NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.1% → 99.7%

Time: 4.7s

Alternatives: 13

Speedup: 2.0×

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: 78.1% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.1%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. frac-sub N/A

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

   6. associate-*r/ N/A

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

   7. lower-/.f64 N/A

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

3. Applied rewrites 87.3%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. associate-/l* N/A

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

   5. frac-2neg-rev N/A

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

   6. lift--.f64 N/A

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

   7. sub-negate-rev N/A

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

   8. lift--.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. distribute-rgt-neg-out N/A

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

   11. lift--.f64 N/A

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

   12. sub-negate-rev N/A

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

   13. lift--.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. div-flip-rev N/A

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

   16. lift-*.f64 N/A

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

   17. lift-+.f64 N/A

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

   18. lift--.f64 N/A

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

   19. difference-of-squares-rev N/A

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

   20. lift--.f64 N/A

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

   21. flip-+ N/A

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

   22. lift-+.f64 N/A

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

5. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/r* N/A

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

   9. inv-pow N/A

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

   10. frac-times N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. sum-to-mult N/A

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

   14. *-inverses N/A

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

   15. div-add N/A

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

   16. +-commutative N/A

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

   17. lift-+.f64 N/A

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

   18. lift-/.f64 N/A

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

   19. frac-times N/A

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

7. Applied rewrites 99.7%

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

Alternative 2: 99.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.1%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. frac-sub N/A

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

   6. associate-*r/ N/A

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

   7. lower-/.f64 N/A

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

3. Applied rewrites 87.3%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. associate-/l* N/A

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

   5. frac-2neg-rev N/A

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

   6. lift--.f64 N/A

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

   7. sub-negate-rev N/A

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

   8. lift--.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. distribute-rgt-neg-out N/A

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

   11. lift--.f64 N/A

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

   12. sub-negate-rev N/A

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

   13. lift--.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. div-flip-rev N/A

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

   16. lift-*.f64 N/A

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

   17. lift-+.f64 N/A

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

   18. lift--.f64 N/A

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

   19. difference-of-squares-rev N/A

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

   20. lift--.f64 N/A

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

   21. flip-+ N/A

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

   22. lift-+.f64 N/A

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

5. Applied rewrites 99.7%

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

Alternative 3: 99.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.1%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. frac-sub N/A

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

   6. associate-*r/ N/A

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

   7. lower-/.f64 N/A

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

3. Applied rewrites 87.3%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. associate-/l* N/A

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

   5. frac-2neg-rev N/A

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

   6. lift--.f64 N/A

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

   7. sub-negate-rev N/A

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

   8. lift--.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. distribute-rgt-neg-out N/A

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

   11. lift--.f64 N/A

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

   12. sub-negate-rev N/A

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

   13. lift--.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. div-flip-rev N/A

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

   16. lift-*.f64 N/A

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

   17. lift-+.f64 N/A

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

   18. lift--.f64 N/A

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

   19. difference-of-squares-rev N/A

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

   20. lift--.f64 N/A

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

   21. flip-+ N/A

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

   22. lift-+.f64 N/A

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

5. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. times-frac N/A

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

   7. lift-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 N/A

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

   12. lower-/.f64 99.6

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

7. Applied rewrites 99.6%

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

Alternative 4: 99.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 2.05 \cdot 10^{+104}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{\left(\left(b + a\right) \cdot b\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{a} \cdot 0.5}{b}}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 2.05e+104)
   (/ (* PI 0.5) (* (* (+ b a) b) a))
   (/ (/ (* (/ PI a) 0.5) b) b)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 2.05e+104) {
		tmp = (((double) M_PI) * 0.5) / (((b + a) * b) * a);
	} else {
		tmp = (((((double) M_PI) / a) * 0.5) / b) / b;
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 2.05e+104) {
		tmp = (Math.PI * 0.5) / (((b + a) * b) * a);
	} else {
		tmp = (((Math.PI / a) * 0.5) / b) / b;
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 2.05e+104:
		tmp = (math.pi * 0.5) / (((b + a) * b) * a)
	else:
		tmp = (((math.pi / a) * 0.5) / b) / b
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 2.05e+104)
		tmp = Float64(Float64(pi * 0.5) / Float64(Float64(Float64(b + a) * b) * a));
	else
		tmp = Float64(Float64(Float64(Float64(pi / a) * 0.5) / b) / b);
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2.05e+104)
		tmp = (pi * 0.5) / (((b + a) * b) * a);
	else
		tmp = (((pi / a) * 0.5) / b) / b;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[b, 2.05e+104], N[(N[(Pi * 0.5), $MachinePrecision] / N[(N[(N[(b + a), $MachinePrecision] * b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(Pi / a), $MachinePrecision] * 0.5), $MachinePrecision] / b), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.04999999999999992e104

   1. Initial program 78.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. difference-of-squares N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-lft-identity N/A

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

      16. /-rgt-identity N/A

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

      17. mult-flip N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 N/A

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

      20. metadata-eval N/A

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

      21. *-lft-identity N/A

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

      22. *-rgt-identity N/A

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

      23. lower--.f64 88.0

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

   3. Applied rewrites 88.0%

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

   5. Applied rewrites 93.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 93.1

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 93.1

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

   7. Applied rewrites 93.1%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 93.1

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 93.1

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

   9. Applied rewrites 93.1%

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

   if 2.04999999999999992e104 < b

   1. Initial program 78.1%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-sub N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 87.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. frac-2neg-rev N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-rgt-neg-out N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lift--.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. div-flip-rev N/A

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

      16. lift-*.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. lift--.f64 N/A

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

      19. difference-of-squares-rev N/A

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

      20. lift--.f64 N/A

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

      21. flip-+ N/A

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

      22. lift-+.f64 N/A

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

   5. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. inv-pow N/A

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

      10. frac-times N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. sum-to-mult N/A

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

      14. *-inverses N/A

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

      15. div-add N/A

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

      16. +-commutative N/A

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

      17. lift-+.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. frac-times N/A

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in a around 0

      \[\leadsto \frac{\frac{\frac{\pi}{a} \cdot \frac{1}{2}}{b}}{\color{blue}{b}} \]
   9. Step-by-step derivation

   10. Applied rewrites 62.5%

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

Alternative 5: 99.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 8.5 \cdot 10^{+77}:\\ \;\;\;\;\frac{0.5}{\left(\left(a + b\right) \cdot b\right) \cdot a} \cdot \pi\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{a} \cdot 0.5}{b}}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 8.5e+77)
   (* (/ 0.5 (* (* (+ a b) b) a)) PI)
   (/ (/ (* (/ PI a) 0.5) b) b)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 8.5e+77) {
		tmp = (0.5 / (((a + b) * b) * a)) * ((double) M_PI);
	} else {
		tmp = (((((double) M_PI) / a) * 0.5) / b) / b;
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 8.5e+77) {
		tmp = (0.5 / (((a + b) * b) * a)) * Math.PI;
	} else {
		tmp = (((Math.PI / a) * 0.5) / b) / b;
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 8.5e+77:
		tmp = (0.5 / (((a + b) * b) * a)) * math.pi
	else:
		tmp = (((math.pi / a) * 0.5) / b) / b
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 8.5e+77)
		tmp = Float64(Float64(0.5 / Float64(Float64(Float64(a + b) * b) * a)) * pi);
	else
		tmp = Float64(Float64(Float64(Float64(pi / a) * 0.5) / b) / b);
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 8.5e+77)
		tmp = (0.5 / (((a + b) * b) * a)) * pi;
	else
		tmp = (((pi / a) * 0.5) / b) / b;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[b, 8.5e+77], N[(N[(0.5 / N[(N[(N[(a + b), $MachinePrecision] * b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision], N[(N[(N[(N[(Pi / a), $MachinePrecision] * 0.5), $MachinePrecision] / b), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.50000000000000018e77

   1. Initial program 78.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. difference-of-squares N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-lft-identity N/A

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

      16. /-rgt-identity N/A

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

      17. mult-flip N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 N/A

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

      20. metadata-eval N/A

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

      21. *-lft-identity N/A

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

      22. *-rgt-identity N/A

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

      23. lower--.f64 88.0

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

   3. Applied rewrites 88.0%

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

   5. Applied rewrites 93.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 93.1

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 93.1

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

   7. Applied rewrites 93.1%

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

   if 8.50000000000000018e77 < b

   1. Initial program 78.1%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-sub N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 87.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. frac-2neg-rev N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-rgt-neg-out N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lift--.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. div-flip-rev N/A

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

      16. lift-*.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. lift--.f64 N/A

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

      19. difference-of-squares-rev N/A

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

      20. lift--.f64 N/A

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

      21. flip-+ N/A

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

      22. lift-+.f64 N/A

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

   5. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. inv-pow N/A

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

      10. frac-times N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. sum-to-mult N/A

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

      14. *-inverses N/A

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

      15. div-add N/A

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

      16. +-commutative N/A

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

      17. lift-+.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. frac-times N/A

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in a around 0

      \[\leadsto \frac{\frac{\frac{\pi}{a} \cdot \frac{1}{2}}{b}}{\color{blue}{b}} \]
   9. Step-by-step derivation

   10. Applied rewrites 62.5%

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

Alternative 6: 93.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.1%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. mult-flip-rev N/A

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

   4. lift-/.f64 N/A

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

   5. mult-flip N/A

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

   6. lift--.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. difference-of-squares N/A

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

   10. times-frac N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 N/A

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

   15. *-lft-identity N/A

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

   16. /-rgt-identity N/A

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

   17. mult-flip N/A

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

   18. metadata-eval N/A

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

   19. lower-/.f64 N/A

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

   20. metadata-eval N/A

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

   21. *-lft-identity N/A

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

   22. *-rgt-identity N/A

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

   23. lower--.f64 88.0

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

3. Applied rewrites 88.0%

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

5. Applied rewrites 93.0%

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

Alternative 7: 83.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-23}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{\left(a \cdot a\right) \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{a} \cdot 0.5}{b}}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -2e-23) (/ (* PI 0.5) (* (* a a) b)) (/ (/ (* (/ PI a) 0.5) b) b)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -2e-23) {
		tmp = (((double) M_PI) * 0.5) / ((a * a) * b);
	} else {
		tmp = (((((double) M_PI) / a) * 0.5) / b) / b;
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -2e-23) {
		tmp = (Math.PI * 0.5) / ((a * a) * b);
	} else {
		tmp = (((Math.PI / a) * 0.5) / b) / b;
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -2e-23:
		tmp = (math.pi * 0.5) / ((a * a) * b)
	else:
		tmp = (((math.pi / a) * 0.5) / b) / b
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -2e-23)
		tmp = Float64(Float64(pi * 0.5) / Float64(Float64(a * a) * b));
	else
		tmp = Float64(Float64(Float64(Float64(pi / a) * 0.5) / b) / b);
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2e-23)
		tmp = (pi * 0.5) / ((a * a) * b);
	else
		tmp = (((pi / a) * 0.5) / b) / 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, -2e-23], N[(N[(Pi * 0.5), $MachinePrecision] / N[(N[(a * a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(Pi / a), $MachinePrecision] * 0.5), $MachinePrecision] / b), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.99999999999999992e-23

   1. Initial program 78.1%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-sub N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 87.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. frac-2neg-rev N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-rgt-neg-out N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lift--.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. div-flip-rev N/A

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

      16. lift-*.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. lift--.f64 N/A

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

      19. difference-of-squares-rev N/A

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

      20. lift--.f64 N/A

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

      21. flip-+ N/A

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

      22. lift-+.f64 N/A

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

   5. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-/.f64 93.1

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 93.1

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 93.1

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 93.1

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-+.f64 93.1

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

   7. Applied rewrites 93.1%

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

   8. Taylor expanded in a around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 57.4

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

   10. Applied rewrites 57.4%

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

   if -1.99999999999999992e-23 < a

   1. Initial program 78.1%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-sub N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 87.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. frac-2neg-rev N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-rgt-neg-out N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lift--.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. div-flip-rev N/A

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

      16. lift-*.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. lift--.f64 N/A

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

      19. difference-of-squares-rev N/A

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

      20. lift--.f64 N/A

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

      21. flip-+ N/A

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

      22. lift-+.f64 N/A

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

   5. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. inv-pow N/A

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

      10. frac-times N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. sum-to-mult N/A

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

      14. *-inverses N/A

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

      15. div-add N/A

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

      16. +-commutative N/A

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

      17. lift-+.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. frac-times N/A

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in a around 0

      \[\leadsto \frac{\frac{\frac{\pi}{a} \cdot \frac{1}{2}}{b}}{\color{blue}{b}} \]
   9. Step-by-step derivation

   10. Applied rewrites 62.5%

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

Alternative 8: 83.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-23}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{\left(a \cdot a\right) \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{b} \cdot \pi}{b \cdot a}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -2e-23) (/ (* PI 0.5) (* (* a a) b)) (/ (* (/ 0.5 b) PI) (* b a))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -2e-23) {
		tmp = (((double) M_PI) * 0.5) / ((a * a) * b);
	} else {
		tmp = ((0.5 / b) * ((double) M_PI)) / (b * a);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -2e-23) {
		tmp = (Math.PI * 0.5) / ((a * a) * b);
	} else {
		tmp = ((0.5 / b) * Math.PI) / (b * a);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -2e-23:
		tmp = (math.pi * 0.5) / ((a * a) * b)
	else:
		tmp = ((0.5 / b) * math.pi) / (b * a)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -2e-23)
		tmp = Float64(Float64(pi * 0.5) / Float64(Float64(a * a) * b));
	else
		tmp = Float64(Float64(Float64(0.5 / b) * pi) / 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 <= -2e-23)
		tmp = (pi * 0.5) / ((a * a) * b);
	else
		tmp = ((0.5 / b) * pi) / (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, -2e-23], N[(N[(Pi * 0.5), $MachinePrecision] / N[(N[(a * a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 / b), $MachinePrecision] * Pi), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.99999999999999992e-23

   1. Initial program 78.1%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-sub N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 87.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. frac-2neg-rev N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-rgt-neg-out N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lift--.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. div-flip-rev N/A

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

      16. lift-*.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. lift--.f64 N/A

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

      19. difference-of-squares-rev N/A

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

      20. lift--.f64 N/A

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

      21. flip-+ N/A

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

      22. lift-+.f64 N/A

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

   5. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-/.f64 93.1

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 93.1

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 93.1

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 93.1

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-+.f64 93.1

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

   7. Applied rewrites 93.1%

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

   8. Taylor expanded in a around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 57.4

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

   10. Applied rewrites 57.4%

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

   if -1.99999999999999992e-23 < a

   1. Initial program 78.1%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-sub N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 87.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. frac-2neg-rev N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-rgt-neg-out N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lift--.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. div-flip-rev N/A

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

      16. lift-*.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. lift--.f64 N/A

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

      19. difference-of-squares-rev N/A

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

      20. lift--.f64 N/A

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

      21. flip-+ N/A

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

      22. lift-+.f64 N/A

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

   5. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/r* N/A

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

      14. associate-*l/ N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 93.5

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 93.5

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 93.5

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

   7. Applied rewrites 93.5%

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

   8. Taylor expanded in a around 0

      \[\leadsto \frac{\frac{\frac{1}{2}}{b} \cdot \pi}{\color{blue}{b} \cdot a} \]
   9. Step-by-step derivation

   10. Applied rewrites 62.5%

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

Alternative 9: 83.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-23}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{\left(a \cdot a\right) \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{0.5}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -2e-23) (/ (* PI 0.5) (* (* a a) b)) (* (/ PI b) (/ 0.5 (* a b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -2e-23) {
		tmp = (((double) M_PI) * 0.5) / ((a * a) * b);
	} else {
		tmp = (((double) M_PI) / b) * (0.5 / (a * b));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -2e-23) {
		tmp = (Math.PI * 0.5) / ((a * a) * b);
	} else {
		tmp = (Math.PI / b) * (0.5 / (a * b));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -2e-23:
		tmp = (math.pi * 0.5) / ((a * a) * b)
	else:
		tmp = (math.pi / b) * (0.5 / (a * b))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -2e-23)
		tmp = Float64(Float64(pi * 0.5) / Float64(Float64(a * a) * b));
	else
		tmp = Float64(Float64(pi / b) * Float64(0.5 / Float64(a * b)));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2e-23)
		tmp = (pi * 0.5) / ((a * a) * b);
	else
		tmp = (pi / b) * (0.5 / (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, -2e-23], N[(N[(Pi * 0.5), $MachinePrecision] / N[(N[(a * a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / b), $MachinePrecision] * N[(0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.99999999999999992e-23

   1. Initial program 78.1%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-sub N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 87.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. frac-2neg-rev N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-rgt-neg-out N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lift--.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. div-flip-rev N/A

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

      16. lift-*.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. lift--.f64 N/A

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

      19. difference-of-squares-rev N/A

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

      20. lift--.f64 N/A

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

      21. flip-+ N/A

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

      22. lift-+.f64 N/A

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

   5. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-/.f64 93.1

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 93.1

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 93.1

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 93.1

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-+.f64 93.1

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

   7. Applied rewrites 93.1%

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

   8. Taylor expanded in a around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 57.4

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

   10. Applied rewrites 57.4%

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

   if -1.99999999999999992e-23 < a

   1. Initial program 78.1%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-sub N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 87.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. frac-2neg-rev N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-rgt-neg-out N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lift--.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. div-flip-rev N/A

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

      16. lift-*.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. lift--.f64 N/A

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

      19. difference-of-squares-rev N/A

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

      20. lift--.f64 N/A

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

      21. flip-+ N/A

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

      22. lift-+.f64 N/A

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

   5. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lift-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 99.6

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in a around 0

      \[\leadsto \frac{\pi}{\color{blue}{b}} \cdot \frac{\frac{1}{2}}{a \cdot b} \]
   9. Step-by-step derivation

   10. Applied rewrites 62.5%

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

Alternative 10: 83.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-23}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{\left(a \cdot a\right) \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{\left(b \cdot a\right) \cdot b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -2e-23) (/ (* PI 0.5) (* (* a a) b)) (/ (* PI 0.5) (* (* b a) b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -2e-23) {
		tmp = (((double) M_PI) * 0.5) / ((a * a) * b);
	} else {
		tmp = (((double) M_PI) * 0.5) / ((b * a) * b);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -2e-23) {
		tmp = (Math.PI * 0.5) / ((a * a) * b);
	} else {
		tmp = (Math.PI * 0.5) / ((b * a) * b);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -2e-23:
		tmp = (math.pi * 0.5) / ((a * a) * b)
	else:
		tmp = (math.pi * 0.5) / ((b * a) * b)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -2e-23)
		tmp = Float64(Float64(pi * 0.5) / Float64(Float64(a * a) * b));
	else
		tmp = Float64(Float64(pi * 0.5) / Float64(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 <= -2e-23)
		tmp = (pi * 0.5) / ((a * a) * b);
	else
		tmp = (pi * 0.5) / ((b * a) * b);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -2e-23], N[(N[(Pi * 0.5), $MachinePrecision] / N[(N[(a * a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * 0.5), $MachinePrecision] / N[(N[(b * a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.99999999999999992e-23

   1. Initial program 78.1%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-sub N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 87.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. frac-2neg-rev N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-rgt-neg-out N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lift--.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. div-flip-rev N/A

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

      16. lift-*.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. lift--.f64 N/A

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

      19. difference-of-squares-rev N/A

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

      20. lift--.f64 N/A

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

      21. flip-+ N/A

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

      22. lift-+.f64 N/A

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

   5. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-/.f64 93.1

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 93.1

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 93.1

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 93.1

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-+.f64 93.1

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

   7. Applied rewrites 93.1%

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

   8. Taylor expanded in a around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 57.4

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

   10. Applied rewrites 57.4%

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

   if -1.99999999999999992e-23 < a

   1. Initial program 78.1%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-sub N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 87.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. frac-2neg-rev N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-rgt-neg-out N/A

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

      11. lift--.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{a - b}{\left(a + b\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\left(b - a\right)}\right)\right)}}{a \cdot b} \]

      12. sub-negate-rev N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{a - b}{\left(a + b\right) \cdot \color{blue}{\left(a - b\right)}}}{a \cdot b} \]

      13. lift--.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{a - b}{\left(a + b\right) \cdot \color{blue}{\left(a - b\right)}}}{a \cdot b} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{a - b}{\color{blue}{\left(a + b\right) \cdot \left(a - b\right)}}}{a \cdot b} \]

      15. div-flip-rev N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \color{blue}{\frac{1}{\frac{\left(a + b\right) \cdot \left(a - b\right)}{a - b}}}}{a \cdot b} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{1}{\frac{\color{blue}{\left(a + b\right) \cdot \left(a - b\right)}}{a - b}}}{a \cdot b} \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{1}{\frac{\color{blue}{\left(a + b\right)} \cdot \left(a - b\right)}{a - b}}}{a \cdot b} \]

      18. lift--.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{1}{\frac{\left(a + b\right) \cdot \color{blue}{\left(a - b\right)}}{a - b}}}{a \cdot b} \]

      19. difference-of-squares-rev N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{1}{\frac{\color{blue}{a \cdot a - b \cdot b}}{a - b}}}{a \cdot b} \]

      20. lift--.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{1}{\frac{a \cdot a - b \cdot b}{\color{blue}{a - b}}}}{a \cdot b} \]

      21. flip-+ N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{1}{\color{blue}{a + b}}}{a \cdot b} \]

      22. lift-+.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{1}{\color{blue}{a + b}}}{a \cdot b} \]

   5. Applied rewrites 99.7%

      \[\leadsto \frac{\color{blue}{\frac{0.5 \cdot \pi}{b + a}}}{a \cdot b} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{1}{2} \cdot \pi}{b + a}}{a \cdot b}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2} \cdot \pi}{b + a}}}{a \cdot b} \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \pi}{\left(b + a\right) \cdot \left(a \cdot b\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \pi}{\left(b + a\right) \cdot \color{blue}{\left(a \cdot b\right)}} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \pi}{\color{blue}{\left(\left(b + a\right) \cdot a\right) \cdot b}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \pi}{\color{blue}{\left(a \cdot \left(b + a\right)\right)} \cdot b} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \pi}{\color{blue}{\left(a \cdot \left(b + a\right)\right)} \cdot b} \]

      8. *-commutative N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \pi}{\color{blue}{b \cdot \left(a \cdot \left(b + a\right)\right)}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \pi}{\color{blue}{b \cdot \left(a \cdot \left(b + a\right)\right)}} \]

      10. lower-/.f64 93.1

          \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{b \cdot \left(a \cdot \left(b + a\right)\right)}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \pi}}{b \cdot \left(a \cdot \left(b + a\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\pi \cdot \frac{1}{2}}}{b \cdot \left(a \cdot \left(b + a\right)\right)} \]

      13. lower-*.f64 93.1

          \[\leadsto \frac{\color{blue}{\pi \cdot 0.5}}{b \cdot \left(a \cdot \left(b + a\right)\right)} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\pi \cdot \frac{1}{2}}{\color{blue}{b \cdot \left(a \cdot \left(b + a\right)\right)}} \]

      15. *-commutative N/A

          \[\leadsto \frac{\pi \cdot \frac{1}{2}}{\color{blue}{\left(a \cdot \left(b + a\right)\right) \cdot b}} \]

      16. lower-*.f64 93.1

          \[\leadsto \frac{\pi \cdot 0.5}{\color{blue}{\left(a \cdot \left(b + a\right)\right) \cdot b}} \]

      17. lift-*.f64 N/A

          \[\leadsto \frac{\pi \cdot \frac{1}{2}}{\color{blue}{\left(a \cdot \left(b + a\right)\right)} \cdot b} \]

      18. *-commutative N/A

          \[\leadsto \frac{\pi \cdot \frac{1}{2}}{\color{blue}{\left(\left(b + a\right) \cdot a\right)} \cdot b} \]

      19. lower-*.f64 93.1

          \[\leadsto \frac{\pi \cdot 0.5}{\color{blue}{\left(\left(b + a\right) \cdot a\right)} \cdot b} \]

      20. lift-+.f64 N/A

          \[\leadsto \frac{\pi \cdot \frac{1}{2}}{\left(\color{blue}{\left(b + a\right)} \cdot a\right) \cdot b} \]

      21. +-commutative N/A

          \[\leadsto \frac{\pi \cdot \frac{1}{2}}{\left(\color{blue}{\left(a + b\right)} \cdot a\right) \cdot b} \]

      22. lower-+.f64 93.1

          \[\leadsto \frac{\pi \cdot 0.5}{\left(\color{blue}{\left(a + b\right)} \cdot a\right) \cdot b} \]

   7. Applied rewrites 93.1%

      \[\leadsto \color{blue}{\frac{\pi \cdot 0.5}{\left(\left(a + b\right) \cdot a\right) \cdot b}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \frac{\pi \cdot \frac{1}{2}}{\left(\color{blue}{b} \cdot a\right) \cdot b} \]
   9. Step-by-step derivation

   10. Applied rewrites 62.3%

       \[\leadsto \frac{\pi \cdot 0.5}{\left(\color{blue}{b} \cdot a\right) \cdot b} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 77.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-23}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{\left(a \cdot a\right) \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -2e-23) (/ (* PI 0.5) (* (* a a) b)) (* (/ PI (* (* b b) a)) 0.5)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -2e-23) {
		tmp = (((double) M_PI) * 0.5) / ((a * a) * b);
	} else {
		tmp = (((double) M_PI) / ((b * b) * a)) * 0.5;
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -2e-23) {
		tmp = (Math.PI * 0.5) / ((a * a) * b);
	} else {
		tmp = (Math.PI / ((b * b) * a)) * 0.5;
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -2e-23:
		tmp = (math.pi * 0.5) / ((a * a) * b)
	else:
		tmp = (math.pi / ((b * b) * a)) * 0.5
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -2e-23)
		tmp = Float64(Float64(pi * 0.5) / Float64(Float64(a * a) * b));
	else
		tmp = Float64(Float64(pi / Float64(Float64(b * b) * a)) * 0.5);
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2e-23)
		tmp = (pi * 0.5) / ((a * a) * b);
	else
		tmp = (pi / ((b * b) * a)) * 0.5;
	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, -2e-23], N[(N[(Pi * 0.5), $MachinePrecision] / N[(N[(a * a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / N[(N[(b * b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.99999999999999992e-23

   1. Initial program 78.1%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)} \]

      2. lift--.f64 N/A

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \color{blue}{\left(\frac{1}{a} - \frac{1}{b}\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\color{blue}{\frac{1}{a}} - \frac{1}{b}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \color{blue}{\frac{1}{b}}\right) \]

      5. frac-sub N/A

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \color{blue}{\frac{1 \cdot b - a \cdot 1}{a \cdot b}} \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(1 \cdot b - a \cdot 1\right)}{a \cdot b}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(1 \cdot b - a \cdot 1\right)}{a \cdot b}} \]

   3. Applied rewrites 87.3%

      \[\leadsto \color{blue}{\frac{\frac{0.5 \cdot \pi}{\left(a + b\right) \cdot \left(b - a\right)} \cdot \left(b - a\right)}{a \cdot b}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2} \cdot \pi}{\left(a + b\right) \cdot \left(b - a\right)} \cdot \left(b - a\right)}}{a \cdot b} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2} \cdot \pi}{\left(a + b\right) \cdot \left(b - a\right)}} \cdot \left(b - a\right)}{a \cdot b} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \left(b - a\right)}{\left(a + b\right) \cdot \left(b - a\right)}}}{a \cdot b} \]

      4. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{b - a}{\left(a + b\right) \cdot \left(b - a\right)}}}{a \cdot b} \]

      5. frac-2neg-rev N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(b - a\right)\right)}{\mathsf{neg}\left(\left(a + b\right) \cdot \left(b - a\right)\right)}}}{a \cdot b} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{\mathsf{neg}\left(\color{blue}{\left(b - a\right)}\right)}{\mathsf{neg}\left(\left(a + b\right) \cdot \left(b - a\right)\right)}}{a \cdot b} \]

      7. sub-negate-rev N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{\color{blue}{a - b}}{\mathsf{neg}\left(\left(a + b\right) \cdot \left(b - a\right)\right)}}{a \cdot b} \]

      8. lift--.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{\color{blue}{a - b}}{\mathsf{neg}\left(\left(a + b\right) \cdot \left(b - a\right)\right)}}{a \cdot b} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{a - b}{\mathsf{neg}\left(\color{blue}{\left(a + b\right) \cdot \left(b - a\right)}\right)}}{a \cdot b} \]

      10. distribute-rgt-neg-out N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{a - b}{\color{blue}{\left(a + b\right) \cdot \left(\mathsf{neg}\left(\left(b - a\right)\right)\right)}}}{a \cdot b} \]

      11. lift--.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{a - b}{\left(a + b\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\left(b - a\right)}\right)\right)}}{a \cdot b} \]

      12. sub-negate-rev N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{a - b}{\left(a + b\right) \cdot \color{blue}{\left(a - b\right)}}}{a \cdot b} \]

      13. lift--.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{a - b}{\left(a + b\right) \cdot \color{blue}{\left(a - b\right)}}}{a \cdot b} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{a - b}{\color{blue}{\left(a + b\right) \cdot \left(a - b\right)}}}{a \cdot b} \]

      15. div-flip-rev N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \color{blue}{\frac{1}{\frac{\left(a + b\right) \cdot \left(a - b\right)}{a - b}}}}{a \cdot b} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{1}{\frac{\color{blue}{\left(a + b\right) \cdot \left(a - b\right)}}{a - b}}}{a \cdot b} \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{1}{\frac{\color{blue}{\left(a + b\right)} \cdot \left(a - b\right)}{a - b}}}{a \cdot b} \]

      18. lift--.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{1}{\frac{\left(a + b\right) \cdot \color{blue}{\left(a - b\right)}}{a - b}}}{a \cdot b} \]

      19. difference-of-squares-rev N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{1}{\frac{\color{blue}{a \cdot a - b \cdot b}}{a - b}}}{a \cdot b} \]

      20. lift--.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{1}{\frac{a \cdot a - b \cdot b}{\color{blue}{a - b}}}}{a \cdot b} \]

      21. flip-+ N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{1}{\color{blue}{a + b}}}{a \cdot b} \]

      22. lift-+.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{1}{\color{blue}{a + b}}}{a \cdot b} \]

   5. Applied rewrites 99.7%

      \[\leadsto \frac{\color{blue}{\frac{0.5 \cdot \pi}{b + a}}}{a \cdot b} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{1}{2} \cdot \pi}{b + a}}{a \cdot b}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2} \cdot \pi}{b + a}}}{a \cdot b} \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \pi}{\left(b + a\right) \cdot \left(a \cdot b\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \pi}{\left(b + a\right) \cdot \color{blue}{\left(a \cdot b\right)}} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \pi}{\color{blue}{\left(\left(b + a\right) \cdot a\right) \cdot b}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \pi}{\color{blue}{\left(a \cdot \left(b + a\right)\right)} \cdot b} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \pi}{\color{blue}{\left(a \cdot \left(b + a\right)\right)} \cdot b} \]

      8. *-commutative N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \pi}{\color{blue}{b \cdot \left(a \cdot \left(b + a\right)\right)}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \pi}{\color{blue}{b \cdot \left(a \cdot \left(b + a\right)\right)}} \]

      10. lower-/.f64 93.1

          \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{b \cdot \left(a \cdot \left(b + a\right)\right)}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \pi}}{b \cdot \left(a \cdot \left(b + a\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\pi \cdot \frac{1}{2}}}{b \cdot \left(a \cdot \left(b + a\right)\right)} \]

      13. lower-*.f64 93.1

          \[\leadsto \frac{\color{blue}{\pi \cdot 0.5}}{b \cdot \left(a \cdot \left(b + a\right)\right)} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\pi \cdot \frac{1}{2}}{\color{blue}{b \cdot \left(a \cdot \left(b + a\right)\right)}} \]

      15. *-commutative N/A

          \[\leadsto \frac{\pi \cdot \frac{1}{2}}{\color{blue}{\left(a \cdot \left(b + a\right)\right) \cdot b}} \]

      16. lower-*.f64 93.1

          \[\leadsto \frac{\pi \cdot 0.5}{\color{blue}{\left(a \cdot \left(b + a\right)\right) \cdot b}} \]

      17. lift-*.f64 N/A

          \[\leadsto \frac{\pi \cdot \frac{1}{2}}{\color{blue}{\left(a \cdot \left(b + a\right)\right)} \cdot b} \]

      18. *-commutative N/A

          \[\leadsto \frac{\pi \cdot \frac{1}{2}}{\color{blue}{\left(\left(b + a\right) \cdot a\right)} \cdot b} \]

      19. lower-*.f64 93.1

          \[\leadsto \frac{\pi \cdot 0.5}{\color{blue}{\left(\left(b + a\right) \cdot a\right)} \cdot b} \]

      20. lift-+.f64 N/A

          \[\leadsto \frac{\pi \cdot \frac{1}{2}}{\left(\color{blue}{\left(b + a\right)} \cdot a\right) \cdot b} \]

      21. +-commutative N/A

          \[\leadsto \frac{\pi \cdot \frac{1}{2}}{\left(\color{blue}{\left(a + b\right)} \cdot a\right) \cdot b} \]

      22. lower-+.f64 93.1

          \[\leadsto \frac{\pi \cdot 0.5}{\left(\color{blue}{\left(a + b\right)} \cdot a\right) \cdot b} \]

   7. Applied rewrites 93.1%

      \[\leadsto \color{blue}{\frac{\pi \cdot 0.5}{\left(\left(a + b\right) \cdot a\right) \cdot b}} \]

   8. Taylor expanded in a around inf

      \[\leadsto \frac{\pi \cdot \frac{1}{2}}{\color{blue}{{a}^{2}} \cdot b} \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\pi \cdot \frac{1}{2}}{\left(a \cdot \color{blue}{a}\right) \cdot b} \]

      2. lower-*.f64 57.4

         \[\leadsto \frac{\pi \cdot 0.5}{\left(a \cdot \color{blue}{a}\right) \cdot b} \]

   10. Applied rewrites 57.4%

       \[\leadsto \frac{\pi \cdot 0.5}{\color{blue}{\left(a \cdot a\right)} \cdot b} \]

   if -1.99999999999999992e-23 < a

   1. Initial program 78.1%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)} \]

      2. lift--.f64 N/A

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \color{blue}{\left(\frac{1}{a} - \frac{1}{b}\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\color{blue}{\frac{1}{a}} - \frac{1}{b}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \color{blue}{\frac{1}{b}}\right) \]

      5. frac-sub N/A

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \color{blue}{\frac{1 \cdot b - a \cdot 1}{a \cdot b}} \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(1 \cdot b - a \cdot 1\right)}{a \cdot b}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(1 \cdot b - a \cdot 1\right)}{a \cdot b}} \]

   3. Applied rewrites 87.3%

      \[\leadsto \color{blue}{\frac{\frac{0.5 \cdot \pi}{\left(a + b\right) \cdot \left(b - a\right)} \cdot \left(b - a\right)}{a \cdot b}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2} \cdot \pi}{\left(a + b\right) \cdot \left(b - a\right)} \cdot \left(b - a\right)}}{a \cdot b} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2} \cdot \pi}{\left(a + b\right) \cdot \left(b - a\right)}} \cdot \left(b - a\right)}{a \cdot b} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \left(b - a\right)}{\left(a + b\right) \cdot \left(b - a\right)}}}{a \cdot b} \]

      4. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{b - a}{\left(a + b\right) \cdot \left(b - a\right)}}}{a \cdot b} \]

      5. frac-2neg-rev N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(b - a\right)\right)}{\mathsf{neg}\left(\left(a + b\right) \cdot \left(b - a\right)\right)}}}{a \cdot b} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{\mathsf{neg}\left(\color{blue}{\left(b - a\right)}\right)}{\mathsf{neg}\left(\left(a + b\right) \cdot \left(b - a\right)\right)}}{a \cdot b} \]

      7. sub-negate-rev N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{\color{blue}{a - b}}{\mathsf{neg}\left(\left(a + b\right) \cdot \left(b - a\right)\right)}}{a \cdot b} \]

      8. lift--.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{\color{blue}{a - b}}{\mathsf{neg}\left(\left(a + b\right) \cdot \left(b - a\right)\right)}}{a \cdot b} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{a - b}{\mathsf{neg}\left(\color{blue}{\left(a + b\right) \cdot \left(b - a\right)}\right)}}{a \cdot b} \]

      10. distribute-rgt-neg-out N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{a - b}{\color{blue}{\left(a + b\right) \cdot \left(\mathsf{neg}\left(\left(b - a\right)\right)\right)}}}{a \cdot b} \]

      11. lift--.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{a - b}{\left(a + b\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\left(b - a\right)}\right)\right)}}{a \cdot b} \]

      12. sub-negate-rev N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{a - b}{\left(a + b\right) \cdot \color{blue}{\left(a - b\right)}}}{a \cdot b} \]

      13. lift--.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{a - b}{\left(a + b\right) \cdot \color{blue}{\left(a - b\right)}}}{a \cdot b} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{a - b}{\color{blue}{\left(a + b\right) \cdot \left(a - b\right)}}}{a \cdot b} \]

      15. div-flip-rev N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \color{blue}{\frac{1}{\frac{\left(a + b\right) \cdot \left(a - b\right)}{a - b}}}}{a \cdot b} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{1}{\frac{\color{blue}{\left(a + b\right) \cdot \left(a - b\right)}}{a - b}}}{a \cdot b} \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{1}{\frac{\color{blue}{\left(a + b\right)} \cdot \left(a - b\right)}{a - b}}}{a \cdot b} \]

      18. lift--.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{1}{\frac{\left(a + b\right) \cdot \color{blue}{\left(a - b\right)}}{a - b}}}{a \cdot b} \]

      19. difference-of-squares-rev N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{1}{\frac{\color{blue}{a \cdot a - b \cdot b}}{a - b}}}{a \cdot b} \]

      20. lift--.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{1}{\frac{a \cdot a - b \cdot b}{\color{blue}{a - b}}}}{a \cdot b} \]

      21. flip-+ N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{1}{\color{blue}{a + b}}}{a \cdot b} \]

      22. lift-+.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{1}{\color{blue}{a + b}}}{a \cdot b} \]

   5. Applied rewrites 99.7%

      \[\leadsto \frac{\color{blue}{\frac{0.5 \cdot \pi}{b + a}}}{a \cdot b} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{1}{2} \cdot \pi}{b + a}}{a \cdot b}} \]

      2. mult-flip N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \pi}{b + a} \cdot \frac{1}{a \cdot b}} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \pi}{b + a}} \cdot \frac{1}{a \cdot b} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \pi}{b + a} \cdot \frac{1}{\color{blue}{a \cdot b}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \pi}{b + a} \cdot \frac{1}{\color{blue}{b \cdot a}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \pi}{b + a} \cdot \frac{1}{\color{blue}{b \cdot a}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \pi}{b + a} \cdot \frac{1}{\color{blue}{b \cdot a}} \]

      8. associate-/r* N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \pi}{b + a} \cdot \color{blue}{\frac{\frac{1}{b}}{a}} \]

      9. inv-pow N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \pi}{b + a} \cdot \frac{\color{blue}{{b}^{-1}}}{a} \]

      10. frac-times N/A

          \[\leadsto \color{blue}{\frac{\left(\frac{1}{2} \cdot \pi\right) \cdot {b}^{-1}}{\left(b + a\right) \cdot a}} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot {b}^{-1}}{\color{blue}{\left(b + a\right)} \cdot a} \]

      12. +-commutative N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot {b}^{-1}}{\color{blue}{\left(a + b\right)} \cdot a} \]

      13. sum-to-mult N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot {b}^{-1}}{\color{blue}{\left(\left(1 + \frac{b}{a}\right) \cdot a\right)} \cdot a} \]

      14. *-inverses N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot {b}^{-1}}{\left(\left(\color{blue}{\frac{a}{a}} + \frac{b}{a}\right) \cdot a\right) \cdot a} \]

      15. div-add N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot {b}^{-1}}{\left(\color{blue}{\frac{a + b}{a}} \cdot a\right) \cdot a} \]

      16. +-commutative N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot {b}^{-1}}{\left(\frac{\color{blue}{b + a}}{a} \cdot a\right) \cdot a} \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot {b}^{-1}}{\left(\frac{\color{blue}{b + a}}{a} \cdot a\right) \cdot a} \]

      18. lift-/.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot {b}^{-1}}{\left(\color{blue}{\frac{b + a}{a}} \cdot a\right) \cdot a} \]

      19. frac-times N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \pi}{\frac{b + a}{a} \cdot a} \cdot \frac{{b}^{-1}}{a}} \]

   7. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\pi}{a} \cdot 0.5}{b}}{a + b}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a \cdot {b}^{2}}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\mathsf{PI}\left(\right)}{a \cdot {b}^{2}} \cdot \color{blue}{\frac{1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{PI}\left(\right)}{a \cdot {b}^{2}} \cdot \color{blue}{\frac{1}{2}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{PI}\left(\right)}{a \cdot {b}^{2}} \cdot \frac{1}{2} \]

      4. lift-PI.f64 N/A

         \[\leadsto \frac{\pi}{a \cdot {b}^{2}} \cdot \frac{1}{2} \]

      5. *-commutative N/A

         \[\leadsto \frac{\pi}{{b}^{2} \cdot a} \cdot \frac{1}{2} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\pi}{{b}^{2} \cdot a} \cdot \frac{1}{2} \]

      7. unpow2 N/A

         \[\leadsto \frac{\pi}{\left(b \cdot b\right) \cdot a} \cdot \frac{1}{2} \]

      8. lower-*.f64 56.3

         \[\leadsto \frac{\pi}{\left(b \cdot b\right) \cdot a} \cdot 0.5 \]

   10. Applied rewrites 56.3%

       \[\leadsto \color{blue}{\frac{\pi}{\left(b \cdot b\right) \cdot a} \cdot 0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 77.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-23}:\\ \;\;\;\;\frac{0.5}{\left(a \cdot a\right) \cdot b} \cdot \pi\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -2e-23) (* (/ 0.5 (* (* a a) b)) PI) (* (/ PI (* (* b b) a)) 0.5)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -2e-23) {
		tmp = (0.5 / ((a * a) * b)) * ((double) M_PI);
	} else {
		tmp = (((double) M_PI) / ((b * b) * a)) * 0.5;
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -2e-23) {
		tmp = (0.5 / ((a * a) * b)) * Math.PI;
	} else {
		tmp = (Math.PI / ((b * b) * a)) * 0.5;
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -2e-23:
		tmp = (0.5 / ((a * a) * b)) * math.pi
	else:
		tmp = (math.pi / ((b * b) * a)) * 0.5
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -2e-23)
		tmp = Float64(Float64(0.5 / Float64(Float64(a * a) * b)) * pi);
	else
		tmp = Float64(Float64(pi / Float64(Float64(b * b) * a)) * 0.5);
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2e-23)
		tmp = (0.5 / ((a * a) * b)) * pi;
	else
		tmp = (pi / ((b * b) * a)) * 0.5;
	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, -2e-23], N[(N[(0.5 / N[(N[(a * a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision], N[(N[(Pi / N[(N[(b * b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.99999999999999992e-23

   1. Initial program 78.1%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\frac{\pi}{2} \cdot \color{blue}{\frac{1}{b \cdot b - a \cdot a}}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      3. mult-flip-rev N/A

         \[\leadsto \color{blue}{\frac{\frac{\pi}{2}}{b \cdot b - a \cdot a}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\pi}{2}}}{b \cdot b - a \cdot a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      5. mult-flip N/A

         \[\leadsto \frac{\color{blue}{\pi \cdot \frac{1}{2}}}{b \cdot b - a \cdot a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      6. lift--.f64 N/A

         \[\leadsto \frac{\pi \cdot \frac{1}{2}}{\color{blue}{b \cdot b - a \cdot a}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\pi \cdot \frac{1}{2}}{\color{blue}{b \cdot b} - a \cdot a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\pi \cdot \frac{1}{2}}{b \cdot b - \color{blue}{a \cdot a}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      9. difference-of-squares N/A

         \[\leadsto \frac{\pi \cdot \frac{1}{2}}{\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      10. times-frac N/A

          \[\leadsto \color{blue}{\left(\frac{\pi}{b + a} \cdot \frac{\frac{1}{2}}{b - a}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{\pi}{b + a} \cdot \frac{\frac{1}{2}}{b - a}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \left(\color{blue}{\frac{\pi}{b + a}} \cdot \frac{\frac{1}{2}}{b - a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      13. +-commutative N/A

          \[\leadsto \left(\frac{\pi}{\color{blue}{a + b}} \cdot \frac{\frac{1}{2}}{b - a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\frac{\pi}{\color{blue}{a + b}} \cdot \frac{\frac{1}{2}}{b - a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      15. *-lft-identity N/A

          \[\leadsto \left(\frac{\pi}{a + b} \cdot \frac{\frac{1}{2}}{\color{blue}{1 \cdot b} - a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      16. /-rgt-identity N/A

          \[\leadsto \left(\frac{\pi}{a + b} \cdot \frac{\frac{1}{2}}{1 \cdot b - \color{blue}{\frac{a}{1}}}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      17. mult-flip N/A

          \[\leadsto \left(\frac{\pi}{a + b} \cdot \frac{\frac{1}{2}}{1 \cdot b - \color{blue}{a \cdot \frac{1}{1}}}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      18. metadata-eval N/A

          \[\leadsto \left(\frac{\pi}{a + b} \cdot \frac{\frac{1}{2}}{1 \cdot b - a \cdot \color{blue}{1}}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      19. lower-/.f64 N/A

          \[\leadsto \left(\frac{\pi}{a + b} \cdot \color{blue}{\frac{\frac{1}{2}}{1 \cdot b - a \cdot 1}}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      20. metadata-eval N/A

          \[\leadsto \left(\frac{\pi}{a + b} \cdot \frac{\color{blue}{\frac{1}{2}}}{1 \cdot b - a \cdot 1}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      21. *-lft-identity N/A

          \[\leadsto \left(\frac{\pi}{a + b} \cdot \frac{\frac{1}{2}}{\color{blue}{b} - a \cdot 1}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      22. *-rgt-identity N/A

          \[\leadsto \left(\frac{\pi}{a + b} \cdot \frac{\frac{1}{2}}{b - \color{blue}{a}}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      23. lower--.f64 88.0

          \[\leadsto \left(\frac{\pi}{a + b} \cdot \frac{0.5}{\color{blue}{b - a}}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   3. Applied rewrites 88.0%

      \[\leadsto \color{blue}{\left(\frac{\pi}{a + b} \cdot \frac{0.5}{b - a}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   5. Applied rewrites 93.0%

      \[\leadsto \color{blue}{\frac{0.5}{b \cdot \left(a \cdot \left(b + a\right)\right)} \cdot \pi} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{2}}{\color{blue}{b \cdot \left(a \cdot \left(b + a\right)\right)}} \cdot \pi \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\left(a \cdot \left(b + a\right)\right) \cdot b}} \cdot \pi \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\left(a \cdot \left(b + a\right)\right)} \cdot b} \cdot \pi \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\left(\left(b + a\right) \cdot a\right)} \cdot b} \cdot \pi \]

      5. associate-*l* N/A

         \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\left(b + a\right) \cdot \left(a \cdot b\right)}} \cdot \pi \]

      6. *-commutative N/A

         \[\leadsto \frac{\frac{1}{2}}{\left(b + a\right) \cdot \color{blue}{\left(b \cdot a\right)}} \cdot \pi \]

      7. associate-*r* N/A

         \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\left(\left(b + a\right) \cdot b\right) \cdot a}} \cdot \pi \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\left(\left(b + a\right) \cdot b\right) \cdot a}} \cdot \pi \]

      9. lower-*.f64 93.1

         \[\leadsto \frac{0.5}{\color{blue}{\left(\left(b + a\right) \cdot b\right)} \cdot a} \cdot \pi \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{\frac{1}{2}}{\left(\color{blue}{\left(b + a\right)} \cdot b\right) \cdot a} \cdot \pi \]

      11. +-commutative N/A

          \[\leadsto \frac{\frac{1}{2}}{\left(\color{blue}{\left(a + b\right)} \cdot b\right) \cdot a} \cdot \pi \]

      12. lower-+.f64 93.1

          \[\leadsto \frac{0.5}{\left(\color{blue}{\left(a + b\right)} \cdot b\right) \cdot a} \cdot \pi \]

   7. Applied rewrites 93.1%

      \[\leadsto \frac{0.5}{\color{blue}{\left(\left(a + b\right) \cdot b\right) \cdot a}} \cdot \pi \]

   8. Taylor expanded in a around inf

      \[\leadsto \frac{\frac{1}{2}}{\color{blue}{{a}^{2} \cdot b}} \cdot \pi \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{2}}{{a}^{2} \cdot \color{blue}{b}} \cdot \pi \]

      2. unpow2 N/A

         \[\leadsto \frac{\frac{1}{2}}{\left(a \cdot a\right) \cdot b} \cdot \pi \]

      3. lower-*.f64 57.4

         \[\leadsto \frac{0.5}{\left(a \cdot a\right) \cdot b} \cdot \pi \]

   10. Applied rewrites 57.4%

       \[\leadsto \frac{0.5}{\color{blue}{\left(a \cdot a\right) \cdot b}} \cdot \pi \]

   if -1.99999999999999992e-23 < a

   1. Initial program 78.1%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)} \]

      2. lift--.f64 N/A

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \color{blue}{\left(\frac{1}{a} - \frac{1}{b}\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\color{blue}{\frac{1}{a}} - \frac{1}{b}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \color{blue}{\frac{1}{b}}\right) \]

      5. frac-sub N/A

         \[\leadsto \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \color{blue}{\frac{1 \cdot b - a \cdot 1}{a \cdot b}} \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(1 \cdot b - a \cdot 1\right)}{a \cdot b}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(1 \cdot b - a \cdot 1\right)}{a \cdot b}} \]

   3. Applied rewrites 87.3%

      \[\leadsto \color{blue}{\frac{\frac{0.5 \cdot \pi}{\left(a + b\right) \cdot \left(b - a\right)} \cdot \left(b - a\right)}{a \cdot b}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2} \cdot \pi}{\left(a + b\right) \cdot \left(b - a\right)} \cdot \left(b - a\right)}}{a \cdot b} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2} \cdot \pi}{\left(a + b\right) \cdot \left(b - a\right)}} \cdot \left(b - a\right)}{a \cdot b} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \left(b - a\right)}{\left(a + b\right) \cdot \left(b - a\right)}}}{a \cdot b} \]

      4. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{b - a}{\left(a + b\right) \cdot \left(b - a\right)}}}{a \cdot b} \]

      5. frac-2neg-rev N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(b - a\right)\right)}{\mathsf{neg}\left(\left(a + b\right) \cdot \left(b - a\right)\right)}}}{a \cdot b} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{\mathsf{neg}\left(\color{blue}{\left(b - a\right)}\right)}{\mathsf{neg}\left(\left(a + b\right) \cdot \left(b - a\right)\right)}}{a \cdot b} \]

      7. sub-negate-rev N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{\color{blue}{a - b}}{\mathsf{neg}\left(\left(a + b\right) \cdot \left(b - a\right)\right)}}{a \cdot b} \]

      8. lift--.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{\color{blue}{a - b}}{\mathsf{neg}\left(\left(a + b\right) \cdot \left(b - a\right)\right)}}{a \cdot b} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{a - b}{\mathsf{neg}\left(\color{blue}{\left(a + b\right) \cdot \left(b - a\right)}\right)}}{a \cdot b} \]

      10. distribute-rgt-neg-out N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{a - b}{\color{blue}{\left(a + b\right) \cdot \left(\mathsf{neg}\left(\left(b - a\right)\right)\right)}}}{a \cdot b} \]

      11. lift--.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{a - b}{\left(a + b\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\left(b - a\right)}\right)\right)}}{a \cdot b} \]

      12. sub-negate-rev N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{a - b}{\left(a + b\right) \cdot \color{blue}{\left(a - b\right)}}}{a \cdot b} \]

      13. lift--.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{a - b}{\left(a + b\right) \cdot \color{blue}{\left(a - b\right)}}}{a \cdot b} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{a - b}{\color{blue}{\left(a + b\right) \cdot \left(a - b\right)}}}{a \cdot b} \]

      15. div-flip-rev N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \color{blue}{\frac{1}{\frac{\left(a + b\right) \cdot \left(a - b\right)}{a - b}}}}{a \cdot b} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{1}{\frac{\color{blue}{\left(a + b\right) \cdot \left(a - b\right)}}{a - b}}}{a \cdot b} \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{1}{\frac{\color{blue}{\left(a + b\right)} \cdot \left(a - b\right)}{a - b}}}{a \cdot b} \]

      18. lift--.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{1}{\frac{\left(a + b\right) \cdot \color{blue}{\left(a - b\right)}}{a - b}}}{a \cdot b} \]

      19. difference-of-squares-rev N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{1}{\frac{\color{blue}{a \cdot a - b \cdot b}}{a - b}}}{a \cdot b} \]

      20. lift--.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{1}{\frac{a \cdot a - b \cdot b}{\color{blue}{a - b}}}}{a \cdot b} \]

      21. flip-+ N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{1}{\color{blue}{a + b}}}{a \cdot b} \]

      22. lift-+.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{1}{\color{blue}{a + b}}}{a \cdot b} \]

   5. Applied rewrites 99.7%

      \[\leadsto \frac{\color{blue}{\frac{0.5 \cdot \pi}{b + a}}}{a \cdot b} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{1}{2} \cdot \pi}{b + a}}{a \cdot b}} \]

      2. mult-flip N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \pi}{b + a} \cdot \frac{1}{a \cdot b}} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \pi}{b + a}} \cdot \frac{1}{a \cdot b} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \pi}{b + a} \cdot \frac{1}{\color{blue}{a \cdot b}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \pi}{b + a} \cdot \frac{1}{\color{blue}{b \cdot a}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \pi}{b + a} \cdot \frac{1}{\color{blue}{b \cdot a}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \pi}{b + a} \cdot \frac{1}{\color{blue}{b \cdot a}} \]

      8. associate-/r* N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \pi}{b + a} \cdot \color{blue}{\frac{\frac{1}{b}}{a}} \]

      9. inv-pow N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \pi}{b + a} \cdot \frac{\color{blue}{{b}^{-1}}}{a} \]

      10. frac-times N/A

          \[\leadsto \color{blue}{\frac{\left(\frac{1}{2} \cdot \pi\right) \cdot {b}^{-1}}{\left(b + a\right) \cdot a}} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot {b}^{-1}}{\color{blue}{\left(b + a\right)} \cdot a} \]

      12. +-commutative N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot {b}^{-1}}{\color{blue}{\left(a + b\right)} \cdot a} \]

      13. sum-to-mult N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot {b}^{-1}}{\color{blue}{\left(\left(1 + \frac{b}{a}\right) \cdot a\right)} \cdot a} \]

      14. *-inverses N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot {b}^{-1}}{\left(\left(\color{blue}{\frac{a}{a}} + \frac{b}{a}\right) \cdot a\right) \cdot a} \]

      15. div-add N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot {b}^{-1}}{\left(\color{blue}{\frac{a + b}{a}} \cdot a\right) \cdot a} \]

      16. +-commutative N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot {b}^{-1}}{\left(\frac{\color{blue}{b + a}}{a} \cdot a\right) \cdot a} \]

      17. lift-+.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot {b}^{-1}}{\left(\frac{\color{blue}{b + a}}{a} \cdot a\right) \cdot a} \]

      18. lift-/.f64 N/A

          \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot {b}^{-1}}{\left(\color{blue}{\frac{b + a}{a}} \cdot a\right) \cdot a} \]

      19. frac-times N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \pi}{\frac{b + a}{a} \cdot a} \cdot \frac{{b}^{-1}}{a}} \]

   7. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\pi}{a} \cdot 0.5}{b}}{a + b}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a \cdot {b}^{2}}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\mathsf{PI}\left(\right)}{a \cdot {b}^{2}} \cdot \color{blue}{\frac{1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{PI}\left(\right)}{a \cdot {b}^{2}} \cdot \color{blue}{\frac{1}{2}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{PI}\left(\right)}{a \cdot {b}^{2}} \cdot \frac{1}{2} \]

      4. lift-PI.f64 N/A

         \[\leadsto \frac{\pi}{a \cdot {b}^{2}} \cdot \frac{1}{2} \]

      5. *-commutative N/A

         \[\leadsto \frac{\pi}{{b}^{2} \cdot a} \cdot \frac{1}{2} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\pi}{{b}^{2} \cdot a} \cdot \frac{1}{2} \]

      7. unpow2 N/A

         \[\leadsto \frac{\pi}{\left(b \cdot b\right) \cdot a} \cdot \frac{1}{2} \]

      8. lower-*.f64 56.3

         \[\leadsto \frac{\pi}{\left(b \cdot b\right) \cdot a} \cdot 0.5 \]

   10. Applied rewrites 56.3%

       \[\leadsto \color{blue}{\frac{\pi}{\left(b \cdot b\right) \cdot a} \cdot 0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 56.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \frac{\pi}{\left(b \cdot b\right) \cdot a} \cdot 0.5 \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (* (/ PI (* (* b b) a)) 0.5))

C

assert(a < b);
double code(double a, double b) {
	return (((double) M_PI) / ((b * b) * a)) * 0.5;
}

Java

assert a < b;
public static double code(double a, double b) {
	return (Math.PI / ((b * b) * a)) * 0.5;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	return (math.pi / ((b * b) * a)) * 0.5

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(Float64(pi / Float64(Float64(b * b) * a)) * 0.5)
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = (pi / ((b * b) * a)) * 0.5;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[(N[(Pi / N[(N[(b * b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 78.1%

   \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)} \]

   2. lift--.f64 N/A

      \[\leadsto \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \color{blue}{\left(\frac{1}{a} - \frac{1}{b}\right)} \]

   3. lift-/.f64 N/A

      \[\leadsto \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\color{blue}{\frac{1}{a}} - \frac{1}{b}\right) \]

   4. lift-/.f64 N/A

      \[\leadsto \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \color{blue}{\frac{1}{b}}\right) \]

   5. frac-sub N/A

      \[\leadsto \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \color{blue}{\frac{1 \cdot b - a \cdot 1}{a \cdot b}} \]

   6. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(1 \cdot b - a \cdot 1\right)}{a \cdot b}} \]

   7. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(1 \cdot b - a \cdot 1\right)}{a \cdot b}} \]

3. Applied rewrites 87.3%

   \[\leadsto \color{blue}{\frac{\frac{0.5 \cdot \pi}{\left(a + b\right) \cdot \left(b - a\right)} \cdot \left(b - a\right)}{a \cdot b}} \]
4. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2} \cdot \pi}{\left(a + b\right) \cdot \left(b - a\right)} \cdot \left(b - a\right)}}{a \cdot b} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2} \cdot \pi}{\left(a + b\right) \cdot \left(b - a\right)}} \cdot \left(b - a\right)}{a \cdot b} \]

   3. associate-*l/ N/A

      \[\leadsto \frac{\color{blue}{\frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \left(b - a\right)}{\left(a + b\right) \cdot \left(b - a\right)}}}{a \cdot b} \]

   4. associate-/l* N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{b - a}{\left(a + b\right) \cdot \left(b - a\right)}}}{a \cdot b} \]

   5. frac-2neg-rev N/A

      \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(b - a\right)\right)}{\mathsf{neg}\left(\left(a + b\right) \cdot \left(b - a\right)\right)}}}{a \cdot b} \]

   6. lift--.f64 N/A

      \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{\mathsf{neg}\left(\color{blue}{\left(b - a\right)}\right)}{\mathsf{neg}\left(\left(a + b\right) \cdot \left(b - a\right)\right)}}{a \cdot b} \]

   7. sub-negate-rev N/A

      \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{\color{blue}{a - b}}{\mathsf{neg}\left(\left(a + b\right) \cdot \left(b - a\right)\right)}}{a \cdot b} \]

   8. lift--.f64 N/A

      \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{\color{blue}{a - b}}{\mathsf{neg}\left(\left(a + b\right) \cdot \left(b - a\right)\right)}}{a \cdot b} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{a - b}{\mathsf{neg}\left(\color{blue}{\left(a + b\right) \cdot \left(b - a\right)}\right)}}{a \cdot b} \]

   10. distribute-rgt-neg-out N/A

       \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{a - b}{\color{blue}{\left(a + b\right) \cdot \left(\mathsf{neg}\left(\left(b - a\right)\right)\right)}}}{a \cdot b} \]

   11. lift--.f64 N/A

       \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{a - b}{\left(a + b\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\left(b - a\right)}\right)\right)}}{a \cdot b} \]

   12. sub-negate-rev N/A

       \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{a - b}{\left(a + b\right) \cdot \color{blue}{\left(a - b\right)}}}{a \cdot b} \]

   13. lift--.f64 N/A

       \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{a - b}{\left(a + b\right) \cdot \color{blue}{\left(a - b\right)}}}{a \cdot b} \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{a - b}{\color{blue}{\left(a + b\right) \cdot \left(a - b\right)}}}{a \cdot b} \]

   15. div-flip-rev N/A

       \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \color{blue}{\frac{1}{\frac{\left(a + b\right) \cdot \left(a - b\right)}{a - b}}}}{a \cdot b} \]

   16. lift-*.f64 N/A

       \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{1}{\frac{\color{blue}{\left(a + b\right) \cdot \left(a - b\right)}}{a - b}}}{a \cdot b} \]

   17. lift-+.f64 N/A

       \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{1}{\frac{\color{blue}{\left(a + b\right)} \cdot \left(a - b\right)}{a - b}}}{a \cdot b} \]

   18. lift--.f64 N/A

       \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{1}{\frac{\left(a + b\right) \cdot \color{blue}{\left(a - b\right)}}{a - b}}}{a \cdot b} \]

   19. difference-of-squares-rev N/A

       \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{1}{\frac{\color{blue}{a \cdot a - b \cdot b}}{a - b}}}{a \cdot b} \]

   20. lift--.f64 N/A

       \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{1}{\frac{a \cdot a - b \cdot b}{\color{blue}{a - b}}}}{a \cdot b} \]

   21. flip-+ N/A

       \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{1}{\color{blue}{a + b}}}{a \cdot b} \]

   22. lift-+.f64 N/A

       \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{1}{\color{blue}{a + b}}}{a \cdot b} \]

5. Applied rewrites 99.7%

   \[\leadsto \frac{\color{blue}{\frac{0.5 \cdot \pi}{b + a}}}{a \cdot b} \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{2} \cdot \pi}{b + a}}{a \cdot b}} \]

   2. mult-flip N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \pi}{b + a} \cdot \frac{1}{a \cdot b}} \]

   3. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \pi}{b + a}} \cdot \frac{1}{a \cdot b} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \pi}{b + a} \cdot \frac{1}{\color{blue}{a \cdot b}} \]

   5. *-commutative N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \pi}{b + a} \cdot \frac{1}{\color{blue}{b \cdot a}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \pi}{b + a} \cdot \frac{1}{\color{blue}{b \cdot a}} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \pi}{b + a} \cdot \frac{1}{\color{blue}{b \cdot a}} \]

   8. associate-/r* N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \pi}{b + a} \cdot \color{blue}{\frac{\frac{1}{b}}{a}} \]

   9. inv-pow N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \pi}{b + a} \cdot \frac{\color{blue}{{b}^{-1}}}{a} \]

   10. frac-times N/A

       \[\leadsto \color{blue}{\frac{\left(\frac{1}{2} \cdot \pi\right) \cdot {b}^{-1}}{\left(b + a\right) \cdot a}} \]

   11. lift-+.f64 N/A

       \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot {b}^{-1}}{\color{blue}{\left(b + a\right)} \cdot a} \]

   12. +-commutative N/A

       \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot {b}^{-1}}{\color{blue}{\left(a + b\right)} \cdot a} \]

   13. sum-to-mult N/A

       \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot {b}^{-1}}{\color{blue}{\left(\left(1 + \frac{b}{a}\right) \cdot a\right)} \cdot a} \]

   14. *-inverses N/A

       \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot {b}^{-1}}{\left(\left(\color{blue}{\frac{a}{a}} + \frac{b}{a}\right) \cdot a\right) \cdot a} \]

   15. div-add N/A

       \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot {b}^{-1}}{\left(\color{blue}{\frac{a + b}{a}} \cdot a\right) \cdot a} \]

   16. +-commutative N/A

       \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot {b}^{-1}}{\left(\frac{\color{blue}{b + a}}{a} \cdot a\right) \cdot a} \]

   17. lift-+.f64 N/A

       \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot {b}^{-1}}{\left(\frac{\color{blue}{b + a}}{a} \cdot a\right) \cdot a} \]

   18. lift-/.f64 N/A

       \[\leadsto \frac{\left(\frac{1}{2} \cdot \pi\right) \cdot {b}^{-1}}{\left(\color{blue}{\frac{b + a}{a}} \cdot a\right) \cdot a} \]

   19. frac-times N/A

       \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \pi}{\frac{b + a}{a} \cdot a} \cdot \frac{{b}^{-1}}{a}} \]

7. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{\frac{\frac{\pi}{a} \cdot 0.5}{b}}{a + b}} \]

8. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a \cdot {b}^{2}}} \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{a \cdot {b}^{2}} \cdot \color{blue}{\frac{1}{2}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{a \cdot {b}^{2}} \cdot \color{blue}{\frac{1}{2}} \]

   3. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{a \cdot {b}^{2}} \cdot \frac{1}{2} \]

   4. lift-PI.f64 N/A

      \[\leadsto \frac{\pi}{a \cdot {b}^{2}} \cdot \frac{1}{2} \]

   5. *-commutative N/A

      \[\leadsto \frac{\pi}{{b}^{2} \cdot a} \cdot \frac{1}{2} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\pi}{{b}^{2} \cdot a} \cdot \frac{1}{2} \]

   7. unpow2 N/A

      \[\leadsto \frac{\pi}{\left(b \cdot b\right) \cdot a} \cdot \frac{1}{2} \]

   8. lower-*.f64 56.3

      \[\leadsto \frac{\pi}{\left(b \cdot b\right) \cdot a} \cdot 0.5 \]

10. Applied rewrites 56.3%

    \[\leadsto \color{blue}{\frac{\pi}{\left(b \cdot b\right) \cdot a} \cdot 0.5} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025140 
(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))))