NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.3% → 99.3%

Time: 15.0s

Alternatives: 15

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: 78.3% 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: 99.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -6.5 \cdot 10^{+136}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{\pi}{a \cdot b}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot -0.5}{b \cdot \left(a \cdot \left(\left(-a\right) - b\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -6.5e+136)
   (/ (* -0.5 (/ PI (* a b))) (- b a))
   (/ (* PI -0.5) (* b (* a (- (- a) b))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -6.5e+136) {
		tmp = (-0.5 * (((double) M_PI) / (a * b))) / (b - a);
	} else {
		tmp = (((double) M_PI) * -0.5) / (b * (a * (-a - b)));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -6.5e+136) {
		tmp = (-0.5 * (Math.PI / (a * b))) / (b - a);
	} else {
		tmp = (Math.PI * -0.5) / (b * (a * (-a - b)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -6.5e+136:
		tmp = (-0.5 * (math.pi / (a * b))) / (b - a)
	else:
		tmp = (math.pi * -0.5) / (b * (a * (-a - b)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -6.5e+136)
		tmp = Float64(Float64(-0.5 * Float64(pi / Float64(a * b))) / Float64(b - a));
	else
		tmp = Float64(Float64(pi * -0.5) / Float64(b * Float64(a * Float64(Float64(-a) - b))));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -6.5e+136)
		tmp = (-0.5 * (pi / (a * b))) / (b - a);
	else
		tmp = (pi * -0.5) / (b * (a * (-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, -6.5e+136], N[(N[(-0.5 * N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * -0.5), $MachinePrecision] / N[(b * N[(a * N[((-a) - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.4999999999999998e136

   1. Initial program 68.8%

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

      1. associate-*r/ 68.8%

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

      2. *-rgt-identity 68.8%

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

      3. associate-*l/ 68.9%

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

      4. difference-of-squares 84.9%

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

      5. *-commutative 84.9%

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

      6. times-frac 99.8%

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

      7. sub-neg 99.8%

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

      8. distribute-neg-frac 99.8%

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

      9. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. associate-*l/ 99.7%

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

      2. div-inv 99.7%

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

      3. metadata-eval 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in a around inf 99.8%

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

   if -6.4999999999999998e136 < a

   1. Initial program 84.4%

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

      1. *-commutative 84.4%

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

      2. associate-*r* 84.4%

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

      3. associate-*r/ 84.3%

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

      4. associate-/l* 84.3%

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

      5. /-rgt-identity 84.3%

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

      6. associate-/l* 84.4%

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

      7. difference-of-squares 90.5%

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

      8. associate-/l* 90.4%

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

      9. associate-/l* 99.7%

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

      10. associate-*r/ 90.7%

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

      11. sub-neg 90.7%

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

      12. distribute-neg-frac 90.7%

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

      13. metadata-eval 90.7%

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

   3. Simplified 90.7%

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

   5. Taylor expanded in a around inf 62.2%

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

   6. Taylor expanded in b around 0 97.2%

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

      1. associate-*r/ 97.2%

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

   8. Simplified 97.2%

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

      1. frac-2neg 97.2%

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

      2. metadata-eval 97.2%

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

      3. associate-/l/ 96.8%

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

      4. *-commutative 96.8%

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

      5. frac-times 96.6%

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

      6. *-un-lft-identity 96.6%

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

      7. *-commutative 96.6%

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

   10. Applied egg-rr 96.6%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.5 \cdot 10^{+136}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{\pi}{a \cdot b}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot -0.5}{b \cdot \left(a \cdot \left(\left(-a\right) - b\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.9%

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

   1. associate-*r/ 82.9%

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

   2. *-rgt-identity 82.9%

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

   3. associate-*l/ 82.8%

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

   4. difference-of-squares 89.9%

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

   5. *-commutative 89.9%

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

   6. times-frac 99.7%

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

   7. sub-neg 99.7%

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

   8. distribute-neg-frac 99.7%

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

   9. metadata-eval 99.7%

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

3. Simplified 99.7%

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

   1. associate-*l/ 99.7%

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

   2. div-inv 99.7%

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

   3. metadata-eval 99.7%

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

6. Applied egg-rr 99.7%

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

7. Final simplification 99.7%

   \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{a + b}}{b - a} \]
8. Add Preprocessing

Alternative 3: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \frac{\frac{1}{a} + \frac{-1}{b}}{a + b} \cdot \frac{\frac{\pi}{2}}{b - a} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (* (/ (+ (/ 1.0 a) (/ -1.0 b)) (+ a b)) (/ (/ PI 2.0) (- b a))))

C

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

Java

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

Python

[a, b] = sort([a, b])
def code(a, b):
	return (((1.0 / a) + (-1.0 / b)) / (a + b)) * ((math.pi / 2.0) / (b - a))

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(Float64(Float64(Float64(1.0 / a) + Float64(-1.0 / b)) / Float64(a + b)) * Float64(Float64(pi / 2.0) / Float64(b - a)))
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = (((1.0 / a) + (-1.0 / b)) / (a + b)) * ((pi / 2.0) / (b - a));
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[(N[(N[(N[(1.0 / a), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision] / N[(a + b), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi / 2.0), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.9%

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

   1. associate-*r/ 82.9%

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

   2. *-rgt-identity 82.9%

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

   3. associate-*l/ 82.8%

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

   4. difference-of-squares 89.9%

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

   5. *-commutative 89.9%

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

   6. times-frac 99.7%

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

   7. sub-neg 99.7%

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

   8. distribute-neg-frac 99.7%

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

   9. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Final simplification 99.7%

   \[\leadsto \frac{\frac{1}{a} + \frac{-1}{b}}{a + b} \cdot \frac{\frac{\pi}{2}}{b - a} \]
6. Add Preprocessing

Alternative 4: 90.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 3.8 \cdot 10^{-80}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{b} \cdot \left(-0.5 \cdot \frac{\frac{\pi}{a}}{b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 3.8e-80)
   (/ (* 0.5 (/ PI a)) (* a b))
   (* (/ -1.0 b) (* -0.5 (/ (/ PI a) b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 3.8e-80) {
		tmp = (0.5 * (((double) M_PI) / a)) / (a * b);
	} else {
		tmp = (-1.0 / b) * (-0.5 * ((((double) M_PI) / a) / b));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 3.8e-80) {
		tmp = (0.5 * (Math.PI / a)) / (a * b);
	} else {
		tmp = (-1.0 / b) * (-0.5 * ((Math.PI / a) / b));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 3.8e-80:
		tmp = (0.5 * (math.pi / a)) / (a * b)
	else:
		tmp = (-1.0 / b) * (-0.5 * ((math.pi / a) / b))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 3.8e-80)
		tmp = Float64(Float64(0.5 * Float64(pi / a)) / Float64(a * b));
	else
		tmp = Float64(Float64(-1.0 / b) * Float64(-0.5 * Float64(Float64(pi / a) / b)));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 3.8e-80)
		tmp = (0.5 * (pi / a)) / (a * b);
	else
		tmp = (-1.0 / b) * (-0.5 * ((pi / 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[b, 3.8e-80], N[(N[(0.5 * N[(Pi / a), $MachinePrecision]), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / b), $MachinePrecision] * N[(-0.5 * N[(N[(Pi / a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.79999999999999967e-80

   1. Initial program 81.5%

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

      1. associate-*r/ 81.5%

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

      2. *-rgt-identity 81.5%

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

      3. associate-*l/ 81.5%

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

      4. difference-of-squares 88.9%

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

      5. *-commutative 88.9%

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

      6. times-frac 99.7%

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

      7. sub-neg 99.7%

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

      8. distribute-neg-frac 99.7%

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

      9. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in a around inf 73.1%

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

   6. Taylor expanded in b around 0 73.3%

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

      1. associate-*r/ 93.8%

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

   8. Simplified 73.3%

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

      1. associate-/l* 73.3%

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

      2. frac-times 73.3%

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

      3. metadata-eval 73.3%

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

   10. Applied egg-rr 73.3%

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

       1. associate-/r* 73.3%

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

       2. div-inv 73.3%

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

       3. div-inv 73.3%

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

       4. clear-num 73.3%

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

       5. *-commutative 73.3%

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

   12. Applied egg-rr 73.3%

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

       1. associate-*r/ 73.4%

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

       2. *-rgt-identity 73.4%

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

       3. *-commutative 73.4%

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

   14. Simplified 73.4%

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

   if 3.79999999999999967e-80 < b

   1. Initial program 85.2%

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

      1. *-commutative 85.2%

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

      2. associate-*r* 85.3%

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

      3. associate-*r/ 85.2%

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

      4. associate-/l* 85.2%

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

      5. /-rgt-identity 85.2%

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

      6. associate-/l* 85.3%

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

      7. difference-of-squares 91.7%

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

      8. associate-/l* 91.6%

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

      9. associate-/l* 99.7%

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

      10. associate-*r/ 91.6%

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

      11. sub-neg 91.6%

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

      12. distribute-neg-frac 91.6%

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

      13. metadata-eval 91.6%

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

   3. Simplified 91.6%

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

   5. Taylor expanded in a around inf 55.6%

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

   6. Taylor expanded in b around 0 99.7%

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

      1. associate-*r/ 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in a around 0 88.4%

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

       1. associate-/r* 88.5%

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

   11. Simplified 88.5%

       \[\leadsto \frac{-1}{b} \cdot \color{blue}{\left(-0.5 \cdot \frac{\frac{\pi}{a}}{b}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.8 \cdot 10^{-80}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{b} \cdot \left(-0.5 \cdot \frac{\frac{\pi}{a}}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 1.8 \cdot 10^{-80}:\\ \;\;\;\;\frac{\pi \cdot -0.5}{a} \cdot \frac{-1}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{b} \cdot \left(-0.5 \cdot \frac{\frac{\pi}{a}}{b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 1.8e-80)
   (* (/ (* PI -0.5) a) (/ -1.0 (* a b)))
   (* (/ -1.0 b) (* -0.5 (/ (/ PI a) b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 1.8e-80) {
		tmp = ((((double) M_PI) * -0.5) / a) * (-1.0 / (a * b));
	} else {
		tmp = (-1.0 / b) * (-0.5 * ((((double) M_PI) / a) / b));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 1.8e-80) {
		tmp = ((Math.PI * -0.5) / a) * (-1.0 / (a * b));
	} else {
		tmp = (-1.0 / b) * (-0.5 * ((Math.PI / a) / b));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 1.8e-80:
		tmp = ((math.pi * -0.5) / a) * (-1.0 / (a * b))
	else:
		tmp = (-1.0 / b) * (-0.5 * ((math.pi / a) / b))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 1.8e-80)
		tmp = Float64(Float64(Float64(pi * -0.5) / a) * Float64(-1.0 / Float64(a * b)));
	else
		tmp = Float64(Float64(-1.0 / b) * Float64(-0.5 * Float64(Float64(pi / a) / b)));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.8e-80)
		tmp = ((pi * -0.5) / a) * (-1.0 / (a * b));
	else
		tmp = (-1.0 / b) * (-0.5 * ((pi / 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[b, 1.8e-80], N[(N[(N[(Pi * -0.5), $MachinePrecision] / a), $MachinePrecision] * N[(-1.0 / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / b), $MachinePrecision] * N[(-0.5 * N[(N[(Pi / a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.8e-80

   1. Initial program 81.5%

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

      1. associate-*r/ 81.5%

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

      2. *-rgt-identity 81.5%

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

      3. associate-*l/ 81.5%

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

      4. difference-of-squares 88.9%

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

      5. *-commutative 88.9%

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

      6. times-frac 99.7%

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

      7. sub-neg 99.7%

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

      8. distribute-neg-frac 99.7%

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

      9. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in a around inf 73.1%

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

   6. Taylor expanded in b around 0 73.3%

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

      1. associate-*r/ 93.8%

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

   8. Simplified 73.3%

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

   if 1.8e-80 < b

   1. Initial program 85.2%

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

      1. *-commutative 85.2%

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

      2. associate-*r* 85.3%

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

      3. associate-*r/ 85.2%

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

      4. associate-/l* 85.2%

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

      5. /-rgt-identity 85.2%

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

      6. associate-/l* 85.3%

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

      7. difference-of-squares 91.7%

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

      8. associate-/l* 91.6%

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

      9. associate-/l* 99.7%

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

      10. associate-*r/ 91.6%

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

      11. sub-neg 91.6%

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

      12. distribute-neg-frac 91.6%

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

      13. metadata-eval 91.6%

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

   3. Simplified 91.6%

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

   5. Taylor expanded in a around inf 55.6%

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

   6. Taylor expanded in b around 0 99.7%

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

      1. associate-*r/ 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in a around 0 88.4%

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

       1. associate-/r* 88.5%

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

   11. Simplified 88.5%

       \[\leadsto \frac{-1}{b} \cdot \color{blue}{\left(-0.5 \cdot \frac{\frac{\pi}{a}}{b}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.8 \cdot 10^{-80}:\\ \;\;\;\;\frac{\pi \cdot -0.5}{a} \cdot \frac{-1}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{b} \cdot \left(-0.5 \cdot \frac{\frac{\pi}{a}}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 1.35 \cdot 10^{-78}:\\ \;\;\;\;\frac{\pi \cdot -0.5}{a} \cdot \frac{-1}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\frac{a \cdot b}{\frac{\pi}{b - a}}}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 1.35e-78)
   (* (/ (* PI -0.5) a) (/ -1.0 (* a b)))
   (/ 0.5 (/ (* a b) (/ PI (- b a))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 1.35e-78) {
		tmp = ((((double) M_PI) * -0.5) / a) * (-1.0 / (a * b));
	} else {
		tmp = 0.5 / ((a * b) / (((double) M_PI) / (b - a)));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 1.35e-78) {
		tmp = ((Math.PI * -0.5) / a) * (-1.0 / (a * b));
	} else {
		tmp = 0.5 / ((a * b) / (Math.PI / (b - a)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 1.35e-78:
		tmp = ((math.pi * -0.5) / a) * (-1.0 / (a * b))
	else:
		tmp = 0.5 / ((a * b) / (math.pi / (b - a)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 1.35e-78)
		tmp = Float64(Float64(Float64(pi * -0.5) / a) * Float64(-1.0 / Float64(a * b)));
	else
		tmp = Float64(0.5 / Float64(Float64(a * b) / Float64(pi / Float64(b - a))));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.35e-78)
		tmp = ((pi * -0.5) / a) * (-1.0 / (a * b));
	else
		tmp = 0.5 / ((a * b) / (pi / (b - a)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[b, 1.35e-78], N[(N[(N[(Pi * -0.5), $MachinePrecision] / a), $MachinePrecision] * N[(-1.0 / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(N[(a * b), $MachinePrecision] / N[(Pi / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.34999999999999997e-78

   1. Initial program 81.5%

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

      1. associate-*r/ 81.5%

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

      2. *-rgt-identity 81.5%

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

      3. associate-*l/ 81.5%

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

      4. difference-of-squares 88.9%

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

      5. *-commutative 88.9%

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

      6. times-frac 99.7%

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

      7. sub-neg 99.7%

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

      8. distribute-neg-frac 99.7%

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

      9. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in a around inf 73.1%

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

   6. Taylor expanded in b around 0 73.3%

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

      1. associate-*r/ 93.8%

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

   8. Simplified 73.3%

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

   if 1.34999999999999997e-78 < b

   1. Initial program 85.2%

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

      1. associate-*r/ 85.3%

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

      2. *-rgt-identity 85.3%

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

      3. associate-*l/ 85.2%

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

      4. difference-of-squares 91.6%

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

      5. *-commutative 91.6%

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

      6. times-frac 99.7%

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

      7. sub-neg 99.7%

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

      8. distribute-neg-frac 99.7%

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

      9. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. associate-*l/ 99.7%

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

      2. div-inv 99.7%

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

      3. metadata-eval 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in a around 0 94.6%

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

      1. expm1-log1p-u 80.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.5 \cdot \frac{\pi}{a \cdot b}}{b - a}\right)\right)} \]

      2. expm1-udef 58.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.5 \cdot \frac{\pi}{a \cdot b}}{b - a}\right)} - 1} \]

      3. associate-*r/ 58.7%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\frac{0.5 \cdot \pi}{a \cdot b}}}{b - a}\right)} - 1 \]

      4. *-commutative 58.7%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\color{blue}{\pi \cdot 0.5}}{a \cdot b}}{b - a}\right)} - 1 \]

   9. Applied egg-rr 58.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\pi \cdot 0.5}{a \cdot b}}{b - a}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 80.1%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\pi \cdot 0.5}{a \cdot b}}{b - a}\right)\right)} \]

       2. expm1-log1p 94.6%

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

       3. metadata-eval 94.6%

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

       4. distribute-rgt-neg-in 94.6%

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

       5. associate-/l/ 94.6%

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

       6. distribute-rgt-neg-in 94.6%

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

       7. metadata-eval 94.6%

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

       8. *-commutative 94.6%

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

   11. Simplified 94.6%

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

       1. associate-/l* 94.5%

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

       2. div-inv 94.5%

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

       3. *-commutative 94.5%

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

       4. *-commutative 94.5%

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

   13. Applied egg-rr 94.5%

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

       1. associate-*r/ 94.5%

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

       2. metadata-eval 94.5%

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

       3. associate-/l* 94.6%

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

       4. *-commutative 94.6%

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

   15. Simplified 94.6%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.35 \cdot 10^{-78}:\\ \;\;\;\;\frac{\pi \cdot -0.5}{a} \cdot \frac{-1}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\frac{a \cdot b}{\frac{\pi}{b - a}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 8.2 \cdot 10^{+101}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot \left(a + b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\frac{a \cdot b}{\frac{\pi}{b - a}}}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 8.2e+101)
   (/ (* PI 0.5) (* a (* b (+ a b))))
   (/ 0.5 (/ (* a b) (/ PI (- b a))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 8.2e+101) {
		tmp = (((double) M_PI) * 0.5) / (a * (b * (a + b)));
	} else {
		tmp = 0.5 / ((a * b) / (((double) M_PI) / (b - a)));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 8.2e+101) {
		tmp = (Math.PI * 0.5) / (a * (b * (a + b)));
	} else {
		tmp = 0.5 / ((a * b) / (Math.PI / (b - a)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 8.2e+101:
		tmp = (math.pi * 0.5) / (a * (b * (a + b)))
	else:
		tmp = 0.5 / ((a * b) / (math.pi / (b - a)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 8.2e+101)
		tmp = Float64(Float64(pi * 0.5) / Float64(a * Float64(b * Float64(a + b))));
	else
		tmp = Float64(0.5 / Float64(Float64(a * b) / Float64(pi / Float64(b - a))));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 8.2e+101)
		tmp = (pi * 0.5) / (a * (b * (a + b)));
	else
		tmp = 0.5 / ((a * b) / (pi / (b - a)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[b, 8.2e+101], N[(N[(Pi * 0.5), $MachinePrecision] / N[(a * N[(b * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(N[(a * b), $MachinePrecision] / N[(Pi / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.1999999999999999e101

   1. Initial program 85.0%

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

      1. *-commutative 85.0%

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

      2. associate-*r* 85.0%

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

      3. associate-*r/ 85.0%

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

      4. associate-/l* 85.0%

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

      5. /-rgt-identity 85.0%

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

      6. associate-/l* 85.0%

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

      7. difference-of-squares 91.0%

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

      8. associate-/l* 90.9%

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

      9. associate-/l* 99.7%

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

      10. associate-*r/ 91.3%

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

      11. sub-neg 91.3%

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

      12. distribute-neg-frac 91.3%

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

      13. metadata-eval 91.3%

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

   3. Simplified 91.3%

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

   5. Taylor expanded in a around inf 64.3%

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

   6. Taylor expanded in b around 0 94.9%

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

      1. associate-*r/ 94.9%

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

   8. Simplified 94.9%

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

      1. *-commutative 94.9%

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

      2. frac-2neg 94.9%

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

      3. metadata-eval 94.9%

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

      4. un-div-inv 95.0%

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

      5. associate-/l/ 94.6%

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

      6. *-commutative 94.6%

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

      7. times-frac 94.9%

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

   10. Applied egg-rr 94.9%

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

   12. Simplified 96.1%

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

   if 8.1999999999999999e101 < b

   1. Initial program 74.7%

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

      1. associate-*r/ 74.7%

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

      2. *-rgt-identity 74.7%

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

      3. associate-*l/ 74.7%

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

      4. difference-of-squares 85.8%

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

      5. *-commutative 85.8%

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

      6. times-frac 99.8%

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

      7. sub-neg 99.8%

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

      8. distribute-neg-frac 99.8%

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

      9. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. associate-*l/ 99.8%

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

      2. div-inv 99.8%

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

      3. metadata-eval 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in a around 0 99.7%

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

      1. expm1-log1p-u 94.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.5 \cdot \frac{\pi}{a \cdot b}}{b - a}\right)\right)} \]

      2. expm1-udef 66.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.5 \cdot \frac{\pi}{a \cdot b}}{b - a}\right)} - 1} \]

      3. associate-*r/ 66.7%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\frac{0.5 \cdot \pi}{a \cdot b}}}{b - a}\right)} - 1 \]

      4. *-commutative 66.7%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\color{blue}{\pi \cdot 0.5}}{a \cdot b}}{b - a}\right)} - 1 \]

   9. Applied egg-rr 66.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\pi \cdot 0.5}{a \cdot b}}{b - a}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 94.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\pi \cdot 0.5}{a \cdot b}}{b - a}\right)\right)} \]

       2. expm1-log1p 99.7%

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

       3. metadata-eval 99.7%

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

       4. distribute-rgt-neg-in 99.7%

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

       5. associate-/l/ 99.8%

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

       6. distribute-rgt-neg-in 99.8%

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

       7. metadata-eval 99.8%

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

       8. *-commutative 99.8%

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

   11. Simplified 99.8%

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

       1. associate-/l* 99.8%

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

       2. div-inv 99.8%

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

       3. *-commutative 99.8%

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

       4. *-commutative 99.8%

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

   13. Applied egg-rr 99.8%

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

       1. associate-*r/ 99.8%

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

       2. metadata-eval 99.8%

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

       3. associate-/l* 99.8%

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

       4. *-commutative 99.8%

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

   15. Simplified 99.8%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8.2 \cdot 10^{+101}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot \left(a + b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\frac{a \cdot b}{\frac{\pi}{b - a}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 8 \cdot 10^{+151}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot \left(a + b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \frac{\pi}{b}}{a}}{b - a}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 8e+151)
   (/ (* PI 0.5) (* a (* b (+ a b))))
   (/ (/ (* 0.5 (/ PI b)) a) (- b a))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 8e+151) {
		tmp = (((double) M_PI) * 0.5) / (a * (b * (a + b)));
	} else {
		tmp = ((0.5 * (((double) M_PI) / b)) / a) / (b - a);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 8e+151) {
		tmp = (Math.PI * 0.5) / (a * (b * (a + b)));
	} else {
		tmp = ((0.5 * (Math.PI / b)) / a) / (b - a);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 8e+151:
		tmp = (math.pi * 0.5) / (a * (b * (a + b)))
	else:
		tmp = ((0.5 * (math.pi / b)) / a) / (b - a)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 8e+151)
		tmp = Float64(Float64(pi * 0.5) / Float64(a * Float64(b * Float64(a + b))));
	else
		tmp = Float64(Float64(Float64(0.5 * Float64(pi / b)) / a) / Float64(b - a));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 8e+151)
		tmp = (pi * 0.5) / (a * (b * (a + b)));
	else
		tmp = ((0.5 * (pi / b)) / a) / (b - a);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[b, 8e+151], N[(N[(Pi * 0.5), $MachinePrecision] / N[(a * N[(b * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * N[(Pi / b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.00000000000000014e151

   1. Initial program 86.2%

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

      1. *-commutative 86.2%

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

      2. associate-*r* 86.2%

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

      3. associate-*r/ 86.2%

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

      4. associate-/l* 86.2%

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

      5. /-rgt-identity 86.2%

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

      6. associate-/l* 86.2%

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

      7. difference-of-squares 91.7%

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

      8. associate-/l* 91.6%

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

      9. associate-/l* 99.6%

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

      10. associate-*r/ 92.0%

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

      11. sub-neg 92.0%

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

      12. distribute-neg-frac 92.0%

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

      13. metadata-eval 92.0%

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

   3. Simplified 92.0%

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

   5. Taylor expanded in a around inf 62.0%

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

   6. Taylor expanded in b around 0 95.3%

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

      1. associate-*r/ 95.3%

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

   8. Simplified 95.3%

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

      1. *-commutative 95.3%

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

      2. frac-2neg 95.3%

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

      3. metadata-eval 95.3%

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

      4. un-div-inv 95.3%

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

      5. associate-/l/ 95.0%

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

      6. *-commutative 95.0%

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

      7. times-frac 95.3%

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

   10. Applied egg-rr 95.3%

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

   12. Simplified 96.4%

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

   if 8.00000000000000014e151 < b

   1. Initial program 62.3%

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

      1. associate-*r/ 62.3%

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

      2. *-rgt-identity 62.3%

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

      3. associate-*l/ 62.3%

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

      4. difference-of-squares 79.0%

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

      5. *-commutative 79.0%

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

      6. times-frac 100.0%

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

      7. sub-neg 100.0%

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

      8. distribute-neg-frac 100.0%

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

      9. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around inf 78.7%

      \[\leadsto \frac{\frac{\pi}{2}}{b - a} \cdot \color{blue}{\frac{-1}{a \cdot b}} \]
   6. Step-by-step derivation

      1. *-commutative 78.7%

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

      2. clear-num 78.7%

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

      3. frac-times 78.7%

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

      4. metadata-eval 78.7%

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

      5. div-inv 78.7%

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

      6. metadata-eval 78.7%

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

   7. Applied egg-rr 78.7%

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

      1. associate-/r* 78.7%

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

      2. *-commutative 78.7%

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

      3. associate-/l/ 78.7%

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

      4. associate-/r* 79.0%

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

      5. associate-*r/ 79.0%

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

      6. associate-/l* 79.0%

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

      7. *-commutative 79.0%

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

      8. associate-/r* 78.7%

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

      9. *-commutative 78.7%

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

      10. associate-*l/ 78.7%

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

      11. neg-mul-1 78.7%

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

      12. distribute-rgt-neg-in 78.7%

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

      13. metadata-eval 78.7%

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

   9. Simplified 78.7%

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

       1. expm1-log1p-u 78.5%

          \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\pi \cdot -0.5}{a}}{b}\right)\right)}}{b - a} \]

       2. expm1-udef 78.6%

          \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\pi \cdot -0.5}{a}}{b}\right)} - 1}}{b - a} \]

       3. div-inv 78.6%

          \[\leadsto \frac{e^{\mathsf{log1p}\left(\color{blue}{\frac{\pi \cdot -0.5}{a} \cdot \frac{1}{b}}\right)} - 1}{b - a} \]

       4. *-commutative 78.6%

          \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\color{blue}{-0.5 \cdot \pi}}{a} \cdot \frac{1}{b}\right)} - 1}{b - a} \]

       5. associate-/l* 78.6%

          \[\leadsto \frac{e^{\mathsf{log1p}\left(\color{blue}{\frac{-0.5}{\frac{a}{\pi}}} \cdot \frac{1}{b}\right)} - 1}{b - a} \]

       6. frac-times 78.6%

          \[\leadsto \frac{e^{\mathsf{log1p}\left(\color{blue}{\frac{-0.5 \cdot 1}{\frac{a}{\pi} \cdot b}}\right)} - 1}{b - a} \]

       7. metadata-eval 78.6%

          \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\color{blue}{-0.5}}{\frac{a}{\pi} \cdot b}\right)} - 1}{b - a} \]

   11. Applied egg-rr 78.6%

       \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{-0.5}{\frac{a}{\pi} \cdot b}\right)} - 1}}{b - a} \]
   12. Step-by-step derivation

       1. expm1-def 78.5%

          \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.5}{\frac{a}{\pi} \cdot b}\right)\right)}}{b - a} \]

       2. expm1-log1p 78.7%

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

       3. *-commutative 78.7%

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

   13. Simplified 78.7%

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

       1. div-inv 78.7%

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

       2. metadata-eval 78.7%

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

       3. associate-*r/ 78.7%

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

       4. *-commutative 78.7%

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

       5. add-sqr-sqrt 28.2%

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

       6. sqrt-unprod 89.3%

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

       7. sqr-neg 89.3%

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

       8. sqrt-unprod 61.0%

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

       9. add-sqr-sqrt 99.9%

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

       10. clear-num 99.8%

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

       11. times-frac 99.8%

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

       12. *-commutative 99.8%

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

       13. *-un-lft-identity 99.8%

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

       14. *-un-lft-identity 99.8%

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

       15. distribute-rgt-neg-in 99.8%

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

       16. times-frac 100.0%

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

       17. *-commutative 100.0%

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

   15. Applied egg-rr 100.0%

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

       1. associate-*l/ 99.9%

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

       2. *-lft-identity 99.9%

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

       3. neg-mul-1 99.9%

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

       4. times-frac 99.9%

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

       5. metadata-eval 99.9%

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

   17. Simplified 99.9%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8 \cdot 10^{+151}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot \left(a + b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \frac{\pi}{b}}{a}}{b - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.9%

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

   1. *-commutative 82.9%

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

   2. associate-*r* 82.8%

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

   3. associate-*r/ 82.8%

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

   4. associate-/l* 82.8%

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

   5. /-rgt-identity 82.8%

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

   6. associate-/l* 82.9%

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

   7. difference-of-squares 89.9%

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

   8. associate-/l* 89.8%

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

   9. associate-/l* 99.7%

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

   10. associate-*r/ 90.1%

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

   11. sub-neg 90.1%

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

   12. distribute-neg-frac 90.1%

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

   13. metadata-eval 90.1%

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

3. Simplified 90.1%

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

5. Taylor expanded in a around inf 64.4%

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

6. Taylor expanded in b around 0 95.9%

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

   1. associate-*r/ 95.9%

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

8. Simplified 95.9%

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

10. Applied egg-rr 99.7%

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

11. Final simplification 99.7%

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

Alternative 10: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 82.9%

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

   1. *-commutative 82.9%

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

   2. associate-*r* 82.8%

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

   3. associate-*r/ 82.8%

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

   4. associate-/l* 82.8%

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

   5. /-rgt-identity 82.8%

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

   6. associate-/l* 82.9%

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

   7. difference-of-squares 89.9%

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

   8. associate-/l* 89.8%

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

   9. associate-/l* 99.7%

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

   10. associate-*r/ 90.1%

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

   11. sub-neg 90.1%

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

   12. distribute-neg-frac 90.1%

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

   13. metadata-eval 90.1%

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

3. Simplified 90.1%

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

5. Taylor expanded in a around inf 64.4%

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

6. Taylor expanded in b around 0 95.9%

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

   1. associate-*r/ 95.9%

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

8. Simplified 95.9%

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

   1. *-commutative 95.9%

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

   2. frac-2neg 95.9%

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

   3. metadata-eval 95.9%

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

   4. un-div-inv 96.0%

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

   5. associate-/l/ 95.7%

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

   6. *-commutative 95.7%

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

   7. times-frac 95.9%

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

10. Applied egg-rr 95.9%

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

    1. associate-/l* 99.6%

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

    2. +-commutative 99.6%

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

12. Simplified 99.6%

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

13. Final simplification 99.6%

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

Alternative 11: 63.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.9%

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

   1. associate-*r/ 82.9%

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

   2. *-rgt-identity 82.9%

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

   3. associate-*l/ 82.8%

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

   4. difference-of-squares 89.9%

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

   5. *-commutative 89.9%

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

   6. times-frac 99.7%

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

   7. sub-neg 99.7%

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

   8. distribute-neg-frac 99.7%

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

   9. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in a around inf 66.3%

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

6. Taylor expanded in b around 0 60.3%

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

   1. associate-*r/ 95.9%

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

8. Simplified 60.3%

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

   1. *-commutative 60.3%

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

   2. frac-2neg 60.3%

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

   3. metadata-eval 60.3%

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

   4. frac-times 60.6%

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

   5. *-un-lft-identity 60.6%

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

   6. *-commutative 60.6%

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

10. Applied egg-rr 60.6%

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

11. Final simplification 60.6%

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

Alternative 12: 30.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.9%

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

   1. associate-*r/ 82.9%

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

   2. *-rgt-identity 82.9%

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

   3. associate-*l/ 82.8%

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

   4. difference-of-squares 89.9%

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

   5. *-commutative 89.9%

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

   6. times-frac 99.7%

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

   7. sub-neg 99.7%

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

   8. distribute-neg-frac 99.7%

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

   9. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in a around inf 66.3%

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

6. Taylor expanded in b around 0 60.3%

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

   1. associate-*r/ 95.9%

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

8. Simplified 60.3%

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

   1. *-commutative 60.3%

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

   2. frac-2neg 60.3%

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

   3. metadata-eval 60.3%

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

   4. frac-times 60.6%

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

   5. *-un-lft-identity 60.6%

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

   6. *-commutative 60.6%

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

10. Applied egg-rr 60.6%

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

    1. *-commutative 60.6%

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

    2. times-frac 60.3%

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

    3. add-sqr-sqrt 29.3%

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

    4. sqrt-unprod 43.2%

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

    5. sqr-neg 43.2%

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

    6. sqrt-unprod 16.1%

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

    7. add-sqr-sqrt 27.3%

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

    8. *-commutative 27.3%

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

12. Applied egg-rr 27.3%

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

13. Final simplification 27.3%

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

Alternative 13: 30.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.9%

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

   1. associate-*r/ 82.9%

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

   2. *-rgt-identity 82.9%

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

   3. associate-*l/ 82.8%

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

   4. difference-of-squares 89.9%

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

   5. *-commutative 89.9%

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

   6. times-frac 99.7%

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

   7. sub-neg 99.7%

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

   8. distribute-neg-frac 99.7%

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

   9. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in a around inf 66.3%

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

6. Taylor expanded in b around 0 60.3%

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

   1. associate-*r/ 95.9%

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

8. Simplified 60.3%

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

   1. associate-/l* 60.2%

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

   2. frac-times 60.2%

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

   3. metadata-eval 60.2%

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

10. Applied egg-rr 60.2%

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

    1. expm1-log1p-u 47.6%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.5}{\frac{a}{\pi} \cdot \left(a \cdot b\right)}\right)\right)} \]

    2. expm1-udef 43.1%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.5}{\frac{a}{\pi} \cdot \left(a \cdot b\right)}\right)} - 1} \]

    3. frac-2neg 43.1%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{-0.5}{-\frac{a}{\pi} \cdot \left(a \cdot b\right)}}\right)} - 1 \]

    4. metadata-eval 43.1%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{-0.5}}{-\frac{a}{\pi} \cdot \left(a \cdot b\right)}\right)} - 1 \]

    5. *-commutative 43.1%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{-0.5}{-\color{blue}{\left(a \cdot b\right) \cdot \frac{a}{\pi}}}\right)} - 1 \]

    6. distribute-lft-neg-in 43.1%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{-0.5}{\color{blue}{\left(-a \cdot b\right) \cdot \frac{a}{\pi}}}\right)} - 1 \]

    7. add-sqr-sqrt 21.7%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{-0.5}{\color{blue}{\left(\sqrt{-a \cdot b} \cdot \sqrt{-a \cdot b}\right)} \cdot \frac{a}{\pi}}\right)} - 1 \]

    8. sqrt-unprod 34.0%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{-0.5}{\color{blue}{\sqrt{\left(-a \cdot b\right) \cdot \left(-a \cdot b\right)}} \cdot \frac{a}{\pi}}\right)} - 1 \]

    9. sqr-neg 34.0%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{-0.5}{\sqrt{\color{blue}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)}} \cdot \frac{a}{\pi}}\right)} - 1 \]

    10. sqrt-unprod 16.8%

        \[\leadsto e^{\mathsf{log1p}\left(\frac{-0.5}{\color{blue}{\left(\sqrt{a \cdot b} \cdot \sqrt{a \cdot b}\right)} \cdot \frac{a}{\pi}}\right)} - 1 \]

    11. add-sqr-sqrt 31.2%

        \[\leadsto e^{\mathsf{log1p}\left(\frac{-0.5}{\color{blue}{\left(a \cdot b\right)} \cdot \frac{a}{\pi}}\right)} - 1 \]

    12. *-commutative 31.2%

        \[\leadsto e^{\mathsf{log1p}\left(\frac{-0.5}{\color{blue}{\frac{a}{\pi} \cdot \left(a \cdot b\right)}}\right)} - 1 \]

    13. *-commutative 31.2%

        \[\leadsto e^{\mathsf{log1p}\left(\frac{-0.5}{\frac{a}{\pi} \cdot \color{blue}{\left(b \cdot a\right)}}\right)} - 1 \]

12. Applied egg-rr 31.2%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-0.5}{\frac{a}{\pi} \cdot \left(b \cdot a\right)}\right)} - 1} \]
13. Step-by-step derivation

    1. expm1-def 25.6%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.5}{\frac{a}{\pi} \cdot \left(b \cdot a\right)}\right)\right)} \]

    2. expm1-log1p 27.3%

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

    3. *-commutative 27.3%

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

14. Simplified 27.3%

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

15. Final simplification 27.3%

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

Alternative 14: 63.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.9%

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

   1. associate-*r/ 82.9%

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

   2. *-rgt-identity 82.9%

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

   3. associate-*l/ 82.8%

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

   4. difference-of-squares 89.9%

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

   5. *-commutative 89.9%

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

   6. times-frac 99.7%

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

   7. sub-neg 99.7%

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

   8. distribute-neg-frac 99.7%

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

   9. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in a around inf 66.3%

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

6. Taylor expanded in b around 0 60.3%

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

   1. associate-*r/ 95.9%

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

8. Simplified 60.3%

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

   1. associate-/l* 60.2%

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

   2. frac-times 60.2%

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

   3. metadata-eval 60.2%

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

10. Applied egg-rr 60.2%

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

11. Final simplification 60.2%

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

Alternative 15: 63.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.9%

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

   1. associate-*r/ 82.9%

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

   2. *-rgt-identity 82.9%

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

   3. associate-*l/ 82.8%

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

   4. difference-of-squares 89.9%

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

   5. *-commutative 89.9%

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

   6. times-frac 99.7%

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

   7. sub-neg 99.7%

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

   8. distribute-neg-frac 99.7%

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

   9. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in a around inf 66.3%

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

6. Taylor expanded in b around 0 60.3%

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

   1. associate-*r/ 95.9%

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

8. Simplified 60.3%

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

   1. associate-/l* 60.2%

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

   2. frac-times 60.2%

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

   3. metadata-eval 60.2%

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

10. Applied egg-rr 60.2%

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

    1. associate-*l/ 60.5%

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

    2. *-commutative 60.5%

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

12. Applied egg-rr 60.5%

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

13. Final simplification 60.5%

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

Reproduce

herbie shell --seed 2024011 
(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))))