NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.9% → 99.6%

Time: 13.5s

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: 78.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{+115}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{b \cdot \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{a} \cdot \frac{0.5}{b}}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 5e+115)
   (* (/ PI a) (/ 0.5 (* b (+ b a))))
   (/ (* (/ PI a) (/ 0.5 b)) b)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 5e+115) {
		tmp = (((double) M_PI) / a) * (0.5 / (b * (b + a)));
	} else {
		tmp = ((((double) M_PI) / a) * (0.5 / b)) / b;
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 5e+115) {
		tmp = (Math.PI / a) * (0.5 / (b * (b + a)));
	} else {
		tmp = ((Math.PI / a) * (0.5 / b)) / b;
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 5e+115:
		tmp = (math.pi / a) * (0.5 / (b * (b + a)))
	else:
		tmp = ((math.pi / a) * (0.5 / b)) / b
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 5e+115)
		tmp = Float64(Float64(pi / a) * Float64(0.5 / Float64(b * Float64(b + a))));
	else
		tmp = Float64(Float64(Float64(pi / a) * Float64(0.5 / b)) / b);
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 5e+115)
		tmp = (pi / a) * (0.5 / (b * (b + a)));
	else
		tmp = ((pi / a) * (0.5 / b)) / b;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[b, 5e+115], N[(N[(Pi / a), $MachinePrecision] * N[(0.5 / N[(b * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi / a), $MachinePrecision] * N[(0.5 / b), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.00000000000000008e115

   1. Initial program 81.6%

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

      1. div-inv 81.6%

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

      2. expm1-log1p-u 58.2%

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

      3. expm1-udef 43.4%

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

   3. Applied egg-rr 48.5%

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

      1. expm1-def 74.3%

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

      2. expm1-log1p 99.6%

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

      3. associate-*r/ 99.6%

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

      4. associate-*r/ 99.6%

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

      5. *-commutative 99.6%

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

      6. +-commutative 99.6%

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

      7. *-commutative 99.6%

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

   5. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-*r/ 99.6%

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

      4. associate-*l* 99.6%

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

   7. Applied egg-rr 99.6%

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

      1. associate-*r/ 99.6%

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

      2. metadata-eval 99.6%

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

      3. +-commutative 99.6%

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

   9. Simplified 99.6%

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

       1. times-frac 96.6%

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

       2. associate-/l/ 96.1%

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

   11. Applied egg-rr 96.1%

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

   if 5.00000000000000008e115 < b

   1. Initial program 79.3%

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

      1. div-inv 79.4%

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

      2. expm1-log1p-u 79.4%

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

      3. expm1-udef 56.4%

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

   3. Applied egg-rr 63.9%

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

      1. expm1-def 99.8%

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

      2. expm1-log1p 99.8%

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

      3. associate-*r/ 99.7%

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

      4. associate-*r/ 99.7%

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

      5. *-commutative 99.7%

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

      6. +-commutative 99.7%

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

      7. *-commutative 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in a around 0 99.7%

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

      1. *-commutative 99.7%

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

      2. times-frac 99.8%

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

   8. Applied egg-rr 99.8%

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

      1. associate-/l/ 99.8%

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

      2. associate-*l/ 99.7%

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

      3. associate-/r* 99.9%

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

      4. associate-*l/ 99.9%

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

   10. Applied egg-rr 99.9%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{+115}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{b \cdot \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{a} \cdot \frac{0.5}{b}}{b}\\ \end{array} \]

Alternative 2: 99.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -2.1 \cdot 10^{+139}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{\frac{0.5}{b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{\frac{0.5}{b}}{a \cdot \left(b + a\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -2.1e+139)
   (* (/ PI a) (/ (/ 0.5 b) a))
   (* PI (/ (/ 0.5 b) (* a (+ b a))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -2.1e+139) {
		tmp = (((double) M_PI) / a) * ((0.5 / b) / a);
	} else {
		tmp = ((double) M_PI) * ((0.5 / b) / (a * (b + a)));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -2.1e+139) {
		tmp = (Math.PI / a) * ((0.5 / b) / a);
	} else {
		tmp = Math.PI * ((0.5 / b) / (a * (b + a)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -2.1e+139:
		tmp = (math.pi / a) * ((0.5 / b) / a)
	else:
		tmp = math.pi * ((0.5 / b) / (a * (b + a)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -2.1e+139)
		tmp = Float64(Float64(pi / a) * Float64(Float64(0.5 / b) / a));
	else
		tmp = Float64(pi * Float64(Float64(0.5 / b) / Float64(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 (a <= -2.1e+139)
		tmp = (pi / a) * ((0.5 / b) / a);
	else
		tmp = pi * ((0.5 / 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[a, -2.1e+139], N[(N[(Pi / a), $MachinePrecision] * N[(N[(0.5 / b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(N[(0.5 / b), $MachinePrecision] / N[(a * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.0999999999999999e139

   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. div-inv 65.1%

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

      2. expm1-log1p-u 62.4%

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

      3. expm1-udef 60.0%

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

   3. Applied egg-rr 70.6%

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

      1. expm1-def 97.2%

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

      2. expm1-log1p 99.9%

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

      3. associate-*r/ 99.8%

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

      4. associate-*r/ 99.7%

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

      5. *-commutative 99.7%

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

      6. +-commutative 99.7%

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

      7. *-commutative 99.7%

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

   5. Simplified 99.7%

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

      1. *-commutative 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-*r/ 99.8%

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

      4. associate-*l* 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. associate-*r/ 99.8%

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

      2. metadata-eval 99.8%

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

      3. +-commutative 99.8%

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

   9. Simplified 99.8%

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

       1. div-inv 99.8%

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

       2. associate-*l* 99.8%

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

       3. *-commutative 99.8%

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

       4. associate-/r* 99.8%

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

       5. times-frac 75.6%

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

       6. div-inv 75.6%

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

       7. *-commutative 75.6%

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

       8. associate-*r/ 75.6%

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

       9. times-frac 99.9%

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

   11. Applied egg-rr 99.9%

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

   12. Taylor expanded in b around 0 99.8%

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

       1. associate-/l/ 99.9%

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

   14. Simplified 99.9%

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

   if -2.0999999999999999e139 < a

   1. Initial program 84.0%

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

      1. associate-*l* 84.0%

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

      2. associate-*l/ 84.0%

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

      3. frac-sub 84.0%

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

      4. *-commutative 84.0%

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

      5. associate-*r/ 84.0%

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

      6. *-un-lft-identity 84.0%

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

      7. *-rgt-identity 84.0%

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

      8. associate-/r/ 84.1%

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

      9. flip-+ 99.6%

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

   3. Applied egg-rr 99.6%

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

      1. associate-/l* 99.0%

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

      2. associate-/l/ 99.0%

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

      3. associate-*r* 96.0%

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

      4. +-commutative 96.0%

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

      5. associate-/r/ 96.0%

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

      6. metadata-eval 96.0%

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

      7. div-inv 96.0%

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

      8. associate-/r* 96.0%

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

      9. metadata-eval 96.0%

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

      10. +-commutative 96.0%

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

   5. Applied egg-rr 96.0%

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

      1. associate-/r* 96.4%

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

      2. +-commutative 96.4%

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

   7. Simplified 96.4%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.1 \cdot 10^{+139}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{\frac{0.5}{b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{\frac{0.5}{b}}{a \cdot \left(b + a\right)}\\ \end{array} \]

Alternative 3: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.2%

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

   1. associate-*l* 81.2%

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

   2. associate-*l/ 81.2%

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

   3. frac-sub 81.2%

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

   4. *-commutative 81.2%

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

   5. associate-*r/ 81.2%

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

   6. *-un-lft-identity 81.2%

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

   7. *-rgt-identity 81.2%

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

   8. associate-/r/ 81.3%

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

   9. flip-+ 99.6%

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

3. Applied egg-rr 99.6%

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

4. Final simplification 99.6%

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

Alternative 4: 89.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{-7}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{\frac{0.5}{b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b} \cdot \frac{\frac{\pi}{b}}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -9e-7) (* (/ PI a) (/ (/ 0.5 b) a)) (* (/ 0.5 b) (/ (/ PI b) a))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -9e-7) {
		tmp = (((double) M_PI) / a) * ((0.5 / b) / a);
	} else {
		tmp = (0.5 / b) * ((((double) M_PI) / b) / a);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -9e-7) {
		tmp = (Math.PI / a) * ((0.5 / b) / a);
	} else {
		tmp = (0.5 / b) * ((Math.PI / b) / a);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -9e-7:
		tmp = (math.pi / a) * ((0.5 / b) / a)
	else:
		tmp = (0.5 / b) * ((math.pi / b) / a)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -9e-7)
		tmp = Float64(Float64(pi / a) * Float64(Float64(0.5 / b) / a));
	else
		tmp = Float64(Float64(0.5 / b) * Float64(Float64(pi / b) / a));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -9e-7)
		tmp = (pi / a) * ((0.5 / b) / a);
	else
		tmp = (0.5 / b) * ((pi / b) / a);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -9e-7], N[(N[(Pi / a), $MachinePrecision] * N[(N[(0.5 / b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / b), $MachinePrecision] * N[(N[(Pi / b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -8.99999999999999959e-7

   1. Initial program 81.6%

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

      1. div-inv 81.6%

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

      2. expm1-log1p-u 66.1%

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

      3. expm1-udef 53.2%

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

   3. Applied egg-rr 58.6%

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

      1. expm1-def 84.2%

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

      2. expm1-log1p 99.7%

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

      3. associate-*r/ 99.6%

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

      4. associate-*r/ 99.7%

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

      5. *-commutative 99.7%

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

      6. +-commutative 99.7%

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

      7. *-commutative 99.7%

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

   5. Simplified 99.7%

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

      1. *-commutative 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-*r/ 99.6%

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

      4. associate-*l* 99.6%

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

   7. Applied egg-rr 99.6%

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

      1. associate-*r/ 99.6%

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

      2. metadata-eval 99.6%

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

      3. +-commutative 99.6%

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

   9. Simplified 99.6%

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

       1. div-inv 99.6%

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

       2. associate-*l* 99.6%

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

       3. *-commutative 99.6%

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

       4. associate-/r* 99.6%

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

       5. times-frac 87.1%

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

       6. div-inv 87.1%

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

       7. *-commutative 87.1%

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

       8. associate-*r/ 87.1%

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

       9. times-frac 99.7%

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

   11. Applied egg-rr 99.7%

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

   12. Taylor expanded in b around 0 92.0%

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

       1. associate-/l/ 91.9%

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

   14. Simplified 91.9%

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

   if -8.99999999999999959e-7 < a

   1. Initial program 81.1%

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

      1. div-inv 81.1%

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

      2. expm1-log1p-u 59.6%

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

      3. expm1-udef 42.3%

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

   3. Applied egg-rr 47.8%

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

      1. expm1-def 75.9%

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

      2. expm1-log1p 99.6%

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

      3. associate-*r/ 99.6%

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

      4. associate-*r/ 99.5%

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

      5. *-commutative 99.5%

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

      6. +-commutative 99.5%

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

      7. *-commutative 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in a around 0 70.6%

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

      1. *-commutative 70.6%

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

      2. times-frac 70.7%

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

   8. Applied egg-rr 70.7%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{-7}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{\frac{0.5}{b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b} \cdot \frac{\frac{\pi}{b}}{a}\\ \end{array} \]

Alternative 5: 89.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -4.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\pi \cdot \frac{0.5}{a}}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b} \cdot \frac{\frac{\pi}{b}}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -4.2e-6) (/ (* PI (/ 0.5 a)) (* b a)) (* (/ 0.5 b) (/ (/ PI b) a))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -4.2e-6) {
		tmp = (((double) M_PI) * (0.5 / a)) / (b * a);
	} else {
		tmp = (0.5 / b) * ((((double) M_PI) / b) / a);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -4.2e-6) {
		tmp = (Math.PI * (0.5 / a)) / (b * a);
	} else {
		tmp = (0.5 / b) * ((Math.PI / b) / a);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -4.2e-6:
		tmp = (math.pi * (0.5 / a)) / (b * a)
	else:
		tmp = (0.5 / b) * ((math.pi / b) / a)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -4.2e-6)
		tmp = Float64(Float64(pi * Float64(0.5 / a)) / Float64(b * a));
	else
		tmp = Float64(Float64(0.5 / b) * Float64(Float64(pi / b) / a));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -4.2e-6)
		tmp = (pi * (0.5 / a)) / (b * a);
	else
		tmp = (0.5 / b) * ((pi / b) / a);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -4.2e-6], N[(N[(Pi * N[(0.5 / a), $MachinePrecision]), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / b), $MachinePrecision] * N[(N[(Pi / b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.1999999999999996e-6

   1. Initial program 81.6%

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

      1. div-inv 81.6%

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

      2. expm1-log1p-u 66.1%

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

      3. expm1-udef 53.2%

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

   3. Applied egg-rr 58.6%

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

      1. expm1-def 84.2%

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

      2. expm1-log1p 99.7%

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

      3. associate-*r/ 99.6%

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

      4. associate-*r/ 99.7%

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

      5. *-commutative 99.7%

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

      6. +-commutative 99.7%

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

      7. *-commutative 99.7%

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

   5. Simplified 99.7%

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

      1. *-commutative 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-*r/ 99.6%

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

      4. associate-*l* 99.6%

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

   7. Applied egg-rr 99.6%

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

      1. associate-*r/ 99.6%

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

      2. metadata-eval 99.6%

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

      3. +-commutative 99.6%

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

   9. Simplified 99.6%

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

   10. Taylor expanded in a around inf 91.9%

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

   if -4.1999999999999996e-6 < a

   1. Initial program 81.1%

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

      1. div-inv 81.1%

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

      2. expm1-log1p-u 59.6%

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

      3. expm1-udef 42.3%

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

   3. Applied egg-rr 47.8%

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

      1. expm1-def 75.9%

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

      2. expm1-log1p 99.6%

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

      3. associate-*r/ 99.6%

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

      4. associate-*r/ 99.5%

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

      5. *-commutative 99.5%

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

      6. +-commutative 99.5%

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

      7. *-commutative 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in a around 0 70.6%

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

      1. *-commutative 70.6%

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

      2. times-frac 70.7%

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

   8. Applied egg-rr 70.7%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\pi \cdot \frac{0.5}{a}}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b} \cdot \frac{\frac{\pi}{b}}{a}\\ \end{array} \]

Alternative 6: 89.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1.25 \cdot 10^{-6}:\\ \;\;\;\;\frac{\pi \cdot \frac{0.5}{a}}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot \frac{0.5}{b}}{b \cdot a}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -1.25e-6)
   (/ (* PI (/ 0.5 a)) (* b a))
   (/ (* PI (/ 0.5 b)) (* b a))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -1.25e-6) {
		tmp = (((double) M_PI) * (0.5 / a)) / (b * a);
	} else {
		tmp = (((double) M_PI) * (0.5 / b)) / (b * a);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -1.25e-6) {
		tmp = (Math.PI * (0.5 / a)) / (b * a);
	} else {
		tmp = (Math.PI * (0.5 / b)) / (b * a);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -1.25e-6:
		tmp = (math.pi * (0.5 / a)) / (b * a)
	else:
		tmp = (math.pi * (0.5 / b)) / (b * a)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -1.25e-6)
		tmp = Float64(Float64(pi * Float64(0.5 / a)) / Float64(b * a));
	else
		tmp = Float64(Float64(pi * Float64(0.5 / b)) / 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 <= -1.25e-6)
		tmp = (pi * (0.5 / a)) / (b * a);
	else
		tmp = (pi * (0.5 / b)) / (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, -1.25e-6], N[(N[(Pi * N[(0.5 / a), $MachinePrecision]), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * N[(0.5 / b), $MachinePrecision]), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.2500000000000001e-6

   1. Initial program 81.6%

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

      1. div-inv 81.6%

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

      2. expm1-log1p-u 66.1%

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

      3. expm1-udef 53.2%

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

   3. Applied egg-rr 58.6%

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

      1. expm1-def 84.2%

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

      2. expm1-log1p 99.7%

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

      3. associate-*r/ 99.6%

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

      4. associate-*r/ 99.7%

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

      5. *-commutative 99.7%

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

      6. +-commutative 99.7%

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

      7. *-commutative 99.7%

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

   5. Simplified 99.7%

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

      1. *-commutative 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-*r/ 99.6%

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

      4. associate-*l* 99.6%

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

   7. Applied egg-rr 99.6%

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

      1. associate-*r/ 99.6%

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

      2. metadata-eval 99.6%

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

      3. +-commutative 99.6%

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

   9. Simplified 99.6%

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

   10. Taylor expanded in a around inf 91.9%

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

   if -1.2500000000000001e-6 < a

   1. Initial program 81.1%

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

      1. div-inv 81.1%

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

      2. expm1-log1p-u 59.6%

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

      3. expm1-udef 42.3%

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

   3. Applied egg-rr 47.8%

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

      1. expm1-def 75.9%

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

      2. expm1-log1p 99.6%

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

      3. associate-*r/ 99.6%

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

      4. associate-*r/ 99.5%

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

      5. *-commutative 99.5%

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

      6. +-commutative 99.5%

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

      7. *-commutative 99.5%

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

   5. Simplified 99.5%

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

      1. *-commutative 99.5%

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

      2. +-commutative 99.5%

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

      3. associate-*r/ 99.6%

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

      4. associate-*l* 99.6%

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

   7. Applied egg-rr 99.6%

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

      1. associate-*r/ 99.6%

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

      2. metadata-eval 99.6%

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

      3. +-commutative 99.6%

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

   9. Simplified 99.6%

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

   10. Taylor expanded in a around 0 70.7%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.25 \cdot 10^{-6}:\\ \;\;\;\;\frac{\pi \cdot \frac{0.5}{a}}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot \frac{0.5}{b}}{b \cdot a}\\ \end{array} \]

Alternative 7: 89.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\pi \cdot \frac{0.5}{a}}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{b}}{a \cdot \frac{b}{\pi}}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -1.8e-7) (/ (* PI (/ 0.5 a)) (* b a)) (/ (/ 0.5 b) (* a (/ b PI)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -1.8e-7) {
		tmp = (((double) M_PI) * (0.5 / a)) / (b * a);
	} else {
		tmp = (0.5 / b) / (a * (b / ((double) M_PI)));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -1.8e-7) {
		tmp = (Math.PI * (0.5 / a)) / (b * a);
	} else {
		tmp = (0.5 / b) / (a * (b / Math.PI));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -1.8e-7:
		tmp = (math.pi * (0.5 / a)) / (b * a)
	else:
		tmp = (0.5 / b) / (a * (b / math.pi))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -1.8e-7)
		tmp = Float64(Float64(pi * Float64(0.5 / a)) / Float64(b * a));
	else
		tmp = Float64(Float64(0.5 / b) / Float64(a * Float64(b / pi)));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.8e-7)
		tmp = (pi * (0.5 / a)) / (b * a);
	else
		tmp = (0.5 / b) / (a * (b / pi));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -1.8e-7], N[(N[(Pi * N[(0.5 / a), $MachinePrecision]), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / b), $MachinePrecision] / N[(a * N[(b / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.79999999999999997e-7

   1. Initial program 81.6%

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

      1. div-inv 81.6%

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

      2. expm1-log1p-u 66.1%

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

      3. expm1-udef 53.2%

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

   3. Applied egg-rr 58.6%

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

      1. expm1-def 84.2%

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

      2. expm1-log1p 99.7%

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

      3. associate-*r/ 99.6%

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

      4. associate-*r/ 99.7%

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

      5. *-commutative 99.7%

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

      6. +-commutative 99.7%

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

      7. *-commutative 99.7%

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

   5. Simplified 99.7%

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

      1. *-commutative 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-*r/ 99.6%

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

      4. associate-*l* 99.6%

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

   7. Applied egg-rr 99.6%

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

      1. associate-*r/ 99.6%

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

      2. metadata-eval 99.6%

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

      3. +-commutative 99.6%

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

   9. Simplified 99.6%

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

   10. Taylor expanded in a around inf 91.9%

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

   if -1.79999999999999997e-7 < a

   1. Initial program 81.1%

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

      1. div-inv 81.1%

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

      2. expm1-log1p-u 59.6%

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

      3. expm1-udef 42.3%

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

   3. Applied egg-rr 47.8%

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

      1. expm1-def 75.9%

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

      2. expm1-log1p 99.6%

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

      3. associate-*r/ 99.6%

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

      4. associate-*r/ 99.5%

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

      5. *-commutative 99.5%

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

      6. +-commutative 99.5%

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

      7. *-commutative 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in a around 0 70.6%

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

      1. *-commutative 70.6%

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

      2. times-frac 70.7%

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

   8. Applied egg-rr 70.7%

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

      1. clear-num 70.6%

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

      2. associate-*l/ 70.7%

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

      3. *-un-lft-identity 70.7%

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

      4. div-inv 70.7%

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

      5. clear-num 70.7%

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

   10. Applied egg-rr 70.7%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\pi \cdot \frac{0.5}{a}}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{b}}{a \cdot \frac{b}{\pi}}\\ \end{array} \]

Alternative 8: 89.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{\pi \cdot 0.5}{a}}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{b}}{a \cdot \frac{b}{\pi}}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -9e-8) (/ (/ (* PI 0.5) a) (* b a)) (/ (/ 0.5 b) (* a (/ b PI)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -9e-8) {
		tmp = ((((double) M_PI) * 0.5) / a) / (b * a);
	} else {
		tmp = (0.5 / b) / (a * (b / ((double) M_PI)));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -9e-8) {
		tmp = ((Math.PI * 0.5) / a) / (b * a);
	} else {
		tmp = (0.5 / b) / (a * (b / Math.PI));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -9e-8:
		tmp = ((math.pi * 0.5) / a) / (b * a)
	else:
		tmp = (0.5 / b) / (a * (b / math.pi))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -9e-8)
		tmp = Float64(Float64(Float64(pi * 0.5) / a) / Float64(b * a));
	else
		tmp = Float64(Float64(0.5 / b) / Float64(a * Float64(b / pi)));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -9e-8)
		tmp = ((pi * 0.5) / a) / (b * a);
	else
		tmp = (0.5 / b) / (a * (b / pi));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -9e-8], N[(N[(N[(Pi * 0.5), $MachinePrecision] / a), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / b), $MachinePrecision] / N[(a * N[(b / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -8.99999999999999986e-8

   1. Initial program 81.6%

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

      1. div-inv 81.6%

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

      2. expm1-log1p-u 66.1%

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

      3. expm1-udef 53.2%

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

   3. Applied egg-rr 58.6%

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

      1. expm1-def 84.2%

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

      2. expm1-log1p 99.7%

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

      3. associate-*r/ 99.6%

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

      4. associate-*r/ 99.7%

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

      5. *-commutative 99.7%

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

      6. +-commutative 99.7%

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

      7. *-commutative 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in a around inf 92.0%

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

      1. associate-*r/ 92.0%

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

      2. *-commutative 92.0%

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

   8. Simplified 92.0%

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

   if -8.99999999999999986e-8 < a

   1. Initial program 81.1%

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

      1. div-inv 81.1%

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

      2. expm1-log1p-u 59.6%

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

      3. expm1-udef 42.3%

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

   3. Applied egg-rr 47.8%

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

      1. expm1-def 75.9%

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

      2. expm1-log1p 99.6%

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

      3. associate-*r/ 99.6%

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

      4. associate-*r/ 99.5%

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

      5. *-commutative 99.5%

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

      6. +-commutative 99.5%

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

      7. *-commutative 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in a around 0 70.6%

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

      1. *-commutative 70.6%

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

      2. times-frac 70.7%

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

   8. Applied egg-rr 70.7%

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

      1. clear-num 70.6%

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

      2. associate-*l/ 70.7%

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

      3. *-un-lft-identity 70.7%

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

      4. div-inv 70.7%

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

      5. clear-num 70.7%

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

   10. Applied egg-rr 70.7%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{\pi \cdot 0.5}{a}}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{b}}{a \cdot \frac{b}{\pi}}\\ \end{array} \]

Alternative 9: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.2%

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

   1. div-inv 81.3%

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

   2. expm1-log1p-u 61.5%

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

   3. expm1-udef 45.4%

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

3. Applied egg-rr 50.9%

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

   1. expm1-def 78.3%

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

   2. expm1-log1p 99.6%

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

   3. associate-*r/ 99.6%

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

   4. associate-*r/ 99.6%

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

   5. *-commutative 99.6%

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

   6. +-commutative 99.6%

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

   7. *-commutative 99.6%

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

5. Simplified 99.6%

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

   1. *-commutative 99.6%

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

   2. +-commutative 99.6%

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

   3. associate-*r/ 99.6%

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

   4. associate-*l* 99.6%

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

7. Applied egg-rr 99.6%

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

   1. associate-*r/ 99.6%

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

   2. metadata-eval 99.6%

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

   3. +-commutative 99.6%

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

9. Simplified 99.6%

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

10. Final simplification 99.6%

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

Alternative 10: 63.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 81.2%

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

   1. div-inv 81.3%

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

   2. expm1-log1p-u 61.5%

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

   3. expm1-udef 45.4%

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

3. Applied egg-rr 50.9%

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

   1. expm1-def 78.3%

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

   2. expm1-log1p 99.6%

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

   3. associate-*r/ 99.6%

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

   4. associate-*r/ 99.6%

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

   5. *-commutative 99.6%

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

   6. +-commutative 99.6%

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

   7. *-commutative 99.6%

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

5. Simplified 99.6%

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

   1. *-commutative 99.6%

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

   2. +-commutative 99.6%

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

   3. associate-*r/ 99.6%

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

   4. associate-*l* 99.6%

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

7. Applied egg-rr 99.6%

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

   1. associate-*r/ 99.6%

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

   2. metadata-eval 99.6%

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

   3. +-commutative 99.6%

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

9. Simplified 99.6%

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

    1. div-inv 99.6%

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

    2. associate-*l* 99.6%

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

    3. *-commutative 99.6%

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

    4. associate-/r* 99.5%

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

    5. times-frac 93.3%

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

    6. div-inv 93.3%

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

    7. *-commutative 93.3%

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

    8. associate-*r/ 93.3%

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

    9. times-frac 95.4%

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

11. Applied egg-rr 95.4%

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

12. Taylor expanded in b around 0 63.6%

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

    1. associate-/l/ 63.6%

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

14. Simplified 63.6%

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

15. Final simplification 63.6%

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

Reproduce

herbie shell --seed 2023301 
(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))))