NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.5% → 99.7%

Time: 11.7s

Alternatives: 12

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: 78.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.5%

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

   1. associate-*l* 78.5%

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

   2. *-rgt-identity 78.5%

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

   3. associate-/l* 78.5%

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

   4. metadata-eval 78.5%

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

   5. associate-*l/ 78.5%

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

   6. *-lft-identity 78.5%

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

   7. sub-neg 78.5%

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

   8. distribute-neg-frac 78.5%

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

   9. metadata-eval 78.5%

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

3. Simplified 78.5%

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

   1. *-un-lft-identity 78.5%

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

   2. clear-num 78.4%

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

   3. frac-add 78.4%

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

   4. associate-/r/ 78.5%

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

   5. *-un-lft-identity 78.5%

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

   6. *-commutative 78.5%

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

   7. neg-mul-1 78.5%

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

   8. sub-neg 78.5%

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

   9. flip-+ 99.1%

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

   10. +-commutative 99.1%

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

6. Applied egg-rr 99.1%

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

   1. *-lft-identity 99.1%

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

   2. associate-/r* 99.6%

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

8. Simplified 99.6%

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

   1. *-un-lft-identity 99.6%

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

   2. associate-/l/ 99.1%

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

   3. *-commutative 99.1%

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

10. Applied egg-rr 99.1%

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

    1. *-lft-identity 99.1%

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

    2. associate-/r* 99.6%

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

    3. *-rgt-identity 99.6%

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

    4. associate-*r/ 99.6%

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

    5. associate-*l/ 99.6%

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

    6. *-lft-identity 99.6%

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

12. Simplified 99.6%

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

    1. associate-*r/ 99.7%

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

    2. div-inv 99.8%

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

    3. *-commutative 99.8%

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

    4. associate-*r/ 99.8%

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

    5. associate-/r* 99.8%

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

14. Applied egg-rr 99.8%

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

Alternative 2: 99.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -3 \cdot 10^{+98}:\\ \;\;\;\;\frac{\frac{\pi}{b \cdot \left(a \cdot 2\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(a \cdot \left(a + b\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -3e+98)
   (/ (/ PI (* b (* a 2.0))) a)
   (* PI (/ 0.5 (* b (* a (+ a b)))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -3e+98) {
		tmp = (((double) M_PI) / (b * (a * 2.0))) / a;
	} else {
		tmp = ((double) M_PI) * (0.5 / (b * (a * (a + b))));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -3e+98) {
		tmp = (Math.PI / (b * (a * 2.0))) / a;
	} else {
		tmp = Math.PI * (0.5 / (b * (a * (a + b))));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -3e+98:
		tmp = (math.pi / (b * (a * 2.0))) / a
	else:
		tmp = math.pi * (0.5 / (b * (a * (a + b))))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -3e+98)
		tmp = Float64(Float64(pi / Float64(b * Float64(a * 2.0))) / a);
	else
		tmp = Float64(pi * Float64(0.5 / Float64(b * Float64(a * Float64(a + b)))));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -3e+98)
		tmp = (pi / (b * (a * 2.0))) / a;
	else
		tmp = pi * (0.5 / (b * (a * (a + b))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -3e+98], N[(N[(Pi / N[(b * N[(a * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(Pi * N[(0.5 / N[(b * N[(a * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.0000000000000001e98

   1. Initial program 71.0%

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

      1. *-commutative 71.0%

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

      2. associate-*r* 71.0%

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

      3. associate-*r/ 71.0%

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

      4. associate-*r* 71.0%

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

      5. *-rgt-identity 71.0%

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

      6. sub-neg 71.0%

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

      7. distribute-neg-frac 71.0%

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

      8. metadata-eval 71.0%

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

   3. Simplified 71.0%

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

      1. associate-*l/ 71.0%

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

      2. div-inv 71.0%

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

      3. metadata-eval 71.0%

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

      4. *-commutative 71.0%

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

      5. associate-*r* 71.0%

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

      6. *-commutative 71.0%

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

   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 99.6%

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

      1. associate-/r* 99.8%

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

      2. div-inv 99.7%

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

      3. associate-/r* 99.7%

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

   9. Applied egg-rr 99.7%

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

       1. associate-*l/ 99.8%

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

       2. associate-*r/ 99.8%

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

       3. associate-*r/ 99.8%

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

       4. metadata-eval 99.8%

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

   11. Simplified 99.8%

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

       1. associate-*l/ 99.8%

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

       2. clear-num 99.9%

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

       3. associate-*l/ 99.9%

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

       4. *-un-lft-identity 99.9%

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

       5. div-inv 99.8%

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

       6. clear-num 100.0%

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

       7. div-inv 100.0%

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

       8. metadata-eval 100.0%

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

   13. Applied egg-rr 100.0%

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

   if -3.0000000000000001e98 < a

   1. Initial program 79.8%

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

      1. *-commutative 79.8%

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

      2. associate-*r* 79.7%

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

      3. associate-*r/ 79.7%

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

      4. associate-*r* 79.7%

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

      5. *-rgt-identity 79.7%

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

      6. sub-neg 79.7%

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

      7. distribute-neg-frac 79.7%

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

      8. metadata-eval 79.7%

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

   3. Simplified 79.7%

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

      1. associate-*l/ 79.8%

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

      2. div-inv 79.8%

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

      3. metadata-eval 79.8%

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

      4. *-commutative 79.8%

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

      5. associate-*r* 79.8%

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

      6. *-commutative 79.8%

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

   6. Applied egg-rr 99.1%

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

   7. Taylor expanded in b around 0 88.8%

      \[\leadsto \frac{0.5}{\color{blue}{b \cdot \left(a \cdot b + {a}^{2}\right)}} \cdot \pi \]
   8. Step-by-step derivation

      1. +-commutative 88.8%

         \[\leadsto \frac{0.5}{b \cdot \color{blue}{\left({a}^{2} + a \cdot b\right)}} \cdot \pi \]

      2. unpow2 88.8%

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

      3. distribute-lft-in 96.1%

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

   9. Simplified 96.1%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3 \cdot 10^{+98}:\\ \;\;\;\;\frac{\frac{\pi}{b \cdot \left(a \cdot 2\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(a \cdot \left(a + b\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 7 \cdot 10^{+98}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(b \cdot \left(a + b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{b \cdot \left(a \cdot 2\right)}}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 7e+98)
   (* PI (/ 0.5 (* a (* b (+ a b)))))
   (/ (/ PI (* b (* a 2.0))) b)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 7e+98) {
		tmp = ((double) M_PI) * (0.5 / (a * (b * (a + b))));
	} else {
		tmp = (((double) M_PI) / (b * (a * 2.0))) / b;
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 7e+98) {
		tmp = Math.PI * (0.5 / (a * (b * (a + b))));
	} else {
		tmp = (Math.PI / (b * (a * 2.0))) / b;
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 7e+98:
		tmp = math.pi * (0.5 / (a * (b * (a + b))))
	else:
		tmp = (math.pi / (b * (a * 2.0))) / b
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 7e+98)
		tmp = Float64(pi * Float64(0.5 / Float64(a * Float64(b * Float64(a + b)))));
	else
		tmp = Float64(Float64(pi / Float64(b * Float64(a * 2.0))) / b);
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 7e+98)
		tmp = pi * (0.5 / (a * (b * (a + b))));
	else
		tmp = (pi / (b * (a * 2.0))) / b;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[b, 7e+98], N[(Pi * N[(0.5 / N[(a * N[(b * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / N[(b * N[(a * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7e98

   1. Initial program 82.3%

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

      1. *-commutative 82.3%

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

      2. associate-*r* 82.2%

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

      3. associate-*r/ 82.2%

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

      4. associate-*r* 82.2%

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

      5. *-rgt-identity 82.2%

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

      6. sub-neg 82.2%

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

      7. distribute-neg-frac 82.2%

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

      8. metadata-eval 82.2%

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

   3. Simplified 82.2%

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

      1. associate-*l/ 82.3%

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

      2. div-inv 82.3%

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

      3. metadata-eval 82.3%

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

      4. *-commutative 82.3%

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

      5. associate-*r* 82.3%

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

      6. *-commutative 82.3%

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

   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. Taylor expanded in a around 0 87.6%

      \[\leadsto \frac{0.5}{\color{blue}{a \cdot \left(a \cdot b + {b}^{2}\right)}} \cdot \pi \]
   8. Step-by-step derivation

      1. unpow2 87.6%

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

      2. distribute-rgt-in 95.4%

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

   9. Simplified 95.4%

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

   if 7e98 < b

   1. Initial program 62.5%

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

      1. *-commutative 62.5%

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

      2. associate-*r* 62.5%

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

      3. associate-*r/ 62.5%

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

      4. associate-*r* 62.5%

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

      5. *-rgt-identity 62.5%

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

      6. sub-neg 62.5%

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

      7. distribute-neg-frac 62.5%

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

      8. metadata-eval 62.5%

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

   3. Simplified 62.5%

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

      1. associate-*l/ 62.5%

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

      2. div-inv 62.5%

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

      3. metadata-eval 62.5%

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

      4. *-commutative 62.5%

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

      5. associate-*r* 62.5%

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

      6. *-commutative 62.5%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in a around 0 99.8%

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

      1. associate-*l/ 99.7%

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

      2. *-commutative 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-/r* 99.8%

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

      5. *-commutative 99.8%

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

      6. associate-*l/ 99.7%

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

      7. associate-/r* 99.7%

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

      8. clear-num 99.7%

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

      9. associate-*l/ 99.7%

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

      10. *-un-lft-identity 99.7%

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

      11. div-inv 99.7%

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

      12. clear-num 99.8%

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

      13. div-inv 99.8%

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

      14. metadata-eval 99.8%

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

   9. Applied egg-rr 99.8%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7 \cdot 10^{+98}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(b \cdot \left(a + b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{b \cdot \left(a \cdot 2\right)}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} t_0 := \frac{\pi}{b \cdot \left(a \cdot 2\right)}\\ \mathbf{if}\;a \leq -4 \cdot 10^{-106}:\\ \;\;\;\;\frac{t\_0}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ PI (* b (* a 2.0))))) (if (<= a -4e-106) (/ t_0 a) (/ t_0 b))))

C

assert(a < b);
double code(double a, double b) {
	double t_0 = ((double) M_PI) / (b * (a * 2.0));
	double tmp;
	if (a <= -4e-106) {
		tmp = t_0 / a;
	} else {
		tmp = t_0 / b;
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double t_0 = Math.PI / (b * (a * 2.0));
	double tmp;
	if (a <= -4e-106) {
		tmp = t_0 / a;
	} else {
		tmp = t_0 / b;
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	t_0 = math.pi / (b * (a * 2.0))
	tmp = 0
	if a <= -4e-106:
		tmp = t_0 / a
	else:
		tmp = t_0 / b
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	t_0 = Float64(pi / Float64(b * Float64(a * 2.0)))
	tmp = 0.0
	if (a <= -4e-106)
		tmp = Float64(t_0 / a);
	else
		tmp = Float64(t_0 / b);
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	t_0 = pi / (b * (a * 2.0));
	tmp = 0.0;
	if (a <= -4e-106)
		tmp = t_0 / a;
	else
		tmp = t_0 / b;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := Block[{t$95$0 = N[(Pi / N[(b * N[(a * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -4e-106], N[(t$95$0 / a), $MachinePrecision], N[(t$95$0 / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -3.99999999999999976e-106

   1. Initial program 84.5%

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

      1. *-commutative 84.5%

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

      2. associate-*r* 84.4%

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

      3. associate-*r/ 84.5%

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

      4. associate-*r* 84.5%

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

      5. *-rgt-identity 84.5%

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

      6. sub-neg 84.5%

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

      7. distribute-neg-frac 84.5%

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

      8. metadata-eval 84.5%

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

   3. Simplified 84.5%

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

      1. associate-*l/ 84.5%

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

      2. div-inv 84.5%

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

      3. metadata-eval 84.5%

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

      4. *-commutative 84.5%

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

      5. associate-*r* 84.5%

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

      6. *-commutative 84.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in a around inf 86.0%

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

      1. associate-/r* 85.0%

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

      2. div-inv 85.0%

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

      3. associate-/r* 84.9%

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

   9. Applied egg-rr 84.9%

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

       1. associate-*l/ 85.0%

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

       2. associate-*r/ 85.0%

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

       3. associate-*r/ 85.0%

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

       4. metadata-eval 85.0%

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

   11. Simplified 85.0%

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

       1. associate-*l/ 85.0%

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

       2. clear-num 85.1%

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

       3. associate-*l/ 85.0%

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

       4. *-un-lft-identity 85.0%

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

       5. div-inv 85.0%

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

       6. clear-num 85.1%

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

       7. div-inv 85.1%

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

       8. metadata-eval 85.1%

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

   13. Applied egg-rr 85.1%

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

   if -3.99999999999999976e-106 < a

   1. Initial program 76.2%

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

      1. *-commutative 76.2%

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

      2. associate-*r* 76.2%

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

      3. associate-*r/ 76.2%

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

      4. associate-*r* 76.2%

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

      5. *-rgt-identity 76.2%

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

      6. sub-neg 76.2%

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

      7. distribute-neg-frac 76.2%

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

      8. metadata-eval 76.2%

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

   3. Simplified 76.2%

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

      1. associate-*l/ 76.2%

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

      2. div-inv 76.2%

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

      3. metadata-eval 76.2%

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

      4. *-commutative 76.2%

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

      5. associate-*r* 76.2%

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

      6. *-commutative 76.2%

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

   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. Taylor expanded in a around 0 73.1%

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

      1. associate-*l/ 73.1%

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

      2. *-commutative 73.1%

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

      3. *-commutative 73.1%

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

      4. associate-/r* 73.3%

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

      5. *-commutative 73.3%

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

      6. associate-*l/ 73.3%

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

      7. associate-/r* 73.3%

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

      8. clear-num 73.2%

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

      9. associate-*l/ 73.3%

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

      10. *-un-lft-identity 73.3%

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

      11. div-inv 73.3%

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

      12. clear-num 73.3%

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

      13. div-inv 73.3%

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

      14. metadata-eval 73.3%

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

   9. Applied egg-rr 73.3%

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

Alternative 5: 89.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -4 \cdot 10^{-106}:\\ \;\;\;\;\frac{\frac{\pi}{b \cdot \left(a \cdot 2\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(a \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -4e-106) (/ (/ PI (* b (* a 2.0))) a) (* PI (/ 0.5 (* b (* a b))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -4e-106) {
		tmp = (((double) M_PI) / (b * (a * 2.0))) / a;
	} else {
		tmp = ((double) M_PI) * (0.5 / (b * (a * b)));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -4e-106) {
		tmp = (Math.PI / (b * (a * 2.0))) / a;
	} else {
		tmp = Math.PI * (0.5 / (b * (a * b)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -4e-106:
		tmp = (math.pi / (b * (a * 2.0))) / a
	else:
		tmp = math.pi * (0.5 / (b * (a * b)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -4e-106)
		tmp = Float64(Float64(pi / Float64(b * Float64(a * 2.0))) / a);
	else
		tmp = Float64(pi * Float64(0.5 / Float64(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 <= -4e-106)
		tmp = (pi / (b * (a * 2.0))) / a;
	else
		tmp = pi * (0.5 / (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, -4e-106], N[(N[(Pi / N[(b * N[(a * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(Pi * N[(0.5 / N[(b * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.99999999999999976e-106

   1. Initial program 84.5%

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

      1. *-commutative 84.5%

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

      2. associate-*r* 84.4%

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

      3. associate-*r/ 84.5%

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

      4. associate-*r* 84.5%

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

      5. *-rgt-identity 84.5%

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

      6. sub-neg 84.5%

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

      7. distribute-neg-frac 84.5%

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

      8. metadata-eval 84.5%

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

   3. Simplified 84.5%

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

      1. associate-*l/ 84.5%

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

      2. div-inv 84.5%

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

      3. metadata-eval 84.5%

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

      4. *-commutative 84.5%

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

      5. associate-*r* 84.5%

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

      6. *-commutative 84.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in a around inf 86.0%

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

      1. associate-/r* 85.0%

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

      2. div-inv 85.0%

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

      3. associate-/r* 84.9%

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

   9. Applied egg-rr 84.9%

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

       1. associate-*l/ 85.0%

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

       2. associate-*r/ 85.0%

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

       3. associate-*r/ 85.0%

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

       4. metadata-eval 85.0%

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

   11. Simplified 85.0%

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

       1. associate-*l/ 85.0%

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

       2. clear-num 85.1%

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

       3. associate-*l/ 85.0%

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

       4. *-un-lft-identity 85.0%

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

       5. div-inv 85.0%

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

       6. clear-num 85.1%

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

       7. div-inv 85.1%

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

       8. metadata-eval 85.1%

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

   13. Applied egg-rr 85.1%

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

   if -3.99999999999999976e-106 < a

   1. Initial program 76.2%

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

      1. *-commutative 76.2%

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

      2. associate-*r* 76.2%

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

      3. associate-*r/ 76.2%

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

      4. associate-*r* 76.2%

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

      5. *-rgt-identity 76.2%

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

      6. sub-neg 76.2%

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

      7. distribute-neg-frac 76.2%

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

      8. metadata-eval 76.2%

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

   3. Simplified 76.2%

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

      1. associate-*l/ 76.2%

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

      2. div-inv 76.2%

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

      3. metadata-eval 76.2%

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

      4. *-commutative 76.2%

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

      5. associate-*r* 76.2%

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

      6. *-commutative 76.2%

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

   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. Taylor expanded in a around 0 73.1%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4 \cdot 10^{-106}:\\ \;\;\;\;\frac{\frac{\pi}{b \cdot \left(a \cdot 2\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(a \cdot b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -4 \cdot 10^{-106}:\\ \;\;\;\;\frac{\pi \cdot \frac{0.5}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(a \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -4e-106) (/ (* PI (/ 0.5 a)) (* a b)) (* PI (/ 0.5 (* b (* a b))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -4e-106) {
		tmp = (((double) M_PI) * (0.5 / a)) / (a * b);
	} else {
		tmp = ((double) M_PI) * (0.5 / (b * (a * b)));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -4e-106) {
		tmp = (Math.PI * (0.5 / a)) / (a * b);
	} else {
		tmp = Math.PI * (0.5 / (b * (a * b)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -4e-106:
		tmp = (math.pi * (0.5 / a)) / (a * b)
	else:
		tmp = math.pi * (0.5 / (b * (a * b)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -4e-106)
		tmp = Float64(Float64(pi * Float64(0.5 / a)) / Float64(a * b));
	else
		tmp = Float64(pi * Float64(0.5 / Float64(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 <= -4e-106)
		tmp = (pi * (0.5 / a)) / (a * b);
	else
		tmp = pi * (0.5 / (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, -4e-106], N[(N[(Pi * N[(0.5 / a), $MachinePrecision]), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(0.5 / N[(b * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.99999999999999976e-106

   1. Initial program 84.5%

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

      1. *-commutative 84.5%

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

      2. associate-*r* 84.4%

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

      3. associate-*r/ 84.5%

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

      4. associate-*r* 84.5%

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

      5. *-rgt-identity 84.5%

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

      6. sub-neg 84.5%

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

      7. distribute-neg-frac 84.5%

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

      8. metadata-eval 84.5%

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

   3. Simplified 84.5%

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

      1. associate-*l/ 84.5%

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

      2. div-inv 84.5%

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

      3. metadata-eval 84.5%

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

      4. *-commutative 84.5%

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

      5. associate-*r* 84.5%

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

      6. *-commutative 84.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in a around inf 86.0%

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

      1. associate-/r* 85.0%

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

      2. div-inv 85.0%

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

      3. associate-/r* 84.9%

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

   9. Applied egg-rr 84.9%

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

       1. associate-*l/ 85.0%

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

       2. associate-*r/ 85.0%

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

       3. associate-*r/ 85.0%

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

       4. metadata-eval 85.0%

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

   11. Simplified 85.0%

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

       1. associate-/l/ 85.0%

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

       2. associate-*l/ 85.0%

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

   13. Applied egg-rr 85.0%

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

   if -3.99999999999999976e-106 < a

   1. Initial program 76.2%

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

      1. *-commutative 76.2%

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

      2. associate-*r* 76.2%

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

      3. associate-*r/ 76.2%

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

      4. associate-*r* 76.2%

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

      5. *-rgt-identity 76.2%

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

      6. sub-neg 76.2%

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

      7. distribute-neg-frac 76.2%

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

      8. metadata-eval 76.2%

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

   3. Simplified 76.2%

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

      1. associate-*l/ 76.2%

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

      2. div-inv 76.2%

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

      3. metadata-eval 76.2%

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

      4. *-commutative 76.2%

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

      5. associate-*r* 76.2%

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

      6. *-commutative 76.2%

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

   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. Taylor expanded in a around 0 73.1%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4 \cdot 10^{-106}:\\ \;\;\;\;\frac{\pi \cdot \frac{0.5}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(a \cdot b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -4 \cdot 10^{-106}:\\ \;\;\;\;\pi \cdot \frac{\frac{\frac{0.5}{b}}{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(a \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -4e-106) (* PI (/ (/ (/ 0.5 b) a) a)) (* PI (/ 0.5 (* b (* a b))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -4e-106) {
		tmp = ((double) M_PI) * (((0.5 / b) / a) / a);
	} else {
		tmp = ((double) M_PI) * (0.5 / (b * (a * b)));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -4e-106) {
		tmp = Math.PI * (((0.5 / b) / a) / a);
	} else {
		tmp = Math.PI * (0.5 / (b * (a * b)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -4e-106:
		tmp = math.pi * (((0.5 / b) / a) / a)
	else:
		tmp = math.pi * (0.5 / (b * (a * b)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -4e-106)
		tmp = Float64(pi * Float64(Float64(Float64(0.5 / b) / a) / a));
	else
		tmp = Float64(pi * Float64(0.5 / Float64(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 <= -4e-106)
		tmp = pi * (((0.5 / b) / a) / a);
	else
		tmp = pi * (0.5 / (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, -4e-106], N[(Pi * N[(N[(N[(0.5 / b), $MachinePrecision] / a), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(0.5 / N[(b * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.99999999999999976e-106

   1. Initial program 84.5%

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

      1. *-commutative 84.5%

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

      2. associate-*r* 84.4%

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

      3. associate-*r/ 84.5%

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

      4. associate-*r* 84.5%

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

      5. *-rgt-identity 84.5%

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

      6. sub-neg 84.5%

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

      7. distribute-neg-frac 84.5%

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

      8. metadata-eval 84.5%

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

   3. Simplified 84.5%

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

      1. associate-*l/ 84.5%

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

      2. div-inv 84.5%

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

      3. metadata-eval 84.5%

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

      4. *-commutative 84.5%

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

      5. associate-*r* 84.5%

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

      6. *-commutative 84.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in a around inf 86.0%

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

      1. associate-/r* 85.0%

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

      2. div-inv 85.0%

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

      3. associate-/r* 84.9%

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

   9. Applied egg-rr 84.9%

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

       1. associate-*l/ 85.0%

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

       2. associate-*r/ 85.0%

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

       3. associate-*r/ 85.0%

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

       4. metadata-eval 85.0%

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

   11. Simplified 85.0%

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

   12. Taylor expanded in a around 0 85.0%

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

       1. associate-/l/ 85.1%

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

   14. Simplified 85.1%

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

   if -3.99999999999999976e-106 < a

   1. Initial program 76.2%

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

      1. *-commutative 76.2%

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

      2. associate-*r* 76.2%

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

      3. associate-*r/ 76.2%

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

      4. associate-*r* 76.2%

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

      5. *-rgt-identity 76.2%

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

      6. sub-neg 76.2%

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

      7. distribute-neg-frac 76.2%

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

      8. metadata-eval 76.2%

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

   3. Simplified 76.2%

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

      1. associate-*l/ 76.2%

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

      2. div-inv 76.2%

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

      3. metadata-eval 76.2%

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

      4. *-commutative 76.2%

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

      5. associate-*r* 76.2%

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

      6. *-commutative 76.2%

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

   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. Taylor expanded in a around 0 73.1%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4 \cdot 10^{-106}:\\ \;\;\;\;\pi \cdot \frac{\frac{\frac{0.5}{b}}{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(a \cdot b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -4 \cdot 10^{-106}:\\ \;\;\;\;\pi \cdot \frac{\frac{\frac{0.5}{a}}{b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(a \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -4e-106) (* PI (/ (/ (/ 0.5 a) b) a)) (* PI (/ 0.5 (* b (* a b))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -4e-106) {
		tmp = ((double) M_PI) * (((0.5 / a) / b) / a);
	} else {
		tmp = ((double) M_PI) * (0.5 / (b * (a * b)));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -4e-106) {
		tmp = Math.PI * (((0.5 / a) / b) / a);
	} else {
		tmp = Math.PI * (0.5 / (b * (a * b)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -4e-106:
		tmp = math.pi * (((0.5 / a) / b) / a)
	else:
		tmp = math.pi * (0.5 / (b * (a * b)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -4e-106)
		tmp = Float64(pi * Float64(Float64(Float64(0.5 / a) / b) / a));
	else
		tmp = Float64(pi * Float64(0.5 / Float64(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 <= -4e-106)
		tmp = pi * (((0.5 / a) / b) / a);
	else
		tmp = pi * (0.5 / (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, -4e-106], N[(Pi * N[(N[(N[(0.5 / a), $MachinePrecision] / b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(0.5 / N[(b * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.99999999999999976e-106

   1. Initial program 84.5%

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

      1. *-commutative 84.5%

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

      2. associate-*r* 84.4%

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

      3. associate-*r/ 84.5%

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

      4. associate-*r* 84.5%

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

      5. *-rgt-identity 84.5%

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

      6. sub-neg 84.5%

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

      7. distribute-neg-frac 84.5%

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

      8. metadata-eval 84.5%

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

   3. Simplified 84.5%

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

      1. associate-*l/ 84.5%

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

      2. div-inv 84.5%

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

      3. metadata-eval 84.5%

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

      4. *-commutative 84.5%

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

      5. associate-*r* 84.5%

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

      6. *-commutative 84.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in a around inf 86.0%

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

      1. associate-/r* 85.0%

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

      2. div-inv 85.0%

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

      3. associate-/r* 84.9%

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

   9. Applied egg-rr 84.9%

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

       1. associate-*l/ 85.0%

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

       2. associate-*r/ 85.0%

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

       3. associate-*r/ 85.0%

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

       4. metadata-eval 85.0%

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

   11. Simplified 85.0%

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

   if -3.99999999999999976e-106 < a

   1. Initial program 76.2%

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

      1. *-commutative 76.2%

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

      2. associate-*r* 76.2%

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

      3. associate-*r/ 76.2%

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

      4. associate-*r* 76.2%

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

      5. *-rgt-identity 76.2%

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

      6. sub-neg 76.2%

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

      7. distribute-neg-frac 76.2%

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

      8. metadata-eval 76.2%

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

   3. Simplified 76.2%

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

      1. associate-*l/ 76.2%

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

      2. div-inv 76.2%

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

      3. metadata-eval 76.2%

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

      4. *-commutative 76.2%

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

      5. associate-*r* 76.2%

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

      6. *-commutative 76.2%

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

   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. Taylor expanded in a around 0 73.1%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4 \cdot 10^{-106}:\\ \;\;\;\;\pi \cdot \frac{\frac{\frac{0.5}{a}}{b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(a \cdot b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 89.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -4 \cdot 10^{-106}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(a \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -4e-106) (* PI (/ 0.5 (* a (* a b)))) (* PI (/ 0.5 (* b (* a b))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -4e-106) {
		tmp = ((double) M_PI) * (0.5 / (a * (a * b)));
	} else {
		tmp = ((double) M_PI) * (0.5 / (b * (a * b)));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -4e-106) {
		tmp = Math.PI * (0.5 / (a * (a * b)));
	} else {
		tmp = Math.PI * (0.5 / (b * (a * b)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -4e-106:
		tmp = math.pi * (0.5 / (a * (a * b)))
	else:
		tmp = math.pi * (0.5 / (b * (a * b)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -4e-106)
		tmp = Float64(pi * Float64(0.5 / Float64(a * Float64(a * b))));
	else
		tmp = Float64(pi * Float64(0.5 / Float64(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 <= -4e-106)
		tmp = pi * (0.5 / (a * (a * b)));
	else
		tmp = pi * (0.5 / (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, -4e-106], N[(Pi * N[(0.5 / N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(0.5 / N[(b * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.99999999999999976e-106

   1. Initial program 84.5%

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

      1. *-commutative 84.5%

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

      2. associate-*r* 84.4%

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

      3. associate-*r/ 84.5%

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

      4. associate-*r* 84.5%

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

      5. *-rgt-identity 84.5%

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

      6. sub-neg 84.5%

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

      7. distribute-neg-frac 84.5%

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

      8. metadata-eval 84.5%

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

   3. Simplified 84.5%

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

      1. associate-*l/ 84.5%

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

      2. div-inv 84.5%

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

      3. metadata-eval 84.5%

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

      4. *-commutative 84.5%

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

      5. associate-*r* 84.5%

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

      6. *-commutative 84.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in a around inf 86.0%

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

   if -3.99999999999999976e-106 < a

   1. Initial program 76.2%

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

      1. *-commutative 76.2%

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

      2. associate-*r* 76.2%

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

      3. associate-*r/ 76.2%

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

      4. associate-*r* 76.2%

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

      5. *-rgt-identity 76.2%

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

      6. sub-neg 76.2%

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

      7. distribute-neg-frac 76.2%

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

      8. metadata-eval 76.2%

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

   3. Simplified 76.2%

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

      1. associate-*l/ 76.2%

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

      2. div-inv 76.2%

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

      3. metadata-eval 76.2%

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

      4. *-commutative 76.2%

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

      5. associate-*r* 76.2%

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

      6. *-commutative 76.2%

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

   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. Taylor expanded in a around 0 73.1%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4 \cdot 10^{-106}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(a \cdot b\right)}\\ \end{array} \]
5. 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 78.5%

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

   1. associate-*l* 78.5%

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

   2. *-rgt-identity 78.5%

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

   3. associate-/l* 78.5%

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

   4. metadata-eval 78.5%

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

   5. associate-*l/ 78.5%

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

   6. *-lft-identity 78.5%

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

   7. sub-neg 78.5%

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

   8. distribute-neg-frac 78.5%

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

   9. metadata-eval 78.5%

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

3. Simplified 78.5%

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

   1. metadata-eval 78.5%

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

   2. div-inv 78.5%

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

   3. clear-num 78.5%

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

   4. clear-num 78.4%

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

   5. frac-times 78.4%

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

   6. metadata-eval 78.4%

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

   7. frac-add 78.4%

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

   8. associate-/r/ 78.5%

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

   9. *-un-lft-identity 78.5%

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

   10. *-commutative 78.5%

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

   11. neg-mul-1 78.5%

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

   12. sub-neg 78.5%

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

   13. flip-+ 99.1%

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

   14. +-commutative 99.1%

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

6. Applied egg-rr 99.1%

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

   1. associate-/r* 99.2%

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

   2. associate-/r/ 99.2%

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

   3. metadata-eval 99.2%

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

   4. *-commutative 99.2%

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

   5. times-frac 99.6%

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

8. Simplified 99.6%

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

Alternative 11: 99.0% 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 78.5%

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

   1. *-commutative 78.5%

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

   2. associate-*r* 78.4%

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

   3. associate-*r/ 78.5%

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

   4. associate-*r* 78.5%

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

   5. *-rgt-identity 78.5%

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

   6. sub-neg 78.5%

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

   7. distribute-neg-frac 78.5%

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

   8. metadata-eval 78.5%

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

3. Simplified 78.5%

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

   1. associate-*l/ 78.5%

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

   2. div-inv 78.5%

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

   3. metadata-eval 78.5%

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

   4. *-commutative 78.5%

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

   5. associate-*r* 78.5%

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

   6. *-commutative 78.5%

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

6. Applied egg-rr 99.1%

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

7. Final simplification 99.1%

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

Alternative 12: 63.4% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 78.5%

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

   1. *-commutative 78.5%

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

   2. associate-*r* 78.4%

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

   3. associate-*r/ 78.5%

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

   4. associate-*r* 78.5%

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

   5. *-rgt-identity 78.5%

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

   6. sub-neg 78.5%

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

   7. distribute-neg-frac 78.5%

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

   8. metadata-eval 78.5%

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

3. Simplified 78.5%

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

   1. associate-*l/ 78.5%

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

   2. div-inv 78.5%

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

   3. metadata-eval 78.5%

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

   4. *-commutative 78.5%

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

   5. associate-*r* 78.5%

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

   6. *-commutative 78.5%

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

6. Applied egg-rr 99.1%

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

7. Taylor expanded in a around inf 61.6%

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

8. Final simplification 61.6%

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

Reproduce

herbie shell --seed 2024116 
(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))))