NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.4% → 99.3%

Time: 10.0s

Alternatives: 7

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 1.28 \cdot 10^{+137}:\\ \;\;\;\;\frac{\pi}{a \cdot \left(2 \cdot \left(b \cdot \left(b + a\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{b} \cdot \frac{0.5}{a}}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 1.28e+137)
   (/ PI (* a (* 2.0 (* b (+ b a)))))
   (/ (* (/ PI b) (/ 0.5 a)) b)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 1.28e+137) {
		tmp = ((double) M_PI) / (a * (2.0 * (b * (b + a))));
	} else {
		tmp = ((((double) M_PI) / b) * (0.5 / a)) / b;
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 1.28e+137) {
		tmp = Math.PI / (a * (2.0 * (b * (b + a))));
	} else {
		tmp = ((Math.PI / b) * (0.5 / a)) / b;
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 1.28e+137:
		tmp = math.pi / (a * (2.0 * (b * (b + a))))
	else:
		tmp = ((math.pi / b) * (0.5 / a)) / b
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 1.28e+137)
		tmp = Float64(pi / Float64(a * Float64(2.0 * Float64(b * Float64(b + a)))));
	else
		tmp = Float64(Float64(Float64(pi / b) * Float64(0.5 / a)) / b);
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.28e+137)
		tmp = pi / (a * (2.0 * (b * (b + a))));
	else
		tmp = ((pi / b) * (0.5 / a)) / b;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[b, 1.28e+137], N[(Pi / N[(a * N[(2.0 * N[(b * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi / b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.27999999999999995e137

   1. Initial program 84.9%

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

      1. associate-*l* N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. sub-neg N/A

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

      8. distribute-lft-out N/A

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

      9. div-inv N/A

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

      10. distribute-rgt-neg-out N/A

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

      11. div-inv N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

   4. Applied egg-rr 84.9%

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

   5. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. distribute-lft-out N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. distribute-rgt-out N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. distribute-lft-out N/A

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

      14. *-commutative N/A

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

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

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

      16. distribute-lft-out N/A

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

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

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

      18. unpow2 N/A

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

      19. distribute-rgt-out N/A

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

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

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

      21. +-lowering-+.f64 96.7%

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

   7. Simplified 96.7%

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

   if 1.27999999999999995e137 < b

   1. Initial program 70.0%

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

      1. associate-*l* N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. sub-neg N/A

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

      8. distribute-lft-out N/A

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

      9. div-inv N/A

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

      10. distribute-rgt-neg-out N/A

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

      11. div-inv N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

   4. Applied egg-rr 70.0%

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

   5. Taylor expanded in b around inf

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 75.7%

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

   7. Simplified 75.7%

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

      1. associate-/r* N/A

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

      2. *-commutative N/A

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

      3. div-inv N/A

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

      4. metadata-eval N/A

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

      5. frac-times N/A

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

      6. associate-/r* N/A

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

      7. associate-*l/ N/A

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

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

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

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

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

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

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

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

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

      12. /-lowering-/.f64 99.9%

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

   9. Applied egg-rr 99.9%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.28 \cdot 10^{+137}:\\ \;\;\;\;\frac{\pi}{a \cdot \left(2 \cdot \left(b \cdot \left(b + a\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{b} \cdot \frac{0.5}{a}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -4 \cdot 10^{-47}:\\ \;\;\;\;\frac{\frac{\pi}{\frac{a}{0.5}}}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{\frac{b}{0.5}}}{b \cdot a}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -4e-47) (/ (/ PI (/ a 0.5)) (* b a)) (/ (/ PI (/ b 0.5)) (* b a))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -4e-47) {
		tmp = (((double) M_PI) / (a / 0.5)) / (b * 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 <= -4e-47) {
		tmp = (Math.PI / (a / 0.5)) / (b * 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 <= -4e-47:
		tmp = (math.pi / (a / 0.5)) / (b * 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 <= -4e-47)
		tmp = Float64(Float64(pi / Float64(a / 0.5)) / Float64(b * a));
	else
		tmp = Float64(Float64(pi / Float64(b / 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 <= -4e-47)
		tmp = (pi / (a / 0.5)) / (b * 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, -4e-47], N[(N[(Pi / N[(a / 0.5), $MachinePrecision]), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / N[(b / 0.5), $MachinePrecision]), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.9999999999999999e-47

   1. Initial program 81.8%

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

   3. Taylor expanded in b around 0

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

      1. associate-/r* N/A

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

      2. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 81.0%

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

   5. Simplified 81.0%

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

      1. associate-*r/ N/A

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

      2. associate-/r* N/A

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

      3. metadata-eval N/A

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

      4. associate-/r/ N/A

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

      5. clear-num N/A

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

      6. associate-/l/ N/A

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

      7. *-commutative N/A

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

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

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

      9. associate-/l/ N/A

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

      10. metadata-eval N/A

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

      11. div-inv N/A

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

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

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

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

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

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

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 89.4%

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

   7. Applied egg-rr 89.4%

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

   if -3.9999999999999999e-47 < a

   1. Initial program 83.3%

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

      1. associate-*l* N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. sub-neg N/A

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

      8. distribute-lft-out N/A

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

      9. div-inv N/A

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

      10. distribute-rgt-neg-out N/A

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

      11. div-inv N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

   4. Applied egg-rr 83.3%

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

   5. Taylor expanded in b around inf

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 57.1%

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

   7. Simplified 57.1%

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

      1. *-un-lft-identity N/A

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

      2. times-frac N/A

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

      3. metadata-eval N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-/l/ N/A

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

      7. associate-*r/ N/A

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

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

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

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

      12. div-inv N/A

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

      13. associate-/l/ N/A

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

      14. *-commutative N/A

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

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

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

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

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

      17. *-commutative N/A

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

      18. metadata-eval N/A

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

      19. div-inv N/A

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

      20. /-lowering-/.f64 N/A

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

      21. *-commutative N/A

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

      22. *-lowering-*.f64 64.6%

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

   9. Applied egg-rr 64.6%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4 \cdot 10^{-47}:\\ \;\;\;\;\frac{\frac{\pi}{\frac{a}{0.5}}}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{\frac{b}{0.5}}}{b \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -5.6 \cdot 10^{-47}:\\ \;\;\;\;\frac{\frac{\pi}{\frac{a}{0.5}}}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{b} \cdot \frac{0.5}{a}}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -5.6e-47)
   (/ (/ PI (/ a 0.5)) (* b a))
   (/ (* (/ PI b) (/ 0.5 a)) b)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -5.6e-47) {
		tmp = (((double) M_PI) / (a / 0.5)) / (b * a);
	} else {
		tmp = ((((double) M_PI) / b) * (0.5 / a)) / b;
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -5.6e-47) {
		tmp = (Math.PI / (a / 0.5)) / (b * a);
	} else {
		tmp = ((Math.PI / b) * (0.5 / a)) / b;
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -5.6e-47:
		tmp = (math.pi / (a / 0.5)) / (b * a)
	else:
		tmp = ((math.pi / b) * (0.5 / a)) / b
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -5.6e-47)
		tmp = Float64(Float64(pi / Float64(a / 0.5)) / Float64(b * a));
	else
		tmp = Float64(Float64(Float64(pi / b) * Float64(0.5 / a)) / b);
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -5.6e-47)
		tmp = (pi / (a / 0.5)) / (b * a);
	else
		tmp = ((pi / b) * (0.5 / a)) / b;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -5.6e-47], N[(N[(Pi / N[(a / 0.5), $MachinePrecision]), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi / b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.59999999999999986e-47

   1. Initial program 81.8%

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

   3. Taylor expanded in b around 0

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

      1. associate-/r* N/A

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

      2. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 81.0%

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

   5. Simplified 81.0%

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

      1. associate-*r/ N/A

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

      2. associate-/r* N/A

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

      3. metadata-eval N/A

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

      4. associate-/r/ N/A

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

      5. clear-num N/A

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

      6. associate-/l/ N/A

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

      7. *-commutative N/A

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

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

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

      9. associate-/l/ N/A

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

      10. metadata-eval N/A

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

      11. div-inv N/A

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

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

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

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

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

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

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 89.4%

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

   7. Applied egg-rr 89.4%

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

   if -5.59999999999999986e-47 < a

   1. Initial program 83.3%

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

      1. associate-*l* N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. sub-neg N/A

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

      8. distribute-lft-out N/A

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

      9. div-inv N/A

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

      10. distribute-rgt-neg-out N/A

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

      11. div-inv N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

   4. Applied egg-rr 83.3%

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

   5. Taylor expanded in b around inf

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 57.1%

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

   7. Simplified 57.1%

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

      1. associate-/r* N/A

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

      2. *-commutative N/A

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

      3. div-inv N/A

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

      4. metadata-eval N/A

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

      5. frac-times N/A

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

      6. associate-/r* N/A

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

      7. associate-*l/ N/A

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

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

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

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

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

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

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

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

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

      12. /-lowering-/.f64 64.6%

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

   9. Applied egg-rr 64.6%

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

Alternative 4: 89.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} t_0 := \frac{\pi}{b} \cdot \frac{0.5}{a}\\ \mathbf{if}\;a \leq -5.6 \cdot 10^{-47}:\\ \;\;\;\;\frac{t\_0}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (/ PI b) (/ 0.5 a))))
   (if (<= a -5.6e-47) (/ t_0 a) (/ t_0 b))))

C

assert(a < b);
double code(double a, double b) {
	double t_0 = (((double) M_PI) / b) * (0.5 / a);
	double tmp;
	if (a <= -5.6e-47) {
		tmp = t_0 / a;
	} else {
		tmp = t_0 / b;
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double t_0 = (Math.PI / b) * (0.5 / a);
	double tmp;
	if (a <= -5.6e-47) {
		tmp = t_0 / a;
	} else {
		tmp = t_0 / b;
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	t_0 = (math.pi / b) * (0.5 / a)
	tmp = 0
	if a <= -5.6e-47:
		tmp = t_0 / a
	else:
		tmp = t_0 / b
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	t_0 = Float64(Float64(pi / b) * Float64(0.5 / a))
	tmp = 0.0
	if (a <= -5.6e-47)
		tmp = Float64(t_0 / a);
	else
		tmp = Float64(t_0 / b);
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	t_0 = (pi / b) * (0.5 / a);
	tmp = 0.0;
	if (a <= -5.6e-47)
		tmp = t_0 / a;
	else
		tmp = t_0 / b;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := Block[{t$95$0 = N[(N[(Pi / b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -5.6e-47], N[(t$95$0 / a), $MachinePrecision], N[(t$95$0 / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -5.59999999999999986e-47

   1. Initial program 81.8%

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

   3. Taylor expanded in b around 0

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

      1. associate-/r* N/A

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

      2. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 81.0%

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

   5. Simplified 81.0%

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

      1. associate-*r/ N/A

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

      2. associate-/r* N/A

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

      3. metadata-eval N/A

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

      4. associate-/r/ N/A

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

      5. clear-num N/A

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

      6. associate-/l/ N/A

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

      7. *-commutative N/A

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

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

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

      9. associate-/l/ N/A

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

      10. metadata-eval N/A

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

      11. div-inv N/A

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

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

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

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

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

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

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 89.4%

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

   7. Applied egg-rr 89.4%

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

      1. div-inv N/A

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

      2. times-frac N/A

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

      3. clear-num N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      9. /-lowering-/.f64 89.2%

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

   9. Applied egg-rr 89.2%

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

   if -5.59999999999999986e-47 < a

   1. Initial program 83.3%

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

      1. associate-*l* N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. sub-neg N/A

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

      8. distribute-lft-out N/A

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

      9. div-inv N/A

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

      10. distribute-rgt-neg-out N/A

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

      11. div-inv N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

   4. Applied egg-rr 83.3%

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

   5. Taylor expanded in b around inf

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 57.1%

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

   7. Simplified 57.1%

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

      1. associate-/r* N/A

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

      2. *-commutative N/A

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

      3. div-inv N/A

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

      4. metadata-eval N/A

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

      5. frac-times N/A

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

      6. associate-/r* N/A

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

      7. associate-*l/ N/A

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

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

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

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

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

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

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

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

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

      12. /-lowering-/.f64 64.6%

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

   9. Applied egg-rr 64.6%

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

Alternative 5: 89.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -5.5 \cdot 10^{-47}:\\ \;\;\;\;\frac{\frac{\pi}{b} \cdot \frac{0.5}{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b} \cdot \frac{\pi}{b \cdot a}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -5.5e-47)
   (/ (* (/ PI b) (/ 0.5 a)) a)
   (* (/ 0.5 b) (/ PI (* b a)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -5.5e-47) {
		tmp = ((((double) M_PI) / b) * (0.5 / a)) / a;
	} else {
		tmp = (0.5 / b) * (((double) M_PI) / (b * a));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -5.5e-47) {
		tmp = ((Math.PI / b) * (0.5 / a)) / a;
	} else {
		tmp = (0.5 / b) * (Math.PI / (b * a));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -5.5e-47:
		tmp = ((math.pi / b) * (0.5 / a)) / a
	else:
		tmp = (0.5 / b) * (math.pi / (b * a))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -5.5e-47)
		tmp = Float64(Float64(Float64(pi / b) * Float64(0.5 / a)) / a);
	else
		tmp = Float64(Float64(0.5 / b) * Float64(pi / 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 <= -5.5e-47)
		tmp = ((pi / b) * (0.5 / a)) / a;
	else
		tmp = (0.5 / b) * (pi / (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, -5.5e-47], N[(N[(N[(Pi / b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(0.5 / b), $MachinePrecision] * N[(Pi / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.5000000000000002e-47

   1. Initial program 81.8%

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

   3. Taylor expanded in b around 0

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

      1. associate-/r* N/A

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

      2. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 81.0%

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

   5. Simplified 81.0%

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

      1. associate-*r/ N/A

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

      2. associate-/r* N/A

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

      3. metadata-eval N/A

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

      4. associate-/r/ N/A

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

      5. clear-num N/A

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

      6. associate-/l/ N/A

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

      7. *-commutative N/A

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

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

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

      9. associate-/l/ N/A

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

      10. metadata-eval N/A

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

      11. div-inv N/A

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

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

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

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

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

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

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 89.4%

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

   7. Applied egg-rr 89.4%

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

      1. div-inv N/A

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

      2. times-frac N/A

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

      3. clear-num N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      9. /-lowering-/.f64 89.2%

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

   9. Applied egg-rr 89.2%

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

   if -5.5000000000000002e-47 < a

   1. Initial program 83.3%

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

      1. associate-*l* N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. sub-neg N/A

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

      8. distribute-lft-out N/A

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

      9. div-inv N/A

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

      10. distribute-rgt-neg-out N/A

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

      11. div-inv N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

   4. Applied egg-rr 83.3%

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

   5. Taylor expanded in b around inf

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 57.1%

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

   7. Simplified 57.1%

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

      1. associate-/r* N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. associate-/l/ N/A

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

      6. times-frac N/A

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

      7. associate-/l* N/A

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

      8. un-div-inv N/A

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

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

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

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

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

      11. frac-times N/A

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

      12. *-rgt-identity N/A

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

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

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

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

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 64.5%

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

   9. Applied egg-rr 64.5%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.5 \cdot 10^{-47}:\\ \;\;\;\;\frac{\frac{\pi}{b} \cdot \frac{0.5}{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b} \cdot \frac{\pi}{b \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -2.8 \cdot 10^{-47}:\\ \;\;\;\;\frac{\pi}{a \cdot \left(2 \cdot \left(b \cdot a\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b} \cdot \frac{\pi}{b \cdot a}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -2.8e-47)
   (/ PI (* a (* 2.0 (* b a))))
   (* (/ 0.5 b) (/ PI (* b a)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -2.8e-47) {
		tmp = ((double) M_PI) / (a * (2.0 * (b * a)));
	} else {
		tmp = (0.5 / b) * (((double) M_PI) / (b * a));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -2.8e-47) {
		tmp = Math.PI / (a * (2.0 * (b * a)));
	} else {
		tmp = (0.5 / b) * (Math.PI / (b * a));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -2.8e-47:
		tmp = math.pi / (a * (2.0 * (b * a)))
	else:
		tmp = (0.5 / b) * (math.pi / (b * a))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -2.8e-47)
		tmp = Float64(pi / Float64(a * Float64(2.0 * Float64(b * a))));
	else
		tmp = Float64(Float64(0.5 / b) * Float64(pi / 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 <= -2.8e-47)
		tmp = pi / (a * (2.0 * (b * a)));
	else
		tmp = (0.5 / b) * (pi / (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, -2.8e-47], N[(Pi / N[(a * N[(2.0 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / b), $MachinePrecision] * N[(Pi / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.79999999999999993e-47

   1. Initial program 81.8%

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

      1. associate-*l* N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. sub-neg N/A

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

      8. distribute-lft-out N/A

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

      9. div-inv N/A

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

      10. distribute-rgt-neg-out N/A

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

      11. div-inv N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

   4. Applied egg-rr 81.8%

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

   5. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

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

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

      8. *-lowering-*.f64 88.8%

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

   7. Simplified 88.8%

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

   if -2.79999999999999993e-47 < a

   1. Initial program 83.3%

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

      1. associate-*l* N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. sub-neg N/A

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

      8. distribute-lft-out N/A

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

      9. div-inv N/A

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

      10. distribute-rgt-neg-out N/A

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

      11. div-inv N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

   4. Applied egg-rr 83.3%

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

   5. Taylor expanded in b around inf

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 57.1%

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

   7. Simplified 57.1%

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

      1. associate-/r* N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. associate-/l/ N/A

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

      6. times-frac N/A

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

      7. associate-/l* N/A

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

      8. un-div-inv N/A

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

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

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

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

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

      11. frac-times N/A

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

      12. *-rgt-identity N/A

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

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

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

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

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 64.5%

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

   9. Applied egg-rr 64.5%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.8 \cdot 10^{-47}:\\ \;\;\;\;\frac{\pi}{a \cdot \left(2 \cdot \left(b \cdot a\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b} \cdot \frac{\pi}{b \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.4% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = (0.5 / b) * (pi / (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 / b), $MachinePrecision] * N[(Pi / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.9%

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

   1. associate-*l* N/A

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

   2. div-inv N/A

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

   3. metadata-eval N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. sub-neg N/A

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

   8. distribute-lft-out N/A

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

   9. div-inv N/A

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

   10. distribute-rgt-neg-out N/A

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

   11. div-inv N/A

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

   12. distribute-neg-frac N/A

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

   13. metadata-eval N/A

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

4. Applied egg-rr 82.9%

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

5. Taylor expanded in b around inf

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

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

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

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

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

   3. unpow2 N/A

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

   4. *-lowering-*.f64 54.0%

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

7. Simplified 54.0%

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

   1. associate-/r* N/A

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

   2. div-inv N/A

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

   3. metadata-eval N/A

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

   4. *-commutative N/A

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

   5. associate-/l/ N/A

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

   6. times-frac N/A

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

   7. associate-/l* N/A

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

   8. un-div-inv N/A

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

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

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

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

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

   11. frac-times N/A

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

   12. *-rgt-identity N/A

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

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

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

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

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

   15. *-commutative N/A

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

   16. *-lowering-*.f64 59.3%

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

9. Applied egg-rr 59.3%

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

10. Final simplification 59.3%

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

Reproduce

herbie shell --seed 2024164 
(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))))