NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.8% → 99.6%

Time: 3.5s

Alternatives: 9

Speedup: 1.6×

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: 78.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 1.5 \cdot 10^{+44}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{\left(a + b\right) \cdot b} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{b - a}}{a}}{b + b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 1.5e+44)
   (* (/ (/ PI a) (* (+ a b) b)) 0.5)
   (/ (/ (/ PI (- b a)) a) (+ b b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 1.5e+44) {
		tmp = ((((double) M_PI) / a) / ((a + b) * b)) * 0.5;
	} else {
		tmp = ((((double) M_PI) / (b - a)) / a) / (b + b);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 1.5e+44) {
		tmp = ((Math.PI / a) / ((a + b) * b)) * 0.5;
	} else {
		tmp = ((Math.PI / (b - a)) / a) / (b + b);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 1.5e+44:
		tmp = ((math.pi / a) / ((a + b) * b)) * 0.5
	else:
		tmp = ((math.pi / (b - a)) / a) / (b + b)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 1.5e+44)
		tmp = Float64(Float64(Float64(pi / a) / Float64(Float64(a + b) * b)) * 0.5);
	else
		tmp = Float64(Float64(Float64(pi / Float64(b - a)) / a) / Float64(b + b));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.5e+44)
		tmp = ((pi / a) / ((a + b) * b)) * 0.5;
	else
		tmp = ((pi / (b - a)) / a) / (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, 1.5e+44], N[(N[(N[(Pi / a), $MachinePrecision] / N[(N[(a + b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(Pi / N[(b - a), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / N[(b + b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.49999999999999993e44

   1. Initial program 78.8%

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

      1. 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

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 87.9%

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

   4. Taylor expanded in a around 0

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

   6. Applied rewrites 64.8%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 93.7%

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

   10. Taylor expanded in a around 0

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

       1. lower-/.f64 N/A

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

       2. lower-PI.f64 93.8

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

   12. Applied rewrites 93.8%

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

   if 1.49999999999999993e44 < b

   1. Initial program 78.8%

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

      1. 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. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip-rev N/A

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

      6. associate-*r/ N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. difference-of-squares N/A

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

      11. sub-negate-rev N/A

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

      12. *-lft-identity N/A

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

      13. *-rgt-identity N/A

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

      14. *-commutative N/A

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

      15. *-commutative N/A

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

      16. sub-negate-rev N/A

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

   3. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in a around 0

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

      1. lower-PI.f64 66.3

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

   8. Applied rewrites 66.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 66.7

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

   10. Applied rewrites 66.7%

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

Alternative 2: 99.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 1.08 \cdot 10^{+76}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{\left(a + b\right) \cdot b} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{\left(b + b\right) \cdot a}}{b - a}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 1.08e+76)
   (* (/ (/ PI a) (* (+ a b) b)) 0.5)
   (/ (/ PI (* (+ b b) a)) (- b a))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 1.08e+76) {
		tmp = ((((double) M_PI) / a) / ((a + b) * b)) * 0.5;
	} else {
		tmp = (((double) M_PI) / ((b + b) * a)) / (b - a);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 1.08e+76) {
		tmp = ((Math.PI / a) / ((a + b) * b)) * 0.5;
	} else {
		tmp = (Math.PI / ((b + b) * a)) / (b - a);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 1.08e+76:
		tmp = ((math.pi / a) / ((a + b) * b)) * 0.5
	else:
		tmp = (math.pi / ((b + b) * a)) / (b - a)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 1.08e+76)
		tmp = Float64(Float64(Float64(pi / a) / Float64(Float64(a + b) * b)) * 0.5);
	else
		tmp = Float64(Float64(pi / Float64(Float64(b + b) * a)) / Float64(b - a));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.08e+76)
		tmp = ((pi / a) / ((a + b) * b)) * 0.5;
	else
		tmp = (pi / ((b + b) * a)) / (b - a);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[b, 1.08e+76], N[(N[(N[(Pi / a), $MachinePrecision] / N[(N[(a + b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(Pi / N[(N[(b + b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.07999999999999999e76

   1. Initial program 78.8%

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

      1. 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

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 87.9%

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

   4. Taylor expanded in a around 0

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

   6. Applied rewrites 64.8%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 93.7%

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

   10. Taylor expanded in a around 0

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

       1. lower-/.f64 N/A

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

       2. lower-PI.f64 93.8

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

   12. Applied rewrites 93.8%

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

   if 1.07999999999999999e76 < b

   1. Initial program 78.8%

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

      1. 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. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip-rev N/A

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

      6. associate-*r/ N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. difference-of-squares N/A

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

      11. sub-negate-rev N/A

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

      12. *-lft-identity N/A

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

      13. *-rgt-identity N/A

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

      14. *-commutative N/A

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

      15. *-commutative N/A

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

      16. sub-negate-rev N/A

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

   3. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in a around 0

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

      1. lower-PI.f64 66.3

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

   8. Applied rewrites 66.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 66.6

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

   10. Applied rewrites 66.6%

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

Alternative 3: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.8%

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

   1. 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. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. mult-flip-rev N/A

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

   6. associate-*r/ N/A

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

   7. lift--.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. difference-of-squares N/A

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

   11. sub-negate-rev N/A

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

   12. *-lft-identity N/A

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

   13. *-rgt-identity N/A

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

   14. *-commutative N/A

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

   15. *-commutative N/A

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

   16. sub-negate-rev N/A

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

3. Applied rewrites 99.6%

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

Alternative 4: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 78.8%

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

   1. 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. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. mult-flip-rev N/A

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

   6. associate-*r/ N/A

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

   7. lift--.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. difference-of-squares N/A

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

   11. *-commutative N/A

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

   12. sub-negate-rev N/A

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

   13. *-lft-identity N/A

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

   14. *-rgt-identity N/A

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

   15. *-commutative N/A

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

   16. *-commutative N/A

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

   17. sub-negate-rev N/A

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

3. Applied rewrites 99.6%

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

Alternative 5: 99.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.8%

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

   1. 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. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. mult-flip-rev N/A

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

   6. associate-*r/ N/A

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

   7. lift--.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. difference-of-squares N/A

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

   11. sub-negate-rev N/A

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

   12. *-lft-identity N/A

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

   13. *-rgt-identity N/A

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

   14. *-commutative N/A

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

   15. *-commutative N/A

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

   16. sub-negate-rev N/A

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

3. Applied rewrites 99.6%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

5. Applied rewrites 99.0%

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

Alternative 6: 89.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -3.8 \cdot 10^{-165}:\\ \;\;\;\;\frac{\pi \cdot \frac{-0.5}{a \cdot b}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{\left(b + b\right) \cdot \left(\left(b - a\right) \cdot a\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -3.8e-165)
   (/ (* PI (/ -0.5 (* a b))) (- b a))
   (/ PI (* (+ b b) (* (- b a) a)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -3.8e-165) {
		tmp = (((double) M_PI) * (-0.5 / (a * b))) / (b - a);
	} else {
		tmp = ((double) M_PI) / ((b + b) * ((b - a) * a));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -3.8e-165) {
		tmp = (Math.PI * (-0.5 / (a * b))) / (b - a);
	} else {
		tmp = Math.PI / ((b + b) * ((b - a) * a));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -3.8e-165:
		tmp = (math.pi * (-0.5 / (a * b))) / (b - a)
	else:
		tmp = math.pi / ((b + b) * ((b - a) * a))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -3.8e-165)
		tmp = Float64(Float64(pi * Float64(-0.5 / Float64(a * b))) / Float64(b - a));
	else
		tmp = Float64(pi / Float64(Float64(b + b) * Float64(Float64(b - a) * a)));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -3.8e-165)
		tmp = (pi * (-0.5 / (a * b))) / (b - a);
	else
		tmp = pi / ((b + b) * ((b - a) * a));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -3.8e-165], N[(N[(Pi * N[(-0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision], N[(Pi / N[(N[(b + b), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.80000000000000018e-165

   1. Initial program 78.8%

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

      1. 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. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip-rev N/A

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

      6. associate-*r/ N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. difference-of-squares N/A

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

      11. sub-negate-rev N/A

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

      12. *-lft-identity N/A

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

      13. *-rgt-identity N/A

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

      14. *-commutative N/A

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

      15. *-commutative N/A

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

      16. sub-negate-rev N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-PI.f64 N/A

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

      4. lower-*.f64 67.1

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

   6. Applied rewrites 67.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 67.1

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

   8. Applied rewrites 67.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 67.1

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

   10. Applied rewrites 67.1%

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

   if -3.80000000000000018e-165 < a

   1. Initial program 78.8%

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

      1. 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. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip-rev N/A

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

      6. associate-*r/ N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. difference-of-squares N/A

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

      11. sub-negate-rev N/A

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

      12. *-lft-identity N/A

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

      13. *-rgt-identity N/A

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

      14. *-commutative N/A

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

      15. *-commutative N/A

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

      16. sub-negate-rev N/A

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

   3. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in a around 0

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

      1. lower-PI.f64 66.3

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

   8. Applied rewrites 66.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 66.4

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

   10. Applied rewrites 66.4%

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

Alternative 7: 89.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -3.8 \cdot 10^{-165}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{\pi}{a \cdot b}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{\left(b + b\right) \cdot \left(\left(b - a\right) \cdot a\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -3.8e-165)
   (/ (* -0.5 (/ PI (* a b))) (- b a))
   (/ PI (* (+ b b) (* (- b a) a)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -3.8e-165) {
		tmp = (-0.5 * (((double) M_PI) / (a * b))) / (b - a);
	} else {
		tmp = ((double) M_PI) / ((b + b) * ((b - a) * a));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -3.8e-165) {
		tmp = (-0.5 * (Math.PI / (a * b))) / (b - a);
	} else {
		tmp = Math.PI / ((b + b) * ((b - a) * a));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -3.8e-165:
		tmp = (-0.5 * (math.pi / (a * b))) / (b - a)
	else:
		tmp = math.pi / ((b + b) * ((b - a) * a))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -3.8e-165)
		tmp = Float64(Float64(-0.5 * Float64(pi / Float64(a * b))) / Float64(b - a));
	else
		tmp = Float64(pi / Float64(Float64(b + b) * Float64(Float64(b - a) * a)));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -3.8e-165)
		tmp = (-0.5 * (pi / (a * b))) / (b - a);
	else
		tmp = pi / ((b + b) * ((b - a) * a));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -3.8e-165], N[(N[(-0.5 * N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision], N[(Pi / N[(N[(b + b), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.80000000000000018e-165

   1. Initial program 78.8%

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

      1. 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. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip-rev N/A

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

      6. associate-*r/ N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. difference-of-squares N/A

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

      11. sub-negate-rev N/A

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

      12. *-lft-identity N/A

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

      13. *-rgt-identity N/A

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

      14. *-commutative N/A

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

      15. *-commutative N/A

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

      16. sub-negate-rev N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-PI.f64 N/A

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

      4. lower-*.f64 67.1

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

   6. Applied rewrites 67.1%

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

   if -3.80000000000000018e-165 < a

   1. Initial program 78.8%

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

      1. 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. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip-rev N/A

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

      6. associate-*r/ N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. difference-of-squares N/A

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

      11. sub-negate-rev N/A

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

      12. *-lft-identity N/A

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

      13. *-rgt-identity N/A

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

      14. *-commutative N/A

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

      15. *-commutative N/A

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

      16. sub-negate-rev N/A

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

   3. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in a around 0

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

      1. lower-PI.f64 66.3

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

   8. Applied rewrites 66.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 66.4

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

   10. Applied rewrites 66.4%

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

Alternative 8: 66.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.8%

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

   1. 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. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. mult-flip-rev N/A

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

   6. associate-*r/ N/A

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

   7. lift--.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. difference-of-squares N/A

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

   11. sub-negate-rev N/A

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

   12. *-lft-identity N/A

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

   13. *-rgt-identity N/A

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

   14. *-commutative N/A

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

   15. *-commutative N/A

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

   16. sub-negate-rev N/A

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

3. Applied rewrites 99.6%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

5. Applied rewrites 99.0%

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

6. Taylor expanded in a around 0

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

   1. lower-PI.f64 66.3

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

8. Applied rewrites 66.3%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 66.4

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

10. Applied rewrites 66.4%

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

Alternative 9: 60.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.8%

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

   1. 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. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. mult-flip-rev N/A

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

   6. associate-*r/ N/A

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

   7. lift--.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. difference-of-squares N/A

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

   11. sub-negate-rev N/A

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

   12. *-lft-identity N/A

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

   13. *-rgt-identity N/A

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

   14. *-commutative N/A

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

   15. *-commutative N/A

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

   16. sub-negate-rev N/A

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

3. Applied rewrites 99.6%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

5. Applied rewrites 99.0%

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

6. Taylor expanded in a around 0

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

   1. lower-PI.f64 66.3

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

8. Applied rewrites 66.3%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 60.7

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

10. Applied rewrites 60.7%

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

Reproduce

herbie shell --seed 2025156 
(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))))