NMSE Section 6.1 mentioned, B

Percentage Accurate: 79.0% → 99.7%

Time: 11.0s

Alternatives: 8

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = (((pi / 2.0) / b) / 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[(N[(Pi / 2.0), $MachinePrecision] / b), $MachinePrecision] / a), $MachinePrecision] / N[(b + a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.2%

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

   1. un-div-inv N/A

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

   2. associate-/r* N/A

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

   3. associate-*l/ N/A

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

   4. distribute-lft-out-- N/A

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

   5. div-inv N/A

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

   6. div-inv N/A

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

   7. *-commutative N/A

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

   8. associate-/l/ N/A

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

   9. difference-of-squares N/A

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

   10. associate-/r* N/A

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

   11. *-rgt-identity N/A

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

   12. *-lft-identity N/A

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

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

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

4. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{\pi}{a} - \frac{\pi}{b}}{2}}{b + a}}{b - a}} \]
5. Step-by-step derivation

   1. div-inv N/A

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

   2. metadata-eval N/A

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

   3. associate-*l/ N/A

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

   4. associate-*r/ N/A

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

   5. associate-*l/ N/A

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

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

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in a around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a \cdot b}\right)}, \mathsf{+.f64}\left(a, b\right)\right) \]
8. Step-by-step derivation

   1. associate-/r* N/A

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

   2. associate-*r/ N/A

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

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

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

   4. associate-*r/ N/A

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

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

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

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

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

   7. PI-lowering-PI.f64 99.7%

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

9. Simplified 99.7%

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

    1. associate-/l/ N/A

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

    2. *-commutative N/A

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

    3. times-frac N/A

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

    4. associate-/l* N/A

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

    5. associate-/r/ N/A

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

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

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

    7. associate-/r/ N/A

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

    8. associate-*l/ N/A

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

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

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

    10. metadata-eval N/A

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

    11. metadata-eval N/A

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

    12. div-inv N/A

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

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

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

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

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

    15. metadata-eval 99.7%

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

11. Applied egg-rr 99.7%

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

12. Final simplification 99.7%

    \[\leadsto \frac{\frac{\frac{\frac{\pi}{2}}{b}}{a}}{b + a} \]
13. Add Preprocessing

Alternative 2: 92.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -3.15 \cdot 10^{-43}:\\ \;\;\;\;\frac{\frac{-0.5 \cdot \frac{\pi}{b}}{a}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{\frac{b}{0.5}}}{b \cdot a}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -3.15e-43)
   (/ (/ (* -0.5 (/ PI b)) a) (- b a))
   (/ (/ PI (/ b 0.5)) (* b a))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -3.15e-43) {
		tmp = ((-0.5 * (((double) M_PI) / b)) / 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 <= -3.15e-43) {
		tmp = ((-0.5 * (Math.PI / b)) / 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 <= -3.15e-43:
		tmp = ((-0.5 * (math.pi / b)) / 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 <= -3.15e-43)
		tmp = Float64(Float64(Float64(-0.5 * Float64(pi / b)) / a) / Float64(b - a));
	else
		tmp = Float64(Float64(pi / Float64(b / 0.5)) / Float64(b * a));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -3.15e-43)
		tmp = ((-0.5 * (pi / b)) / 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, -3.15e-43], N[(N[(N[(-0.5 * N[(Pi / b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / N[(b / 0.5), $MachinePrecision]), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.1500000000000001e-43

   1. Initial program 85.7%

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

      1. un-div-inv N/A

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

      2. associate-/r* N/A

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

      3. associate-*l/ N/A

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

      4. distribute-lft-out-- N/A

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

      5. div-inv N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

      8. associate-/l/ N/A

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

      9. difference-of-squares N/A

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

      10. associate-/r* N/A

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

      11. *-rgt-identity N/A

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

      12. *-lft-identity N/A

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

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

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in a around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a \cdot b}\right)}, \mathsf{\_.f64}\left(b, a\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

      8. PI-lowering-PI.f64 86.0%

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

   7. Simplified 86.0%

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

   if -3.1500000000000001e-43 < a

   1. Initial program 73.8%

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

      1. un-div-inv N/A

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

      2. associate-/r* N/A

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

      3. associate-*l/ N/A

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

      4. distribute-lft-out-- N/A

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

      5. div-inv N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

      8. associate-/l/ N/A

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

      9. difference-of-squares N/A

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

      10. associate-/r* N/A

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

      11. *-rgt-identity N/A

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

      12. *-lft-identity N/A

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

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

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

      6. associate-*r/ N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. associate-*r/ N/A

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

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

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

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

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

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

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

      13. PI-lowering-PI.f64 61.8%

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

   7. Simplified 61.8%

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

      1. associate-/l/ N/A

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

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

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

      3. *-commutative N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. associate-/r/ N/A

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

      7. un-div-inv N/A

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

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

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

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

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

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

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

      11. *-lowering-*.f64 72.1%

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

   9. Applied egg-rr 72.1%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.15 \cdot 10^{-43}:\\ \;\;\;\;\frac{\frac{-0.5 \cdot \frac{\pi}{b}}{a}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{\frac{b}{0.5}}}{b \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1.3 \cdot 10^{-42}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{\frac{b}{0.5}}}{b \cdot a}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -1.3e-42)
   (/ (* 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 <= -1.3e-42) {
		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 <= -1.3e-42) {
		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 <= -1.3e-42:
		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 <= -1.3e-42)
		tmp = Float64(Float64(pi * 0.5) / Float64(a * Float64(b * a)));
	else
		tmp = Float64(Float64(pi / Float64(b / 0.5)) / Float64(b * a));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.3e-42)
		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, -1.3e-42], N[(N[(Pi * 0.5), $MachinePrecision] / N[(a * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / N[(b / 0.5), $MachinePrecision]), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.3e-42

   1. Initial program 85.7%

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

   3. Taylor expanded in b around 0

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

      1. associate-/r* N/A

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

      2. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 67.4%

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

   5. Simplified 67.4%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      7. /-lowering-/.f64 67.4%

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

   7. Applied egg-rr 67.4%

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

      1. frac-times N/A

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

      2. metadata-eval N/A

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

      3. distribute-rgt-neg-in N/A

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

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

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

      5. distribute-rgt-neg-in N/A

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

      6. metadata-eval N/A

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

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

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

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

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

      9. associate-*l* N/A

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

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

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

      11. *-lowering-*.f64 79.8%

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

   9. Applied egg-rr 79.8%

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

   if -1.3e-42 < a

   1. Initial program 73.8%

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

      1. un-div-inv N/A

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

      2. associate-/r* N/A

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

      3. associate-*l/ N/A

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

      4. distribute-lft-out-- N/A

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

      5. div-inv N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

      8. associate-/l/ N/A

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

      9. difference-of-squares N/A

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

      10. associate-/r* N/A

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

      11. *-rgt-identity N/A

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

      12. *-lft-identity N/A

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

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

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

      6. associate-*r/ N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. associate-*r/ N/A

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

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

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

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

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

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

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

      13. PI-lowering-PI.f64 61.8%

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

   7. Simplified 61.8%

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

      1. associate-/l/ N/A

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

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

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

      3. *-commutative N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. associate-/r/ N/A

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

      7. un-div-inv N/A

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

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

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

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

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

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

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

      11. *-lowering-*.f64 72.1%

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

   9. Applied egg-rr 72.1%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.3 \cdot 10^{-42}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{\frac{b}{0.5}}}{b \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -4.2 \cdot 10^{-43}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{b \cdot \left(b \cdot a\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -4.2e-43)
   (/ (* 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 <= -4.2e-43) {
		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 <= -4.2e-43) {
		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 <= -4.2e-43:
		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 <= -4.2e-43)
		tmp = Float64(Float64(pi * 0.5) / Float64(a * Float64(b * a)));
	else
		tmp = Float64(Float64(pi * 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 <= -4.2e-43)
		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, -4.2e-43], N[(N[(Pi * 0.5), $MachinePrecision] / N[(a * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * 0.5), $MachinePrecision] / N[(b * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.2000000000000001e-43

   1. Initial program 85.7%

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

   3. Taylor expanded in b around 0

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

      1. associate-/r* N/A

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

      2. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 67.4%

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

   5. Simplified 67.4%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      7. /-lowering-/.f64 67.4%

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

   7. Applied egg-rr 67.4%

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

      1. frac-times N/A

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

      2. metadata-eval N/A

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

      3. distribute-rgt-neg-in N/A

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

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

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

      5. distribute-rgt-neg-in N/A

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

      6. metadata-eval N/A

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

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

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

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

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

      9. associate-*l* N/A

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

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

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

      11. *-lowering-*.f64 79.8%

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

   9. Applied egg-rr 79.8%

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

   if -4.2000000000000001e-43 < a

   1. Initial program 73.8%

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

      1. un-div-inv N/A

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

      2. associate-/r* N/A

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

      3. associate-*l/ N/A

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

      4. distribute-lft-out-- N/A

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

      5. div-inv N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

      8. associate-/l/ N/A

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

      9. difference-of-squares N/A

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

      10. associate-/r* N/A

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

      11. *-rgt-identity N/A

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

      12. *-lft-identity N/A

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

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

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

      6. associate-*r/ N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. associate-*r/ N/A

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

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

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

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

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

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

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

      13. PI-lowering-PI.f64 61.8%

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

   7. Simplified 61.8%

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

      1. associate-/l/ N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. associate-/l/ N/A

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

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

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

      8. distribute-rgt-neg-in N/A

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

      9. metadata-eval N/A

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

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

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

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

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

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

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

      13. *-lowering-*.f64 72.1%

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

   9. Applied egg-rr 72.1%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.2 \cdot 10^{-43}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{b \cdot \left(b \cdot a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -6.6 \cdot 10^{-43}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{b \cdot b}}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -6.6e-43)
   (/ (* PI 0.5) (* a (* b a)))
   (/ (* 0.5 (/ PI (* b b))) a)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -6.6e-43) {
		tmp = (((double) M_PI) * 0.5) / (a * (b * a));
	} else {
		tmp = (0.5 * (((double) M_PI) / (b * b))) / a;
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -6.6e-43) {
		tmp = (Math.PI * 0.5) / (a * (b * a));
	} else {
		tmp = (0.5 * (Math.PI / (b * b))) / a;
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -6.6e-43:
		tmp = (math.pi * 0.5) / (a * (b * a))
	else:
		tmp = (0.5 * (math.pi / (b * b))) / a
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -6.6e-43)
		tmp = Float64(Float64(pi * 0.5) / Float64(a * Float64(b * a)));
	else
		tmp = Float64(Float64(0.5 * Float64(pi / Float64(b * b))) / a);
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -6.6e-43)
		tmp = (pi * 0.5) / (a * (b * a));
	else
		tmp = (0.5 * (pi / (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, -6.6e-43], N[(N[(Pi * 0.5), $MachinePrecision] / N[(a * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(Pi / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.60000000000000031e-43

   1. Initial program 85.7%

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

   3. Taylor expanded in b around 0

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

      1. associate-/r* N/A

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

      2. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 67.4%

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

   5. Simplified 67.4%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      7. /-lowering-/.f64 67.4%

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

   7. Applied egg-rr 67.4%

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

      1. frac-times N/A

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

      2. metadata-eval N/A

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

      3. distribute-rgt-neg-in N/A

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

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

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

      5. distribute-rgt-neg-in N/A

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

      6. metadata-eval N/A

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

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

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

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

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

      9. associate-*l* N/A

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

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

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

      11. *-lowering-*.f64 79.8%

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

   9. Applied egg-rr 79.8%

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

   if -6.60000000000000031e-43 < a

   1. Initial program 73.8%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. associate-/l/ N/A

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

      3. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 61.8%

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

   5. Simplified 61.8%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.6 \cdot 10^{-43}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{b \cdot b}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1.06 \cdot 10^{-42}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{\frac{0.5}{b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{b \cdot b}}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -1.06e-42)
   (* (/ PI a) (/ (/ 0.5 b) a))
   (/ (* 0.5 (/ PI (* b b))) a)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -1.06e-42) {
		tmp = (((double) M_PI) / a) * ((0.5 / b) / a);
	} else {
		tmp = (0.5 * (((double) M_PI) / (b * b))) / a;
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -1.06e-42) {
		tmp = (Math.PI / a) * ((0.5 / b) / a);
	} else {
		tmp = (0.5 * (Math.PI / (b * b))) / a;
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -1.06e-42:
		tmp = (math.pi / a) * ((0.5 / b) / a)
	else:
		tmp = (0.5 * (math.pi / (b * b))) / a
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -1.06e-42)
		tmp = Float64(Float64(pi / a) * Float64(Float64(0.5 / b) / a));
	else
		tmp = Float64(Float64(0.5 * Float64(pi / Float64(b * b))) / a);
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.06e-42)
		tmp = (pi / a) * ((0.5 / b) / a);
	else
		tmp = (0.5 * (pi / (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, -1.06e-42], N[(N[(Pi / a), $MachinePrecision] * N[(N[(0.5 / b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(Pi / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.0600000000000001e-42

   1. Initial program 85.7%

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

   3. Taylor expanded in b around 0

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

      1. associate-/r* N/A

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

      2. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 67.4%

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

   5. Simplified 67.4%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      7. /-lowering-/.f64 67.4%

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

   7. Applied egg-rr 67.4%

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

      1. associate-*l/ N/A

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

      2. times-frac N/A

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

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

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

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

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

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

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

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

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

      7. /-lowering-/.f64 78.6%

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

   9. Applied egg-rr 78.6%

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

   if -1.0600000000000001e-42 < a

   1. Initial program 73.8%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. associate-/l/ N/A

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

      3. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 61.8%

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

   5. Simplified 61.8%

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

Alternative 7: 99.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.2%

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

   1. un-div-inv N/A

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

   2. associate-/r* N/A

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

   3. associate-*l/ N/A

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

   4. distribute-lft-out-- N/A

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

   5. div-inv N/A

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

   6. div-inv N/A

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

   7. *-commutative N/A

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

   8. associate-/l/ N/A

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

   9. difference-of-squares N/A

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

   10. associate-/r* N/A

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

   11. *-rgt-identity N/A

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

   12. *-lft-identity N/A

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

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

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

4. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{\pi}{a} - \frac{\pi}{b}}{2}}{b + a}}{b - a}} \]
5. Step-by-step derivation

   1. div-inv N/A

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

   2. metadata-eval N/A

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

   3. associate-*l/ N/A

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

   4. associate-*r/ N/A

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

   5. associate-*l/ N/A

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

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

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in a around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a \cdot b}\right)}, \mathsf{+.f64}\left(a, b\right)\right) \]
8. Step-by-step derivation

   1. associate-/r* N/A

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

   2. associate-*r/ N/A

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

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

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

   4. associate-*r/ N/A

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

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

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

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

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

   7. PI-lowering-PI.f64 99.7%

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

9. Simplified 99.7%

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

10. Final simplification 99.7%

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

Alternative 8: 63.7% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 77.2%

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

3. Taylor expanded in b around 0

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

   1. associate-/r* N/A

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

   2. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

   7. unpow2 N/A

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

   8. *-lowering-*.f64 50.2%

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

5. Simplified 50.2%

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

   7. /-lowering-/.f64 50.2%

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

7. Applied egg-rr 50.2%

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

   1. associate-*l/ N/A

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

   2. times-frac N/A

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

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

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

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

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

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

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

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

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

   7. /-lowering-/.f64 58.4%

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

9. Applied egg-rr 58.4%

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

Reproduce

herbie shell --seed 2024150 
(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))))