NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.7% → 99.7%

Time: 10.3s

Alternatives: 7

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: 78.7% 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 80.9%

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

   1. *-commutative 80.9%

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

   2. associate-*r* 80.9%

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

   3. associate-*r/ 80.9%

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

   4. associate-*r* 80.9%

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

   5. *-rgt-identity 80.9%

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

   6. sub-neg 80.9%

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

   7. distribute-neg-frac 80.9%

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

   8. metadata-eval 80.9%

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

3. Simplified 80.9%

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

   1. *-un-lft-identity 80.9%

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

   2. difference-of-squares 88.3%

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

   3. times-frac 99.5%

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

6. Applied egg-rr 61.5%

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

   1. associate-*l/ 61.6%

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

   2. *-lft-identity 61.6%

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

   3. associate-*r* 61.6%

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

   4. distribute-rgt-out 61.6%

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

   5. associate-*l/ 61.6%

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

   6. *-lft-identity 61.6%

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

   7. *-commutative 61.6%

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

   8. associate-*r/ 61.6%

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

   9. associate-*l/ 61.6%

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

   10. *-lft-identity 61.6%

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

   11. *-commutative 61.6%

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

   12. associate-*r/ 61.6%

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

   13. fma-define 61.6%

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

   14. associate-*r/ 61.6%

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

   15. +-commutative 61.6%

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

8. Simplified 61.6%

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

9. Taylor expanded in a around 0 99.6%

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

    1. associate-*r/ 99.6%

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

11. Applied egg-rr 99.6%

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

    1. associate-/r* 99.6%

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

    2. associate-*r/ 99.6%

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

    3. associate-*r/ 99.6%

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

13. Simplified 99.6%

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

Alternative 2: 99.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 2.85 \cdot 10^{+147}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{a \cdot \left(b \cdot \left(a + b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{b}}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 2.85e+147)
   (/ (* 0.5 PI) (* a (* b (+ a b))))
   (/ (* 0.5 (/ PI b)) (* a b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 2.85e+147) {
		tmp = (0.5 * ((double) M_PI)) / (a * (b * (a + b)));
	} else {
		tmp = (0.5 * (((double) M_PI) / b)) / (a * b);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 2.85e+147) {
		tmp = (0.5 * Math.PI) / (a * (b * (a + b)));
	} else {
		tmp = (0.5 * (Math.PI / b)) / (a * b);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 2.85e+147:
		tmp = (0.5 * math.pi) / (a * (b * (a + b)))
	else:
		tmp = (0.5 * (math.pi / b)) / (a * b)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 2.85e+147)
		tmp = Float64(Float64(0.5 * pi) / Float64(a * Float64(b * Float64(a + b))));
	else
		tmp = Float64(Float64(0.5 * Float64(pi / b)) / Float64(a * b));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2.85e+147)
		tmp = (0.5 * pi) / (a * (b * (a + b)));
	else
		tmp = (0.5 * (pi / b)) / (a * b);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[b, 2.85e+147], N[(N[(0.5 * Pi), $MachinePrecision] / N[(a * N[(b * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(Pi / b), $MachinePrecision]), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.84999999999999996e147

   1. Initial program 82.6%

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

      1. *-commutative 82.6%

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

      2. associate-*r* 82.6%

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

      3. associate-*r/ 82.6%

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

      4. associate-*r* 82.6%

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

      5. *-rgt-identity 82.6%

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

      6. sub-neg 82.6%

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

      7. distribute-neg-frac 82.6%

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

      8. metadata-eval 82.6%

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

   3. Simplified 82.6%

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

      1. *-commutative 82.6%

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

      2. difference-of-squares 89.4%

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

      3. times-frac 99.6%

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

      4. div-inv 99.6%

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

      5. metadata-eval 99.6%

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

      6. add-sqr-sqrt 55.9%

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

      7. sqrt-unprod 69.2%

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

      8. frac-times 69.2%

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

      9. metadata-eval 69.2%

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

      10. metadata-eval 69.2%

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

      11. frac-times 69.2%

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

      12. sqrt-unprod 23.2%

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

      13. add-sqr-sqrt 58.5%

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

   6. Applied egg-rr 58.5%

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

      1. *-commutative 58.5%

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

      2. +-commutative 58.5%

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

      3. associate-/l* 58.5%

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

      4. +-commutative 58.5%

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

   8. Simplified 58.5%

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

   9. Taylor expanded in b around inf 99.6%

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

       1. *-un-lft-identity 99.6%

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

       2. associate-/r* 99.5%

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

   11. Applied egg-rr 99.5%

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

       1. associate-*r/ 99.5%

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

       2. associate-*l/ 99.4%

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

       3. associate-*r/ 99.5%

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

       4. *-rgt-identity 99.5%

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

   13. Simplified 99.5%

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

       1. associate-/l/ 99.6%

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

       2. associate-*r/ 99.6%

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

       3. *-commutative 99.6%

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

       4. frac-times 99.1%

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

       5. *-un-lft-identity 99.1%

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

       6. associate-*l* 94.4%

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

   15. Applied egg-rr 94.4%

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

   if 2.84999999999999996e147 < b

   1. Initial program 59.4%

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

      1. *-commutative 59.4%

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

      2. associate-*r* 59.4%

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

      3. associate-*r/ 59.4%

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

      4. associate-*r* 59.4%

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

      5. *-rgt-identity 59.4%

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

      6. sub-neg 59.4%

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

      7. distribute-neg-frac 59.4%

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

      8. metadata-eval 59.4%

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

   3. Simplified 59.4%

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

      1. *-commutative 59.4%

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

      2. difference-of-squares 75.2%

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

      3. times-frac 99.8%

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

      4. div-inv 99.8%

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

      5. metadata-eval 99.8%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 99.8%

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

      8. frac-times 99.8%

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

      9. metadata-eval 99.8%

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

      10. metadata-eval 99.8%

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

      11. frac-times 99.8%

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

      12. sqrt-unprod 99.8%

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

      13. add-sqr-sqrt 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. *-commutative 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-/l* 99.7%

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

      4. +-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in b around inf 99.5%

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

   10. Taylor expanded in a around 0 99.5%

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

       1. *-commutative 99.5%

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

       2. div-inv 99.8%

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

       3. associate-*r/ 99.8%

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

       4. *-commutative 99.8%

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

       5. *-un-lft-identity 99.8%

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

       6. times-frac 99.8%

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

       7. metadata-eval 99.8%

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

   12. Applied egg-rr 99.8%

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

Alternative 3: 89.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -7.5 \cdot 10^{-94}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{b}}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -7.5e-94)
   (/ (* 0.5 (/ PI a)) (* a b))
   (/ (* 0.5 (/ PI b)) (* a b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -7.5e-94) {
		tmp = (0.5 * (((double) M_PI) / a)) / (a * b);
	} else {
		tmp = (0.5 * (((double) M_PI) / b)) / (a * b);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -7.5e-94) {
		tmp = (0.5 * (Math.PI / a)) / (a * b);
	} else {
		tmp = (0.5 * (Math.PI / b)) / (a * b);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -7.5e-94:
		tmp = (0.5 * (math.pi / a)) / (a * b)
	else:
		tmp = (0.5 * (math.pi / b)) / (a * b)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -7.5e-94)
		tmp = Float64(Float64(0.5 * Float64(pi / a)) / Float64(a * b));
	else
		tmp = Float64(Float64(0.5 * Float64(pi / b)) / Float64(a * b));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -7.5e-94)
		tmp = (0.5 * (pi / a)) / (a * b);
	else
		tmp = (0.5 * (pi / b)) / (a * b);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -7.5e-94], N[(N[(0.5 * N[(Pi / a), $MachinePrecision]), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(Pi / b), $MachinePrecision]), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -7.5000000000000003e-94

   1. Initial program 86.9%

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

      1. *-commutative 86.9%

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

      2. associate-*r* 86.9%

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

      3. associate-*r/ 86.9%

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

      4. associate-*r* 86.9%

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

      5. *-rgt-identity 86.9%

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

      6. sub-neg 86.9%

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

      7. distribute-neg-frac 86.9%

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

      8. metadata-eval 86.9%

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

   3. Simplified 86.9%

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

      1. *-commutative 86.9%

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

      2. difference-of-squares 96.0%

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

      3. times-frac 99.6%

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

      4. div-inv 99.6%

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

      5. metadata-eval 99.6%

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

      6. add-sqr-sqrt 47.9%

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

      7. sqrt-unprod 58.3%

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

      8. frac-times 58.3%

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

      9. metadata-eval 58.3%

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

      10. metadata-eval 58.3%

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

      11. frac-times 58.3%

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

      12. sqrt-unprod 19.1%

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

      13. add-sqr-sqrt 46.8%

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

   6. Applied egg-rr 46.8%

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

      1. *-commutative 46.8%

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

      2. +-commutative 46.8%

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

      3. associate-/l* 46.8%

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

      4. +-commutative 46.8%

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

   8. Simplified 46.8%

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

   9. Taylor expanded in b around inf 99.6%

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

       1. associate-*l/ 99.6%

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

       2. *-un-lft-identity 99.6%

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

       3. associate-*r/ 99.6%

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

       4. *-commutative 99.6%

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

   11. Applied egg-rr 99.6%

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

   12. Taylor expanded in a around inf 83.9%

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

   if -7.5000000000000003e-94 < a

   1. Initial program 78.3%

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

      1. *-commutative 78.3%

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

      2. associate-*r* 78.3%

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

      3. associate-*r/ 78.3%

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

      4. associate-*r* 78.3%

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

      5. *-rgt-identity 78.3%

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

      6. sub-neg 78.3%

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

      7. distribute-neg-frac 78.3%

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

      8. metadata-eval 78.3%

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

   3. Simplified 78.3%

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

      1. *-commutative 78.3%

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

      2. difference-of-squares 85.0%

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

      3. times-frac 99.6%

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

      4. div-inv 99.6%

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

      5. metadata-eval 99.6%

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

      6. add-sqr-sqrt 53.4%

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

      7. sqrt-unprod 77.2%

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

      8. frac-times 77.2%

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

      9. metadata-eval 77.2%

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

      10. metadata-eval 77.2%

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

      11. frac-times 77.2%

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

      12. sqrt-unprod 33.1%

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

      13. add-sqr-sqrt 67.9%

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

   6. Applied egg-rr 67.9%

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

      1. *-commutative 67.9%

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

      2. +-commutative 67.9%

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

      3. associate-/l* 67.9%

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

      4. +-commutative 67.9%

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

   8. Simplified 67.9%

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

   9. Taylor expanded in b around inf 99.5%

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

   10. Taylor expanded in a around 0 67.9%

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

       1. *-commutative 67.9%

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

       2. div-inv 67.9%

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

       3. associate-*r/ 68.0%

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

       4. *-commutative 68.0%

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

       5. *-un-lft-identity 68.0%

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

       6. times-frac 68.0%

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

       7. metadata-eval 68.0%

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

   12. Applied egg-rr 68.0%

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

Alternative 4: 89.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1.35 \cdot 10^{-94}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{b \cdot \left(a \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -1.35e-94)
   (/ (* 0.5 (/ PI a)) (* a b))
   (/ (* 0.5 PI) (* b (* a b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -1.35e-94) {
		tmp = (0.5 * (((double) M_PI) / a)) / (a * b);
	} else {
		tmp = (0.5 * ((double) M_PI)) / (b * (a * b));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -1.35e-94) {
		tmp = (0.5 * (Math.PI / a)) / (a * b);
	} else {
		tmp = (0.5 * Math.PI) / (b * (a * b));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -1.35e-94:
		tmp = (0.5 * (math.pi / a)) / (a * b)
	else:
		tmp = (0.5 * math.pi) / (b * (a * b))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -1.35e-94)
		tmp = Float64(Float64(0.5 * Float64(pi / a)) / Float64(a * b));
	else
		tmp = Float64(Float64(0.5 * pi) / 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 <= -1.35e-94)
		tmp = (0.5 * (pi / a)) / (a * b);
	else
		tmp = (0.5 * pi) / (b * (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -1.35e-94], N[(N[(0.5 * N[(Pi / a), $MachinePrecision]), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * Pi), $MachinePrecision] / N[(b * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.3500000000000001e-94

   1. Initial program 86.9%

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

      1. *-commutative 86.9%

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

      2. associate-*r* 86.9%

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

      3. associate-*r/ 86.9%

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

      4. associate-*r* 86.9%

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

      5. *-rgt-identity 86.9%

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

      6. sub-neg 86.9%

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

      7. distribute-neg-frac 86.9%

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

      8. metadata-eval 86.9%

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

   3. Simplified 86.9%

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

      1. *-commutative 86.9%

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

      2. difference-of-squares 96.0%

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

      3. times-frac 99.6%

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

      4. div-inv 99.6%

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

      5. metadata-eval 99.6%

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

      6. add-sqr-sqrt 47.9%

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

      7. sqrt-unprod 58.3%

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

      8. frac-times 58.3%

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

      9. metadata-eval 58.3%

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

      10. metadata-eval 58.3%

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

      11. frac-times 58.3%

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

      12. sqrt-unprod 19.1%

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

      13. add-sqr-sqrt 46.8%

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

   6. Applied egg-rr 46.8%

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

      1. *-commutative 46.8%

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

      2. +-commutative 46.8%

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

      3. associate-/l* 46.8%

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

      4. +-commutative 46.8%

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

   8. Simplified 46.8%

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

   9. Taylor expanded in b around inf 99.6%

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

       1. associate-*l/ 99.6%

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

       2. *-un-lft-identity 99.6%

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

       3. associate-*r/ 99.6%

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

       4. *-commutative 99.6%

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

   11. Applied egg-rr 99.6%

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

   12. Taylor expanded in a around inf 83.9%

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

   if -1.3500000000000001e-94 < a

   1. Initial program 78.3%

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

      1. *-commutative 78.3%

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

      2. associate-*r* 78.3%

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

      3. associate-*r/ 78.3%

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

      4. associate-*r* 78.3%

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

      5. *-rgt-identity 78.3%

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

      6. sub-neg 78.3%

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

      7. distribute-neg-frac 78.3%

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

      8. metadata-eval 78.3%

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

   3. Simplified 78.3%

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

      1. *-commutative 78.3%

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

      2. difference-of-squares 85.0%

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

      3. times-frac 99.6%

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

      4. div-inv 99.6%

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

      5. metadata-eval 99.6%

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

      6. add-sqr-sqrt 53.4%

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

      7. sqrt-unprod 77.2%

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

      8. frac-times 77.2%

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

      9. metadata-eval 77.2%

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

      10. metadata-eval 77.2%

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

      11. frac-times 77.2%

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

      12. sqrt-unprod 33.1%

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

      13. add-sqr-sqrt 67.9%

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

   6. Applied egg-rr 67.9%

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

      1. *-commutative 67.9%

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

      2. +-commutative 67.9%

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

      3. associate-/l* 67.9%

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

      4. +-commutative 67.9%

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

   8. Simplified 67.9%

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

   9. Taylor expanded in b around inf 99.5%

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

   10. Taylor expanded in a around 0 67.9%

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

       1. associate-*r/ 68.0%

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

       2. *-commutative 68.0%

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

       3. frac-times 68.0%

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

       4. *-un-lft-identity 68.0%

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

   12. Applied egg-rr 68.0%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.35 \cdot 10^{-94}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{b \cdot \left(a \cdot b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.7% accurate, 1.9× speedup

Math

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

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(pi / Float64(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[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a + b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.9%

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

   1. *-commutative 80.9%

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

   2. associate-*r* 80.9%

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

   3. associate-*r/ 80.9%

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

   4. associate-*r* 80.9%

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

   5. *-rgt-identity 80.9%

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

   6. sub-neg 80.9%

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

   7. distribute-neg-frac 80.9%

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

   8. metadata-eval 80.9%

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

3. Simplified 80.9%

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

   1. *-un-lft-identity 80.9%

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

   2. difference-of-squares 88.3%

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

   3. times-frac 99.5%

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

6. Applied egg-rr 61.5%

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

   1. associate-*l/ 61.6%

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

   2. *-lft-identity 61.6%

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

   3. associate-*r* 61.6%

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

   4. distribute-rgt-out 61.6%

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

   5. associate-*l/ 61.6%

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

   6. *-lft-identity 61.6%

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

   7. *-commutative 61.6%

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

   8. associate-*r/ 61.6%

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

   9. associate-*l/ 61.6%

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

   10. *-lft-identity 61.6%

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

   11. *-commutative 61.6%

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

   12. associate-*r/ 61.6%

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

   13. fma-define 61.6%

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

   14. associate-*r/ 61.6%

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

   15. +-commutative 61.6%

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

8. Simplified 61.6%

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

9. Taylor expanded in a around 0 99.6%

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

Alternative 6: 99.1% accurate, 1.9× speedup

Math

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

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (/ (* 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 * pi) / Float64(Float64(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 * Pi), $MachinePrecision] / N[(N[(a + b), $MachinePrecision] * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.9%

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

   1. *-commutative 80.9%

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

   2. associate-*r* 80.9%

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

   3. associate-*r/ 80.9%

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

   4. associate-*r* 80.9%

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

   5. *-rgt-identity 80.9%

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

   6. sub-neg 80.9%

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

   7. distribute-neg-frac 80.9%

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

   8. metadata-eval 80.9%

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

3. Simplified 80.9%

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

   1. *-commutative 80.9%

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

   2. difference-of-squares 88.3%

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

   3. times-frac 99.6%

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

   4. div-inv 99.6%

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

   5. metadata-eval 99.6%

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

   6. add-sqr-sqrt 51.7%

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

   7. sqrt-unprod 71.5%

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

   8. frac-times 71.5%

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

   9. metadata-eval 71.5%

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

   10. metadata-eval 71.5%

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

   11. frac-times 71.5%

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

   12. sqrt-unprod 28.9%

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

   13. add-sqr-sqrt 61.6%

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

6. Applied egg-rr 61.6%

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

   1. *-commutative 61.6%

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

   2. +-commutative 61.6%

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

   3. associate-/l* 61.5%

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

   4. +-commutative 61.5%

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

8. Simplified 61.5%

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

9. Taylor expanded in b around inf 99.6%

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

    1. associate-*r/ 99.6%

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

    2. +-commutative 99.6%

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

    3. frac-times 99.2%

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

    4. *-un-lft-identity 99.2%

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

    5. *-commutative 99.2%

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

    6. +-commutative 99.2%

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

11. Applied egg-rr 99.2%

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

12. Final simplification 99.2%

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

Alternative 7: 62.2% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 80.9%

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

   1. *-commutative 80.9%

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

   2. associate-*r* 80.9%

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

   3. associate-*r/ 80.9%

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

   4. associate-*r* 80.9%

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

   5. *-rgt-identity 80.9%

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

   6. sub-neg 80.9%

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

   7. distribute-neg-frac 80.9%

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

   8. metadata-eval 80.9%

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

3. Simplified 80.9%

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

   1. *-commutative 80.9%

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

   2. difference-of-squares 88.3%

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

   3. times-frac 99.6%

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

   4. div-inv 99.6%

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

   5. metadata-eval 99.6%

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

   6. add-sqr-sqrt 51.7%

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

   7. sqrt-unprod 71.5%

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

   8. frac-times 71.5%

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

   9. metadata-eval 71.5%

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

   10. metadata-eval 71.5%

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

   11. frac-times 71.5%

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

   12. sqrt-unprod 28.9%

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

   13. add-sqr-sqrt 61.6%

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

6. Applied egg-rr 61.6%

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

   1. *-commutative 61.6%

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

   2. +-commutative 61.6%

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

   3. associate-/l* 61.5%

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

   4. +-commutative 61.5%

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

8. Simplified 61.5%

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

9. Taylor expanded in b around inf 99.6%

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

10. Taylor expanded in a around 0 59.5%

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

    1. associate-*r/ 59.6%

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

    2. *-commutative 59.6%

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

    3. frac-times 59.3%

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

    4. *-un-lft-identity 59.3%

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

12. Applied egg-rr 59.3%

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

13. Final simplification 59.3%

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

Reproduce

herbie shell --seed 2024129 
(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))))