NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.5% → 99.6%

Time: 9.9s

Alternatives: 11

Speedup: 1.9×

• Rival profile vector overflowed, profile may not be complete

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: 99.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \frac{\frac{1}{b \cdot \frac{\frac{a}{0.5}}{\pi}}}{b + a} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (/ (/ 1.0 (* b (/ (/ a 0.5) PI))) (+ b a)))

C

assert(a < b);
double code(double a, double b) {
	return (1.0 / (b * ((a / 0.5) / ((double) M_PI)))) / (b + a);
}

Java

assert a < b;
public static double code(double a, double b) {
	return (1.0 / (b * ((a / 0.5) / Math.PI))) / (b + a);
}

Python

[a, b] = sort([a, b])
def code(a, b):
	return (1.0 / (b * ((a / 0.5) / math.pi))) / (b + a)

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(Float64(1.0 / Float64(b * Float64(Float64(a / 0.5) / pi))) / Float64(b + a))
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = (1.0 / (b * ((a / 0.5) / pi))) / (b + a);
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[(N[(1.0 / N[(b * N[(N[(a / 0.5), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b + a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.0%

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

   1. *-commutative 75.0%

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

   2. associate-*r* 75.0%

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

   3. associate-*r/ 75.0%

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

   4. associate-*r* 75.0%

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

   5. *-rgt-identity 75.0%

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

   6. sub-neg 75.0%

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

   7. distribute-neg-frac 75.0%

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

   8. metadata-eval 75.0%

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

3. Simplified 75.0%

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

   1. associate-*l/ 75.0%

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

   2. div-inv 75.0%

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

   3. metadata-eval 75.0%

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

   4. *-commutative 75.0%

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

   5. associate-*r* 75.0%

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

   6. *-commutative 75.0%

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

6. Applied egg-rr 98.6%

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

   1. associate-*l/ 98.7%

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

   2. *-commutative 98.7%

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

   3. metadata-eval 98.7%

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

   4. div-inv 98.7%

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

   5. clear-num 98.6%

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

   6. div-inv 98.6%

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

   7. metadata-eval 98.6%

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

8. Applied egg-rr 98.6%

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

   1. inv-pow 98.6%

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

   2. associate-/l* 98.6%

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

   3. unpow-prod-down 99.6%

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

   4. inv-pow 99.6%

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

   5. *-commutative 99.6%

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

   6. times-frac 99.6%

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

10. Applied egg-rr 99.6%

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

    1. associate-*l/ 99.7%

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

    2. *-lft-identity 99.7%

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

    3. unpow-1 99.7%

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

    4. associate-*l/ 99.7%

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

    5. *-lft-identity 99.7%

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

    6. times-frac 99.7%

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

    7. /-rgt-identity 99.7%

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

12. Simplified 99.7%

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

13. Final simplification 99.7%

    \[\leadsto \frac{\frac{1}{b \cdot \frac{\frac{a}{0.5}}{\pi}}}{b + a} \]
14. Add Preprocessing

Alternative 2: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1.3 \cdot 10^{+156}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\pi}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{b}}{a \cdot \left(b + a\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -1.3e+156)
   (* (/ 0.5 a) (/ PI (* b a)))
   (* 0.5 (/ (/ PI b) (* a (+ b a))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -1.3e+156) {
		tmp = (0.5 / a) * (((double) M_PI) / (b * a));
	} else {
		tmp = 0.5 * ((((double) M_PI) / b) / (a * (b + a)));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -1.3e+156) {
		tmp = (0.5 / a) * (Math.PI / (b * a));
	} else {
		tmp = 0.5 * ((Math.PI / b) / (a * (b + a)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -1.3e+156:
		tmp = (0.5 / a) * (math.pi / (b * a))
	else:
		tmp = 0.5 * ((math.pi / b) / (a * (b + a)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -1.3e+156)
		tmp = Float64(Float64(0.5 / a) * Float64(pi / Float64(b * a)));
	else
		tmp = Float64(0.5 * Float64(Float64(pi / b) / Float64(a * Float64(b + a))));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.3e+156)
		tmp = (0.5 / a) * (pi / (b * a));
	else
		tmp = 0.5 * ((pi / b) / (a * (b + a)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -1.3e+156], N[(N[(0.5 / a), $MachinePrecision] * N[(Pi / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(Pi / b), $MachinePrecision] / N[(a * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.30000000000000009e156

   1. Initial program 46.9%

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

      1. associate-*l* 46.9%

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

      2. *-rgt-identity 46.9%

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

      3. associate-/l* 46.9%

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

      4. metadata-eval 46.9%

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

      5. associate-*l/ 46.9%

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

      6. *-lft-identity 46.9%

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

      7. sub-neg 46.9%

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

      8. distribute-neg-frac 46.9%

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

      9. metadata-eval 46.9%

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

   3. Simplified 46.9%

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

      1. metadata-eval 46.9%

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

      2. div-inv 46.9%

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

      3. *-commutative 46.9%

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

      4. clear-num 46.9%

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

      5. frac-times 46.9%

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

      6. *-un-lft-identity 46.9%

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

      7. frac-add 46.9%

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

      8. associate-/r/ 46.9%

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

      9. *-un-lft-identity 46.9%

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

      10. *-commutative 46.9%

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

      11. neg-mul-1 46.9%

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

      12. sub-neg 46.9%

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

      13. flip-+ 97.2%

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

      14. +-commutative 97.2%

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

   6. Applied egg-rr 97.2%

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

      1. *-commutative 97.2%

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

      2. associate-/r* 97.2%

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

   8. Simplified 97.2%

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

   9. Taylor expanded in a around inf 97.2%

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

       1. div-inv 97.2%

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

       2. metadata-eval 97.2%

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

       3. *-commutative 97.2%

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

       4. times-frac 99.8%

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

       5. *-commutative 99.8%

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

   11. Applied egg-rr 99.8%

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

   if -1.30000000000000009e156 < a

   1. Initial program 79.3%

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

      1. *-commutative 79.3%

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

      2. associate-*r* 79.3%

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

      3. associate-*r/ 79.3%

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

      4. associate-*r* 79.3%

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

      5. *-rgt-identity 79.3%

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

      6. sub-neg 79.3%

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

      7. distribute-neg-frac 79.3%

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

      8. metadata-eval 79.3%

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

   3. Simplified 79.3%

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

      1. associate-*l/ 79.3%

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

      2. div-inv 79.3%

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

      3. metadata-eval 79.3%

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

      4. *-commutative 79.3%

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

      5. associate-*r* 79.3%

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

      6. *-commutative 79.3%

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

   6. Applied egg-rr 98.8%

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

      1. associate-*l/ 98.9%

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

      2. *-commutative 98.9%

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

      3. metadata-eval 98.9%

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

      4. div-inv 98.9%

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

      5. clear-num 98.8%

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

      6. div-inv 98.8%

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

      7. metadata-eval 98.8%

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

   8. Applied egg-rr 98.8%

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

      1. inv-pow 98.8%

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

      2. associate-/l* 98.8%

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

      3. unpow-prod-down 99.6%

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

      4. inv-pow 99.6%

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

      5. *-commutative 99.6%

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

      6. times-frac 99.6%

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

   10. Applied egg-rr 99.6%

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

       1. associate-*l/ 99.7%

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

       2. *-lft-identity 99.7%

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

       3. unpow-1 99.7%

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

       4. associate-*l/ 99.7%

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

       5. *-lft-identity 99.7%

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

       6. times-frac 99.7%

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

       7. /-rgt-identity 99.7%

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

   12. Simplified 99.7%

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

       1. *-un-lft-identity 99.7%

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

       2. associate-/r* 99.7%

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

       3. div-inv 99.6%

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

       4. clear-num 99.7%

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

       5. div-inv 99.7%

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

       6. clear-num 99.7%

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

       7. +-commutative 99.7%

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

   14. Applied egg-rr 99.7%

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

       1. *-lft-identity 99.7%

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

       2. associate-*l/ 99.6%

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

       3. *-lft-identity 99.6%

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

       4. associate-*r/ 99.6%

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

       5. *-commutative 99.6%

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

       6. associate-/l/ 99.7%

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

       7. associate-/l/ 98.9%

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

       8. associate-/l* 98.9%

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

       9. associate-/l/ 99.7%

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

       10. associate-/r* 99.6%

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

       11. associate-/l/ 96.8%

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

       12. +-commutative 96.8%

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

   16. Simplified 96.8%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.3 \cdot 10^{+156}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\pi}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{b}}{a \cdot \left(b + a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -2.6 \cdot 10^{-45}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\pi}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot \frac{0.5}{b \cdot a}}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -2.6e-45)
   (* (/ 0.5 a) (/ PI (* b a)))
   (/ (* PI (/ 0.5 (* b a))) b)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -2.6e-45) {
		tmp = (0.5 / a) * (((double) M_PI) / (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 <= -2.6e-45) {
		tmp = (0.5 / a) * (Math.PI / (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 <= -2.6e-45:
		tmp = (0.5 / a) * (math.pi / (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 <= -2.6e-45)
		tmp = Float64(Float64(0.5 / a) * Float64(pi / Float64(b * a)));
	else
		tmp = Float64(Float64(pi * Float64(0.5 / Float64(b * a))) / b);
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.6e-45)
		tmp = (0.5 / a) * (pi / (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, -2.6e-45], N[(N[(0.5 / a), $MachinePrecision] * N[(Pi / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * N[(0.5 / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.59999999999999987e-45

   1. Initial program 76.3%

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

      1. associate-*l* 76.2%

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

      2. *-rgt-identity 76.2%

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

      3. associate-/l* 76.2%

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

      4. metadata-eval 76.2%

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

      5. associate-*l/ 76.3%

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

      6. *-lft-identity 76.3%

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

      7. sub-neg 76.3%

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

      8. distribute-neg-frac 76.3%

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

      9. metadata-eval 76.3%

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

   3. Simplified 76.3%

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

      1. metadata-eval 76.3%

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

      2. div-inv 76.3%

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

      3. *-commutative 76.3%

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

      4. clear-num 76.3%

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

      5. frac-times 76.3%

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

      6. *-un-lft-identity 76.3%

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

      7. frac-add 76.3%

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

      8. associate-/r/ 76.4%

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

      9. *-un-lft-identity 76.4%

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

      10. *-commutative 76.4%

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

      11. neg-mul-1 76.4%

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

      12. sub-neg 76.4%

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

      13. flip-+ 98.6%

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

      14. +-commutative 98.6%

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

   6. Applied egg-rr 98.6%

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

      1. *-commutative 98.6%

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

      2. associate-/r* 98.6%

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

   8. Simplified 98.6%

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

   9. Taylor expanded in a around inf 86.3%

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

       1. div-inv 86.3%

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

       2. metadata-eval 86.3%

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

       3. *-commutative 86.3%

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

       4. times-frac 87.4%

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

       5. *-commutative 87.4%

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

   11. Applied egg-rr 87.4%

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

   if -2.59999999999999987e-45 < a

   1. Initial program 74.4%

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

      1. associate-*l* 74.4%

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

      2. *-rgt-identity 74.4%

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

      3. associate-/l* 74.4%

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

      4. metadata-eval 74.4%

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

      5. associate-*l/ 74.5%

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

      6. *-lft-identity 74.5%

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

      7. sub-neg 74.5%

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

      8. distribute-neg-frac 74.5%

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

      9. metadata-eval 74.5%

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

   3. Simplified 74.5%

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

      1. metadata-eval 74.5%

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

      2. div-inv 74.5%

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

      3. *-commutative 74.5%

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

      4. clear-num 73.9%

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

      5. frac-times 74.0%

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

      6. *-un-lft-identity 74.0%

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

      7. frac-add 74.0%

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

      8. associate-/r/ 74.0%

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

      9. *-un-lft-identity 74.0%

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

      10. *-commutative 74.0%

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

      11. neg-mul-1 74.0%

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

      12. sub-neg 74.0%

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

      13. flip-+ 98.7%

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

      14. +-commutative 98.7%

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

   6. Applied egg-rr 98.7%

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

      1. *-commutative 98.7%

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

      2. associate-/r* 98.7%

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

   8. Simplified 98.7%

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

   9. Taylor expanded in a around 0 76.0%

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

       1. div-inv 76.0%

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

       2. metadata-eval 76.0%

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

       3. times-frac 76.1%

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

   11. Applied egg-rr 76.1%

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

       1. associate-*l/ 76.2%

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

       2. *-commutative 76.2%

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

   13. Applied egg-rr 76.2%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.6 \cdot 10^{-45}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\pi}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot \frac{0.5}{b \cdot a}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} t_0 := \frac{\pi}{b \cdot a}\\ \mathbf{if}\;a \leq -2.1 \cdot 10^{-45}:\\ \;\;\;\;\frac{0.5}{a} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{0.5}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ PI (* b a))))
   (if (<= a -2.1e-45) (* (/ 0.5 a) t_0) (* t_0 (/ 0.5 b)))))

C

assert(a < b);
double code(double a, double b) {
	double t_0 = ((double) M_PI) / (b * a);
	double tmp;
	if (a <= -2.1e-45) {
		tmp = (0.5 / a) * t_0;
	} else {
		tmp = t_0 * (0.5 / b);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double t_0 = Math.PI / (b * a);
	double tmp;
	if (a <= -2.1e-45) {
		tmp = (0.5 / a) * t_0;
	} else {
		tmp = t_0 * (0.5 / b);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	t_0 = math.pi / (b * a)
	tmp = 0
	if a <= -2.1e-45:
		tmp = (0.5 / a) * t_0
	else:
		tmp = t_0 * (0.5 / b)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	t_0 = Float64(pi / Float64(b * a))
	tmp = 0.0
	if (a <= -2.1e-45)
		tmp = Float64(Float64(0.5 / a) * t_0);
	else
		tmp = Float64(t_0 * Float64(0.5 / b));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	t_0 = pi / (b * a);
	tmp = 0.0;
	if (a <= -2.1e-45)
		tmp = (0.5 / a) * t_0;
	else
		tmp = t_0 * (0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := Block[{t$95$0 = N[(Pi / N[(b * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -2.1e-45], N[(N[(0.5 / a), $MachinePrecision] * t$95$0), $MachinePrecision], N[(t$95$0 * N[(0.5 / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.09999999999999995e-45

   1. Initial program 76.3%

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

      1. associate-*l* 76.2%

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

      2. *-rgt-identity 76.2%

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

      3. associate-/l* 76.2%

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

      4. metadata-eval 76.2%

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

      5. associate-*l/ 76.3%

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

      6. *-lft-identity 76.3%

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

      7. sub-neg 76.3%

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

      8. distribute-neg-frac 76.3%

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

      9. metadata-eval 76.3%

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

   3. Simplified 76.3%

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

      1. metadata-eval 76.3%

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

      2. div-inv 76.3%

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

      3. *-commutative 76.3%

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

      4. clear-num 76.3%

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

      5. frac-times 76.3%

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

      6. *-un-lft-identity 76.3%

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

      7. frac-add 76.3%

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

      8. associate-/r/ 76.4%

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

      9. *-un-lft-identity 76.4%

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

      10. *-commutative 76.4%

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

      11. neg-mul-1 76.4%

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

      12. sub-neg 76.4%

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

      13. flip-+ 98.6%

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

      14. +-commutative 98.6%

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

   6. Applied egg-rr 98.6%

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

      1. *-commutative 98.6%

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

      2. associate-/r* 98.6%

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

   8. Simplified 98.6%

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

   9. Taylor expanded in a around inf 86.3%

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

       1. div-inv 86.3%

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

       2. metadata-eval 86.3%

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

       3. *-commutative 86.3%

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

       4. times-frac 87.4%

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

       5. *-commutative 87.4%

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

   11. Applied egg-rr 87.4%

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

   if -2.09999999999999995e-45 < a

   1. Initial program 74.4%

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

      1. associate-*l* 74.4%

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

      2. *-rgt-identity 74.4%

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

      3. associate-/l* 74.4%

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

      4. metadata-eval 74.4%

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

      5. associate-*l/ 74.5%

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

      6. *-lft-identity 74.5%

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

      7. sub-neg 74.5%

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

      8. distribute-neg-frac 74.5%

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

      9. metadata-eval 74.5%

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

   3. Simplified 74.5%

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

      1. metadata-eval 74.5%

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

      2. div-inv 74.5%

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

      3. *-commutative 74.5%

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

      4. clear-num 73.9%

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

      5. frac-times 74.0%

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

      6. *-un-lft-identity 74.0%

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

      7. frac-add 74.0%

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

      8. associate-/r/ 74.0%

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

      9. *-un-lft-identity 74.0%

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

      10. *-commutative 74.0%

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

      11. neg-mul-1 74.0%

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

      12. sub-neg 74.0%

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

      13. flip-+ 98.7%

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

      14. +-commutative 98.7%

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

   6. Applied egg-rr 98.7%

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

      1. *-commutative 98.7%

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

      2. associate-/r* 98.7%

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

   8. Simplified 98.7%

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

   9. Taylor expanded in a around 0 76.0%

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

       1. div-inv 76.0%

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

       2. metadata-eval 76.0%

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

       3. *-commutative 76.0%

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

       4. times-frac 76.2%

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

   11. Applied egg-rr 76.2%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.1 \cdot 10^{-45}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\pi}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b \cdot a} \cdot \frac{0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -2.3 \cdot 10^{-45}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b \cdot a} \cdot \frac{0.5}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -2.3e-45)
   (* PI (/ 0.5 (* a (* b a))))
   (* (/ PI (* b a)) (/ 0.5 b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -2.3e-45) {
		tmp = ((double) M_PI) * (0.5 / (a * (b * a)));
	} else {
		tmp = (((double) M_PI) / (b * a)) * (0.5 / b);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -2.3e-45) {
		tmp = Math.PI * (0.5 / (a * (b * a)));
	} else {
		tmp = (Math.PI / (b * a)) * (0.5 / b);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -2.3e-45:
		tmp = math.pi * (0.5 / (a * (b * a)))
	else:
		tmp = (math.pi / (b * a)) * (0.5 / b)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -2.3e-45)
		tmp = Float64(pi * Float64(0.5 / Float64(a * Float64(b * a))));
	else
		tmp = Float64(Float64(pi / Float64(b * a)) * Float64(0.5 / b));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.3e-45)
		tmp = pi * (0.5 / (a * (b * a)));
	else
		tmp = (pi / (b * a)) * (0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -2.3e-45], N[(Pi * N[(0.5 / N[(a * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / N[(b * a), $MachinePrecision]), $MachinePrecision] * N[(0.5 / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.29999999999999992e-45

   1. Initial program 76.3%

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

      1. *-commutative 76.3%

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

      2. associate-*r* 76.3%

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

      3. associate-*r/ 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/ 76.3%

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

      2. div-inv 76.3%

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

      3. metadata-eval 76.3%

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

      4. *-commutative 76.3%

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

      5. associate-*r* 76.3%

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

      6. *-commutative 76.3%

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

   6. Applied egg-rr 98.5%

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

   7. Taylor expanded in a around inf 86.3%

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

   if -2.29999999999999992e-45 < a

   1. Initial program 74.4%

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

      1. associate-*l* 74.4%

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

      2. *-rgt-identity 74.4%

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

      3. associate-/l* 74.4%

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

      4. metadata-eval 74.4%

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

      5. associate-*l/ 74.5%

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

      6. *-lft-identity 74.5%

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

      7. sub-neg 74.5%

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

      8. distribute-neg-frac 74.5%

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

      9. metadata-eval 74.5%

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

   3. Simplified 74.5%

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

      1. metadata-eval 74.5%

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

      2. div-inv 74.5%

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

      3. *-commutative 74.5%

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

      4. clear-num 73.9%

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

      5. frac-times 74.0%

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

      6. *-un-lft-identity 74.0%

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

      7. frac-add 74.0%

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

      8. associate-/r/ 74.0%

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

      9. *-un-lft-identity 74.0%

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

      10. *-commutative 74.0%

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

      11. neg-mul-1 74.0%

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

      12. sub-neg 74.0%

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

      13. flip-+ 98.7%

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

      14. +-commutative 98.7%

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

   6. Applied egg-rr 98.7%

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

      1. *-commutative 98.7%

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

      2. associate-/r* 98.7%

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

   8. Simplified 98.7%

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

   9. Taylor expanded in a around 0 76.0%

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

       1. div-inv 76.0%

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

       2. metadata-eval 76.0%

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

       3. *-commutative 76.0%

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

       4. times-frac 76.2%

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

   11. Applied egg-rr 76.2%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.3 \cdot 10^{-45}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b \cdot a} \cdot \frac{0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1.08 \cdot 10^{-45}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{0.5}{b \cdot a}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -1.08e-45)
   (* PI (/ 0.5 (* a (* b a))))
   (* (/ PI b) (/ 0.5 (* b a)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -1.08e-45) {
		tmp = ((double) M_PI) * (0.5 / (a * (b * a)));
	} else {
		tmp = (((double) M_PI) / b) * (0.5 / (b * a));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -1.08e-45) {
		tmp = Math.PI * (0.5 / (a * (b * a)));
	} else {
		tmp = (Math.PI / b) * (0.5 / (b * a));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -1.08e-45:
		tmp = math.pi * (0.5 / (a * (b * a)))
	else:
		tmp = (math.pi / b) * (0.5 / (b * a))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -1.08e-45)
		tmp = Float64(pi * Float64(0.5 / Float64(a * Float64(b * a))));
	else
		tmp = Float64(Float64(pi / b) * Float64(0.5 / Float64(b * a)));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.08e-45)
		tmp = pi * (0.5 / (a * (b * a)));
	else
		tmp = (pi / b) * (0.5 / (b * a));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -1.08e-45], N[(Pi * N[(0.5 / N[(a * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / b), $MachinePrecision] * N[(0.5 / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.08e-45

   1. Initial program 76.3%

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

      1. *-commutative 76.3%

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

      2. associate-*r* 76.3%

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

      3. associate-*r/ 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/ 76.3%

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

      2. div-inv 76.3%

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

      3. metadata-eval 76.3%

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

      4. *-commutative 76.3%

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

      5. associate-*r* 76.3%

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

      6. *-commutative 76.3%

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

   6. Applied egg-rr 98.5%

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

   7. Taylor expanded in a around inf 86.3%

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

   if -1.08e-45 < a

   1. Initial program 74.4%

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

      1. associate-*l* 74.4%

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

      2. *-rgt-identity 74.4%

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

      3. associate-/l* 74.4%

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

      4. metadata-eval 74.4%

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

      5. associate-*l/ 74.5%

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

      6. *-lft-identity 74.5%

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

      7. sub-neg 74.5%

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

      8. distribute-neg-frac 74.5%

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

      9. metadata-eval 74.5%

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

   3. Simplified 74.5%

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

      1. metadata-eval 74.5%

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

      2. div-inv 74.5%

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

      3. *-commutative 74.5%

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

      4. clear-num 73.9%

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

      5. frac-times 74.0%

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

      6. *-un-lft-identity 74.0%

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

      7. frac-add 74.0%

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

      8. associate-/r/ 74.0%

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

      9. *-un-lft-identity 74.0%

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

      10. *-commutative 74.0%

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

      11. neg-mul-1 74.0%

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

      12. sub-neg 74.0%

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

      13. flip-+ 98.7%

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

      14. +-commutative 98.7%

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

   6. Applied egg-rr 98.7%

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

      1. *-commutative 98.7%

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

      2. associate-/r* 98.7%

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

   8. Simplified 98.7%

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

   9. Taylor expanded in a around 0 76.0%

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

       1. div-inv 76.0%

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

       2. metadata-eval 76.0%

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

       3. times-frac 76.1%

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

   11. Applied egg-rr 76.1%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.08 \cdot 10^{-45}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{0.5}{b \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -2.35 \cdot 10^{-45}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(b \cdot a\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -2.35e-45)
   (* PI (/ 0.5 (* a (* b a))))
   (* PI (/ 0.5 (* b (* b a))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -2.35e-45) {
		tmp = ((double) M_PI) * (0.5 / (a * (b * a)));
	} else {
		tmp = ((double) M_PI) * (0.5 / (b * (b * a)));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -2.35e-45) {
		tmp = Math.PI * (0.5 / (a * (b * a)));
	} else {
		tmp = Math.PI * (0.5 / (b * (b * a)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -2.35e-45:
		tmp = math.pi * (0.5 / (a * (b * a)))
	else:
		tmp = math.pi * (0.5 / (b * (b * a)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -2.35e-45)
		tmp = Float64(pi * Float64(0.5 / Float64(a * Float64(b * a))));
	else
		tmp = Float64(pi * Float64(0.5 / Float64(b * Float64(b * a))));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.35e-45)
		tmp = pi * (0.5 / (a * (b * a)));
	else
		tmp = pi * (0.5 / (b * (b * a)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -2.35e-45], N[(Pi * N[(0.5 / N[(a * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(0.5 / N[(b * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.3499999999999999e-45

   1. Initial program 76.3%

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

      1. *-commutative 76.3%

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

      2. associate-*r* 76.3%

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

      3. associate-*r/ 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/ 76.3%

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

      2. div-inv 76.3%

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

      3. metadata-eval 76.3%

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

      4. *-commutative 76.3%

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

      5. associate-*r* 76.3%

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

      6. *-commutative 76.3%

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

   6. Applied egg-rr 98.5%

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

   7. Taylor expanded in a around inf 86.3%

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

   if -2.3499999999999999e-45 < a

   1. Initial program 74.4%

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

      1. *-commutative 74.4%

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

      2. associate-*r* 74.5%

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

      3. associate-*r/ 74.5%

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

      4. associate-*r* 74.5%

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

      5. *-rgt-identity 74.5%

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

      6. sub-neg 74.5%

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

      7. distribute-neg-frac 74.5%

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

      8. metadata-eval 74.5%

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

   3. Simplified 74.5%

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

      1. associate-*l/ 74.5%

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

      2. div-inv 74.5%

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

      3. metadata-eval 74.5%

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

      4. *-commutative 74.5%

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

      5. associate-*r* 74.5%

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

      6. *-commutative 74.5%

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

   6. Applied egg-rr 98.7%

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

   7. Taylor expanded in a around 0 76.0%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.35 \cdot 10^{-45}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(b \cdot a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.0%

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

   1. *-commutative 75.0%

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

   2. associate-*r* 75.0%

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

   3. associate-*r/ 75.0%

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

   4. associate-*r* 75.0%

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

   5. *-rgt-identity 75.0%

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

   6. sub-neg 75.0%

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

   7. distribute-neg-frac 75.0%

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

   8. metadata-eval 75.0%

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

3. Simplified 75.0%

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

   1. associate-*l/ 75.0%

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

   2. div-inv 75.0%

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

   3. metadata-eval 75.0%

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

   4. *-commutative 75.0%

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

   5. associate-*r* 75.0%

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

   6. *-commutative 75.0%

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

6. Applied egg-rr 98.6%

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

   1. associate-*l/ 98.7%

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

   2. *-commutative 98.7%

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

   3. metadata-eval 98.7%

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

   4. div-inv 98.7%

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

   5. clear-num 98.6%

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

   6. div-inv 98.6%

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

   7. metadata-eval 98.6%

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

8. Applied egg-rr 98.6%

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

   1. inv-pow 98.6%

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

   2. associate-/l* 98.6%

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

   3. unpow-prod-down 99.6%

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

   4. inv-pow 99.6%

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

   5. *-commutative 99.6%

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

   6. times-frac 99.6%

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

10. Applied egg-rr 99.6%

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

    1. associate-*l/ 99.7%

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

    2. *-lft-identity 99.7%

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

    3. unpow-1 99.7%

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

    4. associate-*l/ 99.7%

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

    5. *-lft-identity 99.7%

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

    6. times-frac 99.7%

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

    7. /-rgt-identity 99.7%

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

12. Simplified 99.7%

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

    1. *-un-lft-identity 99.7%

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

    2. associate-/r* 99.6%

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

    3. div-inv 99.6%

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

    4. clear-num 99.7%

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

    5. div-inv 99.7%

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

    6. clear-num 99.7%

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

    7. +-commutative 99.7%

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

14. Applied egg-rr 99.7%

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

    1. *-lft-identity 99.7%

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

    2. associate-*r* 99.7%

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

    3. *-commutative 99.7%

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

    4. associate-*r/ 99.7%

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

    5. *-rgt-identity 99.7%

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

    6. +-commutative 99.7%

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

16. Simplified 99.7%

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

17. Final simplification 99.7%

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

Alternative 9: 99.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.0%

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

   1. *-commutative 75.0%

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

   2. associate-*r* 75.0%

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

   3. associate-*r/ 75.0%

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

   4. associate-*r* 75.0%

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

   5. *-rgt-identity 75.0%

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

   6. sub-neg 75.0%

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

   7. distribute-neg-frac 75.0%

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

   8. metadata-eval 75.0%

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

3. Simplified 75.0%

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

   1. associate-*l/ 75.0%

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

   2. div-inv 75.0%

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

   3. metadata-eval 75.0%

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

   4. *-commutative 75.0%

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

   5. associate-*r* 75.0%

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

   6. *-commutative 75.0%

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

6. Applied egg-rr 98.6%

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

   1. associate-*l/ 98.7%

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

   2. *-commutative 98.7%

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

   3. metadata-eval 98.7%

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

   4. div-inv 98.7%

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

   5. associate-/r* 99.6%

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

   6. div-inv 99.6%

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

   7. metadata-eval 99.6%

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

8. Applied egg-rr 99.6%

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

   1. associate-/l/ 98.7%

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

   2. times-frac 99.6%

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

10. Applied egg-rr 99.6%

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

11. Final simplification 99.6%

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

Alternative 10: 99.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 75.0%

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

   1. *-commutative 75.0%

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

   2. associate-*r* 75.0%

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

   3. associate-*r/ 75.0%

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

   4. associate-*r* 75.0%

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

   5. *-rgt-identity 75.0%

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

   6. sub-neg 75.0%

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

   7. distribute-neg-frac 75.0%

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

   8. metadata-eval 75.0%

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

3. Simplified 75.0%

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

   1. associate-*l/ 75.0%

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

   2. div-inv 75.0%

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

   3. metadata-eval 75.0%

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

   4. *-commutative 75.0%

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

   5. associate-*r* 75.0%

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

   6. *-commutative 75.0%

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

6. Applied egg-rr 98.6%

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

7. Final simplification 98.6%

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

Alternative 11: 62.9% accurate, 2.3× speedup

Math

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

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)))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 75.0%

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

   1. *-commutative 75.0%

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

   2. associate-*r* 75.0%

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

   3. associate-*r/ 75.0%

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

   4. associate-*r* 75.0%

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

   5. *-rgt-identity 75.0%

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

   6. sub-neg 75.0%

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

   7. distribute-neg-frac 75.0%

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

   8. metadata-eval 75.0%

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

3. Simplified 75.0%

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

   1. associate-*l/ 75.0%

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

   2. div-inv 75.0%

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

   3. metadata-eval 75.0%

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

   4. *-commutative 75.0%

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

   5. associate-*r* 75.0%

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

   6. *-commutative 75.0%

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

6. Applied egg-rr 98.6%

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

7. Taylor expanded in a around inf 60.2%

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

8. Final simplification 60.2%

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

Reproduce

herbie shell --seed 2024155 
(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))))