NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.5% → 99.6%

Time: 8.7s

Alternatives: 9

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: 78.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.1%

   \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
2. 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. associate-*r/ N/A

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

   7. lift--.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. difference-of-squares N/A

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

   11. *-commutative N/A

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

   12. *-rgt-identity N/A

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

   13. *-lft-identity N/A

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

   14. times-frac N/A

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

   15. lower-*.f64 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in b around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.7

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

7. Applied rewrites 99.7%

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

9. Applied rewrites 99.7%

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

10. Final simplification 99.7%

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

Alternative 2: 99.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.1%

   \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
2. 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. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

4. Applied rewrites 81.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

6. Applied rewrites 99.3%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/r* N/A

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

   8. lift-/.f64 N/A

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

   9. lower-/.f64 99.7

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

   10. lift-/.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. metadata-eval N/A

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

   13. div-inv N/A

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

   14. associate-/l/ N/A

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

   15. *-commutative N/A

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

   16. lift-*.f64 N/A

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

   17. lower-/.f64 99.7

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

   18. lift-*.f64 N/A

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

   19. *-commutative N/A

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

   20. lower-*.f64 99.7

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

   21. lift-*.f64 N/A

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

   22. *-commutative N/A

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

   23. lower-*.f64 99.7

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

   24. lift-+.f64 N/A

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

8. Applied rewrites 99.7%

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

9. Final simplification 99.7%

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

Alternative 3: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.1%

   \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
2. 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. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

4. Applied rewrites 81.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

6. Applied rewrites 99.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 99.6

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 99.6

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 99.6

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

8. Applied rewrites 99.6%

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

9. Final simplification 99.6%

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

Alternative 4: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.1%

   \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
2. 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. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

4. Applied rewrites 81.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

6. Applied rewrites 99.3%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 99.6

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 99.6

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

8. Applied rewrites 99.6%

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

9. Final simplification 99.6%

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

Alternative 5: 84.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -6.7 \cdot 10^{-74}:\\ \;\;\;\;\frac{0.5}{\left(a \cdot b\right) \cdot a} \cdot \pi\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -6.7e-74)
   (* (/ 0.5 (* (* a b) a)) PI)
   (* (/ PI (* (* b b) a)) 0.5)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -6.7e-74) {
		tmp = (0.5 / ((a * b) * a)) * ((double) M_PI);
	} else {
		tmp = (((double) M_PI) / ((b * b) * a)) * 0.5;
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -6.7e-74) {
		tmp = (0.5 / ((a * b) * a)) * Math.PI;
	} else {
		tmp = (Math.PI / ((b * b) * a)) * 0.5;
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -6.7e-74:
		tmp = (0.5 / ((a * b) * a)) * math.pi
	else:
		tmp = (math.pi / ((b * b) * a)) * 0.5
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -6.7e-74)
		tmp = Float64(Float64(0.5 / Float64(Float64(a * b) * a)) * pi);
	else
		tmp = Float64(Float64(pi / Float64(Float64(b * b) * a)) * 0.5);
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -6.7e-74)
		tmp = (0.5 / ((a * b) * a)) * pi;
	else
		tmp = (pi / ((b * b) * a)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -6.7e-74], N[(N[(0.5 / N[(N[(a * b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision], N[(N[(Pi / N[(N[(b * b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.6999999999999996e-74

   1. Initial program 84.6%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 76.5

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

   5. Applied rewrites 76.5%

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

   7. Applied rewrites 76.6%

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

   9. Applied rewrites 82.2%

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

   if -6.6999999999999996e-74 < a

   1. Initial program 75.8%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 59.0

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

   5. Applied rewrites 59.0%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.7 \cdot 10^{-74}:\\ \;\;\;\;\frac{0.5}{\left(a \cdot b\right) \cdot a} \cdot \pi\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.1% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 78.1%

   \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
2. 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. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

4. Applied rewrites 81.1%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. clear-num N/A

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

   8. un-div-inv N/A

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

   9. lower-/.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. associate-/l* N/A

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

   13. lift-*.f64 N/A

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lift--.f64 N/A

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

   17. difference-of-squares N/A

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

   18. lift--.f64 N/A

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

   19. flip-+ N/A

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

   20. +-commutative N/A

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

   21. lift-+.f64 N/A

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

6. Applied rewrites 99.3%

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

Alternative 7: 99.0% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.1%

   \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
2. 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. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

4. Applied rewrites 81.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

6. Applied rewrites 99.3%

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

Alternative 8: 62.2% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.1%

   \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
2. 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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 59.0

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

5. Applied rewrites 59.0%

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

7. Applied rewrites 59.0%

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

9. Applied rewrites 63.1%

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

10. Final simplification 63.1%

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

Alternative 9: 56.3% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.1%

   \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
2. 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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 59.0

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

5. Applied rewrites 59.0%

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

7. Applied rewrites 59.0%

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

8. Final simplification 59.0%

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

Reproduce

herbie shell --seed 2024235 
(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))))