NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.0% → 99.7%

Time: 10.8s

Alternatives: 10

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: 78.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 7.2 \cdot 10^{+93}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{b \cdot \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{b} \cdot \frac{0.5}{a}}{b - a}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 7.2e+93)
   (/ (* 0.5 (/ PI a)) (* b (+ b a)))
   (/ (* (/ PI b) (/ 0.5 a)) (- b a))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 7.2e+93) {
		tmp = (0.5 * (((double) M_PI) / a)) / (b * (b + a));
	} else {
		tmp = ((((double) M_PI) / b) * (0.5 / a)) / (b - a);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 7.2e+93) {
		tmp = (0.5 * (Math.PI / a)) / (b * (b + a));
	} else {
		tmp = ((Math.PI / b) * (0.5 / a)) / (b - a);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 7.2e+93:
		tmp = (0.5 * (math.pi / a)) / (b * (b + a))
	else:
		tmp = ((math.pi / b) * (0.5 / a)) / (b - a)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 7.2e+93)
		tmp = Float64(Float64(0.5 * Float64(pi / a)) / Float64(b * Float64(b + a)));
	else
		tmp = Float64(Float64(Float64(pi / b) * Float64(0.5 / 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 <= 7.2e+93)
		tmp = (0.5 * (pi / a)) / (b * (b + a));
	else
		tmp = ((pi / b) * (0.5 / 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, 7.2e+93], N[(N[(0.5 * N[(Pi / a), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi / b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.1999999999999998e93

   1. Initial program 80.8%

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

      1. *-commutative 80.8%

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

      2. associate-*r* 80.8%

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

      3. associate-*r/ 80.8%

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

      4. associate-/l* 80.8%

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

      5. /-rgt-identity 80.8%

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

      6. associate-/l* 80.8%

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

      7. difference-of-squares 88.9%

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

      8. associate-/l* 88.9%

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

      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/ 89.7%

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

      11. sub-neg 89.7%

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

      12. distribute-neg-frac 89.7%

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

      13. metadata-eval 89.7%

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

   3. Simplified 89.7%

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

   4. Taylor expanded in a around inf 65.8%

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

   5. Taylor expanded in b around 0 91.6%

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

      1. associate-*r/ 91.6%

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

   7. Simplified 91.6%

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

      1. clear-num 91.0%

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

      2. frac-times 91.0%

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

      3. metadata-eval 91.0%

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

      4. *-un-lft-identity 91.0%

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

      5. times-frac 91.0%

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

      6. metadata-eval 91.0%

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

   9. Applied egg-rr 91.0%

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

       1. associate-*r/ 96.8%

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

       2. associate-/l* 97.5%

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

       3. associate-*r* 97.5%

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

       4. metadata-eval 97.5%

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

       5. +-commutative 97.5%

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

   11. Simplified 97.5%

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

   if 7.1999999999999998e93 < b

   1. Initial program 65.1%

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

      1. associate-*r/ 65.1%

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

      2. *-rgt-identity 65.1%

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

      3. associate-*l/ 65.2%

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

      4. difference-of-squares 80.8%

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

      5. *-commutative 80.8%

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

      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. Step-by-step derivation

      1. associate-*l/ 99.9%

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

      2. div-inv 99.9%

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

      3. metadata-eval 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in a around 0 99.8%

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

      1. associate-*r/ 99.8%

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

      2. *-commutative 99.8%

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

      3. *-commutative 99.8%

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

      4. times-frac 99.8%

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

   8. Simplified 99.8%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.2 \cdot 10^{+93}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{b \cdot \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{b} \cdot \frac{0.5}{a}}{b - a}\\ \end{array} \]

Alternative 2: 90.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 1.65 \cdot 10^{-176}:\\ \;\;\;\;\frac{\pi \cdot -0.5}{a \cdot \left(b \cdot \left(-a\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b \cdot a} \cdot \frac{\pi}{b - a}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 1.65e-176)
   (/ (* PI -0.5) (* a (* b (- a))))
   (* (/ 0.5 (* b a)) (/ PI (- b a)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 1.65e-176) {
		tmp = (((double) M_PI) * -0.5) / (a * (b * -a));
	} else {
		tmp = (0.5 / (b * a)) * (((double) M_PI) / (b - a));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 1.65e-176) {
		tmp = (Math.PI * -0.5) / (a * (b * -a));
	} else {
		tmp = (0.5 / (b * a)) * (Math.PI / (b - a));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 1.65e-176:
		tmp = (math.pi * -0.5) / (a * (b * -a))
	else:
		tmp = (0.5 / (b * a)) * (math.pi / (b - a))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 1.65e-176)
		tmp = Float64(Float64(pi * -0.5) / Float64(a * Float64(b * Float64(-a))));
	else
		tmp = Float64(Float64(0.5 / Float64(b * a)) * 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 <= 1.65e-176)
		tmp = (pi * -0.5) / (a * (b * -a));
	else
		tmp = (0.5 / (b * a)) * (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, 1.65e-176], N[(N[(Pi * -0.5), $MachinePrecision] / N[(a * N[(b * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / N[(b * a), $MachinePrecision]), $MachinePrecision] * N[(Pi / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.65000000000000006e-176

   1. Initial program 73.9%

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

      1. associate-*r/ 73.9%

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

      2. *-rgt-identity 73.9%

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

      3. associate-*l/ 73.9%

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

      4. difference-of-squares 86.6%

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

      5. *-commutative 86.6%

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

      6. times-frac 99.6%

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

      7. sub-neg 99.6%

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

      8. distribute-neg-frac 99.6%

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

      9. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a around inf 80.1%

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

   5. Taylor expanded in b around 0 73.8%

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

      1. associate-*r/ 90.3%

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

   7. Simplified 73.8%

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

      1. *-commutative 73.8%

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

      2. frac-2neg 73.8%

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

      3. metadata-eval 73.8%

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

      4. frac-times 73.1%

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

      5. *-un-lft-identity 73.1%

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

      6. *-commutative 73.1%

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

      7. *-commutative 73.1%

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

   9. Applied egg-rr 73.1%

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

   if 1.65000000000000006e-176 < b

   1. Initial program 82.6%

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

      1. associate-*r/ 82.7%

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

      2. *-rgt-identity 82.7%

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

      3. associate-*l/ 82.6%

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

      4. difference-of-squares 88.4%

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

      5. *-commutative 88.4%

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

      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. 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%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in a around 0 81.6%

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

      1. associate-/r* 81.6%

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

   8. Simplified 81.6%

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

      1. expm1-log1p-u 69.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(\pi \cdot 0.5\right) \cdot \frac{\frac{1}{a}}{b}}{b - a}\right)\right)} \]

      2. expm1-udef 51.5%

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

      3. associate-/l* 51.5%

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

      4. *-commutative 51.5%

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

      5. associate-/l/ 51.5%

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

   10. Applied egg-rr 51.5%

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

       1. expm1-def 69.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.5 \cdot \pi}{\frac{b - a}{\frac{1}{b \cdot a}}}\right)\right)} \]

       2. expm1-log1p 81.3%

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

       3. associate-/r/ 81.6%

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

       4. associate-*r/ 81.6%

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

       5. *-commutative 81.6%

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

       6. associate-*r* 81.6%

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

       7. *-commutative 81.6%

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

       8. associate-*r/ 81.6%

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

       9. metadata-eval 81.6%

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

       10. *-commutative 81.6%

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

   12. Simplified 81.6%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.65 \cdot 10^{-176}:\\ \;\;\;\;\frac{\pi \cdot -0.5}{a \cdot \left(b \cdot \left(-a\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b \cdot a} \cdot \frac{\pi}{b - a}\\ \end{array} \]

Alternative 3: 99.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{+147}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b \cdot \left(\frac{a}{\pi} \cdot \left(b + a\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -1e+147)
   (* (/ PI a) (/ 0.5 (* b a)))
   (/ 0.5 (* b (* (/ a PI) (+ b a))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -1e+147) {
		tmp = (((double) M_PI) / a) * (0.5 / (b * a));
	} else {
		tmp = 0.5 / (b * ((a / ((double) M_PI)) * (b + a)));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -1e+147) {
		tmp = (Math.PI / a) * (0.5 / (b * a));
	} else {
		tmp = 0.5 / (b * ((a / Math.PI) * (b + a)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -1e+147:
		tmp = (math.pi / a) * (0.5 / (b * a))
	else:
		tmp = 0.5 / (b * ((a / math.pi) * (b + a)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -1e+147)
		tmp = Float64(Float64(pi / a) * Float64(0.5 / Float64(b * a)));
	else
		tmp = Float64(0.5 / Float64(b * Float64(Float64(a / 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 (a <= -1e+147)
		tmp = (pi / a) * (0.5 / (b * a));
	else
		tmp = 0.5 / (b * ((a / 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[a, -1e+147], N[(N[(Pi / a), $MachinePrecision] * N[(0.5 / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(b * N[(N[(a / Pi), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -9.9999999999999998e146

   1. Initial program 60.6%

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

      1. associate-*r/ 60.6%

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

      2. *-rgt-identity 60.6%

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

      3. associate-*l/ 60.6%

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

      4. difference-of-squares 75.3%

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

      5. *-commutative 75.3%

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

      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. Taylor expanded in a around inf 99.9%

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

   5. Taylor expanded in b around 0 99.9%

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

      1. associate-*r/ 76.7%

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

   7. Simplified 99.9%

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

      1. associate-/l* 99.9%

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

      2. frac-times 99.8%

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

      3. metadata-eval 99.8%

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

      4. *-commutative 99.8%

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

   9. Applied egg-rr 99.8%

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

       1. associate-*r* 99.7%

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

       2. associate-/l/ 99.8%

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

       3. *-un-lft-identity 99.8%

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

       4. times-frac 99.7%

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

       5. clear-num 99.8%

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

       6. associate-/r* 99.9%

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

       7. *-commutative 99.9%

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

   11. Applied egg-rr 99.9%

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

   if -9.9999999999999998e146 < a

   1. Initial program 80.7%

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

      1. *-commutative 80.7%

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

      2. associate-*r* 80.7%

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

      3. associate-*r/ 80.8%

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

      4. associate-/l* 80.8%

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

      5. /-rgt-identity 80.8%

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

      6. associate-/l* 80.7%

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

      7. difference-of-squares 89.3%

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

      8. associate-/l* 89.3%

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

      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/ 89.9%

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

      11. sub-neg 89.9%

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

      12. distribute-neg-frac 89.9%

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

      13. metadata-eval 89.9%

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

   3. Simplified 89.9%

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

   4. Taylor expanded in a around inf 62.4%

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

   5. Taylor expanded in b around 0 95.5%

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

      1. associate-*r/ 95.5%

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

   7. Simplified 95.5%

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

      1. expm1-log1p-u 71.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-1}{b} \cdot \frac{\frac{-0.5 \cdot \pi}{a}}{b + a}\right)\right)} \]

      2. expm1-udef 50.4%

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

      3. associate-*r/ 50.4%

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

      4. associate-/l* 50.4%

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

      5. frac-times 50.4%

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

      6. metadata-eval 50.4%

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

   9. Applied egg-rr 50.4%

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

       1. expm1-def 75.8%

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

       2. expm1-log1p 99.7%

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

       3. associate-/r* 99.7%

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

       4. +-commutative 99.7%

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

   11. Simplified 99.7%

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

       1. expm1-log1p-u 75.8%

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

       2. expm1-udef 50.4%

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

       3. div-inv 50.4%

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

       4. associate-/l/ 50.4%

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

       5. frac-times 50.4%

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

       6. metadata-eval 50.4%

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

       7. *-commutative 50.4%

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

       8. +-commutative 50.4%

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

   13. Applied egg-rr 50.4%

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

       1. expm1-def 74.9%

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

       2. expm1-log1p 98.8%

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

       3. associate-*l* 95.1%

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

       4. +-commutative 95.1%

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

   15. Simplified 95.1%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{+147}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b \cdot \left(\frac{a}{\pi} \cdot \left(b + a\right)\right)}\\ \end{array} \]

Alternative 4: 99.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 3.6 \cdot 10^{+131}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{b \cdot \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{0.5}{b \cdot a}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 3.6e+131)
   (/ (* 0.5 (/ PI a)) (* b (+ b a)))
   (* (/ PI b) (/ 0.5 (* b a)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 3.6e+131) {
		tmp = (0.5 * (((double) M_PI) / a)) / (b * (b + a));
	} else {
		tmp = (((double) M_PI) / b) * (0.5 / (b * a));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 3.6e+131) {
		tmp = (0.5 * (Math.PI / a)) / (b * (b + a));
	} else {
		tmp = (Math.PI / b) * (0.5 / (b * a));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 3.6e+131:
		tmp = (0.5 * (math.pi / a)) / (b * (b + a))
	else:
		tmp = (math.pi / b) * (0.5 / (b * a))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 3.6e+131)
		tmp = Float64(Float64(0.5 * Float64(pi / a)) / Float64(b * Float64(b + a)));
	else
		tmp = Float64(Float64(pi / b) * Float64(0.5 / 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 <= 3.6e+131)
		tmp = (0.5 * (pi / a)) / (b * (b + a));
	else
		tmp = (pi / b) * (0.5 / (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, 3.6e+131], N[(N[(0.5 * N[(Pi / a), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / b), $MachinePrecision] * N[(0.5 / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.60000000000000031e131

   1. Initial program 81.8%

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

      1. *-commutative 81.8%

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

      2. associate-*r* 81.8%

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

      3. associate-*r/ 81.9%

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

      4. associate-/l* 81.9%

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

      5. /-rgt-identity 81.9%

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

      6. associate-/l* 81.8%

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

      7. difference-of-squares 89.5%

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

      8. associate-/l* 89.5%

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

      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/ 90.3%

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

      11. sub-neg 90.3%

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

      12. distribute-neg-frac 90.3%

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

      13. metadata-eval 90.3%

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

   3. Simplified 90.3%

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

   4. Taylor expanded in a around inf 65.0%

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

   5. Taylor expanded in b around 0 92.0%

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

      1. associate-*r/ 92.0%

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

   7. Simplified 92.0%

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

      1. clear-num 91.5%

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

      2. frac-times 91.5%

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

      3. metadata-eval 91.5%

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

      4. *-un-lft-identity 91.5%

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

      5. times-frac 91.5%

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

      6. metadata-eval 91.5%

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

   9. Applied egg-rr 91.5%

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

       1. associate-*r/ 96.9%

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

       2. associate-/l* 97.7%

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

       3. associate-*r* 97.7%

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

       4. metadata-eval 97.7%

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

       5. +-commutative 97.7%

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

   11. Simplified 97.7%

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

   if 3.60000000000000031e131 < b

   1. Initial program 52.5%

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

      1. associate-*r/ 52.5%

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

      2. *-rgt-identity 52.5%

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

      3. associate-*l/ 52.6%

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

      4. difference-of-squares 73.8%

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

      5. *-commutative 73.8%

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

      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. Step-by-step derivation

      1. associate-*l/ 99.9%

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

      2. div-inv 99.9%

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

      3. metadata-eval 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in a around 0 99.7%

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

      1. associate-/r* 99.9%

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

   8. Simplified 99.9%

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

      1. expm1-log1p-u 96.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(\pi \cdot 0.5\right) \cdot \frac{\frac{1}{a}}{b}}{b - a}\right)\right)} \]

      2. expm1-udef 59.1%

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

      3. associate-/l* 59.1%

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

      4. *-commutative 59.1%

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

      5. associate-/l/ 59.1%

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

   10. Applied egg-rr 59.1%

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

       1. expm1-def 95.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.5 \cdot \pi}{\frac{b - a}{\frac{1}{b \cdot a}}}\right)\right)} \]

       2. expm1-log1p 98.6%

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

       3. associate-/r/ 99.8%

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

       4. associate-*r/ 99.8%

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

       5. *-commutative 99.8%

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

       6. associate-*r* 99.8%

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

       7. *-commutative 99.8%

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

       8. associate-*r/ 99.8%

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

       9. metadata-eval 99.8%

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

       10. *-commutative 99.8%

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

   12. Simplified 99.8%

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

   13. Taylor expanded in b around inf 99.8%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.6 \cdot 10^{+131}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{b \cdot \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{0.5}{b \cdot a}\\ \end{array} \]

Alternative 5: 99.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 10^{+93}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{b \cdot \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{b \cdot a}}{b - a}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 1e+93)
   (/ (* 0.5 (/ PI a)) (* b (+ b a)))
   (/ (* 0.5 (/ PI (* b a))) (- b a))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 1e+93) {
		tmp = (0.5 * (((double) M_PI) / a)) / (b * (b + a));
	} 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 <= 1e+93) {
		tmp = (0.5 * (Math.PI / a)) / (b * (b + a));
	} 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 <= 1e+93:
		tmp = (0.5 * (math.pi / a)) / (b * (b + a))
	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 <= 1e+93)
		tmp = Float64(Float64(0.5 * Float64(pi / a)) / Float64(b * Float64(b + a)));
	else
		tmp = Float64(Float64(0.5 * Float64(pi / Float64(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 <= 1e+93)
		tmp = (0.5 * (pi / a)) / (b * (b + a));
	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, 1e+93], N[(N[(0.5 * N[(Pi / a), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(Pi / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.00000000000000004e93

   1. Initial program 80.8%

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

      1. *-commutative 80.8%

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

      2. associate-*r* 80.8%

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

      3. associate-*r/ 80.8%

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

      4. associate-/l* 80.8%

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

      5. /-rgt-identity 80.8%

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

      6. associate-/l* 80.8%

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

      7. difference-of-squares 88.9%

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

      8. associate-/l* 88.9%

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

      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/ 89.7%

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

      11. sub-neg 89.7%

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

      12. distribute-neg-frac 89.7%

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

      13. metadata-eval 89.7%

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

   3. Simplified 89.7%

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

   4. Taylor expanded in a around inf 65.8%

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

   5. Taylor expanded in b around 0 91.6%

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

      1. associate-*r/ 91.6%

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

   7. Simplified 91.6%

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

      1. clear-num 91.0%

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

      2. frac-times 91.0%

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

      3. metadata-eval 91.0%

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

      4. *-un-lft-identity 91.0%

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

      5. times-frac 91.0%

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

      6. metadata-eval 91.0%

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

   9. Applied egg-rr 91.0%

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

       1. associate-*r/ 96.8%

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

       2. associate-/l* 97.5%

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

       3. associate-*r* 97.5%

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

       4. metadata-eval 97.5%

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

       5. +-commutative 97.5%

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

   11. Simplified 97.5%

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

   if 1.00000000000000004e93 < b

   1. Initial program 65.1%

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

      1. associate-*r/ 65.1%

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

      2. *-rgt-identity 65.1%

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

      3. associate-*l/ 65.2%

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

      4. difference-of-squares 80.8%

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

      5. *-commutative 80.8%

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

      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. Step-by-step derivation

      1. associate-*l/ 99.9%

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

      2. div-inv 99.9%

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

      3. metadata-eval 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in a around 0 99.8%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 10^{+93}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{b \cdot \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{b \cdot a}}{b - a}\\ \end{array} \]

Alternative 6: 89.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1.3 \cdot 10^{-41}:\\ \;\;\;\;\frac{\pi \cdot -0.5}{a \cdot \left(b \cdot \left(-a\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{0.5}{b \cdot a}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -1.3e-41)
   (/ (* PI -0.5) (* a (* b (- a))))
   (* (/ PI b) (/ 0.5 (* b a)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -1.3e-41) {
		tmp = (((double) M_PI) * -0.5) / (a * (b * -a));
	} else {
		tmp = (((double) M_PI) / b) * (0.5 / (b * a));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -1.3e-41) {
		tmp = (Math.PI * -0.5) / (a * (b * -a));
	} else {
		tmp = (Math.PI / b) * (0.5 / (b * a));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -1.3e-41:
		tmp = (math.pi * -0.5) / (a * (b * -a))
	else:
		tmp = (math.pi / b) * (0.5 / (b * a))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -1.3e-41)
		tmp = Float64(Float64(pi * -0.5) / Float64(a * Float64(b * Float64(-a))));
	else
		tmp = Float64(Float64(pi / b) * Float64(0.5 / Float64(b * a)));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.3e-41)
		tmp = (pi * -0.5) / (a * (b * -a));
	else
		tmp = (pi / b) * (0.5 / (b * a));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -1.3e-41], N[(N[(Pi * -0.5), $MachinePrecision] / N[(a * N[(b * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / b), $MachinePrecision] * N[(0.5 / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.3e-41

   1. Initial program 80.8%

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

      1. associate-*r/ 80.8%

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

      2. *-rgt-identity 80.8%

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

      3. associate-*l/ 80.8%

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

      4. difference-of-squares 87.7%

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

      5. *-commutative 87.7%

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

      6. times-frac 99.6%

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

      7. sub-neg 99.6%

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

      8. distribute-neg-frac 99.6%

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

      9. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a around inf 92.6%

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

   5. Taylor expanded in b around 0 82.8%

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

      1. associate-*r/ 88.9%

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

   7. Simplified 82.8%

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

      1. *-commutative 82.8%

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

      2. frac-2neg 82.8%

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

      3. metadata-eval 82.8%

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

      4. frac-times 82.8%

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

      5. *-un-lft-identity 82.8%

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

      6. *-commutative 82.8%

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

      7. *-commutative 82.8%

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

   9. Applied egg-rr 82.8%

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

   if -1.3e-41 < a

   1. Initial program 77.0%

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

      1. associate-*r/ 77.0%

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

      2. *-rgt-identity 77.0%

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

      3. associate-*l/ 77.0%

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

      4. difference-of-squares 87.3%

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

      5. *-commutative 87.3%

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

      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. 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%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in a around 0 65.6%

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

      1. associate-/r* 65.6%

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

   8. Simplified 65.6%

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

      1. expm1-log1p-u 53.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(\pi \cdot 0.5\right) \cdot \frac{\frac{1}{a}}{b}}{b - a}\right)\right)} \]

      2. expm1-udef 39.4%

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

      3. associate-/l* 39.4%

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

      4. *-commutative 39.4%

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

      5. associate-/l/ 39.4%

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

   10. Applied egg-rr 39.4%

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

       1. expm1-def 53.1%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.5 \cdot \pi}{\frac{b - a}{\frac{1}{b \cdot a}}}\right)\right)} \]

       2. expm1-log1p 65.6%

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

       3. associate-/r/ 65.6%

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

       4. associate-*r/ 65.6%

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

       5. *-commutative 65.6%

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

       6. associate-*r* 65.6%

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

       7. *-commutative 65.6%

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

       8. associate-*r/ 65.6%

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

       9. metadata-eval 65.6%

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

       10. *-commutative 65.6%

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

   12. Simplified 65.6%

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

   13. Taylor expanded in b around inf 63.2%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.3 \cdot 10^{-41}:\\ \;\;\;\;\frac{\pi \cdot -0.5}{a \cdot \left(b \cdot \left(-a\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{0.5}{b \cdot a}\\ \end{array} \]

Alternative 7: 89.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} t_0 := \frac{0.5}{b \cdot a}\\ \mathbf{if}\;a \leq -1.2 \cdot 10^{-41}:\\ \;\;\;\;\frac{\pi}{a} \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b} \cdot t_0\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ 0.5 (* b a))))
   (if (<= a -1.2e-41) (* (/ PI a) t_0) (* (/ PI b) t_0))))

C

assert(a < b);
double code(double a, double b) {
	double t_0 = 0.5 / (b * a);
	double tmp;
	if (a <= -1.2e-41) {
		tmp = (((double) M_PI) / a) * t_0;
	} else {
		tmp = (((double) M_PI) / b) * t_0;
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double t_0 = 0.5 / (b * a);
	double tmp;
	if (a <= -1.2e-41) {
		tmp = (Math.PI / a) * t_0;
	} else {
		tmp = (Math.PI / b) * t_0;
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	t_0 = 0.5 / (b * a)
	tmp = 0
	if a <= -1.2e-41:
		tmp = (math.pi / a) * t_0
	else:
		tmp = (math.pi / b) * t_0
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	t_0 = Float64(0.5 / Float64(b * a))
	tmp = 0.0
	if (a <= -1.2e-41)
		tmp = Float64(Float64(pi / a) * t_0);
	else
		tmp = Float64(Float64(pi / b) * t_0);
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	t_0 = 0.5 / (b * a);
	tmp = 0.0;
	if (a <= -1.2e-41)
		tmp = (pi / a) * t_0;
	else
		tmp = (pi / b) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := Block[{t$95$0 = N[(0.5 / N[(b * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.2e-41], N[(N[(Pi / a), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(Pi / b), $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.20000000000000011e-41

   1. Initial program 80.8%

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

      1. associate-*r/ 80.8%

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

      2. *-rgt-identity 80.8%

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

      3. associate-*l/ 80.8%

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

      4. difference-of-squares 87.7%

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

      5. *-commutative 87.7%

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

      6. times-frac 99.6%

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

      7. sub-neg 99.6%

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

      8. distribute-neg-frac 99.6%

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

      9. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a around inf 92.6%

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

   5. Taylor expanded in b around 0 82.8%

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

      1. associate-*r/ 88.9%

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

   7. Simplified 82.8%

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

      1. associate-/l* 82.8%

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

      2. frac-times 82.8%

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

      3. metadata-eval 82.8%

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

      4. *-commutative 82.8%

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

   9. Applied egg-rr 82.8%

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

       1. associate-*r* 82.8%

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

       2. associate-/l/ 82.9%

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

       3. *-un-lft-identity 82.9%

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

       4. times-frac 82.8%

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

       5. clear-num 82.8%

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

       6. associate-/r* 82.8%

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

       7. *-commutative 82.8%

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

   11. Applied egg-rr 82.8%

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

   if -1.20000000000000011e-41 < a

   1. Initial program 77.0%

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

      1. associate-*r/ 77.0%

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

      2. *-rgt-identity 77.0%

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

      3. associate-*l/ 77.0%

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

      4. difference-of-squares 87.3%

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

      5. *-commutative 87.3%

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

      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. 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%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in a around 0 65.6%

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

      1. associate-/r* 65.6%

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

   8. Simplified 65.6%

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

      1. expm1-log1p-u 53.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(\pi \cdot 0.5\right) \cdot \frac{\frac{1}{a}}{b}}{b - a}\right)\right)} \]

      2. expm1-udef 39.4%

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

      3. associate-/l* 39.4%

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

      4. *-commutative 39.4%

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

      5. associate-/l/ 39.4%

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

   10. Applied egg-rr 39.4%

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

       1. expm1-def 53.1%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.5 \cdot \pi}{\frac{b - a}{\frac{1}{b \cdot a}}}\right)\right)} \]

       2. expm1-log1p 65.6%

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

       3. associate-/r/ 65.6%

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

       4. associate-*r/ 65.6%

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

       5. *-commutative 65.6%

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

       6. associate-*r* 65.6%

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

       7. *-commutative 65.6%

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

       8. associate-*r/ 65.6%

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

       9. metadata-eval 65.6%

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

       10. *-commutative 65.6%

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

   12. Simplified 65.6%

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

   13. Taylor expanded in b around inf 63.2%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.2 \cdot 10^{-41}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{0.5}{b \cdot a}\\ \end{array} \]

Alternative 8: 89.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{-41}:\\ \;\;\;\;\frac{\frac{0.5}{a}}{b \cdot \frac{a}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{0.5}{b \cdot a}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -1e-41) (/ (/ 0.5 a) (* b (/ a PI))) (* (/ PI b) (/ 0.5 (* b a)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -1e-41) {
		tmp = (0.5 / a) / (b * (a / ((double) M_PI)));
	} else {
		tmp = (((double) M_PI) / b) * (0.5 / (b * a));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -1e-41) {
		tmp = (0.5 / a) / (b * (a / Math.PI));
	} else {
		tmp = (Math.PI / b) * (0.5 / (b * a));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -1e-41:
		tmp = (0.5 / a) / (b * (a / math.pi))
	else:
		tmp = (math.pi / b) * (0.5 / (b * a))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -1e-41)
		tmp = Float64(Float64(0.5 / a) / Float64(b * Float64(a / pi)));
	else
		tmp = Float64(Float64(pi / b) * Float64(0.5 / Float64(b * a)));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1e-41)
		tmp = (0.5 / a) / (b * (a / pi));
	else
		tmp = (pi / b) * (0.5 / (b * a));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -1e-41], N[(N[(0.5 / a), $MachinePrecision] / N[(b * N[(a / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / b), $MachinePrecision] * N[(0.5 / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.00000000000000001e-41

   1. Initial program 80.8%

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

      1. associate-*r/ 80.8%

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

      2. *-rgt-identity 80.8%

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

      3. associate-*l/ 80.8%

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

      4. difference-of-squares 87.7%

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

      5. *-commutative 87.7%

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

      6. times-frac 99.6%

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

      7. sub-neg 99.6%

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

      8. distribute-neg-frac 99.6%

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

      9. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a around inf 92.6%

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

   5. Taylor expanded in b around 0 82.8%

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

      1. associate-*r/ 88.9%

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

   7. Simplified 82.8%

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

      1. associate-/l* 82.8%

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

      2. associate-/r* 82.8%

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

      3. frac-times 82.9%

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

   9. Applied egg-rr 82.9%

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

       1. associate-*r/ 82.9%

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

       2. metadata-eval 82.9%

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

   11. Simplified 82.9%

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

   if -1.00000000000000001e-41 < a

   1. Initial program 77.0%

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

      1. associate-*r/ 77.0%

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

      2. *-rgt-identity 77.0%

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

      3. associate-*l/ 77.0%

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

      4. difference-of-squares 87.3%

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

      5. *-commutative 87.3%

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

      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. 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%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in a around 0 65.6%

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

      1. associate-/r* 65.6%

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

   8. Simplified 65.6%

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

      1. expm1-log1p-u 53.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(\pi \cdot 0.5\right) \cdot \frac{\frac{1}{a}}{b}}{b - a}\right)\right)} \]

      2. expm1-udef 39.4%

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

      3. associate-/l* 39.4%

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

      4. *-commutative 39.4%

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

      5. associate-/l/ 39.4%

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

   10. Applied egg-rr 39.4%

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

       1. expm1-def 53.1%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.5 \cdot \pi}{\frac{b - a}{\frac{1}{b \cdot a}}}\right)\right)} \]

       2. expm1-log1p 65.6%

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

       3. associate-/r/ 65.6%

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

       4. associate-*r/ 65.6%

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

       5. *-commutative 65.6%

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

       6. associate-*r* 65.6%

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

       7. *-commutative 65.6%

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

       8. associate-*r/ 65.6%

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

       9. metadata-eval 65.6%

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

       10. *-commutative 65.6%

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

   12. Simplified 65.6%

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

   13. Taylor expanded in b around inf 63.2%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{-41}:\\ \;\;\;\;\frac{\frac{0.5}{a}}{b \cdot \frac{a}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{0.5}{b \cdot a}\\ \end{array} \]

Alternative 9: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.0%

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

   1. *-commutative 78.0%

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

   2. associate-*r* 78.0%

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

   3. associate-*r/ 78.1%

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

   4. associate-/l* 78.1%

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

   5. /-rgt-identity 78.1%

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

   6. associate-/l* 78.0%

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

   7. difference-of-squares 87.4%

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

   8. associate-/l* 87.5%

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

   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/ 88.1%

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

   11. sub-neg 88.1%

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

   12. distribute-neg-frac 88.1%

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

   13. metadata-eval 88.1%

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

3. Simplified 88.1%

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

4. Taylor expanded in a around inf 64.3%

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

5. Taylor expanded in b around 0 93.0%

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

   1. associate-*r/ 93.0%

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

7. Simplified 93.0%

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

   1. expm1-log1p-u 72.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-1}{b} \cdot \frac{\frac{-0.5 \cdot \pi}{a}}{b + a}\right)\right)} \]

   2. expm1-udef 53.3%

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

   3. associate-*r/ 53.3%

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

   4. associate-/l* 53.3%

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

   5. frac-times 53.3%

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

   6. metadata-eval 53.3%

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

9. Applied egg-rr 53.3%

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

    1. expm1-def 79.0%

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

    2. expm1-log1p 99.7%

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

    3. associate-/r* 99.7%

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

    4. +-commutative 99.7%

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

11. Simplified 99.7%

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

12. Final simplification 99.7%

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

Alternative 10: 63.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.0%

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

   1. associate-*r/ 78.1%

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

   2. *-rgt-identity 78.1%

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

   3. associate-*l/ 78.1%

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

   4. difference-of-squares 87.5%

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

   5. *-commutative 87.5%

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

   6. times-frac 99.6%

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

   7. sub-neg 99.6%

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

   8. distribute-neg-frac 99.6%

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

   9. metadata-eval 99.6%

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

3. Simplified 99.6%

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

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

5. Applied egg-rr 99.7%

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

6. Taylor expanded in a around 0 64.9%

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

   1. associate-/r* 65.0%

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

8. Simplified 65.0%

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

   1. expm1-log1p-u 56.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(\pi \cdot 0.5\right) \cdot \frac{\frac{1}{a}}{b}}{b - a}\right)\right)} \]

   2. expm1-udef 44.3%

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

   3. associate-/l* 44.3%

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

   4. *-commutative 44.3%

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

   5. associate-/l/ 44.3%

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

10. Applied egg-rr 44.3%

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

    1. expm1-def 55.8%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.5 \cdot \pi}{\frac{b - a}{\frac{1}{b \cdot a}}}\right)\right)} \]

    2. expm1-log1p 64.9%

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

    3. associate-/r/ 65.0%

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

    4. associate-*r/ 65.0%

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

    5. *-commutative 65.0%

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

    6. associate-*r* 65.0%

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

    7. *-commutative 65.0%

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

    8. associate-*r/ 65.0%

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

    9. metadata-eval 65.0%

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

    10. *-commutative 65.0%

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

12. Simplified 65.0%

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

13. Taylor expanded in b around inf 58.9%

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

14. Final simplification 58.9%

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

Reproduce

herbie shell --seed 2023322 
(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))))