NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.8% → 99.6%

Time: 10.2s

Alternatives: 13

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{+104}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a \cdot \left(a + b\right)} \cdot \frac{\pi}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -5e+104)
   (/ (* 0.5 (/ PI a)) (* a b))
   (* (/ 0.5 (* a (+ a b))) (/ PI b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -5e+104) {
		tmp = (0.5 * (((double) M_PI) / a)) / (a * b);
	} else {
		tmp = (0.5 / (a * (a + b))) * (((double) M_PI) / b);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -5e+104) {
		tmp = (0.5 * (Math.PI / a)) / (a * b);
	} else {
		tmp = (0.5 / (a * (a + b))) * (Math.PI / b);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -5e+104:
		tmp = (0.5 * (math.pi / a)) / (a * b)
	else:
		tmp = (0.5 / (a * (a + b))) * (math.pi / b)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -5e+104)
		tmp = Float64(Float64(0.5 * Float64(pi / a)) / Float64(a * b));
	else
		tmp = Float64(Float64(0.5 / Float64(a * Float64(a + b))) * Float64(pi / b));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -5e+104)
		tmp = (0.5 * (pi / a)) / (a * b);
	else
		tmp = (0.5 / (a * (a + b))) * (pi / b);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -5e+104], N[(N[(0.5 * N[(Pi / a), $MachinePrecision]), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / N[(a * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(Pi / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.9999999999999997e104

   1. Initial program 78.2%

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

      1. associate-*l* 78.2%

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

      2. *-rgt-identity 78.2%

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

      3. associate-/l* 78.2%

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

      4. metadata-eval 78.2%

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

      5. associate-*l/ 78.2%

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

      6. *-lft-identity 78.2%

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

      7. sub-neg 78.2%

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

      8. distribute-neg-frac 78.2%

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

      9. metadata-eval 78.2%

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

   3. Simplified 78.2%

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

      1. metadata-eval 78.2%

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

      2. div-inv 78.2%

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

      3. clear-num 78.2%

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

      4. clear-num 78.2%

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

      5. frac-times 78.2%

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

      6. metadata-eval 78.2%

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

      7. frac-add 78.1%

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

      8. associate-/r/ 78.1%

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

      9. *-un-lft-identity 78.1%

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

      10. *-commutative 78.1%

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

      11. neg-mul-1 78.1%

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

      12. sub-neg 78.1%

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

      13. flip-+ 99.7%

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

      14. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. inv-pow 99.7%

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

      2. associate-*r* 99.8%

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

      3. unpow-prod-down 99.7%

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

      4. pow-prod-down 99.8%

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

      5. inv-pow 99.8%

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

      6. inv-pow 99.8%

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

      7. frac-times 99.7%

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

      8. metadata-eval 99.7%

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

   8. Applied egg-rr 99.7%

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

      1. associate-*r/ 99.8%

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

      2. *-rgt-identity 99.8%

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

      3. unpow-1 99.8%

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

      4. *-commutative 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in a around inf 100.0%

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

   if -4.9999999999999997e104 < a

   1. Initial program 78.3%

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

      1. associate-*l* 78.3%

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

      2. *-rgt-identity 78.3%

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

      3. associate-/l* 78.3%

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

      4. metadata-eval 78.3%

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

      5. associate-*l/ 78.3%

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

      6. *-lft-identity 78.3%

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

      7. sub-neg 78.3%

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

      8. distribute-neg-frac 78.3%

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

      9. metadata-eval 78.3%

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

   3. Simplified 78.3%

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

      1. metadata-eval 78.3%

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

      2. div-inv 78.3%

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

      3. clear-num 78.3%

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

      4. clear-num 78.0%

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

      5. frac-times 78.0%

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

      6. metadata-eval 78.0%

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

      7. frac-add 78.0%

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

      8. associate-/r/ 78.0%

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

      9. *-un-lft-identity 78.0%

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

      10. *-commutative 78.0%

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

      11. neg-mul-1 78.0%

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

      12. sub-neg 78.0%

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

      13. flip-+ 98.9%

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

      14. +-commutative 98.9%

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

   6. Applied egg-rr 98.9%

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

      1. associate-*l/ 98.9%

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

      2. clear-num 98.9%

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

      3. *-un-lft-identity 98.9%

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

      4. times-frac 98.9%

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

      5. metadata-eval 98.9%

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

      6. metadata-eval 98.9%

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

      7. times-frac 98.9%

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

      8. *-un-lft-identity 98.9%

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

      9. associate-*r* 93.7%

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

      10. times-frac 94.3%

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

   8. Applied egg-rr 94.3%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{+104}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a \cdot \left(a + b\right)} \cdot \frac{\pi}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{+108}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b} \cdot \frac{\frac{\pi}{a}}{a + b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -1e+108)
   (/ (* 0.5 (/ PI a)) (* a b))
   (* (/ 0.5 b) (/ (/ PI a) (+ a b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -1e+108) {
		tmp = (0.5 * (((double) M_PI) / a)) / (a * b);
	} else {
		tmp = (0.5 / b) * ((((double) M_PI) / a) / (a + b));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -1e+108) {
		tmp = (0.5 * (Math.PI / a)) / (a * b);
	} else {
		tmp = (0.5 / b) * ((Math.PI / a) / (a + b));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -1e+108:
		tmp = (0.5 * (math.pi / a)) / (a * b)
	else:
		tmp = (0.5 / b) * ((math.pi / a) / (a + b))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -1e+108)
		tmp = Float64(Float64(0.5 * Float64(pi / a)) / Float64(a * b));
	else
		tmp = Float64(Float64(0.5 / b) * Float64(Float64(pi / a) / Float64(a + b)));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1e+108)
		tmp = (0.5 * (pi / a)) / (a * b);
	else
		tmp = (0.5 / b) * ((pi / a) / (a + b));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -1e+108], N[(N[(0.5 * N[(Pi / a), $MachinePrecision]), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / b), $MachinePrecision] * N[(N[(Pi / a), $MachinePrecision] / N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1e108

   1. Initial program 77.6%

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

      1. associate-*l* 77.6%

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

      2. *-rgt-identity 77.6%

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

      3. associate-/l* 77.6%

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

      4. metadata-eval 77.6%

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

      5. associate-*l/ 77.6%

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

      6. *-lft-identity 77.6%

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

      7. sub-neg 77.6%

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

      8. distribute-neg-frac 77.6%

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

      9. metadata-eval 77.6%

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

   3. Simplified 77.6%

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

      1. metadata-eval 77.6%

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

      2. div-inv 77.6%

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

      3. clear-num 77.6%

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

      4. clear-num 77.6%

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

      5. frac-times 77.6%

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

      6. metadata-eval 77.6%

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

      7. frac-add 77.6%

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

      8. associate-/r/ 77.6%

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

      9. *-un-lft-identity 77.6%

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

      10. *-commutative 77.6%

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

      11. neg-mul-1 77.6%

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

      12. sub-neg 77.6%

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

      13. flip-+ 99.8%

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

      14. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. inv-pow 99.8%

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

      2. associate-*r* 99.8%

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

      3. unpow-prod-down 99.8%

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

      4. pow-prod-down 99.8%

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

      5. inv-pow 99.8%

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

      6. inv-pow 99.8%

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

      7. frac-times 99.8%

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

      8. metadata-eval 99.8%

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

   8. Applied egg-rr 99.8%

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

      1. associate-*r/ 99.9%

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

      2. *-rgt-identity 99.9%

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

      3. unpow-1 99.9%

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

      4. *-commutative 99.9%

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

   10. Simplified 99.9%

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

   11. Taylor expanded in a around inf 100.0%

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

   if -1e108 < a

   1. Initial program 78.4%

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

      1. associate-*l* 78.4%

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

      2. *-rgt-identity 78.4%

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

      3. associate-/l* 78.4%

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

      4. metadata-eval 78.4%

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

      5. associate-*l/ 78.4%

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

      6. *-lft-identity 78.4%

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

      7. sub-neg 78.4%

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

      8. distribute-neg-frac 78.4%

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

      9. metadata-eval 78.4%

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

   3. Simplified 78.4%

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

      1. metadata-eval 78.4%

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

      2. div-inv 78.4%

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

      3. clear-num 78.4%

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

      4. clear-num 78.1%

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

      5. frac-times 78.1%

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

      6. metadata-eval 78.1%

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

      7. frac-add 78.1%

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

      8. associate-/r/ 78.1%

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

      9. *-un-lft-identity 78.1%

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

      10. *-commutative 78.1%

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

      11. neg-mul-1 78.1%

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

      12. sub-neg 78.1%

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

      13. flip-+ 98.9%

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

      14. +-commutative 98.9%

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

   6. Applied egg-rr 98.9%

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

      1. associate-*l/ 98.9%

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

      2. clear-num 98.9%

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

      3. *-un-lft-identity 98.9%

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

      4. times-frac 98.9%

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

      5. metadata-eval 98.9%

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

      6. metadata-eval 98.9%

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

      7. times-frac 98.9%

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

      8. *-un-lft-identity 98.9%

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

      9. *-commutative 98.9%

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

      10. *-un-lft-identity 98.9%

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

      11. associate-*r* 93.8%

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

      12. times-frac 94.3%

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

   8. Applied egg-rr 94.3%

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

      1. associate-*l/ 94.5%

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

      2. *-commutative 94.5%

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

      3. associate-*r/ 94.5%

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

      4. *-lft-identity 94.5%

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

      5. associate-*r/ 94.5%

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

      6. *-commutative 94.5%

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

      7. associate-/l* 94.3%

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

      8. metadata-eval 94.3%

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

      9. associate-*r/ 94.3%

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

      10. associate-*l/ 94.3%

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

      11. *-commutative 94.3%

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

      12. associate-*r/ 94.3%

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

      13. metadata-eval 94.3%

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

      14. associate-/l/ 95.2%

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

   10. Simplified 95.2%

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

Alternative 3: 90.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -3.1 \cdot 10^{-27}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{b}}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -3.1e-27)
   (/ (* 0.5 (/ PI a)) (* a b))
   (/ (* 0.5 (/ PI b)) (* a b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -3.1e-27) {
		tmp = (0.5 * (((double) M_PI) / a)) / (a * b);
	} else {
		tmp = (0.5 * (((double) M_PI) / b)) / (a * b);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -3.1e-27) {
		tmp = (0.5 * (Math.PI / a)) / (a * b);
	} else {
		tmp = (0.5 * (Math.PI / b)) / (a * b);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -3.1e-27:
		tmp = (0.5 * (math.pi / a)) / (a * b)
	else:
		tmp = (0.5 * (math.pi / b)) / (a * b)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -3.1e-27)
		tmp = Float64(Float64(0.5 * Float64(pi / a)) / Float64(a * b));
	else
		tmp = Float64(Float64(0.5 * Float64(pi / b)) / Float64(a * b));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -3.1e-27)
		tmp = (0.5 * (pi / a)) / (a * b);
	else
		tmp = (0.5 * (pi / b)) / (a * b);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -3.1e-27], N[(N[(0.5 * N[(Pi / a), $MachinePrecision]), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(Pi / b), $MachinePrecision]), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.0999999999999998e-27

   1. Initial program 86.4%

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

      1. associate-*l* 86.3%

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

      2. *-rgt-identity 86.3%

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

      3. associate-/l* 86.3%

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

      4. metadata-eval 86.3%

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

      5. associate-*l/ 86.3%

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

      6. *-lft-identity 86.3%

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

      7. sub-neg 86.3%

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

      8. distribute-neg-frac 86.3%

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

      9. metadata-eval 86.3%

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

   3. Simplified 86.3%

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

      1. metadata-eval 86.3%

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

      2. div-inv 86.3%

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

      3. clear-num 86.3%

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

      4. clear-num 86.3%

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

      5. frac-times 86.4%

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

      6. metadata-eval 86.4%

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

      7. frac-add 86.5%

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

      8. associate-/r/ 86.4%

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

      9. *-un-lft-identity 86.4%

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

      10. *-commutative 86.4%

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

      11. neg-mul-1 86.4%

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

      12. sub-neg 86.4%

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

      13. flip-+ 99.7%

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

      14. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. inv-pow 99.7%

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

      2. associate-*r* 99.7%

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

      3. unpow-prod-down 99.7%

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

      4. pow-prod-down 99.7%

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

      5. inv-pow 99.7%

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

      6. inv-pow 99.7%

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

      7. frac-times 99.7%

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

      8. metadata-eval 99.7%

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

   8. Applied egg-rr 99.7%

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

      1. associate-*r/ 99.8%

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

      2. *-rgt-identity 99.8%

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

      3. unpow-1 99.8%

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

      4. *-commutative 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in a around inf 86.1%

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

   if -3.0999999999999998e-27 < a

   1. Initial program 75.5%

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

      1. associate-*l* 75.5%

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

      2. *-rgt-identity 75.5%

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

      3. associate-/l* 75.5%

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

      4. metadata-eval 75.5%

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

      5. associate-*l/ 75.6%

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

      6. *-lft-identity 75.6%

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

      7. sub-neg 75.6%

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

      8. distribute-neg-frac 75.6%

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

      9. metadata-eval 75.6%

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

   3. Simplified 75.6%

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

      1. metadata-eval 75.6%

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

      2. div-inv 75.6%

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

      3. clear-num 75.6%

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

      4. clear-num 75.2%

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

      5. frac-times 75.2%

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

      6. metadata-eval 75.2%

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

      7. frac-add 75.1%

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

      8. associate-/r/ 75.2%

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

      9. *-un-lft-identity 75.2%

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

      10. *-commutative 75.2%

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

      11. neg-mul-1 75.2%

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

      12. sub-neg 75.2%

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

      13. flip-+ 98.8%

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

      14. +-commutative 98.8%

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

   6. Applied egg-rr 98.8%

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

      1. inv-pow 98.8%

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

      2. associate-*r* 98.8%

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

      3. unpow-prod-down 99.5%

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

      4. pow-prod-down 99.5%

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

      5. inv-pow 99.5%

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

      6. inv-pow 99.5%

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

      7. frac-times 99.5%

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

      8. metadata-eval 99.5%

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

   8. Applied egg-rr 99.5%

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

      1. associate-*r/ 99.5%

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

      2. *-rgt-identity 99.5%

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

      3. unpow-1 99.5%

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

      4. *-commutative 99.5%

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

   10. Simplified 99.5%

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

   11. Taylor expanded in a around 0 70.3%

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

Alternative 4: 90.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -5.9 \cdot 10^{-27}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a \cdot b} \cdot \frac{0.5}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -5.9e-27)
   (/ (* 0.5 (/ PI a)) (* a b))
   (* (/ PI (* a b)) (/ 0.5 b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -5.9e-27) {
		tmp = (0.5 * (((double) M_PI) / a)) / (a * b);
	} else {
		tmp = (((double) M_PI) / (a * b)) * (0.5 / b);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -5.9e-27) {
		tmp = (0.5 * (Math.PI / a)) / (a * b);
	} else {
		tmp = (Math.PI / (a * b)) * (0.5 / b);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -5.9e-27:
		tmp = (0.5 * (math.pi / a)) / (a * b)
	else:
		tmp = (math.pi / (a * b)) * (0.5 / b)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -5.9e-27)
		tmp = Float64(Float64(0.5 * Float64(pi / a)) / Float64(a * b));
	else
		tmp = Float64(Float64(pi / Float64(a * b)) * Float64(0.5 / b));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -5.9e-27)
		tmp = (0.5 * (pi / a)) / (a * b);
	else
		tmp = (pi / (a * b)) * (0.5 / 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, -5.9e-27], N[(N[(0.5 * N[(Pi / a), $MachinePrecision]), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(0.5 / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.8999999999999998e-27

   1. Initial program 86.4%

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

      1. associate-*l* 86.3%

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

      2. *-rgt-identity 86.3%

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

      3. associate-/l* 86.3%

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

      4. metadata-eval 86.3%

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

      5. associate-*l/ 86.3%

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

      6. *-lft-identity 86.3%

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

      7. sub-neg 86.3%

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

      8. distribute-neg-frac 86.3%

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

      9. metadata-eval 86.3%

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

   3. Simplified 86.3%

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

      1. metadata-eval 86.3%

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

      2. div-inv 86.3%

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

      3. clear-num 86.3%

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

      4. clear-num 86.3%

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

      5. frac-times 86.4%

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

      6. metadata-eval 86.4%

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

      7. frac-add 86.5%

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

      8. associate-/r/ 86.4%

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

      9. *-un-lft-identity 86.4%

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

      10. *-commutative 86.4%

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

      11. neg-mul-1 86.4%

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

      12. sub-neg 86.4%

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

      13. flip-+ 99.7%

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

      14. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. inv-pow 99.7%

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

      2. associate-*r* 99.7%

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

      3. unpow-prod-down 99.7%

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

      4. pow-prod-down 99.7%

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

      5. inv-pow 99.7%

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

      6. inv-pow 99.7%

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

      7. frac-times 99.7%

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

      8. metadata-eval 99.7%

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

   8. Applied egg-rr 99.7%

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

      1. associate-*r/ 99.8%

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

      2. *-rgt-identity 99.8%

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

      3. unpow-1 99.8%

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

      4. *-commutative 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in a around inf 86.1%

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

   if -5.8999999999999998e-27 < a

   1. Initial program 75.5%

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

      1. associate-*l* 75.5%

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

      2. *-rgt-identity 75.5%

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

      3. associate-/l* 75.5%

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

      4. metadata-eval 75.5%

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

      5. associate-*l/ 75.6%

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

      6. *-lft-identity 75.6%

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

      7. sub-neg 75.6%

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

      8. distribute-neg-frac 75.6%

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

      9. metadata-eval 75.6%

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

   3. Simplified 75.6%

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

      1. metadata-eval 75.6%

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

      2. div-inv 75.6%

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

      3. clear-num 75.6%

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

      4. clear-num 75.2%

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

      5. frac-times 75.2%

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

      6. metadata-eval 75.2%

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

      7. frac-add 75.1%

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

      8. associate-/r/ 75.2%

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

      9. *-un-lft-identity 75.2%

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

      10. *-commutative 75.2%

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

      11. neg-mul-1 75.2%

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

      12. sub-neg 75.2%

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

      13. flip-+ 98.8%

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

      14. +-commutative 98.8%

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

   6. Applied egg-rr 98.8%

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

      1. associate-*l/ 98.8%

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

      2. clear-num 98.8%

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

      3. *-un-lft-identity 98.8%

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

      4. times-frac 98.8%

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

      5. metadata-eval 98.8%

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

      6. metadata-eval 98.8%

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

      7. times-frac 98.8%

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

      8. *-un-lft-identity 98.8%

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

      9. *-commutative 98.8%

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

      10. *-commutative 98.8%

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

      11. times-frac 99.5%

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

   8. Applied egg-rr 99.5%

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

   9. Taylor expanded in a around 0 70.2%

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

Alternative 5: 90.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -3.1 \cdot 10^{-27}:\\ \;\;\;\;\frac{0.5}{\frac{a \cdot \left(a \cdot b\right)}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a \cdot b} \cdot \frac{0.5}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -3.1e-27)
   (/ 0.5 (/ (* a (* a b)) PI))
   (* (/ PI (* a b)) (/ 0.5 b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -3.1e-27) {
		tmp = 0.5 / ((a * (a * b)) / ((double) M_PI));
	} else {
		tmp = (((double) M_PI) / (a * b)) * (0.5 / b);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -3.1e-27) {
		tmp = 0.5 / ((a * (a * b)) / Math.PI);
	} else {
		tmp = (Math.PI / (a * b)) * (0.5 / b);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -3.1e-27:
		tmp = 0.5 / ((a * (a * b)) / math.pi)
	else:
		tmp = (math.pi / (a * b)) * (0.5 / b)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -3.1e-27)
		tmp = Float64(0.5 / Float64(Float64(a * Float64(a * b)) / pi));
	else
		tmp = Float64(Float64(pi / Float64(a * b)) * Float64(0.5 / b));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -3.1e-27)
		tmp = 0.5 / ((a * (a * b)) / pi);
	else
		tmp = (pi / (a * b)) * (0.5 / 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, -3.1e-27], N[(0.5 / N[(N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(0.5 / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.0999999999999998e-27

   1. Initial program 86.4%

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

      1. associate-*l* 86.3%

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

      2. *-rgt-identity 86.3%

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

      3. associate-/l* 86.3%

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

      4. metadata-eval 86.3%

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

      5. associate-*l/ 86.3%

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

      6. *-lft-identity 86.3%

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

      7. sub-neg 86.3%

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

      8. distribute-neg-frac 86.3%

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

      9. metadata-eval 86.3%

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

   3. Simplified 86.3%

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

      1. metadata-eval 86.3%

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

      2. div-inv 86.3%

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

      3. clear-num 86.3%

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

      4. clear-num 86.3%

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

      5. frac-times 86.4%

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

      6. metadata-eval 86.4%

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

      7. frac-add 86.5%

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

      8. associate-/r/ 86.4%

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

      9. *-un-lft-identity 86.4%

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

      10. *-commutative 86.4%

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

      11. neg-mul-1 86.4%

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

      12. sub-neg 86.4%

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

      13. flip-+ 99.7%

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

      14. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. inv-pow 99.7%

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

      2. associate-*l/ 99.7%

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

      3. *-un-lft-identity 99.7%

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

      4. times-frac 99.7%

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

      5. metadata-eval 99.7%

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

   8. Applied egg-rr 99.7%

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

      1. unpow-1 99.7%

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

      2. associate-/r* 99.7%

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

      3. metadata-eval 99.7%

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

      4. *-commutative 99.7%

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

   10. Simplified 99.7%

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

   11. Taylor expanded in a around inf 86.0%

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

   if -3.0999999999999998e-27 < a

   1. Initial program 75.5%

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

      1. associate-*l* 75.5%

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

      2. *-rgt-identity 75.5%

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

      3. associate-/l* 75.5%

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

      4. metadata-eval 75.5%

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

      5. associate-*l/ 75.6%

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

      6. *-lft-identity 75.6%

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

      7. sub-neg 75.6%

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

      8. distribute-neg-frac 75.6%

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

      9. metadata-eval 75.6%

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

   3. Simplified 75.6%

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

      1. metadata-eval 75.6%

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

      2. div-inv 75.6%

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

      3. clear-num 75.6%

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

      4. clear-num 75.2%

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

      5. frac-times 75.2%

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

      6. metadata-eval 75.2%

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

      7. frac-add 75.1%

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

      8. associate-/r/ 75.2%

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

      9. *-un-lft-identity 75.2%

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

      10. *-commutative 75.2%

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

      11. neg-mul-1 75.2%

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

      12. sub-neg 75.2%

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

      13. flip-+ 98.8%

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

      14. +-commutative 98.8%

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

   6. Applied egg-rr 98.8%

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

      1. associate-*l/ 98.8%

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

      2. clear-num 98.8%

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

      3. *-un-lft-identity 98.8%

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

      4. times-frac 98.8%

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

      5. metadata-eval 98.8%

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

      6. metadata-eval 98.8%

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

      7. times-frac 98.8%

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

      8. *-un-lft-identity 98.8%

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

      9. *-commutative 98.8%

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

      10. *-commutative 98.8%

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

      11. times-frac 99.5%

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

   8. Applied egg-rr 99.5%

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

   9. Taylor expanded in a around 0 70.2%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.1 \cdot 10^{-27}:\\ \;\;\;\;\frac{0.5}{\frac{a \cdot \left(a \cdot b\right)}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a \cdot b} \cdot \frac{0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{-27}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a \cdot b} \cdot \frac{0.5}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -5e-27) (* PI (/ 0.5 (* a (* a b)))) (* (/ PI (* a b)) (/ 0.5 b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -5e-27) {
		tmp = ((double) M_PI) * (0.5 / (a * (a * b)));
	} else {
		tmp = (((double) M_PI) / (a * b)) * (0.5 / b);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -5e-27) {
		tmp = Math.PI * (0.5 / (a * (a * b)));
	} else {
		tmp = (Math.PI / (a * b)) * (0.5 / b);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -5e-27:
		tmp = math.pi * (0.5 / (a * (a * b)))
	else:
		tmp = (math.pi / (a * b)) * (0.5 / b)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -5e-27)
		tmp = Float64(pi * Float64(0.5 / Float64(a * Float64(a * b))));
	else
		tmp = Float64(Float64(pi / Float64(a * b)) * Float64(0.5 / b));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -5e-27)
		tmp = pi * (0.5 / (a * (a * b)));
	else
		tmp = (pi / (a * b)) * (0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -5e-27], N[(Pi * N[(0.5 / N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(0.5 / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.0000000000000002e-27

   1. Initial program 86.4%

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

      1. *-commutative 86.4%

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

      2. associate-*r* 86.4%

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

      3. associate-*r/ 86.4%

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

      4. associate-*r* 86.4%

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

      5. *-rgt-identity 86.4%

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

      6. sub-neg 86.4%

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

      7. distribute-neg-frac 86.4%

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

      8. metadata-eval 86.4%

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

   3. Simplified 86.4%

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

      1. *-commutative 86.4%

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

      2. associate-*r/ 86.3%

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

      3. div-inv 86.3%

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

      4. metadata-eval 86.3%

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

      5. associate-*l* 86.3%

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

      6. *-commutative 86.3%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in a around inf 85.9%

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

   if -5.0000000000000002e-27 < a

   1. Initial program 75.5%

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

      1. associate-*l* 75.5%

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

      2. *-rgt-identity 75.5%

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

      3. associate-/l* 75.5%

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

      4. metadata-eval 75.5%

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

      5. associate-*l/ 75.6%

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

      6. *-lft-identity 75.6%

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

      7. sub-neg 75.6%

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

      8. distribute-neg-frac 75.6%

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

      9. metadata-eval 75.6%

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

   3. Simplified 75.6%

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

      1. metadata-eval 75.6%

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

      2. div-inv 75.6%

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

      3. clear-num 75.6%

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

      4. clear-num 75.2%

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

      5. frac-times 75.2%

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

      6. metadata-eval 75.2%

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

      7. frac-add 75.1%

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

      8. associate-/r/ 75.2%

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

      9. *-un-lft-identity 75.2%

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

      10. *-commutative 75.2%

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

      11. neg-mul-1 75.2%

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

      12. sub-neg 75.2%

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

      13. flip-+ 98.8%

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

      14. +-commutative 98.8%

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

   6. Applied egg-rr 98.8%

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

      1. associate-*l/ 98.8%

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

      2. clear-num 98.8%

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

      3. *-un-lft-identity 98.8%

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

      4. times-frac 98.8%

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

      5. metadata-eval 98.8%

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

      6. metadata-eval 98.8%

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

      7. times-frac 98.8%

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

      8. *-un-lft-identity 98.8%

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

      9. *-commutative 98.8%

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

      10. *-commutative 98.8%

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

      11. times-frac 99.5%

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

   8. Applied egg-rr 99.5%

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

   9. Taylor expanded in a around 0 70.2%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{-27}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a \cdot b} \cdot \frac{0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -5.4 \cdot 10^{-27}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b} \cdot \frac{\frac{\pi}{a}}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -5.4e-27)
   (* PI (/ 0.5 (* a (* a b))))
   (* (/ 0.5 b) (/ (/ PI a) b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -5.4e-27) {
		tmp = ((double) M_PI) * (0.5 / (a * (a * b)));
	} else {
		tmp = (0.5 / b) * ((((double) M_PI) / a) / b);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -5.4e-27) {
		tmp = Math.PI * (0.5 / (a * (a * b)));
	} else {
		tmp = (0.5 / b) * ((Math.PI / a) / b);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -5.4e-27:
		tmp = math.pi * (0.5 / (a * (a * b)))
	else:
		tmp = (0.5 / b) * ((math.pi / a) / b)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -5.4e-27)
		tmp = Float64(pi * Float64(0.5 / Float64(a * Float64(a * b))));
	else
		tmp = Float64(Float64(0.5 / b) * Float64(Float64(pi / a) / b));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -5.4e-27)
		tmp = pi * (0.5 / (a * (a * b)));
	else
		tmp = (0.5 / b) * ((pi / a) / b);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -5.4e-27], N[(Pi * N[(0.5 / N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / b), $MachinePrecision] * N[(N[(Pi / a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.39999999999999978e-27

   1. Initial program 86.4%

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

      1. *-commutative 86.4%

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

      2. associate-*r* 86.4%

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

      3. associate-*r/ 86.4%

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

      4. associate-*r* 86.4%

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

      5. *-rgt-identity 86.4%

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

      6. sub-neg 86.4%

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

      7. distribute-neg-frac 86.4%

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

      8. metadata-eval 86.4%

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

   3. Simplified 86.4%

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

      1. *-commutative 86.4%

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

      2. associate-*r/ 86.3%

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

      3. div-inv 86.3%

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

      4. metadata-eval 86.3%

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

      5. associate-*l* 86.3%

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

      6. *-commutative 86.3%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in a around inf 85.9%

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

   if -5.39999999999999978e-27 < a

   1. Initial program 75.5%

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

      1. associate-*l* 75.5%

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

      2. *-rgt-identity 75.5%

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

      3. associate-/l* 75.5%

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

      4. metadata-eval 75.5%

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

      5. associate-*l/ 75.6%

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

      6. *-lft-identity 75.6%

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

      7. sub-neg 75.6%

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

      8. distribute-neg-frac 75.6%

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

      9. metadata-eval 75.6%

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

   3. Simplified 75.6%

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

      1. metadata-eval 75.6%

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

      2. div-inv 75.6%

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

      3. clear-num 75.6%

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

      4. clear-num 75.2%

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

      5. frac-times 75.2%

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

      6. metadata-eval 75.2%

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

      7. frac-add 75.1%

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

      8. associate-/r/ 75.2%

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

      9. *-un-lft-identity 75.2%

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

      10. *-commutative 75.2%

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

      11. neg-mul-1 75.2%

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

      12. sub-neg 75.2%

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

      13. flip-+ 98.8%

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

      14. +-commutative 98.8%

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

   6. Applied egg-rr 98.8%

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

      1. associate-*l/ 98.8%

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

      2. clear-num 98.8%

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

      3. *-un-lft-identity 98.8%

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

      4. times-frac 98.8%

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

      5. metadata-eval 98.8%

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

      6. metadata-eval 98.8%

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

      7. times-frac 98.8%

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

      8. *-un-lft-identity 98.8%

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

      9. *-commutative 98.8%

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

      10. *-un-lft-identity 98.8%

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

      11. associate-*r* 93.0%

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

      12. times-frac 93.6%

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

   8. Applied egg-rr 93.6%

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

      1. associate-*l/ 93.8%

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

      2. *-commutative 93.8%

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

      3. associate-*r/ 93.8%

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

      4. *-lft-identity 93.8%

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

      5. associate-*r/ 93.8%

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

      6. *-commutative 93.8%

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

      7. associate-/l* 93.6%

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

      8. metadata-eval 93.6%

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

      9. associate-*r/ 93.6%

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

      10. associate-*l/ 93.6%

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

      11. *-commutative 93.6%

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

      12. associate-*r/ 93.6%

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

      13. metadata-eval 93.6%

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

      14. associate-/l/ 94.6%

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

   10. Simplified 94.6%

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

   11. Taylor expanded in a around 0 70.2%

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

       1. associate-/r* 70.2%

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

   13. Simplified 70.2%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.4 \cdot 10^{-27}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b} \cdot \frac{\frac{\pi}{a}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -5.9 \cdot 10^{-27}:\\ \;\;\;\;\frac{0.5}{b} \cdot \frac{\frac{\pi}{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b} \cdot \frac{\frac{\pi}{a}}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -5.9e-27)
   (* (/ 0.5 b) (/ (/ PI a) a))
   (* (/ 0.5 b) (/ (/ PI a) b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -5.9e-27) {
		tmp = (0.5 / b) * ((((double) M_PI) / a) / a);
	} else {
		tmp = (0.5 / b) * ((((double) M_PI) / a) / b);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -5.9e-27) {
		tmp = (0.5 / b) * ((Math.PI / a) / a);
	} else {
		tmp = (0.5 / b) * ((Math.PI / a) / b);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -5.9e-27:
		tmp = (0.5 / b) * ((math.pi / a) / a)
	else:
		tmp = (0.5 / b) * ((math.pi / a) / b)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -5.9e-27)
		tmp = Float64(Float64(0.5 / b) * Float64(Float64(pi / a) / a));
	else
		tmp = Float64(Float64(0.5 / b) * Float64(Float64(pi / a) / b));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -5.9e-27)
		tmp = (0.5 / b) * ((pi / a) / a);
	else
		tmp = (0.5 / b) * ((pi / a) / b);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -5.9e-27], N[(N[(0.5 / b), $MachinePrecision] * N[(N[(Pi / a), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / b), $MachinePrecision] * N[(N[(Pi / a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.8999999999999998e-27

   1. Initial program 86.4%

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

      1. associate-*l* 86.3%

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

      2. *-rgt-identity 86.3%

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

      3. associate-/l* 86.3%

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

      4. metadata-eval 86.3%

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

      5. associate-*l/ 86.3%

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

      6. *-lft-identity 86.3%

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

      7. sub-neg 86.3%

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

      8. distribute-neg-frac 86.3%

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

      9. metadata-eval 86.3%

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

   3. Simplified 86.3%

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

      1. metadata-eval 86.3%

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

      2. div-inv 86.3%

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

      3. clear-num 86.3%

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

      4. clear-num 86.3%

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

      5. frac-times 86.4%

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

      6. metadata-eval 86.4%

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

      7. frac-add 86.5%

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

      8. associate-/r/ 86.4%

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

      9. *-un-lft-identity 86.4%

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

      10. *-commutative 86.4%

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

      11. neg-mul-1 86.4%

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

      12. sub-neg 86.4%

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

      13. flip-+ 99.7%

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

      14. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. associate-*l/ 99.7%

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

      2. clear-num 99.7%

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

      3. *-un-lft-identity 99.7%

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

      4. times-frac 99.7%

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

      5. metadata-eval 99.7%

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

      6. metadata-eval 99.7%

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

      7. times-frac 99.7%

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

      8. *-un-lft-identity 99.7%

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

      9. *-commutative 99.7%

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

      10. *-un-lft-identity 99.7%

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

      11. associate-*r* 92.6%

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

      12. times-frac 92.5%

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

   8. Applied egg-rr 92.5%

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

      1. associate-*l/ 92.6%

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

      2. *-commutative 92.6%

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

      3. associate-*r/ 92.6%

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

      4. *-lft-identity 92.6%

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

      5. associate-*r/ 92.6%

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

      6. *-commutative 92.6%

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

      7. associate-/l* 92.6%

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

      8. metadata-eval 92.6%

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

      9. associate-*r/ 92.6%

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

      10. associate-*l/ 92.6%

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

      11. *-commutative 92.6%

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

      12. associate-*r/ 92.6%

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

      13. metadata-eval 92.6%

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

      14. associate-/l/ 92.6%

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

   10. Simplified 92.6%

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

   11. Taylor expanded in a around inf 78.8%

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

   if -5.8999999999999998e-27 < a

   1. Initial program 75.5%

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

      1. associate-*l* 75.5%

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

      2. *-rgt-identity 75.5%

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

      3. associate-/l* 75.5%

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

      4. metadata-eval 75.5%

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

      5. associate-*l/ 75.6%

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

      6. *-lft-identity 75.6%

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

      7. sub-neg 75.6%

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

      8. distribute-neg-frac 75.6%

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

      9. metadata-eval 75.6%

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

   3. Simplified 75.6%

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

      1. metadata-eval 75.6%

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

      2. div-inv 75.6%

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

      3. clear-num 75.6%

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

      4. clear-num 75.2%

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

      5. frac-times 75.2%

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

      6. metadata-eval 75.2%

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

      7. frac-add 75.1%

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

      8. associate-/r/ 75.2%

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

      9. *-un-lft-identity 75.2%

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

      10. *-commutative 75.2%

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

      11. neg-mul-1 75.2%

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

      12. sub-neg 75.2%

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

      13. flip-+ 98.8%

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

      14. +-commutative 98.8%

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

   6. Applied egg-rr 98.8%

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

      1. associate-*l/ 98.8%

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

      2. clear-num 98.8%

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

      3. *-un-lft-identity 98.8%

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

      4. times-frac 98.8%

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

      5. metadata-eval 98.8%

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

      6. metadata-eval 98.8%

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

      7. times-frac 98.8%

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

      8. *-un-lft-identity 98.8%

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

      9. *-commutative 98.8%

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

      10. *-un-lft-identity 98.8%

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

      11. associate-*r* 93.0%

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

      12. times-frac 93.6%

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

   8. Applied egg-rr 93.6%

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

      1. associate-*l/ 93.8%

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

      2. *-commutative 93.8%

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

      3. associate-*r/ 93.8%

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

      4. *-lft-identity 93.8%

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

      5. associate-*r/ 93.8%

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

      6. *-commutative 93.8%

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

      7. associate-/l* 93.6%

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

      8. metadata-eval 93.6%

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

      9. associate-*r/ 93.6%

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

      10. associate-*l/ 93.6%

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

      11. *-commutative 93.6%

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

      12. associate-*r/ 93.6%

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

      13. metadata-eval 93.6%

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

      14. associate-/l/ 94.6%

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

   10. Simplified 94.6%

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

   11. Taylor expanded in a around 0 70.2%

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

       1. associate-/r* 70.2%

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

   13. Simplified 70.2%

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

Alternative 9: 99.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.3%

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

   1. associate-*l* 78.3%

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

   2. *-rgt-identity 78.3%

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

   3. associate-/l* 78.3%

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

   4. metadata-eval 78.3%

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

   5. associate-*l/ 78.3%

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

   6. *-lft-identity 78.3%

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

   7. sub-neg 78.3%

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

   8. distribute-neg-frac 78.3%

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

   9. metadata-eval 78.3%

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

3. Simplified 78.3%

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

   1. metadata-eval 78.3%

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

   2. div-inv 78.3%

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

   3. clear-num 78.3%

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

   4. clear-num 78.0%

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

   5. frac-times 78.0%

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

   6. metadata-eval 78.0%

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

   7. frac-add 78.0%

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

   8. associate-/r/ 78.0%

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

   9. *-un-lft-identity 78.0%

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

   10. *-commutative 78.0%

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

   11. neg-mul-1 78.0%

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

   12. sub-neg 78.0%

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

   13. flip-+ 99.0%

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

   14. +-commutative 99.0%

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

6. Applied egg-rr 99.0%

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

   1. inv-pow 99.0%

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

   2. associate-*r* 99.0%

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

   3. unpow-prod-down 99.5%

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

   4. pow-prod-down 99.5%

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

   5. inv-pow 99.5%

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

   6. inv-pow 99.5%

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

   7. frac-times 99.5%

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

   8. metadata-eval 99.5%

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

8. Applied egg-rr 99.5%

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

   1. associate-*r/ 99.6%

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

   2. *-rgt-identity 99.6%

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

   3. unpow-1 99.6%

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

   4. *-commutative 99.6%

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

10. Simplified 99.6%

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

Alternative 10: 99.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.3%

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

   1. associate-*l* 78.3%

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

   2. *-rgt-identity 78.3%

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

   3. associate-/l* 78.3%

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

   4. metadata-eval 78.3%

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

   5. associate-*l/ 78.3%

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

   6. *-lft-identity 78.3%

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

   7. sub-neg 78.3%

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

   8. distribute-neg-frac 78.3%

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

   9. metadata-eval 78.3%

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

3. Simplified 78.3%

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

   1. metadata-eval 78.3%

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

   2. div-inv 78.3%

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

   3. clear-num 78.3%

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

   4. clear-num 78.0%

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

   5. frac-times 78.0%

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

   6. metadata-eval 78.0%

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

   7. frac-add 78.0%

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

   8. associate-/r/ 78.0%

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

   9. *-un-lft-identity 78.0%

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

   10. *-commutative 78.0%

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

   11. neg-mul-1 78.0%

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

   12. sub-neg 78.0%

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

   13. flip-+ 99.0%

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

   14. +-commutative 99.0%

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

6. Applied egg-rr 99.0%

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

   1. associate-*l/ 99.1%

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

   2. clear-num 99.0%

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

   3. *-un-lft-identity 99.0%

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

   4. times-frac 99.0%

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

   5. metadata-eval 99.0%

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

   6. metadata-eval 99.0%

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

   7. times-frac 99.0%

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

   8. *-un-lft-identity 99.0%

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

   9. *-commutative 99.0%

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

   10. times-frac 99.6%

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

8. Applied egg-rr 99.6%

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

Alternative 11: 99.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.3%

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

   1. associate-*l* 78.3%

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

   2. *-rgt-identity 78.3%

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

   3. associate-/l* 78.3%

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

   4. metadata-eval 78.3%

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

   5. associate-*l/ 78.3%

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

   6. *-lft-identity 78.3%

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

   7. sub-neg 78.3%

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

   8. distribute-neg-frac 78.3%

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

   9. metadata-eval 78.3%

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

3. Simplified 78.3%

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

   1. metadata-eval 78.3%

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

   2. div-inv 78.3%

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

   3. clear-num 78.3%

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

   4. clear-num 78.0%

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

   5. frac-times 78.0%

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

   6. metadata-eval 78.0%

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

   7. frac-add 78.0%

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

   8. associate-/r/ 78.0%

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

   9. *-un-lft-identity 78.0%

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

   10. *-commutative 78.0%

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

   11. neg-mul-1 78.0%

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

   12. sub-neg 78.0%

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

   13. flip-+ 99.0%

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

   14. +-commutative 99.0%

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

6. Applied egg-rr 99.0%

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

   1. associate-*l/ 99.1%

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

   2. clear-num 99.0%

      \[\leadsto \color{blue}{\frac{\pi}{2 \cdot \left(\left(a + b\right) \cdot \left(a \cdot b\right)\right)}} \]

   3. *-un-lft-identity 99.0%

      \[\leadsto \frac{\color{blue}{1 \cdot \pi}}{2 \cdot \left(\left(a + b\right) \cdot \left(a \cdot b\right)\right)} \]

   4. times-frac 99.0%

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\pi}{\left(a + b\right) \cdot \left(a \cdot b\right)}} \]

   5. metadata-eval 99.0%

      \[\leadsto \color{blue}{0.5} \cdot \frac{\pi}{\left(a + b\right) \cdot \left(a \cdot b\right)} \]

   6. metadata-eval 99.0%

      \[\leadsto \color{blue}{\frac{0.5}{1}} \cdot \frac{\pi}{\left(a + b\right) \cdot \left(a \cdot b\right)} \]

   7. times-frac 99.0%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{1 \cdot \left(\left(a + b\right) \cdot \left(a \cdot b\right)\right)}} \]

   8. *-un-lft-identity 99.0%

      \[\leadsto \frac{0.5 \cdot \pi}{\color{blue}{\left(a + b\right) \cdot \left(a \cdot b\right)}} \]

   9. *-commutative 99.0%

      \[\leadsto \frac{\color{blue}{\pi \cdot 0.5}}{\left(a + b\right) \cdot \left(a \cdot b\right)} \]

   10. *-commutative 99.0%

       \[\leadsto \frac{\pi \cdot 0.5}{\color{blue}{\left(a \cdot b\right) \cdot \left(a + b\right)}} \]

   11. times-frac 99.6%

       \[\leadsto \color{blue}{\frac{\pi}{a \cdot b} \cdot \frac{0.5}{a + b}} \]

8. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{\frac{\pi}{a \cdot b} \cdot \frac{0.5}{a + b}} \]
9. Add Preprocessing

Alternative 12: 99.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \pi \cdot \frac{0.5}{\left(a + b\right) \cdot \left(a \cdot b\right)} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (* PI (/ 0.5 (* (+ a b) (* a b)))))

C

assert(a < b);
double code(double a, double b) {
	return ((double) M_PI) * (0.5 / ((a + b) * (a * b)));
}

Java

assert a < b;
public static double code(double a, double b) {
	return Math.PI * (0.5 / ((a + b) * (a * b)));
}

Python

[a, b] = sort([a, b])
def code(a, b):
	return math.pi * (0.5 / ((a + b) * (a * b)))

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(pi * Float64(0.5 / Float64(Float64(a + b) * Float64(a * b))))
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = pi * (0.5 / ((a + b) * (a * b)));
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[(Pi * N[(0.5 / N[(N[(a + b), $MachinePrecision] * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.3%

   \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
2. Step-by-step derivation

   1. *-commutative 78.3%

      \[\leadsto \color{blue}{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right)} \]

   2. associate-*r* 78.3%

      \[\leadsto \color{blue}{\left(\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \frac{\pi}{2}\right) \cdot \frac{1}{b \cdot b - a \cdot a}} \]

   3. associate-*r/ 78.3%

      \[\leadsto \color{blue}{\frac{\left(\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \frac{\pi}{2}\right) \cdot 1}{b \cdot b - a \cdot a}} \]

   4. associate-*r* 78.3%

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \left(\frac{\pi}{2} \cdot 1\right)}}{b \cdot b - a \cdot a} \]

   5. *-rgt-identity 78.3%

      \[\leadsto \frac{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \color{blue}{\frac{\pi}{2}}}{b \cdot b - a \cdot a} \]

   6. sub-neg 78.3%

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{a} + \left(-\frac{1}{b}\right)\right)} \cdot \frac{\pi}{2}}{b \cdot b - a \cdot a} \]

   7. distribute-neg-frac 78.3%

      \[\leadsto \frac{\left(\frac{1}{a} + \color{blue}{\frac{-1}{b}}\right) \cdot \frac{\pi}{2}}{b \cdot b - a \cdot a} \]

   8. metadata-eval 78.3%

      \[\leadsto \frac{\left(\frac{1}{a} + \frac{\color{blue}{-1}}{b}\right) \cdot \frac{\pi}{2}}{b \cdot b - a \cdot a} \]

3. Simplified 78.3%

   \[\leadsto \color{blue}{\frac{\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \frac{\pi}{2}}{b \cdot b - a \cdot a}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-commutative 78.3%

      \[\leadsto \frac{\color{blue}{\frac{\pi}{2} \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)}}{b \cdot b - a \cdot a} \]

   2. associate-*r/ 78.3%

      \[\leadsto \color{blue}{\frac{\pi}{2} \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b \cdot b - a \cdot a}} \]

   3. div-inv 78.3%

      \[\leadsto \color{blue}{\left(\pi \cdot \frac{1}{2}\right)} \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b \cdot b - a \cdot a} \]

   4. metadata-eval 78.3%

      \[\leadsto \left(\pi \cdot \color{blue}{0.5}\right) \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b \cdot b - a \cdot a} \]

   5. associate-*l* 78.3%

      \[\leadsto \color{blue}{\pi \cdot \left(0.5 \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b \cdot b - a \cdot a}\right)} \]

   6. *-commutative 78.3%

      \[\leadsto \color{blue}{\left(0.5 \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b \cdot b - a \cdot a}\right) \cdot \pi} \]

6. Applied egg-rr 99.0%

   \[\leadsto \color{blue}{\frac{0.5}{\left(a + b\right) \cdot \left(a \cdot b\right)} \cdot \pi} \]

7. Final simplification 99.0%

   \[\leadsto \pi \cdot \frac{0.5}{\left(a + b\right) \cdot \left(a \cdot b\right)} \]
8. Add Preprocessing

Alternative 13: 56.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \frac{0.5}{b} \cdot \frac{\frac{\pi}{a}}{a} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (* (/ 0.5 b) (/ (/ PI a) a)))

C

assert(a < b);
double code(double a, double b) {
	return (0.5 / b) * ((((double) M_PI) / a) / a);
}

Java

assert a < b;
public static double code(double a, double b) {
	return (0.5 / b) * ((Math.PI / a) / a);
}

Python

[a, b] = sort([a, b])
def code(a, b):
	return (0.5 / b) * ((math.pi / a) / a)

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(Float64(0.5 / b) * Float64(Float64(pi / a) / a))
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = (0.5 / b) * ((pi / a) / 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[(N[(Pi / a), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.3%

   \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
2. Step-by-step derivation

   1. associate-*l* 78.3%

      \[\leadsto \color{blue}{\frac{\pi}{2} \cdot \left(\frac{1}{b \cdot b - a \cdot a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\right)} \]

   2. *-rgt-identity 78.3%

      \[\leadsto \frac{\color{blue}{\pi \cdot 1}}{2} \cdot \left(\frac{1}{b \cdot b - a \cdot a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\right) \]

   3. associate-/l* 78.3%

      \[\leadsto \color{blue}{\left(\pi \cdot \frac{1}{2}\right)} \cdot \left(\frac{1}{b \cdot b - a \cdot a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\right) \]

   4. metadata-eval 78.3%

      \[\leadsto \left(\pi \cdot \color{blue}{0.5}\right) \cdot \left(\frac{1}{b \cdot b - a \cdot a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\right) \]

   5. associate-*l/ 78.3%

      \[\leadsto \left(\pi \cdot 0.5\right) \cdot \color{blue}{\frac{1 \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{b \cdot b - a \cdot a}} \]

   6. *-lft-identity 78.3%

      \[\leadsto \left(\pi \cdot 0.5\right) \cdot \frac{\color{blue}{\frac{1}{a} - \frac{1}{b}}}{b \cdot b - a \cdot a} \]

   7. sub-neg 78.3%

      \[\leadsto \left(\pi \cdot 0.5\right) \cdot \frac{\color{blue}{\frac{1}{a} + \left(-\frac{1}{b}\right)}}{b \cdot b - a \cdot a} \]

   8. distribute-neg-frac 78.3%

      \[\leadsto \left(\pi \cdot 0.5\right) \cdot \frac{\frac{1}{a} + \color{blue}{\frac{-1}{b}}}{b \cdot b - a \cdot a} \]

   9. metadata-eval 78.3%

      \[\leadsto \left(\pi \cdot 0.5\right) \cdot \frac{\frac{1}{a} + \frac{\color{blue}{-1}}{b}}{b \cdot b - a \cdot a} \]

3. Simplified 78.3%

   \[\leadsto \color{blue}{\left(\pi \cdot 0.5\right) \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b \cdot b - a \cdot a}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. metadata-eval 78.3%

      \[\leadsto \left(\pi \cdot \color{blue}{\frac{1}{2}}\right) \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b \cdot b - a \cdot a} \]

   2. div-inv 78.3%

      \[\leadsto \color{blue}{\frac{\pi}{2}} \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b \cdot b - a \cdot a} \]

   3. clear-num 78.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{2}{\pi}}} \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b \cdot b - a \cdot a} \]

   4. clear-num 78.0%

      \[\leadsto \frac{1}{\frac{2}{\pi}} \cdot \color{blue}{\frac{1}{\frac{b \cdot b - a \cdot a}{\frac{1}{a} + \frac{-1}{b}}}} \]

   5. frac-times 78.0%

      \[\leadsto \color{blue}{\frac{1 \cdot 1}{\frac{2}{\pi} \cdot \frac{b \cdot b - a \cdot a}{\frac{1}{a} + \frac{-1}{b}}}} \]

   6. metadata-eval 78.0%

      \[\leadsto \frac{\color{blue}{1}}{\frac{2}{\pi} \cdot \frac{b \cdot b - a \cdot a}{\frac{1}{a} + \frac{-1}{b}}} \]

   7. frac-add 78.0%

      \[\leadsto \frac{1}{\frac{2}{\pi} \cdot \frac{b \cdot b - a \cdot a}{\color{blue}{\frac{1 \cdot b + a \cdot -1}{a \cdot b}}}} \]

   8. associate-/r/ 78.0%

      \[\leadsto \frac{1}{\frac{2}{\pi} \cdot \color{blue}{\left(\frac{b \cdot b - a \cdot a}{1 \cdot b + a \cdot -1} \cdot \left(a \cdot b\right)\right)}} \]

   9. *-un-lft-identity 78.0%

      \[\leadsto \frac{1}{\frac{2}{\pi} \cdot \left(\frac{b \cdot b - a \cdot a}{\color{blue}{b} + a \cdot -1} \cdot \left(a \cdot b\right)\right)} \]

   10. *-commutative 78.0%

       \[\leadsto \frac{1}{\frac{2}{\pi} \cdot \left(\frac{b \cdot b - a \cdot a}{b + \color{blue}{-1 \cdot a}} \cdot \left(a \cdot b\right)\right)} \]

   11. neg-mul-1 78.0%

       \[\leadsto \frac{1}{\frac{2}{\pi} \cdot \left(\frac{b \cdot b - a \cdot a}{b + \color{blue}{\left(-a\right)}} \cdot \left(a \cdot b\right)\right)} \]

   12. sub-neg 78.0%

       \[\leadsto \frac{1}{\frac{2}{\pi} \cdot \left(\frac{b \cdot b - a \cdot a}{\color{blue}{b - a}} \cdot \left(a \cdot b\right)\right)} \]

   13. flip-+ 99.0%

       \[\leadsto \frac{1}{\frac{2}{\pi} \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(a \cdot b\right)\right)} \]

   14. +-commutative 99.0%

       \[\leadsto \frac{1}{\frac{2}{\pi} \cdot \left(\color{blue}{\left(a + b\right)} \cdot \left(a \cdot b\right)\right)} \]

6. Applied egg-rr 99.0%

   \[\leadsto \color{blue}{\frac{1}{\frac{2}{\pi} \cdot \left(\left(a + b\right) \cdot \left(a \cdot b\right)\right)}} \]
7. Step-by-step derivation

   1. associate-*l/ 99.1%

      \[\leadsto \frac{1}{\color{blue}{\frac{2 \cdot \left(\left(a + b\right) \cdot \left(a \cdot b\right)\right)}{\pi}}} \]

   2. clear-num 99.0%

      \[\leadsto \color{blue}{\frac{\pi}{2 \cdot \left(\left(a + b\right) \cdot \left(a \cdot b\right)\right)}} \]

   3. *-un-lft-identity 99.0%

      \[\leadsto \frac{\color{blue}{1 \cdot \pi}}{2 \cdot \left(\left(a + b\right) \cdot \left(a \cdot b\right)\right)} \]

   4. times-frac 99.0%

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\pi}{\left(a + b\right) \cdot \left(a \cdot b\right)}} \]

   5. metadata-eval 99.0%

      \[\leadsto \color{blue}{0.5} \cdot \frac{\pi}{\left(a + b\right) \cdot \left(a \cdot b\right)} \]

   6. metadata-eval 99.0%

      \[\leadsto \color{blue}{\frac{0.5}{1}} \cdot \frac{\pi}{\left(a + b\right) \cdot \left(a \cdot b\right)} \]

   7. times-frac 99.0%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{1 \cdot \left(\left(a + b\right) \cdot \left(a \cdot b\right)\right)}} \]

   8. *-un-lft-identity 99.0%

      \[\leadsto \frac{0.5 \cdot \pi}{\color{blue}{\left(a + b\right) \cdot \left(a \cdot b\right)}} \]

   9. *-commutative 99.0%

      \[\leadsto \frac{\color{blue}{\pi \cdot 0.5}}{\left(a + b\right) \cdot \left(a \cdot b\right)} \]

   10. *-un-lft-identity 99.0%

       \[\leadsto \frac{\color{blue}{1 \cdot \left(\pi \cdot 0.5\right)}}{\left(a + b\right) \cdot \left(a \cdot b\right)} \]

   11. associate-*r* 92.9%

       \[\leadsto \frac{1 \cdot \left(\pi \cdot 0.5\right)}{\color{blue}{\left(\left(a + b\right) \cdot a\right) \cdot b}} \]

   12. times-frac 93.4%

       \[\leadsto \color{blue}{\frac{1}{\left(a + b\right) \cdot a} \cdot \frac{\pi \cdot 0.5}{b}} \]

8. Applied egg-rr 93.4%

   \[\leadsto \color{blue}{\frac{1}{\left(a + b\right) \cdot a} \cdot \frac{\pi \cdot 0.5}{b}} \]
9. Step-by-step derivation

   1. associate-*l/ 93.5%

      \[\leadsto \color{blue}{\frac{1 \cdot \frac{\pi \cdot 0.5}{b}}{\left(a + b\right) \cdot a}} \]

   2. *-commutative 93.5%

      \[\leadsto \frac{1 \cdot \frac{\color{blue}{0.5 \cdot \pi}}{b}}{\left(a + b\right) \cdot a} \]

   3. associate-*r/ 93.5%

      \[\leadsto \frac{1 \cdot \color{blue}{\left(0.5 \cdot \frac{\pi}{b}\right)}}{\left(a + b\right) \cdot a} \]

   4. *-lft-identity 93.5%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \frac{\pi}{b}}}{\left(a + b\right) \cdot a} \]

   5. associate-*r/ 93.5%

      \[\leadsto \frac{\color{blue}{\frac{0.5 \cdot \pi}{b}}}{\left(a + b\right) \cdot a} \]

   6. *-commutative 93.5%

      \[\leadsto \frac{\frac{\color{blue}{\pi \cdot 0.5}}{b}}{\left(a + b\right) \cdot a} \]

   7. associate-/l* 93.3%

      \[\leadsto \frac{\color{blue}{\pi \cdot \frac{0.5}{b}}}{\left(a + b\right) \cdot a} \]

   8. metadata-eval 93.3%

      \[\leadsto \frac{\pi \cdot \frac{\color{blue}{0.5 \cdot 1}}{b}}{\left(a + b\right) \cdot a} \]

   9. associate-*r/ 93.3%

      \[\leadsto \frac{\pi \cdot \color{blue}{\left(0.5 \cdot \frac{1}{b}\right)}}{\left(a + b\right) \cdot a} \]

   10. associate-*l/ 93.3%

       \[\leadsto \color{blue}{\frac{\pi}{\left(a + b\right) \cdot a} \cdot \left(0.5 \cdot \frac{1}{b}\right)} \]

   11. *-commutative 93.3%

       \[\leadsto \color{blue}{\left(0.5 \cdot \frac{1}{b}\right) \cdot \frac{\pi}{\left(a + b\right) \cdot a}} \]

   12. associate-*r/ 93.3%

       \[\leadsto \color{blue}{\frac{0.5 \cdot 1}{b}} \cdot \frac{\pi}{\left(a + b\right) \cdot a} \]

   13. metadata-eval 93.3%

       \[\leadsto \frac{\color{blue}{0.5}}{b} \cdot \frac{\pi}{\left(a + b\right) \cdot a} \]

   14. associate-/l/ 94.1%

       \[\leadsto \frac{0.5}{b} \cdot \color{blue}{\frac{\frac{\pi}{a}}{a + b}} \]

10. Simplified 94.1%

    \[\leadsto \color{blue}{\frac{0.5}{b} \cdot \frac{\frac{\pi}{a}}{a + b}} \]

11. Taylor expanded in a around inf 57.0%

    \[\leadsto \frac{0.5}{b} \cdot \frac{\frac{\pi}{a}}{\color{blue}{a}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024139 
(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))))