NMSE Section 6.1 mentioned, B

Percentage Accurate: 79.1% → 97.2%

Time: 3.5s

Alternatives: 8

Speedup: 1.8×

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: 79.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} t_0 := \left(a \cdot b\right) \cdot 2\\ \mathbf{if}\;a \leq -1.95 \cdot 10^{+157}:\\ \;\;\;\;\frac{\frac{\pi + -1 \cdot \frac{b \cdot \pi}{a}}{a}}{t\_0}\\ \mathbf{elif}\;a \leq -4.7 \cdot 10^{-176}:\\ \;\;\;\;\frac{\left(1 \cdot b - a \cdot 1\right) \cdot \left(\pi \cdot \frac{1}{\left(b + a\right) \cdot \left(b - a\right)}\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{b}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (* a b) 2.0)))
   (if (<= a -1.95e+157)
     (/ (/ (+ PI (* -1.0 (/ (* b PI) a))) a) t_0)
     (if (<= a -4.7e-176)
       (/ (* (- (* 1.0 b) (* a 1.0)) (* PI (/ 1.0 (* (+ b a) (- b a))))) t_0)
       (/ (/ PI b) t_0)))))

C

assert(a < b);
double code(double a, double b) {
	double t_0 = (a * b) * 2.0;
	double tmp;
	if (a <= -1.95e+157) {
		tmp = ((((double) M_PI) + (-1.0 * ((b * ((double) M_PI)) / a))) / a) / t_0;
	} else if (a <= -4.7e-176) {
		tmp = (((1.0 * b) - (a * 1.0)) * (((double) M_PI) * (1.0 / ((b + a) * (b - a))))) / t_0;
	} else {
		tmp = (((double) M_PI) / b) / t_0;
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double t_0 = (a * b) * 2.0;
	double tmp;
	if (a <= -1.95e+157) {
		tmp = ((Math.PI + (-1.0 * ((b * Math.PI) / a))) / a) / t_0;
	} else if (a <= -4.7e-176) {
		tmp = (((1.0 * b) - (a * 1.0)) * (Math.PI * (1.0 / ((b + a) * (b - a))))) / t_0;
	} else {
		tmp = (Math.PI / b) / t_0;
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	t_0 = (a * b) * 2.0
	tmp = 0
	if a <= -1.95e+157:
		tmp = ((math.pi + (-1.0 * ((b * math.pi) / a))) / a) / t_0
	elif a <= -4.7e-176:
		tmp = (((1.0 * b) - (a * 1.0)) * (math.pi * (1.0 / ((b + a) * (b - a))))) / t_0
	else:
		tmp = (math.pi / b) / t_0
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	t_0 = Float64(Float64(a * b) * 2.0)
	tmp = 0.0
	if (a <= -1.95e+157)
		tmp = Float64(Float64(Float64(pi + Float64(-1.0 * Float64(Float64(b * pi) / a))) / a) / t_0);
	elseif (a <= -4.7e-176)
		tmp = Float64(Float64(Float64(Float64(1.0 * b) - Float64(a * 1.0)) * Float64(pi * Float64(1.0 / Float64(Float64(b + a) * Float64(b - a))))) / t_0);
	else
		tmp = Float64(Float64(pi / b) / t_0);
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	t_0 = (a * b) * 2.0;
	tmp = 0.0;
	if (a <= -1.95e+157)
		tmp = ((pi + (-1.0 * ((b * pi) / a))) / a) / t_0;
	elseif (a <= -4.7e-176)
		tmp = (((1.0 * b) - (a * 1.0)) * (pi * (1.0 / ((b + a) * (b - a))))) / t_0;
	else
		tmp = (pi / b) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := Block[{t$95$0 = N[(N[(a * b), $MachinePrecision] * 2.0), $MachinePrecision]}, If[LessEqual[a, -1.95e+157], N[(N[(N[(Pi + N[(-1.0 * N[(N[(b * Pi), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[a, -4.7e-176], N[(N[(N[(N[(1.0 * b), $MachinePrecision] - N[(a * 1.0), $MachinePrecision]), $MachinePrecision] * N[(Pi * N[(1.0 / N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(Pi / b), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.94999999999999985e157

   1. Initial program 79.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. frac-sub N/A

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

      14. associate-*l/ N/A

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

      15. frac-times N/A

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

   3. Applied rewrites 87.6%

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

   4. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{PI}\left(\right) + -1 \cdot \frac{b \cdot \mathsf{PI}\left(\right)}{a}}{a}}{\left(a \cdot b\right) \cdot 2} \]

      3. lift-PI.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 56.6

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

   6. Applied rewrites 56.6%

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

   if -1.94999999999999985e157 < a < -4.69999999999999984e-176

   1. Initial program 79.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. frac-sub N/A

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

      14. associate-*l/ N/A

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

      15. frac-times N/A

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

   3. Applied rewrites 87.6%

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

   if -4.69999999999999984e-176 < a

   1. Initial program 79.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. frac-sub N/A

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

      14. associate-*l/ N/A

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

      15. frac-times N/A

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

   3. Applied rewrites 87.6%

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

   4. Taylor expanded in a around 0

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

      1. lift-/.f64 N/A

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

      2. lift-PI.f64 62.6

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

   6. Applied rewrites 62.6%

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

Alternative 2: 97.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} t_0 := \left(a \cdot b\right) \cdot 2\\ \mathbf{if}\;a \leq -1.95 \cdot 10^{+157}:\\ \;\;\;\;\frac{\frac{\pi + -1 \cdot \frac{b \cdot \pi}{a}}{a}}{t\_0}\\ \mathbf{elif}\;a \leq -5 \cdot 10^{-176}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{\left(b + a\right) \cdot \left(b - a\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{b}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (* a b) 2.0)))
   (if (<= a -1.95e+157)
     (/ (/ (+ PI (* -1.0 (/ (* b PI) a))) a) t_0)
     (if (<= a -5e-176)
       (* (/ (* 0.5 PI) (* (+ b a) (- b a))) (- (/ 1.0 a) (/ 1.0 b)))
       (/ (/ PI b) t_0)))))

C

assert(a < b);
double code(double a, double b) {
	double t_0 = (a * b) * 2.0;
	double tmp;
	if (a <= -1.95e+157) {
		tmp = ((((double) M_PI) + (-1.0 * ((b * ((double) M_PI)) / a))) / a) / t_0;
	} else if (a <= -5e-176) {
		tmp = ((0.5 * ((double) M_PI)) / ((b + a) * (b - a))) * ((1.0 / a) - (1.0 / b));
	} else {
		tmp = (((double) M_PI) / b) / t_0;
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double t_0 = (a * b) * 2.0;
	double tmp;
	if (a <= -1.95e+157) {
		tmp = ((Math.PI + (-1.0 * ((b * Math.PI) / a))) / a) / t_0;
	} else if (a <= -5e-176) {
		tmp = ((0.5 * Math.PI) / ((b + a) * (b - a))) * ((1.0 / a) - (1.0 / b));
	} else {
		tmp = (Math.PI / b) / t_0;
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	t_0 = (a * b) * 2.0
	tmp = 0
	if a <= -1.95e+157:
		tmp = ((math.pi + (-1.0 * ((b * math.pi) / a))) / a) / t_0
	elif a <= -5e-176:
		tmp = ((0.5 * math.pi) / ((b + a) * (b - a))) * ((1.0 / a) - (1.0 / b))
	else:
		tmp = (math.pi / b) / t_0
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	t_0 = Float64(Float64(a * b) * 2.0)
	tmp = 0.0
	if (a <= -1.95e+157)
		tmp = Float64(Float64(Float64(pi + Float64(-1.0 * Float64(Float64(b * pi) / a))) / a) / t_0);
	elseif (a <= -5e-176)
		tmp = Float64(Float64(Float64(0.5 * pi) / Float64(Float64(b + a) * Float64(b - a))) * Float64(Float64(1.0 / a) - Float64(1.0 / b)));
	else
		tmp = Float64(Float64(pi / b) / t_0);
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	t_0 = (a * b) * 2.0;
	tmp = 0.0;
	if (a <= -1.95e+157)
		tmp = ((pi + (-1.0 * ((b * pi) / a))) / a) / t_0;
	elseif (a <= -5e-176)
		tmp = ((0.5 * pi) / ((b + a) * (b - a))) * ((1.0 / a) - (1.0 / b));
	else
		tmp = (pi / b) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := Block[{t$95$0 = N[(N[(a * b), $MachinePrecision] * 2.0), $MachinePrecision]}, If[LessEqual[a, -1.95e+157], N[(N[(N[(Pi + N[(-1.0 * N[(N[(b * Pi), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[a, -5e-176], N[(N[(N[(0.5 * Pi), $MachinePrecision] / N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / a), $MachinePrecision] - N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / b), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.94999999999999985e157

   1. Initial program 79.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. frac-sub N/A

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

      14. associate-*l/ N/A

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

      15. frac-times N/A

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

   3. Applied rewrites 87.6%

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

   4. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{PI}\left(\right) + -1 \cdot \frac{b \cdot \mathsf{PI}\left(\right)}{a}}{a}}{\left(a \cdot b\right) \cdot 2} \]

      3. lift-PI.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 56.6

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

   6. Applied rewrites 56.6%

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

   if -1.94999999999999985e157 < a < -5e-176

   1. Initial program 79.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. mult-flip-rev N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. mult-flip N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-PI.f64 N/A

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

      15. difference-of-squares N/A

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

      16. lower-*.f64 N/A

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

      17. lower-+.f64 N/A

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

      18. lower--.f64 87.7

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

   3. Applied rewrites 87.7%

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

   if -5e-176 < a

   1. Initial program 79.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. frac-sub N/A

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

      14. associate-*l/ N/A

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

      15. frac-times N/A

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

   3. Applied rewrites 87.6%

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

   4. Taylor expanded in a around 0

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

      1. lift-/.f64 N/A

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

      2. lift-PI.f64 62.6

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

   6. Applied rewrites 62.6%

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

Alternative 3: 92.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} t_0 := \left(a \cdot b\right) \cdot 2\\ \mathbf{if}\;a \leq -1.95 \cdot 10^{+157}:\\ \;\;\;\;\frac{\frac{\pi + -1 \cdot \frac{b \cdot \pi}{a}}{a}}{t\_0}\\ \mathbf{elif}\;a \leq -2.5 \cdot 10^{-106}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{\left(b + a\right) \cdot \left(b - a\right)} \cdot \frac{-1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{b}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (* a b) 2.0)))
   (if (<= a -1.95e+157)
     (/ (/ (+ PI (* -1.0 (/ (* b PI) a))) a) t_0)
     (if (<= a -2.5e-106)
       (* (/ (* PI 0.5) (* (+ b a) (- b a))) (/ -1.0 b))
       (/ (/ PI b) t_0)))))

C

assert(a < b);
double code(double a, double b) {
	double t_0 = (a * b) * 2.0;
	double tmp;
	if (a <= -1.95e+157) {
		tmp = ((((double) M_PI) + (-1.0 * ((b * ((double) M_PI)) / a))) / a) / t_0;
	} else if (a <= -2.5e-106) {
		tmp = ((((double) M_PI) * 0.5) / ((b + a) * (b - a))) * (-1.0 / b);
	} else {
		tmp = (((double) M_PI) / b) / t_0;
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double t_0 = (a * b) * 2.0;
	double tmp;
	if (a <= -1.95e+157) {
		tmp = ((Math.PI + (-1.0 * ((b * Math.PI) / a))) / a) / t_0;
	} else if (a <= -2.5e-106) {
		tmp = ((Math.PI * 0.5) / ((b + a) * (b - a))) * (-1.0 / b);
	} else {
		tmp = (Math.PI / b) / t_0;
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	t_0 = (a * b) * 2.0
	tmp = 0
	if a <= -1.95e+157:
		tmp = ((math.pi + (-1.0 * ((b * math.pi) / a))) / a) / t_0
	elif a <= -2.5e-106:
		tmp = ((math.pi * 0.5) / ((b + a) * (b - a))) * (-1.0 / b)
	else:
		tmp = (math.pi / b) / t_0
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	t_0 = Float64(Float64(a * b) * 2.0)
	tmp = 0.0
	if (a <= -1.95e+157)
		tmp = Float64(Float64(Float64(pi + Float64(-1.0 * Float64(Float64(b * pi) / a))) / a) / t_0);
	elseif (a <= -2.5e-106)
		tmp = Float64(Float64(Float64(pi * 0.5) / Float64(Float64(b + a) * Float64(b - a))) * Float64(-1.0 / b));
	else
		tmp = Float64(Float64(pi / b) / t_0);
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	t_0 = (a * b) * 2.0;
	tmp = 0.0;
	if (a <= -1.95e+157)
		tmp = ((pi + (-1.0 * ((b * pi) / a))) / a) / t_0;
	elseif (a <= -2.5e-106)
		tmp = ((pi * 0.5) / ((b + a) * (b - a))) * (-1.0 / b);
	else
		tmp = (pi / b) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := Block[{t$95$0 = N[(N[(a * b), $MachinePrecision] * 2.0), $MachinePrecision]}, If[LessEqual[a, -1.95e+157], N[(N[(N[(Pi + N[(-1.0 * N[(N[(b * Pi), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[a, -2.5e-106], N[(N[(N[(Pi * 0.5), $MachinePrecision] / N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / b), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.94999999999999985e157

   1. Initial program 79.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. frac-sub N/A

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

      14. associate-*l/ N/A

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

      15. frac-times N/A

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

   3. Applied rewrites 87.6%

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

   4. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{PI}\left(\right) + -1 \cdot \frac{b \cdot \mathsf{PI}\left(\right)}{a}}{a}}{\left(a \cdot b\right) \cdot 2} \]

      3. lift-PI.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 56.6

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

   6. Applied rewrites 56.6%

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

   if -1.94999999999999985e157 < a < -2.49999999999999991e-106

   1. Initial program 79.1%

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

   2. Taylor expanded in a around inf

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

      1. lower-/.f64 54.9

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

   4. Applied rewrites 54.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares-rev N/A

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

      7. mult-flip-rev N/A

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

      8. lower-/.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. lift-PI.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lift--.f64 N/A

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

      17. lift-*.f64 63.5

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

   6. Applied rewrites 63.5%

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

   if -2.49999999999999991e-106 < a

   1. Initial program 79.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. frac-sub N/A

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

      14. associate-*l/ N/A

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

      15. frac-times N/A

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

   3. Applied rewrites 87.6%

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

   4. Taylor expanded in a around 0

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

      1. lift-/.f64 N/A

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

      2. lift-PI.f64 62.6

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

   6. Applied rewrites 62.6%

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

Alternative 4: 86.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 1.15 \cdot 10^{-166}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{b}}{a \cdot a}\\ \mathbf{elif}\;b \leq 3.6 \cdot 10^{+60}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{\left(b + a\right) \cdot \left(b - a\right)} \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{b}}{\left(a \cdot b\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 1.15e-166)
   (/ (* 0.5 (/ PI b)) (* a a))
   (if (<= b 3.6e+60)
     (* (/ (* 0.5 PI) (* (+ b a) (- b a))) (/ 1.0 a))
     (/ (/ PI b) (* (* a b) 2.0)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 1.15e-166) {
		tmp = (0.5 * (((double) M_PI) / b)) / (a * a);
	} else if (b <= 3.6e+60) {
		tmp = ((0.5 * ((double) M_PI)) / ((b + a) * (b - a))) * (1.0 / a);
	} else {
		tmp = (((double) M_PI) / b) / ((a * b) * 2.0);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 1.15e-166) {
		tmp = (0.5 * (Math.PI / b)) / (a * a);
	} else if (b <= 3.6e+60) {
		tmp = ((0.5 * Math.PI) / ((b + a) * (b - a))) * (1.0 / a);
	} else {
		tmp = (Math.PI / b) / ((a * b) * 2.0);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 1.15e-166:
		tmp = (0.5 * (math.pi / b)) / (a * a)
	elif b <= 3.6e+60:
		tmp = ((0.5 * math.pi) / ((b + a) * (b - a))) * (1.0 / a)
	else:
		tmp = (math.pi / b) / ((a * b) * 2.0)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 1.15e-166)
		tmp = Float64(Float64(0.5 * Float64(pi / b)) / Float64(a * a));
	elseif (b <= 3.6e+60)
		tmp = Float64(Float64(Float64(0.5 * pi) / Float64(Float64(b + a) * Float64(b - a))) * Float64(1.0 / a));
	else
		tmp = Float64(Float64(pi / b) / Float64(Float64(a * b) * 2.0));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.15e-166)
		tmp = (0.5 * (pi / b)) / (a * a);
	elseif (b <= 3.6e+60)
		tmp = ((0.5 * pi) / ((b + a) * (b - a))) * (1.0 / a);
	else
		tmp = (pi / b) / ((a * b) * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[b, 1.15e-166], N[(N[(0.5 * N[(Pi / b), $MachinePrecision]), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.6e+60], N[(N[(N[(0.5 * Pi), $MachinePrecision] / N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / a), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / b), $MachinePrecision] / N[(N[(a * b), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.14999999999999999e-166

   1. Initial program 79.1%

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

   2. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{PI}\left(\right)}{b} \cdot \frac{1}{2} + \frac{-1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a}}{{a}^{2}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{PI}\left(\right)}{b}, \frac{1}{2}, \frac{-1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a}\right)}{{\color{blue}{a}}^{2}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{PI}\left(\right)}{b}, \frac{1}{2}, \frac{-1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a}\right)}{{a}^{2}} \]

      6. lift-PI.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\pi}{b}, \frac{1}{2}, \frac{-1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a}\right)}{{a}^{2}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\pi}{b}, \frac{1}{2}, \frac{\mathsf{PI}\left(\right)}{a} \cdot \frac{-1}{2}\right)}{{a}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\pi}{b}, \frac{1}{2}, \frac{\mathsf{PI}\left(\right)}{a} \cdot \frac{-1}{2}\right)}{{a}^{2}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\pi}{b}, \frac{1}{2}, \frac{\mathsf{PI}\left(\right)}{a} \cdot \frac{-1}{2}\right)}{{a}^{2}} \]

      10. lift-PI.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\pi}{b}, \frac{1}{2}, \frac{\pi}{a} \cdot \frac{-1}{2}\right)}{{a}^{2}} \]

      11. pow2 N/A

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

      12. lift-*.f64 49.2

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

   4. Applied rewrites 49.2%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{b}}{a \cdot a} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{b}}{a \cdot a} \]

      3. lift-PI.f64 56.1

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

   7. Applied rewrites 56.1%

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

   if 1.14999999999999999e-166 < b < 3.59999999999999968e60

   1. Initial program 79.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. mult-flip-rev N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. mult-flip N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-PI.f64 N/A

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

      15. difference-of-squares N/A

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

      16. lower-*.f64 N/A

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

      17. lower-+.f64 N/A

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

      18. lower--.f64 87.7

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

   3. Applied rewrites 87.7%

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

   4. Taylor expanded in a around 0

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

      1. lift-/.f64 63.7

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

   6. Applied rewrites 63.7%

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

   if 3.59999999999999968e60 < b

   1. Initial program 79.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. frac-sub N/A

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

      14. associate-*l/ N/A

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

      15. frac-times N/A

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

   3. Applied rewrites 87.6%

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

   4. Taylor expanded in a around 0

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

      1. lift-/.f64 N/A

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

      2. lift-PI.f64 62.6

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

   6. Applied rewrites 62.6%

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

Alternative 5: 85.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -2.5 \cdot 10^{-106}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{\left(b + a\right) \cdot \left(b - a\right)} \cdot \frac{-1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{b}}{\left(a \cdot b\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -2.5e-106)
   (* (/ (* PI 0.5) (* (+ b a) (- b a))) (/ -1.0 b))
   (/ (/ PI b) (* (* a b) 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -2.5e-106) {
		tmp = ((((double) M_PI) * 0.5) / ((b + a) * (b - a))) * (-1.0 / b);
	} else {
		tmp = (((double) M_PI) / b) / ((a * b) * 2.0);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -2.5e-106) {
		tmp = ((Math.PI * 0.5) / ((b + a) * (b - a))) * (-1.0 / b);
	} else {
		tmp = (Math.PI / b) / ((a * b) * 2.0);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -2.5e-106:
		tmp = ((math.pi * 0.5) / ((b + a) * (b - a))) * (-1.0 / b)
	else:
		tmp = (math.pi / b) / ((a * b) * 2.0)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -2.5e-106)
		tmp = Float64(Float64(Float64(pi * 0.5) / Float64(Float64(b + a) * Float64(b - a))) * Float64(-1.0 / b));
	else
		tmp = Float64(Float64(pi / b) / Float64(Float64(a * b) * 2.0));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.5e-106)
		tmp = ((pi * 0.5) / ((b + a) * (b - a))) * (-1.0 / b);
	else
		tmp = (pi / b) / ((a * b) * 2.0);
	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, -2.5e-106], N[(N[(N[(Pi * 0.5), $MachinePrecision] / N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / b), $MachinePrecision] / N[(N[(a * b), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.49999999999999991e-106

   1. Initial program 79.1%

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

   2. Taylor expanded in a around inf

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

      1. lower-/.f64 54.9

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

   4. Applied rewrites 54.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares-rev N/A

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

      7. mult-flip-rev N/A

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

      8. lower-/.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. lift-PI.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lift--.f64 N/A

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

      17. lift-*.f64 63.5

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

   6. Applied rewrites 63.5%

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

   if -2.49999999999999991e-106 < a

   1. Initial program 79.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. frac-sub N/A

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

      14. associate-*l/ N/A

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

      15. frac-times N/A

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

   3. Applied rewrites 87.6%

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

   4. Taylor expanded in a around 0

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

      1. lift-/.f64 N/A

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

      2. lift-PI.f64 62.6

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

   6. Applied rewrites 62.6%

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

Alternative 6: 83.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1.6 \cdot 10^{-59}:\\ \;\;\;\;\frac{\pi}{\left(a \cdot a\right) \cdot b} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{b}}{\left(a \cdot b\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -1.6e-59)
   (* (/ PI (* (* a a) b)) 0.5)
   (/ (/ PI b) (* (* a b) 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -1.6e-59) {
		tmp = (((double) M_PI) / ((a * a) * b)) * 0.5;
	} else {
		tmp = (((double) M_PI) / b) / ((a * b) * 2.0);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -1.6e-59) {
		tmp = (Math.PI / ((a * a) * b)) * 0.5;
	} else {
		tmp = (Math.PI / b) / ((a * b) * 2.0);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -1.6e-59:
		tmp = (math.pi / ((a * a) * b)) * 0.5
	else:
		tmp = (math.pi / b) / ((a * b) * 2.0)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -1.6e-59)
		tmp = Float64(Float64(pi / Float64(Float64(a * a) * b)) * 0.5);
	else
		tmp = Float64(Float64(pi / b) / Float64(Float64(a * b) * 2.0));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.6e-59)
		tmp = (pi / ((a * a) * b)) * 0.5;
	else
		tmp = (pi / b) / ((a * b) * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -1.6e-59], N[(N[(Pi / N[(N[(a * a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(Pi / b), $MachinePrecision] / N[(N[(a * b), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.6e-59

   1. Initial program 79.1%

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

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{{a}^{2} \cdot b}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{PI}\left(\right)}{{a}^{2} \cdot b} \cdot \frac{1}{2} \]

      4. lift-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 56.2

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

   4. Applied rewrites 56.2%

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

   if -1.6e-59 < a

   1. Initial program 79.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. frac-sub N/A

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

      14. associate-*l/ N/A

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

      15. frac-times N/A

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

   3. Applied rewrites 87.6%

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

   4. Taylor expanded in a around 0

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

      1. lift-/.f64 N/A

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

      2. lift-PI.f64 62.6

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

   6. Applied rewrites 62.6%

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

Alternative 7: 77.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -6.8 \cdot 10^{-60}:\\ \;\;\;\;\frac{\pi}{\left(a \cdot a\right) \cdot b} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -6.8e-60)
   (* (/ PI (* (* a a) b)) 0.5)
   (* (/ PI (* (* b b) a)) 0.5)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -6.8e-60) {
		tmp = (((double) M_PI) / ((a * a) * b)) * 0.5;
	} else {
		tmp = (((double) M_PI) / ((b * b) * a)) * 0.5;
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -6.8e-60) {
		tmp = (Math.PI / ((a * a) * b)) * 0.5;
	} else {
		tmp = (Math.PI / ((b * b) * a)) * 0.5;
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -6.8e-60:
		tmp = (math.pi / ((a * a) * b)) * 0.5
	else:
		tmp = (math.pi / ((b * b) * a)) * 0.5
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -6.8e-60)
		tmp = Float64(Float64(pi / Float64(Float64(a * a) * b)) * 0.5);
	else
		tmp = Float64(Float64(pi / Float64(Float64(b * b) * a)) * 0.5);
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -6.8e-60)
		tmp = (pi / ((a * a) * b)) * 0.5;
	else
		tmp = (pi / ((b * b) * a)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -6.8e-60], N[(N[(Pi / N[(N[(a * a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(Pi / N[(N[(b * b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.80000000000000013e-60

   1. Initial program 79.1%

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

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{{a}^{2} \cdot b}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{PI}\left(\right)}{{a}^{2} \cdot b} \cdot \frac{1}{2} \]

      4. lift-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 56.2

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

   4. Applied rewrites 56.2%

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

   if -6.80000000000000013e-60 < a

   1. Initial program 79.1%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a \cdot {b}^{2}}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{PI}\left(\right)}{a \cdot {b}^{2}} \cdot \frac{1}{2} \]

      4. lift-PI.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 56.8

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

   4. Applied rewrites 56.8%

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

Alternative 8: 56.2% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

2. Taylor expanded in a around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{{a}^{2} \cdot b}} \]
3. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{{a}^{2} \cdot b} \cdot \frac{1}{2} \]

   4. lift-PI.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 56.2

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

4. Applied rewrites 56.2%

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

Reproduce

herbie shell --seed 2025134 
(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))))