NMSE Section 6.1 mentioned, B

Percentage Accurate: 77.8% → 97.3%

Time: 8.7s

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: 77.8% 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: 97.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 4.2 \cdot 10^{-171}:\\ \;\;\;\;\frac{\pi}{b - a} \cdot \frac{\frac{-0.5}{b}}{b + a}\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{+84}:\\ \;\;\;\;\left(\frac{\pi \cdot \frac{1}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\pi \cdot \left(\frac{1}{b \cdot a} \cdot \frac{1}{b}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 4.2e-171)
   (* (/ PI (- b a)) (/ (/ -0.5 b) (+ b a)))
   (if (<= b 2.5e+84)
     (* (* (/ (* PI (/ 1.0 (+ b a))) (- b a)) 0.5) (+ (/ 1.0 a) (/ -1.0 b)))
     (* 0.5 (* PI (* (/ 1.0 (* b a)) (/ 1.0 b)))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 4.2e-171) {
		tmp = (((double) M_PI) / (b - a)) * ((-0.5 / b) / (b + a));
	} else if (b <= 2.5e+84) {
		tmp = (((((double) M_PI) * (1.0 / (b + a))) / (b - a)) * 0.5) * ((1.0 / a) + (-1.0 / b));
	} else {
		tmp = 0.5 * (((double) M_PI) * ((1.0 / (b * a)) * (1.0 / b)));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 4.2e-171) {
		tmp = (Math.PI / (b - a)) * ((-0.5 / b) / (b + a));
	} else if (b <= 2.5e+84) {
		tmp = (((Math.PI * (1.0 / (b + a))) / (b - a)) * 0.5) * ((1.0 / a) + (-1.0 / b));
	} else {
		tmp = 0.5 * (Math.PI * ((1.0 / (b * a)) * (1.0 / b)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 4.2e-171:
		tmp = (math.pi / (b - a)) * ((-0.5 / b) / (b + a))
	elif b <= 2.5e+84:
		tmp = (((math.pi * (1.0 / (b + a))) / (b - a)) * 0.5) * ((1.0 / a) + (-1.0 / b))
	else:
		tmp = 0.5 * (math.pi * ((1.0 / (b * a)) * (1.0 / b)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 4.2e-171)
		tmp = Float64(Float64(pi / Float64(b - a)) * Float64(Float64(-0.5 / b) / Float64(b + a)));
	elseif (b <= 2.5e+84)
		tmp = Float64(Float64(Float64(Float64(pi * Float64(1.0 / Float64(b + a))) / Float64(b - a)) * 0.5) * Float64(Float64(1.0 / a) + Float64(-1.0 / b)));
	else
		tmp = Float64(0.5 * Float64(pi * Float64(Float64(1.0 / Float64(b * a)) * Float64(1.0 / b))));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 4.2e-171)
		tmp = (pi / (b - a)) * ((-0.5 / b) / (b + a));
	elseif (b <= 2.5e+84)
		tmp = (((pi * (1.0 / (b + a))) / (b - a)) * 0.5) * ((1.0 / a) + (-1.0 / b));
	else
		tmp = 0.5 * (pi * ((1.0 / (b * a)) * (1.0 / 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, 4.2e-171], N[(N[(Pi / N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.5 / b), $MachinePrecision] / N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.5e+84], N[(N[(N[(N[(Pi * N[(1.0 / N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(1.0 / a), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(Pi * N[(N[(1.0 / N[(b * a), $MachinePrecision]), $MachinePrecision] * N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 4.2e-171

   1. Initial program 77.8%

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

      1. times-frac 77.9%

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

      2. *-commutative 77.9%

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

      3. times-frac 77.9%

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

      4. difference-of-squares 88.8%

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

      5. associate-/r* 89.3%

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

      6. metadata-eval 89.3%

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

      7. sub-neg 89.3%

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

      8. distribute-neg-frac 89.3%

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

      9. metadata-eval 89.3%

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

   3. Simplified 89.3%

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

   4. Taylor expanded in a around inf 65.1%

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

      1. expm1-log1p-u 45.3%

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

      2. expm1-udef 41.2%

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

      3. associate-*l* 41.2%

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

      4. associate-/l/ 41.2%

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

   6. Applied egg-rr 41.2%

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

      1. expm1-def 45.3%

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

      2. expm1-log1p 65.2%

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

      3. associate-*l/ 65.2%

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

      4. times-frac 71.6%

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

      5. associate-*r/ 71.6%

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

      6. metadata-eval 71.6%

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

      7. +-commutative 71.6%

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

   8. Simplified 71.6%

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

   if 4.2e-171 < b < 2.5e84

   1. Initial program 95.5%

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

      1. times-frac 95.5%

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

      2. *-commutative 95.5%

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

      3. times-frac 95.5%

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

      4. difference-of-squares 95.5%

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

      5. associate-/r* 95.5%

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

      6. metadata-eval 95.5%

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

      7. sub-neg 95.5%

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

      8. distribute-neg-frac 95.5%

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

      9. metadata-eval 95.5%

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

   3. Simplified 95.5%

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

      1. div-inv 95.6%

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

   5. Applied egg-rr 95.6%

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

   if 2.5e84 < b

   1. Initial program 68.6%

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

      1. *-commutative 68.6%

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

      2. associate-/r/ 68.6%

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

      3. associate-*l/ 68.6%

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

      4. *-commutative 68.6%

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

      5. associate-/r/ 68.6%

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

      6. times-frac 68.5%

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

   3. Simplified 68.7%

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

   4. Taylor expanded in b around inf 84.5%

      \[\leadsto \color{blue}{\frac{\pi}{a \cdot {b}^{2}}} \cdot 0.5 \]
   5. Step-by-step derivation

      1. unpow2 84.5%

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

   6. Simplified 84.5%

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

      1. div-inv 84.7%

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

   8. Applied egg-rr 84.7%

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

      1. inv-pow 84.7%

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

      2. associate-*r* 99.5%

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

      3. unpow-prod-down 99.6%

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

      4. inv-pow 99.6%

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

      5. *-commutative 99.6%

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

      6. inv-pow 99.6%

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

   10. Applied egg-rr 99.6%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.2 \cdot 10^{-171}:\\ \;\;\;\;\frac{\pi}{b - a} \cdot \frac{\frac{-0.5}{b}}{b + a}\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{+84}:\\ \;\;\;\;\left(\frac{\pi \cdot \frac{1}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\pi \cdot \left(\frac{1}{b \cdot a} \cdot \frac{1}{b}\right)\right)\\ \end{array} \]

Alternative 2: 97.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 3.8 \cdot 10^{-171}:\\ \;\;\;\;\frac{\pi}{b - a} \cdot \frac{\frac{-0.5}{b}}{b + a}\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{+84}:\\ \;\;\;\;\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \left(0.5 \cdot \frac{\frac{\pi}{b + a}}{b - a}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\pi \cdot \left(\frac{1}{b \cdot a} \cdot \frac{1}{b}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 3.8e-171)
   (* (/ PI (- b a)) (/ (/ -0.5 b) (+ b a)))
   (if (<= b 2.5e+84)
     (* (+ (/ 1.0 a) (/ -1.0 b)) (* 0.5 (/ (/ PI (+ b a)) (- b a))))
     (* 0.5 (* PI (* (/ 1.0 (* b a)) (/ 1.0 b)))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 3.8e-171) {
		tmp = (((double) M_PI) / (b - a)) * ((-0.5 / b) / (b + a));
	} else if (b <= 2.5e+84) {
		tmp = ((1.0 / a) + (-1.0 / b)) * (0.5 * ((((double) M_PI) / (b + a)) / (b - a)));
	} else {
		tmp = 0.5 * (((double) M_PI) * ((1.0 / (b * a)) * (1.0 / b)));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 3.8e-171) {
		tmp = (Math.PI / (b - a)) * ((-0.5 / b) / (b + a));
	} else if (b <= 2.5e+84) {
		tmp = ((1.0 / a) + (-1.0 / b)) * (0.5 * ((Math.PI / (b + a)) / (b - a)));
	} else {
		tmp = 0.5 * (Math.PI * ((1.0 / (b * a)) * (1.0 / b)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 3.8e-171:
		tmp = (math.pi / (b - a)) * ((-0.5 / b) / (b + a))
	elif b <= 2.5e+84:
		tmp = ((1.0 / a) + (-1.0 / b)) * (0.5 * ((math.pi / (b + a)) / (b - a)))
	else:
		tmp = 0.5 * (math.pi * ((1.0 / (b * a)) * (1.0 / b)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 3.8e-171)
		tmp = Float64(Float64(pi / Float64(b - a)) * Float64(Float64(-0.5 / b) / Float64(b + a)));
	elseif (b <= 2.5e+84)
		tmp = Float64(Float64(Float64(1.0 / a) + Float64(-1.0 / b)) * Float64(0.5 * Float64(Float64(pi / Float64(b + a)) / Float64(b - a))));
	else
		tmp = Float64(0.5 * Float64(pi * Float64(Float64(1.0 / Float64(b * a)) * Float64(1.0 / b))));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 3.8e-171)
		tmp = (pi / (b - a)) * ((-0.5 / b) / (b + a));
	elseif (b <= 2.5e+84)
		tmp = ((1.0 / a) + (-1.0 / b)) * (0.5 * ((pi / (b + a)) / (b - a)));
	else
		tmp = 0.5 * (pi * ((1.0 / (b * a)) * (1.0 / b)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[b, 3.8e-171], N[(N[(Pi / N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.5 / b), $MachinePrecision] / N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.5e+84], N[(N[(N[(1.0 / a), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[(N[(Pi / N[(b + a), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(Pi * N[(N[(1.0 / N[(b * a), $MachinePrecision]), $MachinePrecision] * N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 3.80000000000000021e-171

   1. Initial program 77.8%

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

      1. times-frac 77.9%

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

      2. *-commutative 77.9%

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

      3. times-frac 77.9%

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

      4. difference-of-squares 88.8%

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

      5. associate-/r* 89.3%

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

      6. metadata-eval 89.3%

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

      7. sub-neg 89.3%

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

      8. distribute-neg-frac 89.3%

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

      9. metadata-eval 89.3%

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

   3. Simplified 89.3%

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

   4. Taylor expanded in a around inf 65.1%

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

      1. expm1-log1p-u 45.3%

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

      2. expm1-udef 41.2%

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

      3. associate-*l* 41.2%

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

      4. associate-/l/ 41.2%

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

   6. Applied egg-rr 41.2%

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

      1. expm1-def 45.3%

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

      2. expm1-log1p 65.2%

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

      3. associate-*l/ 65.2%

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

      4. times-frac 71.6%

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

      5. associate-*r/ 71.6%

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

      6. metadata-eval 71.6%

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

      7. +-commutative 71.6%

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

   8. Simplified 71.6%

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

   if 3.80000000000000021e-171 < b < 2.5e84

   1. Initial program 95.5%

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

      1. times-frac 95.5%

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

      2. *-commutative 95.5%

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

      3. times-frac 95.5%

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

      4. difference-of-squares 95.5%

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

      5. associate-/r* 95.5%

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

      6. metadata-eval 95.5%

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

      7. sub-neg 95.5%

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

      8. distribute-neg-frac 95.5%

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

      9. metadata-eval 95.5%

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

   3. Simplified 95.5%

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

   if 2.5e84 < b

   1. Initial program 68.6%

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

      1. *-commutative 68.6%

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

      2. associate-/r/ 68.6%

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

      3. associate-*l/ 68.6%

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

      4. *-commutative 68.6%

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

      5. associate-/r/ 68.6%

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

      6. times-frac 68.5%

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

   3. Simplified 68.7%

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

   4. Taylor expanded in b around inf 84.5%

      \[\leadsto \color{blue}{\frac{\pi}{a \cdot {b}^{2}}} \cdot 0.5 \]
   5. Step-by-step derivation

      1. unpow2 84.5%

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

   6. Simplified 84.5%

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

      1. div-inv 84.7%

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

   8. Applied egg-rr 84.7%

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

      1. inv-pow 84.7%

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

      2. associate-*r* 99.5%

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

      3. unpow-prod-down 99.6%

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

      4. inv-pow 99.6%

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

      5. *-commutative 99.6%

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

      6. inv-pow 99.6%

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

   10. Applied egg-rr 99.6%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.8 \cdot 10^{-171}:\\ \;\;\;\;\frac{\pi}{b - a} \cdot \frac{\frac{-0.5}{b}}{b + a}\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{+84}:\\ \;\;\;\;\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \left(0.5 \cdot \frac{\frac{\pi}{b + a}}{b - a}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\pi \cdot \left(\frac{1}{b \cdot a} \cdot \frac{1}{b}\right)\right)\\ \end{array} \]

Alternative 3: 97.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 2.9 \cdot 10^{-171}:\\ \;\;\;\;\frac{\pi}{b - a} \cdot \frac{\frac{-0.5}{b}}{b + a}\\ \mathbf{elif}\;b \leq 8 \cdot 10^{+123}:\\ \;\;\;\;\left(0.5 \cdot \frac{\frac{\pi}{b + a}}{b - a}\right) \cdot \frac{b - a}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\pi \cdot \left(\frac{1}{b \cdot a} \cdot \frac{1}{b}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 2.9e-171)
   (* (/ PI (- b a)) (/ (/ -0.5 b) (+ b a)))
   (if (<= b 8e+123)
     (* (* 0.5 (/ (/ PI (+ b a)) (- b a))) (/ (- b a) (* b a)))
     (* 0.5 (* PI (* (/ 1.0 (* b a)) (/ 1.0 b)))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 2.9e-171) {
		tmp = (((double) M_PI) / (b - a)) * ((-0.5 / b) / (b + a));
	} else if (b <= 8e+123) {
		tmp = (0.5 * ((((double) M_PI) / (b + a)) / (b - a))) * ((b - a) / (b * a));
	} else {
		tmp = 0.5 * (((double) M_PI) * ((1.0 / (b * a)) * (1.0 / b)));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 2.9e-171) {
		tmp = (Math.PI / (b - a)) * ((-0.5 / b) / (b + a));
	} else if (b <= 8e+123) {
		tmp = (0.5 * ((Math.PI / (b + a)) / (b - a))) * ((b - a) / (b * a));
	} else {
		tmp = 0.5 * (Math.PI * ((1.0 / (b * a)) * (1.0 / b)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 2.9e-171:
		tmp = (math.pi / (b - a)) * ((-0.5 / b) / (b + a))
	elif b <= 8e+123:
		tmp = (0.5 * ((math.pi / (b + a)) / (b - a))) * ((b - a) / (b * a))
	else:
		tmp = 0.5 * (math.pi * ((1.0 / (b * a)) * (1.0 / b)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 2.9e-171)
		tmp = Float64(Float64(pi / Float64(b - a)) * Float64(Float64(-0.5 / b) / Float64(b + a)));
	elseif (b <= 8e+123)
		tmp = Float64(Float64(0.5 * Float64(Float64(pi / Float64(b + a)) / Float64(b - a))) * Float64(Float64(b - a) / Float64(b * a)));
	else
		tmp = Float64(0.5 * Float64(pi * Float64(Float64(1.0 / Float64(b * a)) * Float64(1.0 / b))));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2.9e-171)
		tmp = (pi / (b - a)) * ((-0.5 / b) / (b + a));
	elseif (b <= 8e+123)
		tmp = (0.5 * ((pi / (b + a)) / (b - a))) * ((b - a) / (b * a));
	else
		tmp = 0.5 * (pi * ((1.0 / (b * a)) * (1.0 / 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, 2.9e-171], N[(N[(Pi / N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.5 / b), $MachinePrecision] / N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8e+123], N[(N[(0.5 * N[(N[(Pi / N[(b + a), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(Pi * N[(N[(1.0 / N[(b * a), $MachinePrecision]), $MachinePrecision] * N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 2.8999999999999999e-171

   1. Initial program 77.8%

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

      1. times-frac 77.9%

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

      2. *-commutative 77.9%

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

      3. times-frac 77.9%

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

      4. difference-of-squares 88.8%

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

      5. associate-/r* 89.3%

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

      6. metadata-eval 89.3%

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

      7. sub-neg 89.3%

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

      8. distribute-neg-frac 89.3%

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

      9. metadata-eval 89.3%

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

   3. Simplified 89.3%

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

   4. Taylor expanded in a around inf 65.1%

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

      1. expm1-log1p-u 45.3%

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

      2. expm1-udef 41.2%

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

      3. associate-*l* 41.2%

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

      4. associate-/l/ 41.2%

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

   6. Applied egg-rr 41.2%

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

      1. expm1-def 45.3%

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

      2. expm1-log1p 65.2%

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

      3. associate-*l/ 65.2%

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

      4. times-frac 71.6%

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

      5. associate-*r/ 71.6%

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

      6. metadata-eval 71.6%

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

      7. +-commutative 71.6%

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

   8. Simplified 71.6%

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

   if 2.8999999999999999e-171 < b < 7.99999999999999982e123

   1. Initial program 96.2%

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

      1. times-frac 96.2%

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

      2. *-commutative 96.2%

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

      3. times-frac 96.2%

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

      4. difference-of-squares 96.2%

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

      5. associate-/r* 96.2%

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

      6. metadata-eval 96.2%

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

      7. sub-neg 96.2%

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

      8. distribute-neg-frac 96.2%

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

      9. metadata-eval 96.2%

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

   3. Simplified 96.2%

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

      1. frac-add 96.3%

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

      2. *-un-lft-identity 96.3%

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

   5. Applied egg-rr 96.3%

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

      1. *-commutative 96.3%

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

      2. neg-mul-1 96.3%

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

      3. sub-neg 96.3%

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

   7. Simplified 96.3%

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

   if 7.99999999999999982e123 < b

   1. Initial program 57.2%

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

      1. *-commutative 57.2%

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

      2. associate-/r/ 57.3%

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

      3. associate-*l/ 57.2%

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

      4. *-commutative 57.2%

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

      5. associate-/r/ 57.2%

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

      6. times-frac 57.2%

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

   3. Simplified 57.3%

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

   4. Taylor expanded in b around inf 79.4%

      \[\leadsto \color{blue}{\frac{\pi}{a \cdot {b}^{2}}} \cdot 0.5 \]
   5. Step-by-step derivation

      1. unpow2 79.4%

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

   6. Simplified 79.4%

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

      1. div-inv 79.5%

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

   8. Applied egg-rr 79.5%

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

      1. inv-pow 79.5%

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

      2. associate-*r* 99.8%

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

      3. unpow-prod-down 99.7%

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

      4. inv-pow 99.7%

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

      5. *-commutative 99.7%

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

      6. inv-pow 99.7%

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

   10. Applied egg-rr 99.7%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.9 \cdot 10^{-171}:\\ \;\;\;\;\frac{\pi}{b - a} \cdot \frac{\frac{-0.5}{b}}{b + a}\\ \mathbf{elif}\;b \leq 8 \cdot 10^{+123}:\\ \;\;\;\;\left(0.5 \cdot \frac{\frac{\pi}{b + a}}{b - a}\right) \cdot \frac{b - a}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\pi \cdot \left(\frac{1}{b \cdot a} \cdot \frac{1}{b}\right)\right)\\ \end{array} \]

Alternative 4: 84.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -4.6 \cdot 10^{-68}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{b \cdot \left(a \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\pi \cdot \left(\frac{1}{b \cdot a} \cdot \frac{1}{b}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -4.6e-68)
   (/ (* PI 0.5) (* b (* a a)))
   (* 0.5 (* PI (* (/ 1.0 (* b a)) (/ 1.0 b))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -4.6e-68) {
		tmp = (((double) M_PI) * 0.5) / (b * (a * a));
	} else {
		tmp = 0.5 * (((double) M_PI) * ((1.0 / (b * a)) * (1.0 / b)));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -4.6e-68) {
		tmp = (Math.PI * 0.5) / (b * (a * a));
	} else {
		tmp = 0.5 * (Math.PI * ((1.0 / (b * a)) * (1.0 / b)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -4.6e-68:
		tmp = (math.pi * 0.5) / (b * (a * a))
	else:
		tmp = 0.5 * (math.pi * ((1.0 / (b * a)) * (1.0 / b)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -4.6e-68)
		tmp = Float64(Float64(pi * 0.5) / Float64(b * Float64(a * a)));
	else
		tmp = Float64(0.5 * Float64(pi * Float64(Float64(1.0 / Float64(b * a)) * Float64(1.0 / b))));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -4.6e-68)
		tmp = (pi * 0.5) / (b * (a * a));
	else
		tmp = 0.5 * (pi * ((1.0 / (b * a)) * (1.0 / b)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -4.6e-68], N[(N[(Pi * 0.5), $MachinePrecision] / N[(b * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(Pi * N[(N[(1.0 / N[(b * a), $MachinePrecision]), $MachinePrecision] * N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.59999999999999994e-68

   1. Initial program 79.4%

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

   2. Taylor expanded in b around 0 67.5%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{{a}^{2} \cdot b}} \]
   3. Step-by-step derivation

      1. associate-*r/ 67.5%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{{a}^{2} \cdot b}} \]

      2. unpow2 67.5%

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

   4. Simplified 67.5%

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

   if -4.59999999999999994e-68 < a

   1. Initial program 79.8%

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

      1. *-commutative 79.8%

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

      2. associate-/r/ 79.8%

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

      3. associate-*l/ 79.8%

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

      4. *-commutative 79.8%

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

      5. associate-/r/ 79.8%

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

      6. times-frac 79.8%

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

   3. Simplified 79.8%

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

   4. Taylor expanded in b around inf 70.7%

      \[\leadsto \color{blue}{\frac{\pi}{a \cdot {b}^{2}}} \cdot 0.5 \]
   5. Step-by-step derivation

      1. unpow2 70.7%

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

   6. Simplified 70.7%

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

      1. div-inv 70.6%

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

   8. Applied egg-rr 70.6%

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

      1. inv-pow 70.6%

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

      2. associate-*r* 77.4%

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

      3. unpow-prod-down 77.0%

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

      4. inv-pow 77.0%

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

      5. *-commutative 77.0%

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

      6. inv-pow 77.0%

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

   10. Applied egg-rr 77.0%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.6 \cdot 10^{-68}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{b \cdot \left(a \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\pi \cdot \left(\frac{1}{b \cdot a} \cdot \frac{1}{b}\right)\right)\\ \end{array} \]

Alternative 5: 92.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -7.4 \cdot 10^{-96}:\\ \;\;\;\;\frac{\pi}{b - a} \cdot \frac{\frac{-0.5}{b}}{b + a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\pi \cdot \left(\frac{1}{b \cdot a} \cdot \frac{1}{b}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -7.4e-96)
   (* (/ PI (- b a)) (/ (/ -0.5 b) (+ b a)))
   (* 0.5 (* PI (* (/ 1.0 (* b a)) (/ 1.0 b))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -7.4e-96) {
		tmp = (((double) M_PI) / (b - a)) * ((-0.5 / b) / (b + a));
	} else {
		tmp = 0.5 * (((double) M_PI) * ((1.0 / (b * a)) * (1.0 / b)));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -7.4e-96) {
		tmp = (Math.PI / (b - a)) * ((-0.5 / b) / (b + a));
	} else {
		tmp = 0.5 * (Math.PI * ((1.0 / (b * a)) * (1.0 / b)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -7.4e-96:
		tmp = (math.pi / (b - a)) * ((-0.5 / b) / (b + a))
	else:
		tmp = 0.5 * (math.pi * ((1.0 / (b * a)) * (1.0 / b)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -7.4e-96)
		tmp = Float64(Float64(pi / Float64(b - a)) * Float64(Float64(-0.5 / b) / Float64(b + a)));
	else
		tmp = Float64(0.5 * Float64(pi * Float64(Float64(1.0 / Float64(b * a)) * Float64(1.0 / b))));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -7.4e-96)
		tmp = (pi / (b - a)) * ((-0.5 / b) / (b + a));
	else
		tmp = 0.5 * (pi * ((1.0 / (b * a)) * (1.0 / b)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -7.4e-96], N[(N[(Pi / N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.5 / b), $MachinePrecision] / N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(Pi * N[(N[(1.0 / N[(b * a), $MachinePrecision]), $MachinePrecision] * N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -7.39999999999999972e-96

   1. Initial program 79.8%

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

      1. times-frac 80.0%

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

      2. *-commutative 80.0%

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

      3. times-frac 80.0%

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

      4. difference-of-squares 89.5%

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

      5. associate-/r* 89.5%

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

      6. metadata-eval 89.5%

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

      7. sub-neg 89.5%

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

      8. distribute-neg-frac 89.5%

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

      9. metadata-eval 89.5%

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

   3. Simplified 89.5%

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

   4. Taylor expanded in a around inf 74.0%

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

      1. expm1-log1p-u 53.5%

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

      2. expm1-udef 46.4%

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

      3. associate-*l* 46.4%

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

      4. associate-/l/ 46.4%

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

   6. Applied egg-rr 46.4%

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

      1. expm1-def 53.5%

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

      2. expm1-log1p 74.1%

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

      3. associate-*l/ 74.1%

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

      4. times-frac 82.9%

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

      5. associate-*r/ 82.9%

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

      6. metadata-eval 82.9%

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

      7. +-commutative 82.9%

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

   8. Simplified 82.9%

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

   if -7.39999999999999972e-96 < a

   1. Initial program 79.6%

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

      1. *-commutative 79.6%

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

      2. associate-/r/ 79.6%

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

      3. associate-*l/ 79.6%

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

      4. *-commutative 79.6%

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

      5. associate-/r/ 79.6%

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

      6. times-frac 79.6%

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

   3. Simplified 79.6%

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

   4. Taylor expanded in b around inf 70.3%

      \[\leadsto \color{blue}{\frac{\pi}{a \cdot {b}^{2}}} \cdot 0.5 \]
   5. Step-by-step derivation

      1. unpow2 70.3%

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

   6. Simplified 70.3%

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

      1. div-inv 70.3%

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

   8. Applied egg-rr 70.3%

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

      1. inv-pow 70.3%

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

      2. associate-*r* 77.2%

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

      3. unpow-prod-down 76.7%

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

      4. inv-pow 76.7%

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

      5. *-commutative 76.7%

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

      6. inv-pow 76.7%

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

   10. Applied egg-rr 76.7%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7.4 \cdot 10^{-96}:\\ \;\;\;\;\frac{\pi}{b - a} \cdot \frac{\frac{-0.5}{b}}{b + a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\pi \cdot \left(\frac{1}{b \cdot a} \cdot \frac{1}{b}\right)\right)\\ \end{array} \]

Alternative 6: 78.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -4.5 \cdot 10^{-68}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{0.5}{a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -4.5e-68)
   (* (/ PI b) (/ 0.5 (* a a)))
   (* (/ PI a) (/ 0.5 (* b b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -4.5e-68) {
		tmp = (((double) M_PI) / b) * (0.5 / (a * 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 (a <= -4.5e-68) {
		tmp = (Math.PI / b) * (0.5 / (a * 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 a <= -4.5e-68:
		tmp = (math.pi / b) * (0.5 / (a * 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 (a <= -4.5e-68)
		tmp = Float64(Float64(pi / b) * Float64(0.5 / Float64(a * a)));
	else
		tmp = Float64(Float64(pi / a) * Float64(0.5 / Float64(b * b)));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -4.5e-68)
		tmp = (pi / b) * (0.5 / (a * 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[a, -4.5e-68], N[(N[(Pi / b), $MachinePrecision] * N[(0.5 / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / a), $MachinePrecision] * N[(0.5 / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.49999999999999999e-68

   1. Initial program 79.4%

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

      1. times-frac 79.5%

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

      2. *-commutative 79.5%

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

      3. times-frac 79.5%

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

      4. difference-of-squares 89.3%

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

      5. associate-/r* 89.2%

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

      6. metadata-eval 89.2%

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

      7. sub-neg 89.2%

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

      8. distribute-neg-frac 89.2%

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

      9. metadata-eval 89.2%

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

   3. Simplified 89.2%

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

      1. frac-add 89.2%

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

      2. *-un-lft-identity 89.2%

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

   5. Applied egg-rr 89.2%

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

      1. *-commutative 89.2%

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

      2. neg-mul-1 89.2%

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

      3. sub-neg 89.2%

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

   7. Simplified 89.2%

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

      1. div-inv 89.2%

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

   9. Applied egg-rr 89.2%

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

   10. Taylor expanded in b around 0 67.5%

       \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{{a}^{2} \cdot b}} \]
   11. Step-by-step derivation

       1. associate-*r/ 67.5%

          \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{{a}^{2} \cdot b}} \]

       2. *-commutative 67.5%

          \[\leadsto \frac{\color{blue}{\pi \cdot 0.5}}{{a}^{2} \cdot b} \]

       3. *-commutative 67.5%

          \[\leadsto \frac{\pi \cdot 0.5}{\color{blue}{b \cdot {a}^{2}}} \]

       4. times-frac 67.5%

          \[\leadsto \color{blue}{\frac{\pi}{b} \cdot \frac{0.5}{{a}^{2}}} \]

       5. unpow2 67.5%

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

   12. Simplified 67.5%

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

   if -4.49999999999999999e-68 < a

   1. Initial program 79.8%

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

      1. times-frac 79.8%

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

      2. *-commutative 79.8%

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

      3. times-frac 79.8%

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

      4. difference-of-squares 89.6%

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

      5. associate-/r* 90.7%

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

      6. metadata-eval 90.7%

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

      7. sub-neg 90.7%

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

      8. distribute-neg-frac 90.7%

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

      9. metadata-eval 90.7%

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

   3. Simplified 90.7%

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

      1. div-inv 90.7%

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

   5. Applied egg-rr 90.7%

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

   6. Taylor expanded in b around inf 70.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{a \cdot {b}^{2}}} \]
   7. Step-by-step derivation

      1. associate-*r/ 70.7%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{a \cdot {b}^{2}}} \]

      2. *-commutative 70.7%

         \[\leadsto \frac{\color{blue}{\pi \cdot 0.5}}{a \cdot {b}^{2}} \]

      3. times-frac 70.2%

         \[\leadsto \color{blue}{\frac{\pi}{a} \cdot \frac{0.5}{{b}^{2}}} \]

      4. unpow2 70.2%

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

   8. Simplified 70.2%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.5 \cdot 10^{-68}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{0.5}{a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{b \cdot b}\\ \end{array} \]

Alternative 7: 78.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -4.6 \cdot 10^{-68}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{\frac{0.5}{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -4.6e-68)
   (* (/ PI b) (/ (/ 0.5 a) a))
   (* (/ PI a) (/ 0.5 (* b b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -4.6e-68) {
		tmp = (((double) M_PI) / b) * ((0.5 / a) / 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 (a <= -4.6e-68) {
		tmp = (Math.PI / b) * ((0.5 / a) / 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 a <= -4.6e-68:
		tmp = (math.pi / b) * ((0.5 / a) / 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 (a <= -4.6e-68)
		tmp = Float64(Float64(pi / b) * Float64(Float64(0.5 / a) / a));
	else
		tmp = Float64(Float64(pi / a) * Float64(0.5 / Float64(b * b)));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -4.6e-68)
		tmp = (pi / b) * ((0.5 / a) / 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[a, -4.6e-68], N[(N[(Pi / b), $MachinePrecision] * N[(N[(0.5 / a), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / a), $MachinePrecision] * N[(0.5 / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.59999999999999994e-68

   1. Initial program 79.4%

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

      1. times-frac 79.5%

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

      2. *-commutative 79.5%

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

      3. times-frac 79.5%

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

      4. difference-of-squares 89.3%

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

      5. associate-/r* 89.2%

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

      6. metadata-eval 89.2%

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

      7. sub-neg 89.2%

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

      8. distribute-neg-frac 89.2%

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

      9. metadata-eval 89.2%

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

   3. Simplified 89.2%

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

      1. frac-add 89.2%

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

      2. *-un-lft-identity 89.2%

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

   5. Applied egg-rr 89.2%

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

      1. *-commutative 89.2%

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

      2. neg-mul-1 89.2%

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

      3. sub-neg 89.2%

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

   7. Simplified 89.2%

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

      1. div-inv 89.2%

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

   9. Applied egg-rr 89.2%

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

   10. Taylor expanded in b around 0 67.5%

       \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{{a}^{2} \cdot b}} \]
   11. Step-by-step derivation

       1. associate-*r/ 67.5%

          \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{{a}^{2} \cdot b}} \]

       2. *-commutative 67.5%

          \[\leadsto \frac{\color{blue}{\pi \cdot 0.5}}{{a}^{2} \cdot b} \]

       3. *-commutative 67.5%

          \[\leadsto \frac{\pi \cdot 0.5}{\color{blue}{b \cdot {a}^{2}}} \]

       4. times-frac 67.5%

          \[\leadsto \color{blue}{\frac{\pi}{b} \cdot \frac{0.5}{{a}^{2}}} \]

       5. unpow2 67.5%

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

   12. Simplified 67.5%

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

   13. Taylor expanded in a around 0 67.5%

       \[\leadsto \frac{\pi}{b} \cdot \color{blue}{\frac{0.5}{{a}^{2}}} \]
   14. Step-by-step derivation

       1. unpow2 67.5%

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

       2. associate-/r* 67.5%

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

   15. Simplified 67.5%

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

   if -4.59999999999999994e-68 < a

   1. Initial program 79.8%

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

      1. times-frac 79.8%

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

      2. *-commutative 79.8%

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

      3. times-frac 79.8%

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

      4. difference-of-squares 89.6%

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

      5. associate-/r* 90.7%

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

      6. metadata-eval 90.7%

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

      7. sub-neg 90.7%

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

      8. distribute-neg-frac 90.7%

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

      9. metadata-eval 90.7%

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

   3. Simplified 90.7%

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

      1. div-inv 90.7%

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

   5. Applied egg-rr 90.7%

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

   6. Taylor expanded in b around inf 70.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{a \cdot {b}^{2}}} \]
   7. Step-by-step derivation

      1. associate-*r/ 70.7%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{a \cdot {b}^{2}}} \]

      2. *-commutative 70.7%

         \[\leadsto \frac{\color{blue}{\pi \cdot 0.5}}{a \cdot {b}^{2}} \]

      3. times-frac 70.2%

         \[\leadsto \color{blue}{\frac{\pi}{a} \cdot \frac{0.5}{{b}^{2}}} \]

      4. unpow2 70.2%

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

   8. Simplified 70.2%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.6 \cdot 10^{-68}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{\frac{0.5}{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{b \cdot b}\\ \end{array} \]

Alternative 8: 78.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -4.5 \cdot 10^{-68}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{b \cdot \left(a \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -4.5e-68)
   (/ (* PI 0.5) (* b (* a a)))
   (* (/ PI a) (/ 0.5 (* b b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -4.5e-68) {
		tmp = (((double) M_PI) * 0.5) / (b * (a * 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 (a <= -4.5e-68) {
		tmp = (Math.PI * 0.5) / (b * (a * 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 a <= -4.5e-68:
		tmp = (math.pi * 0.5) / (b * (a * 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 (a <= -4.5e-68)
		tmp = Float64(Float64(pi * 0.5) / Float64(b * Float64(a * a)));
	else
		tmp = Float64(Float64(pi / a) * Float64(0.5 / Float64(b * b)));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -4.5e-68)
		tmp = (pi * 0.5) / (b * (a * 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[a, -4.5e-68], N[(N[(Pi * 0.5), $MachinePrecision] / N[(b * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / a), $MachinePrecision] * N[(0.5 / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.49999999999999999e-68

   1. Initial program 79.4%

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

   2. Taylor expanded in b around 0 67.5%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{{a}^{2} \cdot b}} \]
   3. Step-by-step derivation

      1. associate-*r/ 67.5%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{{a}^{2} \cdot b}} \]

      2. unpow2 67.5%

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

   4. Simplified 67.5%

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

   if -4.49999999999999999e-68 < a

   1. Initial program 79.8%

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

      1. times-frac 79.8%

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

      2. *-commutative 79.8%

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

      3. times-frac 79.8%

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

      4. difference-of-squares 89.6%

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

      5. associate-/r* 90.7%

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

      6. metadata-eval 90.7%

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

      7. sub-neg 90.7%

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

      8. distribute-neg-frac 90.7%

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

      9. metadata-eval 90.7%

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

   3. Simplified 90.7%

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

      1. div-inv 90.7%

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

   5. Applied egg-rr 90.7%

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

   6. Taylor expanded in b around inf 70.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{a \cdot {b}^{2}}} \]
   7. Step-by-step derivation

      1. associate-*r/ 70.7%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{a \cdot {b}^{2}}} \]

      2. *-commutative 70.7%

         \[\leadsto \frac{\color{blue}{\pi \cdot 0.5}}{a \cdot {b}^{2}} \]

      3. times-frac 70.2%

         \[\leadsto \color{blue}{\frac{\pi}{a} \cdot \frac{0.5}{{b}^{2}}} \]

      4. unpow2 70.2%

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

   8. Simplified 70.2%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.5 \cdot 10^{-68}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{b \cdot \left(a \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{b \cdot b}\\ \end{array} \]

Alternative 9: 78.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -2.7 \cdot 10^{-69}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{b \cdot \left(a \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{b} \cdot \frac{\pi}{b}}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -2.7e-69)
   (/ (* PI 0.5) (* b (* a a)))
   (/ (* (/ 0.5 b) (/ PI b)) a)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -2.7e-69) {
		tmp = (((double) M_PI) * 0.5) / (b * (a * 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 <= -2.7e-69) {
		tmp = (Math.PI * 0.5) / (b * (a * 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 <= -2.7e-69:
		tmp = (math.pi * 0.5) / (b * (a * 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 <= -2.7e-69)
		tmp = Float64(Float64(pi * 0.5) / Float64(b * Float64(a * a)));
	else
		tmp = Float64(Float64(Float64(0.5 / b) * 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 <= -2.7e-69)
		tmp = (pi * 0.5) / (b * (a * 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, -2.7e-69], N[(N[(Pi * 0.5), $MachinePrecision] / N[(b * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 / b), $MachinePrecision] * N[(Pi / b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.6999999999999997e-69

   1. Initial program 79.4%

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

   2. Taylor expanded in b around 0 67.5%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{{a}^{2} \cdot b}} \]
   3. Step-by-step derivation

      1. associate-*r/ 67.5%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{{a}^{2} \cdot b}} \]

      2. unpow2 67.5%

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

   4. Simplified 67.5%

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

   if -2.6999999999999997e-69 < a

   1. Initial program 79.8%

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

   2. Taylor expanded in b around inf 70.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{a \cdot {b}^{2}}} \]
   3. Step-by-step derivation

      1. associate-*r/ 70.7%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{a \cdot {b}^{2}}} \]

      2. *-commutative 70.7%

         \[\leadsto \frac{\color{blue}{\pi \cdot 0.5}}{a \cdot {b}^{2}} \]

      3. *-commutative 70.7%

         \[\leadsto \frac{\pi \cdot 0.5}{\color{blue}{{b}^{2} \cdot a}} \]

      4. associate-/r* 70.2%

         \[\leadsto \color{blue}{\frac{\frac{\pi \cdot 0.5}{{b}^{2}}}{a}} \]

      5. *-commutative 70.2%

         \[\leadsto \frac{\frac{\color{blue}{0.5 \cdot \pi}}{{b}^{2}}}{a} \]

      6. unpow2 70.2%

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

   4. Simplified 70.2%

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

      1. times-frac 71.3%

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

   6. Applied egg-rr 71.3%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.7 \cdot 10^{-69}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{b \cdot \left(a \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{b} \cdot \frac{\pi}{b}}{a}\\ \end{array} \]

Alternative 10: 57.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.7%

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

   1. times-frac 79.7%

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

   2. *-commutative 79.7%

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

   3. times-frac 79.7%

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

   4. difference-of-squares 89.5%

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

   5. associate-/r* 90.2%

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

   6. metadata-eval 90.2%

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

   7. sub-neg 90.2%

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

   8. distribute-neg-frac 90.2%

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

   9. metadata-eval 90.2%

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

3. Simplified 90.2%

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

   1. div-inv 90.3%

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

5. Applied egg-rr 90.3%

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

6. Taylor expanded in b around inf 64.5%

   \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{a \cdot {b}^{2}}} \]
7. Step-by-step derivation

   1. associate-*r/ 64.5%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{a \cdot {b}^{2}}} \]

   2. *-commutative 64.5%

      \[\leadsto \frac{\color{blue}{\pi \cdot 0.5}}{a \cdot {b}^{2}} \]

   3. times-frac 63.7%

      \[\leadsto \color{blue}{\frac{\pi}{a} \cdot \frac{0.5}{{b}^{2}}} \]

   4. unpow2 63.7%

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

8. Simplified 63.7%

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

9. Final simplification 63.7%

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

Reproduce

herbie shell --seed 2023229 
(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))))