NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.8% → 99.6%

Time: 10.6s

Alternatives: 8

Speedup: 2.4×

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, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lift--.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. frac-sub N/A

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

   9. frac-times N/A

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

   10. lower-/.f64 N/A

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

4. Applied rewrites 86.1%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r/ N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-+.f64 N/A

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

   7. lift--.f64 N/A

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

   8. difference-of-squares N/A

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

   9. lift--.f64 N/A

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

   10. flip-+ N/A

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

   11. lift-+.f64 N/A

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

   12. clear-num N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 99.7

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

6. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. clear-num N/A

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

   6. associate-*l/ N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-/r* N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f64 99.7

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

8. Applied rewrites 99.7%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. times-frac N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 99.7

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

10. Applied rewrites 99.7%

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

Alternative 2: 98.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1.26 \cdot 10^{+79}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(a \cdot \left(b + a\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -1.26e+79)
   (/ (* PI 0.5) (* a (* b a)))
   (* PI (/ 0.5 (* b (* a (+ b a)))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -1.26e+79) {
		tmp = (((double) M_PI) * 0.5) / (a * (b * a));
	} else {
		tmp = ((double) M_PI) * (0.5 / (b * (a * (b + a))));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -1.26e+79) {
		tmp = (Math.PI * 0.5) / (a * (b * a));
	} else {
		tmp = Math.PI * (0.5 / (b * (a * (b + a))));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -1.26e+79:
		tmp = (math.pi * 0.5) / (a * (b * a))
	else:
		tmp = math.pi * (0.5 / (b * (a * (b + a))))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -1.26e+79)
		tmp = Float64(Float64(pi * 0.5) / Float64(a * Float64(b * a)));
	else
		tmp = Float64(pi * Float64(0.5 / Float64(b * Float64(a * Float64(b + a)))));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.26e+79)
		tmp = (pi * 0.5) / (a * (b * a));
	else
		tmp = pi * (0.5 / (b * (a * (b + a))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -1.26e+79], N[(N[(Pi * 0.5), $MachinePrecision] / N[(a * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(0.5 / N[(b * N[(a * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.26e79

   1. Initial program 64.0%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 99.4

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

   5. Applied rewrites 99.4%

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

   if -1.26e79 < a

   1. Initial program 77.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. frac-sub N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 86.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. difference-of-squares N/A

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

      9. lift--.f64 N/A

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

      10. flip-+ N/A

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

      11. lift-+.f64 N/A

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

      12. clear-num N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 99.7

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

   6. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 99.6

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

   8. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/l/ N/A

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

      8. *-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. distribute-rgt-out N/A

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

      15. lower-*.f64 N/A

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

      16. distribute-rgt-out N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 N/A

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

      19. lower-*.f64 95.5

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

   10. Applied rewrites 95.5%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.26 \cdot 10^{+79}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(a \cdot \left(b + a\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lift--.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. frac-sub N/A

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

   9. frac-times N/A

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

   10. lower-/.f64 N/A

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

4. Applied rewrites 86.1%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r/ N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-+.f64 N/A

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

   7. lift--.f64 N/A

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

   8. difference-of-squares N/A

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

   9. lift--.f64 N/A

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

   10. flip-+ N/A

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

   11. lift-+.f64 N/A

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

   12. clear-num N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 99.7

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

6. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. clear-num N/A

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

   6. associate-*l/ N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-/r* N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f64 99.7

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

8. Applied rewrites 99.7%

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

Alternative 4: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 74.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. un-div-inv N/A

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

   6. lift-/.f64 N/A

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

   7. div-inv N/A

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

   8. lift--.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. difference-of-squares N/A

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

   12. times-frac N/A

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

   13. associate-*r* N/A

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

   14. lower-*.f64 N/A

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

4. Applied rewrites 99.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. frac-times N/A

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

   6. lift-/.f64 N/A

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

   7. frac-times N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. times-frac N/A

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

6. Applied rewrites 99.7%

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

7. Final simplification 99.7%

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

Alternative 5: 83.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 3.5 \cdot 10^{-97}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 3.5e-97) (/ (* PI 0.5) (* a (* b a))) (/ (* PI 0.5) (* a (* b b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 3.5e-97) {
		tmp = (((double) M_PI) * 0.5) / (a * (b * a));
	} else {
		tmp = (((double) M_PI) * 0.5) / (a * (b * b));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 3.5e-97) {
		tmp = (Math.PI * 0.5) / (a * (b * a));
	} else {
		tmp = (Math.PI * 0.5) / (a * (b * b));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 3.5e-97:
		tmp = (math.pi * 0.5) / (a * (b * a))
	else:
		tmp = (math.pi * 0.5) / (a * (b * b))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 3.5e-97)
		tmp = Float64(Float64(pi * 0.5) / Float64(a * Float64(b * a)));
	else
		tmp = Float64(Float64(pi * 0.5) / Float64(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 <= 3.5e-97)
		tmp = (pi * 0.5) / (a * (b * a));
	else
		tmp = (pi * 0.5) / (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, 3.5e-97], N[(N[(Pi * 0.5), $MachinePrecision] / N[(a * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * 0.5), $MachinePrecision] / N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.50000000000000019e-97

   1. Initial program 75.1%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 75.9

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

   5. Applied rewrites 75.9%

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

   if 3.50000000000000019e-97 < b

   1. Initial program 74.1%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 65.0

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

   5. Applied rewrites 65.0%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.5 \cdot 10^{-97}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.0% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = pi / ((b + a) * ((b * a) * 2.0));
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[(Pi / N[(N[(b + a), $MachinePrecision] * N[(N[(b * a), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lift--.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. frac-sub N/A

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

   9. frac-times N/A

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

   10. lower-/.f64 N/A

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

4. Applied rewrites 86.1%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r/ N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-+.f64 N/A

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

   7. lift--.f64 N/A

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

   8. difference-of-squares N/A

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

   9. lift--.f64 N/A

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

   10. flip-+ N/A

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

   11. lift-+.f64 N/A

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

   12. clear-num N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 99.7

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

6. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. clear-num N/A

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

   6. frac-times N/A

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

   7. *-rgt-identity N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 99.2

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

8. Applied rewrites 99.2%

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

9. Final simplification 99.2%

   \[\leadsto \frac{\pi}{\left(b + a\right) \cdot \left(\left(b \cdot a\right) \cdot 2\right)} \]
10. Add Preprocessing

Alternative 7: 63.2% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.8%

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

3. Taylor expanded in b around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 72.6

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

5. Applied rewrites 72.6%

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

6. Final simplification 72.6%

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

Alternative 8: 63.2% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.8%

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

3. Taylor expanded in b around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 72.6

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

5. Applied rewrites 72.6%

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

7. Applied rewrites 72.5%

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

Reproduce

herbie shell --seed 2024221 
(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))))