NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.1% → 99.7%

Time: 9.5s

Alternatives: 8

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.1× speedup

Math

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

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (/ (/ (/ PI 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(Float64(pi / a) / b) / Float64(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[(N[(Pi / a), $MachinePrecision] / b), $MachinePrecision] / N[(2.0 * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.0%

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

   1. associate-*r/ 79.0%

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

   2. *-rgt-identity 79.0%

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

   3. associate-*l/ 79.0%

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

   4. difference-of-squares 85.2%

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

   5. times-frac 99.7%

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

   6. associate-/l/ 99.7%

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

   7. sub-neg 99.7%

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

   8. distribute-neg-frac 99.7%

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

   9. metadata-eval 99.7%

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

3. Simplified 99.7%

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

   1. associate-*l/ 99.6%

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

   2. add-sqr-sqrt 49.0%

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

   3. sqrt-unprod 76.4%

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

   4. frac-times 76.4%

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

   5. metadata-eval 76.4%

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

   6. metadata-eval 76.4%

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

   7. frac-times 76.4%

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

   8. sqrt-unprod 34.8%

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

   9. add-sqr-sqrt 66.0%

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

   10. *-commutative 66.0%

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

5. Applied egg-rr 66.0%

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

6. Taylor expanded in a around 0 99.7%

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

   1. associate-/r* 99.7%

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

8. Simplified 99.7%

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

9. Final simplification 99.7%

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

Alternative 2: 90.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -8.5 \cdot 10^{-47}:\\ \;\;\;\;\frac{0.5}{\left(a \cdot b\right) \cdot \frac{a}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b \cdot \left(a \cdot \frac{b}{\pi}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -8.5e-47)
   (/ 0.5 (* (* a b) (/ a PI)))
   (/ 0.5 (* b (* a (/ b PI))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -8.5e-47) {
		tmp = 0.5 / ((a * b) * (a / ((double) M_PI)));
	} else {
		tmp = 0.5 / (b * (a * (b / ((double) M_PI))));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -8.5e-47) {
		tmp = 0.5 / ((a * b) * (a / Math.PI));
	} else {
		tmp = 0.5 / (b * (a * (b / Math.PI)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -8.5e-47:
		tmp = 0.5 / ((a * b) * (a / math.pi))
	else:
		tmp = 0.5 / (b * (a * (b / math.pi)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -8.5e-47)
		tmp = Float64(0.5 / Float64(Float64(a * b) * Float64(a / pi)));
	else
		tmp = Float64(0.5 / Float64(b * Float64(a * Float64(b / pi))));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -8.5e-47)
		tmp = 0.5 / ((a * b) * (a / pi));
	else
		tmp = 0.5 / (b * (a * (b / pi)));
	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, -8.5e-47], N[(0.5 / N[(N[(a * b), $MachinePrecision] * N[(a / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(b * N[(a * N[(b / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -8.4999999999999999e-47

   1. Initial program 82.0%

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

      1. associate-*r/ 82.1%

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

      2. *-rgt-identity 82.1%

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

      3. associate-*l/ 82.0%

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

      4. difference-of-squares 89.4%

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

      5. times-frac 99.7%

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

      6. associate-/l/ 99.7%

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

      7. sub-neg 99.7%

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

      8. distribute-neg-frac 99.7%

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

      9. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in a around 0 99.6%

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

      1. associate-/r* 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in b around 0 84.7%

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

      1. associate-*r/ 84.7%

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

   9. Simplified 84.7%

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

       1. associate-/l* 84.6%

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

       2. associate-/r* 84.6%

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

       3. frac-times 82.7%

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

       4. metadata-eval 82.7%

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

   11. Applied egg-rr 82.7%

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

   if -8.4999999999999999e-47 < a

   1. Initial program 77.9%

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

      1. associate-*r/ 77.9%

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

      2. *-rgt-identity 77.9%

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

      3. associate-*l/ 77.9%

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

      4. difference-of-squares 83.7%

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

      5. times-frac 99.6%

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

      6. associate-/l/ 99.6%

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

      7. sub-neg 99.6%

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

      8. distribute-neg-frac 99.6%

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

      9. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a around 0 99.7%

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

      1. associate-/r* 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in b around inf 68.9%

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

      1. *-commutative 68.9%

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

      2. associate-/l/ 68.9%

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

      3. *-commutative 68.9%

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

      4. associate-*r/ 68.9%

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

      5. *-commutative 68.9%

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

      6. frac-times 68.7%

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

      7. *-un-lft-identity 68.7%

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

      8. times-frac 68.9%

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

      9. clear-num 68.8%

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

      10. frac-times 68.7%

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

      11. metadata-eval 68.7%

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

      12. *-un-lft-identity 68.7%

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

      13. times-frac 68.7%

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

      14. /-rgt-identity 68.7%

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

   9. Applied egg-rr 68.7%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -8.5 \cdot 10^{-47}:\\ \;\;\;\;\frac{0.5}{\left(a \cdot b\right) \cdot \frac{a}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b \cdot \left(a \cdot \frac{b}{\pi}\right)}\\ \end{array} \]

Alternative 3: 90.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-46}:\\ \;\;\;\;\frac{0.5}{\left(a \cdot b\right) \cdot \frac{a}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{a}}{b} \cdot 0.5}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -2e-46) (/ 0.5 (* (* a b) (/ a PI))) (/ (* (/ (/ PI a) b) 0.5) b)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -2e-46) {
		tmp = 0.5 / ((a * b) * (a / ((double) M_PI)));
	} else {
		tmp = (((((double) M_PI) / a) / b) * 0.5) / b;
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -2e-46) {
		tmp = 0.5 / ((a * b) * (a / Math.PI));
	} else {
		tmp = (((Math.PI / a) / b) * 0.5) / b;
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -2e-46:
		tmp = 0.5 / ((a * b) * (a / math.pi))
	else:
		tmp = (((math.pi / a) / b) * 0.5) / b
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -2e-46)
		tmp = Float64(0.5 / Float64(Float64(a * b) * Float64(a / pi)));
	else
		tmp = Float64(Float64(Float64(Float64(pi / a) / b) * 0.5) / b);
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2e-46)
		tmp = 0.5 / ((a * b) * (a / pi));
	else
		tmp = (((pi / a) / b) * 0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -2e-46], N[(0.5 / N[(N[(a * b), $MachinePrecision] * N[(a / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(Pi / a), $MachinePrecision] / b), $MachinePrecision] * 0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.00000000000000005e-46

   1. Initial program 82.0%

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

      1. associate-*r/ 82.1%

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

      2. *-rgt-identity 82.1%

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

      3. associate-*l/ 82.0%

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

      4. difference-of-squares 89.4%

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

      5. times-frac 99.7%

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

      6. associate-/l/ 99.7%

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

      7. sub-neg 99.7%

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

      8. distribute-neg-frac 99.7%

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

      9. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in a around 0 99.6%

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

      1. associate-/r* 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in b around 0 84.7%

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

      1. associate-*r/ 84.7%

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

   9. Simplified 84.7%

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

       1. associate-/l* 84.6%

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

       2. associate-/r* 84.6%

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

       3. frac-times 82.7%

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

       4. metadata-eval 82.7%

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

   11. Applied egg-rr 82.7%

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

   if -2.00000000000000005e-46 < a

   1. Initial program 77.9%

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

      1. associate-*r/ 77.9%

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

      2. *-rgt-identity 77.9%

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

      3. associate-*l/ 77.9%

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

      4. difference-of-squares 83.7%

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

      5. times-frac 99.6%

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

      6. associate-/l/ 99.6%

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

      7. sub-neg 99.6%

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

      8. distribute-neg-frac 99.6%

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

      9. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a around 0 99.7%

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

      1. associate-/r* 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in b around inf 68.9%

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

      1. *-commutative 68.9%

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

      2. associate-/l/ 68.9%

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

      3. *-commutative 68.9%

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

      4. associate-*r/ 68.9%

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

      5. *-commutative 68.9%

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

      6. frac-times 68.7%

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

      7. *-un-lft-identity 68.7%

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

      8. associate-/r* 68.9%

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

      9. *-commutative 68.9%

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

      10. *-un-lft-identity 68.9%

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

      11. times-frac 68.9%

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

      12. metadata-eval 68.9%

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

      13. associate-/r* 68.9%

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

   9. Applied egg-rr 68.9%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-46}:\\ \;\;\;\;\frac{0.5}{\left(a \cdot b\right) \cdot \frac{a}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{a}}{b} \cdot 0.5}{b}\\ \end{array} \]

Alternative 4: 90.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-46}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{2 \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{a}}{b} \cdot 0.5}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -2e-46) (/ (/ PI a) (* 2.0 (* a b))) (/ (* (/ (/ PI a) b) 0.5) b)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -2e-46) {
		tmp = (((double) M_PI) / a) / (2.0 * (a * b));
	} else {
		tmp = (((((double) M_PI) / a) / b) * 0.5) / b;
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -2e-46) {
		tmp = (Math.PI / a) / (2.0 * (a * b));
	} else {
		tmp = (((Math.PI / a) / b) * 0.5) / b;
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -2e-46:
		tmp = (math.pi / a) / (2.0 * (a * b))
	else:
		tmp = (((math.pi / a) / b) * 0.5) / b
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -2e-46)
		tmp = Float64(Float64(pi / a) / Float64(2.0 * Float64(a * b)));
	else
		tmp = Float64(Float64(Float64(Float64(pi / a) / b) * 0.5) / b);
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2e-46)
		tmp = (pi / a) / (2.0 * (a * b));
	else
		tmp = (((pi / a) / b) * 0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -2e-46], N[(N[(Pi / a), $MachinePrecision] / N[(2.0 * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(Pi / a), $MachinePrecision] / b), $MachinePrecision] * 0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.00000000000000005e-46

   1. Initial program 82.0%

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

      1. associate-*r/ 82.1%

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

      2. *-rgt-identity 82.1%

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

      3. associate-*l/ 82.0%

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

      4. difference-of-squares 89.4%

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

      5. times-frac 99.7%

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

      6. associate-/l/ 99.7%

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

      7. sub-neg 99.7%

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

      8. distribute-neg-frac 99.7%

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

      9. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in a around 0 99.6%

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

      1. expm1-log1p-u 87.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\pi}{\left(b + a\right) \cdot 2} \cdot \frac{1}{a \cdot b}\right)\right)} \]

      2. expm1-udef 51.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\pi}{\left(b + a\right) \cdot 2} \cdot \frac{1}{a \cdot b}\right)} - 1} \]

      3. un-div-inv 51.4%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{\pi}{\left(b + a\right) \cdot 2}}{a \cdot b}}\right)} - 1 \]

      4. +-commutative 51.4%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\pi}{\color{blue}{\left(a + b\right)} \cdot 2}}{a \cdot b}\right)} - 1 \]

   6. Applied egg-rr 51.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\pi}{\left(a + b\right) \cdot 2}}{a \cdot b}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 87.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\pi}{\left(a + b\right) \cdot 2}}{a \cdot b}\right)\right)} \]

      2. expm1-log1p 99.6%

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

      3. associate-/r* 97.6%

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

      4. *-rgt-identity 97.6%

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

      5. times-frac 99.6%

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

      6. associate-/r* 99.7%

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

      7. times-frac 99.6%

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

      8. associate-*l* 99.6%

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

      9. associate-*r/ 99.6%

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

      10. *-rgt-identity 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in a around inf 84.6%

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

       1. *-commutative 84.6%

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

   11. Simplified 84.6%

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

   if -2.00000000000000005e-46 < a

   1. Initial program 77.9%

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

      1. associate-*r/ 77.9%

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

      2. *-rgt-identity 77.9%

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

      3. associate-*l/ 77.9%

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

      4. difference-of-squares 83.7%

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

      5. times-frac 99.6%

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

      6. associate-/l/ 99.6%

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

      7. sub-neg 99.6%

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

      8. distribute-neg-frac 99.6%

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

      9. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a around 0 99.7%

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

      1. associate-/r* 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in b around inf 68.9%

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

      1. *-commutative 68.9%

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

      2. associate-/l/ 68.9%

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

      3. *-commutative 68.9%

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

      4. associate-*r/ 68.9%

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

      5. *-commutative 68.9%

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

      6. frac-times 68.7%

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

      7. *-un-lft-identity 68.7%

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

      8. associate-/r* 68.9%

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

      9. *-commutative 68.9%

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

      10. *-un-lft-identity 68.9%

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

      11. times-frac 68.9%

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

      12. metadata-eval 68.9%

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

      13. associate-/r* 68.9%

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

   9. Applied egg-rr 68.9%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-46}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{2 \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{a}}{b} \cdot 0.5}{b}\\ \end{array} \]

Alternative 5: 90.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1.9 \cdot 10^{-46}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{2 \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \frac{\pi}{b}}{a}}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -1.9e-46)
   (/ (/ PI a) (* 2.0 (* a b)))
   (/ (/ (* 0.5 (/ PI b)) a) b)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -1.9e-46) {
		tmp = (((double) M_PI) / a) / (2.0 * (a * b));
	} else {
		tmp = ((0.5 * (((double) M_PI) / b)) / a) / b;
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -1.9e-46) {
		tmp = (Math.PI / a) / (2.0 * (a * b));
	} else {
		tmp = ((0.5 * (Math.PI / b)) / a) / b;
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -1.9e-46:
		tmp = (math.pi / a) / (2.0 * (a * b))
	else:
		tmp = ((0.5 * (math.pi / b)) / a) / b
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -1.9e-46)
		tmp = Float64(Float64(pi / a) / Float64(2.0 * Float64(a * b)));
	else
		tmp = Float64(Float64(Float64(0.5 * Float64(pi / b)) / a) / b);
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.9e-46)
		tmp = (pi / a) / (2.0 * (a * b));
	else
		tmp = ((0.5 * (pi / b)) / a) / b;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -1.9e-46], N[(N[(Pi / a), $MachinePrecision] / N[(2.0 * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * N[(Pi / b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.8999999999999998e-46

   1. Initial program 82.0%

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

      1. associate-*r/ 82.1%

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

      2. *-rgt-identity 82.1%

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

      3. associate-*l/ 82.0%

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

      4. difference-of-squares 89.4%

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

      5. times-frac 99.7%

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

      6. associate-/l/ 99.7%

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

      7. sub-neg 99.7%

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

      8. distribute-neg-frac 99.7%

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

      9. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in a around 0 99.6%

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

      1. expm1-log1p-u 87.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\pi}{\left(b + a\right) \cdot 2} \cdot \frac{1}{a \cdot b}\right)\right)} \]

      2. expm1-udef 51.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\pi}{\left(b + a\right) \cdot 2} \cdot \frac{1}{a \cdot b}\right)} - 1} \]

      3. un-div-inv 51.4%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{\pi}{\left(b + a\right) \cdot 2}}{a \cdot b}}\right)} - 1 \]

      4. +-commutative 51.4%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\pi}{\color{blue}{\left(a + b\right)} \cdot 2}}{a \cdot b}\right)} - 1 \]

   6. Applied egg-rr 51.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\pi}{\left(a + b\right) \cdot 2}}{a \cdot b}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 87.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\pi}{\left(a + b\right) \cdot 2}}{a \cdot b}\right)\right)} \]

      2. expm1-log1p 99.6%

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

      3. associate-/r* 97.6%

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

      4. *-rgt-identity 97.6%

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

      5. times-frac 99.6%

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

      6. associate-/r* 99.7%

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

      7. times-frac 99.6%

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

      8. associate-*l* 99.6%

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

      9. associate-*r/ 99.6%

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

      10. *-rgt-identity 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in a around inf 84.6%

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

       1. *-commutative 84.6%

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

   11. Simplified 84.6%

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

   if -1.8999999999999998e-46 < a

   1. Initial program 77.9%

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

      1. associate-*r/ 77.9%

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

      2. *-rgt-identity 77.9%

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

      3. associate-*l/ 77.9%

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

      4. difference-of-squares 83.7%

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

      5. times-frac 99.6%

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

      6. associate-/l/ 99.6%

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

      7. sub-neg 99.6%

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

      8. distribute-neg-frac 99.6%

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

      9. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a around 0 99.7%

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

      1. associate-/r* 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in b around inf 68.9%

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

      1. associate-*r/ 68.9%

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

      2. un-div-inv 69.0%

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

   9. Applied egg-rr 69.0%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.9 \cdot 10^{-46}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{2 \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \frac{\pi}{b}}{a}}{b}\\ \end{array} \]

Alternative 6: 98.9% accurate, 1.1× speedup

Math

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

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (/ PI (* (* a b) (* 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(pi / Float64(Float64(a * b) * Float64(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[(Pi / N[(N[(a * b), $MachinePrecision] * N[(2.0 * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.0%

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

   1. associate-*r/ 79.0%

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

   2. *-rgt-identity 79.0%

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

   3. associate-*l/ 79.0%

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

   4. difference-of-squares 85.2%

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

   5. times-frac 99.7%

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

   6. associate-/l/ 99.7%

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

   7. sub-neg 99.7%

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

   8. distribute-neg-frac 99.7%

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

   9. metadata-eval 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in a around 0 99.6%

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

   1. expm1-log1p-u 77.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\pi}{\left(b + a\right) \cdot 2} \cdot \frac{1}{a \cdot b}\right)\right)} \]

   2. expm1-udef 48.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\pi}{\left(b + a\right) \cdot 2} \cdot \frac{1}{a \cdot b}\right)} - 1} \]

   3. un-div-inv 48.4%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{\pi}{\left(b + a\right) \cdot 2}}{a \cdot b}}\right)} - 1 \]

   4. +-commutative 48.4%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\pi}{\color{blue}{\left(a + b\right)} \cdot 2}}{a \cdot b}\right)} - 1 \]

6. Applied egg-rr 48.4%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\pi}{\left(a + b\right) \cdot 2}}{a \cdot b}\right)} - 1} \]
7. Step-by-step derivation

   1. expm1-def 77.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\pi}{\left(a + b\right) \cdot 2}}{a \cdot b}\right)\right)} \]

   2. expm1-log1p 99.7%

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

   3. associate-/l/ 98.7%

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

   4. *-commutative 98.7%

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

8. Simplified 98.7%

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

9. Final simplification 98.7%

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

Alternative 7: 99.7% accurate, 1.1× speedup

Math

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

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (/ (/ PI (* 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(a * b)) / Float64(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[(a * b), $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.0%

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

   1. associate-*r/ 79.0%

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

   2. *-rgt-identity 79.0%

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

   3. associate-*l/ 79.0%

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

   4. difference-of-squares 85.2%

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

   5. times-frac 99.7%

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

   6. associate-/l/ 99.7%

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

   7. sub-neg 99.7%

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

   8. distribute-neg-frac 99.7%

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

   9. metadata-eval 99.7%

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

3. Simplified 99.7%

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

   1. associate-*l/ 99.6%

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

   2. add-sqr-sqrt 49.0%

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

   3. sqrt-unprod 76.4%

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

   4. frac-times 76.4%

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

   5. metadata-eval 76.4%

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

   6. metadata-eval 76.4%

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

   7. frac-times 76.4%

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

   8. sqrt-unprod 34.8%

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

   9. add-sqr-sqrt 66.0%

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

   10. *-commutative 66.0%

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

5. Applied egg-rr 66.0%

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

6. Taylor expanded in a around 0 99.7%

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

7. Final simplification 99.7%

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

Alternative 8: 62.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.0%

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

   1. associate-*r/ 79.0%

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

   2. *-rgt-identity 79.0%

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

   3. associate-*l/ 79.0%

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

   4. difference-of-squares 85.2%

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

   5. times-frac 99.7%

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

   6. associate-/l/ 99.7%

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

   7. sub-neg 99.7%

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

   8. distribute-neg-frac 99.7%

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

   9. metadata-eval 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in a around 0 99.6%

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

   1. associate-/r* 99.6%

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

6. Simplified 99.6%

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

7. Taylor expanded in b around inf 60.2%

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

   1. *-commutative 60.2%

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

   2. associate-/l/ 60.2%

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

   3. *-commutative 60.2%

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

   4. associate-*r/ 60.2%

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

   5. *-commutative 60.2%

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

   6. frac-times 60.4%

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

   7. *-un-lft-identity 60.4%

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

   8. times-frac 60.2%

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

   9. clear-num 60.2%

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

   10. frac-times 60.3%

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

   11. metadata-eval 60.3%

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

   12. *-un-lft-identity 60.3%

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

   13. times-frac 60.4%

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

   14. /-rgt-identity 60.4%

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

9. Applied egg-rr 60.4%

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

10. Final simplification 60.4%

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

Reproduce

herbie shell --seed 2023314 
(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))))