NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.9% → 99.7%

Time: 10.8s

Alternatives: 5

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 78.9%

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

   1. un-div-inv 79.0%

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

   2. difference-of-squares 89.1%

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

   3. associate-/r* 89.8%

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

   4. div-inv 89.8%

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

   5. metadata-eval 89.8%

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

4. Applied egg-rr 89.8%

   \[\leadsto \color{blue}{\frac{\frac{\pi \cdot 0.5}{b + a}}{b - a}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
5. Step-by-step derivation

   1. associate-*l/ 99.7%

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

   2. associate-/l* 99.6%

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

6. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{\frac{\left(\pi \cdot \frac{0.5}{b + a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{b - a}} \]
7. Step-by-step derivation

   1. associate-/l* 99.6%

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

   2. associate-*r/ 99.6%

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

   3. *-commutative 99.6%

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

   4. +-commutative 99.6%

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

   5. sub-neg 99.6%

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

   6. distribute-neg-frac 99.6%

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

   7. metadata-eval 99.6%

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

8. Simplified 99.6%

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

   1. associate-*l/ 99.7%

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

10. Applied egg-rr 99.7%

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

11. Taylor expanded in a around 0 99.7%

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

    1. associate-/r* 99.8%

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

13. Simplified 99.8%

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

Alternative 2: 70.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -3.05 \cdot 10^{+103}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{\left(a \cdot b\right) \cdot \left(b - a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a \cdot b} \cdot \frac{\pi}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -3.05e+103)
   (/ (* 0.5 PI) (* (* a b) (- b a)))
   (* (/ 0.5 (* a b)) (/ PI b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -3.05e+103) {
		tmp = (0.5 * ((double) M_PI)) / ((a * b) * (b - a));
	} else {
		tmp = (0.5 / (a * b)) * (((double) M_PI) / b);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -3.05e+103) {
		tmp = (0.5 * Math.PI) / ((a * b) * (b - a));
	} else {
		tmp = (0.5 / (a * b)) * (Math.PI / b);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -3.05e+103:
		tmp = (0.5 * math.pi) / ((a * b) * (b - a))
	else:
		tmp = (0.5 / (a * b)) * (math.pi / b)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -3.05e+103)
		tmp = Float64(Float64(0.5 * pi) / Float64(Float64(a * b) * Float64(b - a)));
	else
		tmp = Float64(Float64(0.5 / Float64(a * b)) * Float64(pi / b));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -3.05e+103)
		tmp = (0.5 * pi) / ((a * b) * (b - a));
	else
		tmp = (0.5 / (a * b)) * (pi / b);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -3.05e+103], N[(N[(0.5 * Pi), $MachinePrecision] / N[(N[(a * b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(Pi / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.0500000000000001e103

   1. Initial program 62.1%

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

      1. *-commutative 62.1%

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

      2. associate-*r* 62.1%

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

      3. associate-*r/ 62.1%

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

      4. associate-*r* 62.1%

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

      5. *-rgt-identity 62.1%

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

      6. sub-neg 62.1%

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

      7. distribute-neg-frac 62.1%

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

      8. metadata-eval 62.1%

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

   3. Simplified 62.1%

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

      1. difference-of-squares 80.3%

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

      2. times-frac 99.8%

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

      3. add-sqr-sqrt 47.5%

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

      4. sqrt-unprod 57.7%

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

      5. frac-times 57.7%

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

      6. metadata-eval 57.7%

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

      7. metadata-eval 57.7%

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

      8. frac-times 57.7%

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

      9. sqrt-unprod 32.5%

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

      10. add-sqr-sqrt 62.5%

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

      11. div-inv 62.5%

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

      12. metadata-eval 62.5%

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

   6. Applied egg-rr 62.5%

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

      1. +-commutative 62.5%

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

      2. +-commutative 62.5%

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

      3. *-commutative 62.5%

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

   8. Simplified 62.5%

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

   9. Taylor expanded in b around 0 62.5%

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

       1. frac-times 62.6%

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

       2. *-un-lft-identity 62.6%

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

   11. Applied egg-rr 62.6%

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

   if -3.0500000000000001e103 < a

   1. Initial program 82.4%

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

      1. *-commutative 82.4%

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

      2. associate-*r* 82.4%

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

      3. associate-*r/ 82.5%

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

      4. associate-*r* 82.5%

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

      5. *-rgt-identity 82.5%

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

      6. sub-neg 82.5%

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

      7. distribute-neg-frac 82.5%

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

      8. metadata-eval 82.5%

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

   3. Simplified 82.5%

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

      1. difference-of-squares 91.0%

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

      2. times-frac 99.6%

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

      3. add-sqr-sqrt 50.2%

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

      4. sqrt-unprod 81.1%

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

      5. frac-times 81.0%

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

      6. metadata-eval 81.0%

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

      7. metadata-eval 81.0%

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

      8. frac-times 81.1%

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

      9. sqrt-unprod 34.5%

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

      10. add-sqr-sqrt 71.5%

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

      11. div-inv 71.5%

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

      12. metadata-eval 71.5%

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

   6. Applied egg-rr 71.5%

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

      1. +-commutative 71.5%

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

      2. +-commutative 71.5%

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

      3. *-commutative 71.5%

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

   8. Simplified 71.5%

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

   9. Taylor expanded in b around 0 71.5%

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

       1. frac-times 70.9%

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

       2. *-un-lft-identity 70.9%

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

   11. Applied egg-rr 70.9%

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

       1. times-frac 71.5%

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

   13. Simplified 71.5%

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

   14. Taylor expanded in b around inf 72.8%

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

Alternative 3: 70.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -3.75 \cdot 10^{+99}:\\ \;\;\;\;\frac{\frac{\pi}{a} \cdot -0.5}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a \cdot b} \cdot \frac{\pi}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -3.75e+99)
   (/ (* (/ PI a) -0.5) (* a b))
   (* (/ 0.5 (* a b)) (/ PI b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -3.75e+99) {
		tmp = ((((double) M_PI) / a) * -0.5) / (a * b);
	} else {
		tmp = (0.5 / (a * b)) * (((double) M_PI) / b);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -3.75e+99) {
		tmp = ((Math.PI / a) * -0.5) / (a * b);
	} else {
		tmp = (0.5 / (a * b)) * (Math.PI / b);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -3.75e+99:
		tmp = ((math.pi / a) * -0.5) / (a * b)
	else:
		tmp = (0.5 / (a * b)) * (math.pi / b)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -3.75e+99)
		tmp = Float64(Float64(Float64(pi / a) * -0.5) / Float64(a * b));
	else
		tmp = Float64(Float64(0.5 / Float64(a * b)) * Float64(pi / b));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -3.75e+99)
		tmp = ((pi / a) * -0.5) / (a * b);
	else
		tmp = (0.5 / (a * b)) * (pi / b);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -3.75e+99], N[(N[(N[(Pi / a), $MachinePrecision] * -0.5), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(Pi / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.74999999999999982e99

   1. Initial program 62.1%

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

      1. *-commutative 62.1%

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

      2. associate-*r* 62.1%

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

      3. associate-*r/ 62.1%

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

      4. associate-*r* 62.1%

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

      5. *-rgt-identity 62.1%

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

      6. sub-neg 62.1%

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

      7. distribute-neg-frac 62.1%

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

      8. metadata-eval 62.1%

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

   3. Simplified 62.1%

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

      1. difference-of-squares 80.3%

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

      2. times-frac 99.8%

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

      3. add-sqr-sqrt 47.5%

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

      4. sqrt-unprod 57.7%

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

      5. frac-times 57.7%

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

      6. metadata-eval 57.7%

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

      7. metadata-eval 57.7%

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

      8. frac-times 57.7%

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

      9. sqrt-unprod 32.5%

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

      10. add-sqr-sqrt 62.5%

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

      11. div-inv 62.5%

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

      12. metadata-eval 62.5%

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

   6. Applied egg-rr 62.5%

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

      1. +-commutative 62.5%

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

      2. +-commutative 62.5%

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

      3. *-commutative 62.5%

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

   8. Simplified 62.5%

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

   9. Taylor expanded in b around 0 62.5%

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

       1. associate-*l/ 62.5%

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

       2. *-un-lft-identity 62.5%

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

   11. Applied egg-rr 62.5%

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

   12. Taylor expanded in b around 0 62.5%

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

   if -3.74999999999999982e99 < a

   1. Initial program 82.4%

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

      1. *-commutative 82.4%

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

      2. associate-*r* 82.4%

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

      3. associate-*r/ 82.5%

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

      4. associate-*r* 82.5%

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

      5. *-rgt-identity 82.5%

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

      6. sub-neg 82.5%

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

      7. distribute-neg-frac 82.5%

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

      8. metadata-eval 82.5%

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

   3. Simplified 82.5%

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

      1. difference-of-squares 91.0%

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

      2. times-frac 99.6%

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

      3. add-sqr-sqrt 50.2%

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

      4. sqrt-unprod 81.1%

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

      5. frac-times 81.0%

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

      6. metadata-eval 81.0%

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

      7. metadata-eval 81.0%

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

      8. frac-times 81.1%

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

      9. sqrt-unprod 34.5%

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

      10. add-sqr-sqrt 71.5%

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

      11. div-inv 71.5%

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

      12. metadata-eval 71.5%

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

   6. Applied egg-rr 71.5%

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

      1. +-commutative 71.5%

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

      2. +-commutative 71.5%

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

      3. *-commutative 71.5%

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

   8. Simplified 71.5%

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

   9. Taylor expanded in b around 0 71.5%

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

       1. frac-times 70.9%

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

       2. *-un-lft-identity 70.9%

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

   11. Applied egg-rr 70.9%

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

       1. times-frac 71.5%

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

   13. Simplified 71.5%

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

   14. Taylor expanded in b around inf 72.8%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.75 \cdot 10^{+99}:\\ \;\;\;\;\frac{\frac{\pi}{a} \cdot -0.5}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a \cdot b} \cdot \frac{\pi}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 78.9%

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

   1. un-div-inv 79.0%

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

   2. difference-of-squares 89.1%

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

   3. associate-/r* 89.8%

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

   4. div-inv 89.8%

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

   5. metadata-eval 89.8%

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

4. Applied egg-rr 89.8%

   \[\leadsto \color{blue}{\frac{\frac{\pi \cdot 0.5}{b + a}}{b - a}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
5. Step-by-step derivation

   1. associate-*l/ 99.7%

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

   2. associate-/l* 99.6%

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

6. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{\frac{\left(\pi \cdot \frac{0.5}{b + a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{b - a}} \]
7. Step-by-step derivation

   1. associate-/l* 99.6%

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

   2. associate-*r/ 99.6%

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

   3. *-commutative 99.6%

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

   4. +-commutative 99.6%

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

   5. sub-neg 99.6%

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

   6. distribute-neg-frac 99.6%

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

   7. metadata-eval 99.6%

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

8. Simplified 99.6%

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

   1. associate-*l/ 99.7%

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

10. Applied egg-rr 99.7%

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

11. Taylor expanded in a around 0 99.7%

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

Alternative 5: 62.3% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

   1. *-commutative 78.9%

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

   2. associate-*r* 78.9%

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

   3. associate-*r/ 79.0%

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

   4. associate-*r* 79.0%

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

   5. *-rgt-identity 79.0%

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

   6. sub-neg 79.0%

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

   7. distribute-neg-frac 79.0%

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

   8. metadata-eval 79.0%

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

3. Simplified 79.0%

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

   1. difference-of-squares 89.1%

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

   2. times-frac 99.6%

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

   3. add-sqr-sqrt 49.8%

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

   4. sqrt-unprod 77.0%

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

   5. frac-times 77.0%

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

   6. metadata-eval 77.0%

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

   7. metadata-eval 77.0%

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

   8. frac-times 77.0%

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

   9. sqrt-unprod 34.2%

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

   10. add-sqr-sqrt 70.0%

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

   11. div-inv 70.0%

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

   12. metadata-eval 70.0%

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

6. Applied egg-rr 70.0%

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

   1. +-commutative 70.0%

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

   2. +-commutative 70.0%

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

   3. *-commutative 70.0%

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

8. Simplified 70.0%

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

9. Taylor expanded in b around 0 70.0%

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

    1. frac-times 69.5%

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

    2. *-un-lft-identity 69.5%

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

11. Applied egg-rr 69.5%

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

    1. times-frac 70.0%

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

13. Simplified 70.0%

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

14. Taylor expanded in b around inf 67.6%

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

Reproduce

herbie shell --seed 2024160 
(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))))