-rgt-identity N/A

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

   3. Applied rewrites 88.6%

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

   4. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-PI.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 57.2

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

   6. Applied rewrites 57.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lower-/.f64 57.2

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 57.2

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 57.2

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. lower-*.f64 57.2

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

   8. Applied rewrites 57.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 63.0

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 63.0

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

   10. Applied rewrites 63.0%

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

Alternative 6: 88.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -7 \cdot 10^{-94}:\\ \;\;\;\;\frac{\pi}{a \cdot b} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{\left(a \cdot b\right) \cdot b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -7e-94) (* (/ PI (* a b)) (/ 0.5 a)) (* 0.5 (/ PI (* (* a b) b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -7e-94) {
		tmp = (((double) M_PI) / (a * b)) * (0.5 / a);
	} else {
		tmp = 0.5 * (((double) M_PI) / ((a * b) * b));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -7e-94) {
		tmp = (Math.PI / (a * b)) * (0.5 / a);
	} else {
		tmp = 0.5 * (Math.PI / ((a * b) * b));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -7e-94:
		tmp = (math.pi / (a * b)) * (0.5 / a)
	else:
		tmp = 0.5 * (math.pi / ((a * b) * b))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -7e-94)
		tmp = Float64(Float64(pi / Float64(a * b)) * Float64(0.5 / a));
	else
		tmp = Float64(0.5 * Float64(pi / Float64(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 <= -7e-94)
		tmp = (pi / (a * b)) * (0.5 / a);
	else
		tmp = 0.5 * (pi / ((a * b) * b));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -7e-94], N[(N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(Pi / N[(N[(a * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.99999999999999996e-94

   1. Initial program 78.8%

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

   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. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-PI.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 56.7

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

   4. Applied rewrites 56.7%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 62.3

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 62.3

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

   6. Applied rewrites 62.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 62.4

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

   8. Applied rewrites 62.4%

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

   if -6.99999999999999996e-94 < a

   1. Initial program 78.8%

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

      1. 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. mult-flip-rev N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. difference-of-squares N/A

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

      8. associate-/r* N/A

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

      9. *-lft-identity N/A

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

      10. /-rgt-identity N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. div-flip N/A

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

      17. associate-/r/ N/A

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

      18. lower-*.f64 N/A

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

      19. metadata-eval N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. *-lft-identity N/A

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

      23. *-rgt-identity N/A

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

   3. Applied rewrites 88.6%

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

   4. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-PI.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 57.2

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

   6. Applied rewrites 57.2%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 62.8

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 62.8

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

   8. Applied rewrites 62.8%

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

Alternative 7: 88.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -7 \cdot 10^{-94}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{\left(a \cdot b\right) \cdot b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -7e-94) (* (/ PI a) (/ 0.5 (* a b))) (* 0.5 (/ PI (* (* a b) b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -7e-94) {
		tmp = (((double) M_PI) / a) * (0.5 / (a * b));
	} else {
		tmp = 0.5 * (((double) M_PI) / ((a * b) * b));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -7e-94) {
		tmp = (Math.PI / a) * (0.5 / (a * b));
	} else {
		tmp = 0.5 * (Math.PI / ((a * b) * b));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -7e-94:
		tmp = (math.pi / a) * (0.5 / (a * b))
	else:
		tmp = 0.5 * (math.pi / ((a * b) * b))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -7e-94)
		tmp = Float64(Float64(pi / a) * Float64(0.5 / Float64(a * b)));
	else
		tmp = Float64(0.5 * Float64(pi / Float64(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 <= -7e-94)
		tmp = (pi / a) * (0.5 / (a * b));
	else
		tmp = 0.5 * (pi / ((a * b) * b));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -7e-94], N[(N[(Pi / a), $MachinePrecision] * N[(0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(Pi / N[(N[(a * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.99999999999999996e-94

   1. Initial program 78.8%

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

   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. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-PI.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 56.7

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

   4. Applied rewrites 56.7%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 62.3

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 62.3

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

   6. Applied rewrites 62.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 62.4

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 62.4

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

   8. Applied rewrites 62.4%

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

   if -6.99999999999999996e-94 < a

   1. Initial program 78.8%

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

      1. 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. mult-flip-rev N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. difference-of-squares N/A

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

      8. associate-/r* N/A

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

      9. *-lft-identity N/A

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

      10. /-rgt-identity N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. div-flip N/A

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

      17. associate-/r/ N/A

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

      18. lower-*.f64 N/A

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

      19. metadata-eval N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. *-lft-identity N/A

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

      23. *-rgt-identity N/A

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

   3. Applied rewrites 88.6%

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

   4. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-PI.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 57.2

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

   6. Applied rewrites 57.2%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 62.8

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 62.8

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

   8. Applied rewrites 62.8%

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

Alternative 8: 88.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -7 \cdot 10^{-94}:\\ \;\;\;\;\frac{0.5}{\left(b \cdot a\right) \cdot a} \cdot \pi\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{\left(a \cdot b\right) \cdot b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -7e-94) (* (/ 0.5 (* (* b a) a)) PI) (* 0.5 (/ PI (* (* a b) b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -7e-94) {
		tmp = (0.5 / ((b * a) * a)) * ((double) M_PI);
	} else {
		tmp = 0.5 * (((double) M_PI) / ((a * b) * b));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -7e-94) {
		tmp = (0.5 / ((b * a) * a)) * Math.PI;
	} else {
		tmp = 0.5 * (Math.PI / ((a * b) * b));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -7e-94:
		tmp = (0.5 / ((b * a) * a)) * math.pi
	else:
		tmp = 0.5 * (math.pi / ((a * b) * b))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -7e-94)
		tmp = Float64(Float64(0.5 / Float64(Float64(b * a) * a)) * pi);
	else
		tmp = Float64(0.5 * Float64(pi / Float64(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 <= -7e-94)
		tmp = (0.5 / ((b * a) * a)) * pi;
	else
		tmp = 0.5 * (pi / ((a * b) * b));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -7e-94], N[(N[(0.5 / N[(N[(b * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision], N[(0.5 * N[(Pi / N[(N[(a * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.99999999999999996e-94

   1. Initial program 78.8%

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

   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. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-PI.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 56.7

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

   4. Applied rewrites 56.7%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 62.3

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 62.3

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

   6. Applied rewrites 62.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l/ N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 56.7

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

   8. Applied rewrites 56.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 62.3

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 62.3

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

   10. Applied rewrites 62.3%

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

   if -6.99999999999999996e-94 < a

   1. Initial program 78.8%

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

      1. 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. mult-flip-rev N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. difference-of-squares N/A

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

      8. associate-/r* N/A

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

      9. *-lft-identity N/A

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

      10. /-rgt-identity N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. div-flip N/A

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

      17. associate-/r/ N/A

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

      18. lower-*.f64 N/A

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

      19. metadata-eval N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. *-lft-identity N/A

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

      23. *-rgt-identity N/A

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

   3. Applied rewrites 88.6%

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

   4. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-PI.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 57.2

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

   6. Applied rewrites 57.2%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 62.8

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 62.8

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

   8. Applied rewrites 62.8%

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

Alternative 9: 62.3% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.8%

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

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. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-PI.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-pow.f64 56.7

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

4. Applied rewrites 56.7%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. lower-*.f64 62.3

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 62.3

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

6. Applied rewrites 62.3%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*l/ N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 56.7

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

8. Applied rewrites 56.7%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 62.3

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 62.3

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

10. Applied rewrites 62.3%

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

Alternative 10: 62.3% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.8%

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

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. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-PI.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-pow.f64 56.7

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

4. Applied rewrites 56.7%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. lower-*.f64 62.3

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 62.3

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

6. Applied rewrites 62.3%

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

Reproduce

herbie shell --seed 2025156 
(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))))