NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.8% → 99.6%

Time: 4.5s

Alternatives: 10

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: 78.8% 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: 99.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{+157}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{b} \cdot 0.5}{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{\left(a + b\right) \cdot \left(-a\right)} \cdot \frac{-1}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -9e+157)
   (/ (/ (* (/ PI b) 0.5) a) a)
   (* (/ (* PI 0.5) (* (+ a b) (- a))) (/ -1.0 b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -9e+157) {
		tmp = (((((double) M_PI) / b) * 0.5) / a) / a;
	} else {
		tmp = ((((double) M_PI) * 0.5) / ((a + b) * -a)) * (-1.0 / b);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -9e+157) {
		tmp = (((Math.PI / b) * 0.5) / a) / a;
	} else {
		tmp = ((Math.PI * 0.5) / ((a + b) * -a)) * (-1.0 / b);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -9e+157:
		tmp = (((math.pi / b) * 0.5) / a) / a
	else:
		tmp = ((math.pi * 0.5) / ((a + b) * -a)) * (-1.0 / b)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -9e+157)
		tmp = Float64(Float64(Float64(Float64(pi / b) * 0.5) / a) / a);
	else
		tmp = Float64(Float64(Float64(pi * 0.5) / Float64(Float64(a + b) * Float64(-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 <= -9e+157)
		tmp = (((pi / b) * 0.5) / a) / a;
	else
		tmp = ((pi * 0.5) / ((a + 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, -9e+157], N[(N[(N[(N[(Pi / b), $MachinePrecision] * 0.5), $MachinePrecision] / a), $MachinePrecision] / a), $MachinePrecision], N[(N[(N[(Pi * 0.5), $MachinePrecision] / N[(N[(a + b), $MachinePrecision] * (-a)), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -8.9999999999999997e157

   1. Initial program 78.8%

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

   2. Taylor expanded in 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 49.4

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

   4. Applied rewrites 49.4%

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

   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 56.6

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

   7. Applied rewrites 56.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 62.7

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

      6. lift-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-PI.f64 62.7

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

   9. Applied rewrites 62.7%

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

   if -8.9999999999999997e157 < a

   1. Initial program 78.8%

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

      1. 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. difference-of-squares N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lift-PI.f64 N/A

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

      19. lower--.f64 88.4

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

   3. Applied rewrites 88.4%

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

   4. Taylor expanded in a around inf

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

      1. lower-/.f64 64.5

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

   6. Applied rewrites 64.5%

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

   7. Taylor expanded in a around inf

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

      1. lower-*.f64 93.8

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

   9. Applied rewrites 93.8%

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

       1. lift-*.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. lift-PI.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. frac-times N/A

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

       7. lower-/.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. lift-PI.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. +-commutative N/A

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

       13. lower-+.f64 93.6

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

       14. lift-*.f64 N/A

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

       15. mul-1-neg N/A

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

       16. lower-neg.f64 93.6

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

   11. Applied rewrites 93.6%

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

Alternative 2: 99.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.8%

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

   1. 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. difference-of-squares N/A

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

   13. times-frac N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-+.f64 N/A

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

   17. lower-/.f64 N/A

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

   18. lift-PI.f64 N/A

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

   19. lower--.f64 88.4

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

3. Applied rewrites 88.4%

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

4. Taylor expanded in a around inf

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

   1. lower-/.f64 64.5

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

6. Applied rewrites 64.5%

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

7. Taylor expanded in a around inf

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

   1. lower-*.f64 93.8

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

9. Applied rewrites 93.8%

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

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-+.f64 N/A

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

    4. lift-/.f64 N/A

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

    5. associate-*l* N/A

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

    6. lower-*.f64 N/A

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

    7. lift-/.f64 N/A

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

    8. +-commutative N/A

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

    9. lower-+.f64 N/A

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

    10. lower-*.f64 99.6

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

    11. lift-*.f64 N/A

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

    12. mul-1-neg N/A

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

    13. lower-neg.f64 99.6

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

11. Applied rewrites 99.6%

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

Alternative 3: 92.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -2.05 \cdot 10^{+68}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{b} \cdot 0.5}{a}}{a}\\ \mathbf{elif}\;a \leq -4.5 \cdot 10^{-72}:\\ \;\;\;\;\left(\frac{0.5}{a} \cdot \frac{\pi}{b - a}\right) \cdot \frac{-1}{b}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.5}{b} \cdot \frac{\pi}{-1 \cdot a}\right) \cdot \frac{-1}{b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -2.05e+68)
   (/ (/ (* (/ PI b) 0.5) a) a)
   (if (<= a -4.5e-72)
     (* (* (/ 0.5 a) (/ PI (- b a))) (/ -1.0 b))
     (* (* (/ 0.5 b) (/ PI (* -1.0 a))) (/ -1.0 b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -2.05e+68) {
		tmp = (((((double) M_PI) / b) * 0.5) / a) / a;
	} else if (a <= -4.5e-72) {
		tmp = ((0.5 / a) * (((double) M_PI) / (b - a))) * (-1.0 / b);
	} else {
		tmp = ((0.5 / b) * (((double) M_PI) / (-1.0 * a))) * (-1.0 / b);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -2.05e+68) {
		tmp = (((Math.PI / b) * 0.5) / a) / a;
	} else if (a <= -4.5e-72) {
		tmp = ((0.5 / a) * (Math.PI / (b - a))) * (-1.0 / b);
	} else {
		tmp = ((0.5 / b) * (Math.PI / (-1.0 * a))) * (-1.0 / b);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -2.05e+68:
		tmp = (((math.pi / b) * 0.5) / a) / a
	elif a <= -4.5e-72:
		tmp = ((0.5 / a) * (math.pi / (b - a))) * (-1.0 / b)
	else:
		tmp = ((0.5 / b) * (math.pi / (-1.0 * a))) * (-1.0 / b)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -2.05e+68)
		tmp = Float64(Float64(Float64(Float64(pi / b) * 0.5) / a) / a);
	elseif (a <= -4.5e-72)
		tmp = Float64(Float64(Float64(0.5 / a) * Float64(pi / Float64(b - a))) * Float64(-1.0 / b));
	else
		tmp = Float64(Float64(Float64(0.5 / b) * Float64(pi / Float64(-1.0 * 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 <= -2.05e+68)
		tmp = (((pi / b) * 0.5) / a) / a;
	elseif (a <= -4.5e-72)
		tmp = ((0.5 / a) * (pi / (b - a))) * (-1.0 / b);
	else
		tmp = ((0.5 / b) * (pi / (-1.0 * 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, -2.05e+68], N[(N[(N[(N[(Pi / b), $MachinePrecision] * 0.5), $MachinePrecision] / a), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[a, -4.5e-72], N[(N[(N[(0.5 / a), $MachinePrecision] * N[(Pi / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 / b), $MachinePrecision] * N[(Pi / N[(-1.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.05e68

   1. Initial program 78.8%

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

   2. Taylor expanded in 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 49.4

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

   4. Applied rewrites 49.4%

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

   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 56.6

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

   7. Applied rewrites 56.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 62.7

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

      6. lift-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-PI.f64 62.7

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

   9. Applied rewrites 62.7%

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

   if -2.05e68 < a < -4.5e-72

   1. Initial program 78.8%

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

      1. 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. difference-of-squares N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lift-PI.f64 N/A

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

      19. lower--.f64 88.4

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

   3. Applied rewrites 88.4%

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

   4. Taylor expanded in a around inf

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

      1. lower-/.f64 64.5

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

   6. Applied rewrites 64.5%

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

   7. Taylor expanded in a around inf

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

      1. lower-/.f64 61.4

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

   9. Applied rewrites 61.4%

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

   if -4.5e-72 < a

   1. Initial program 78.8%

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

      1. 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. difference-of-squares N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lift-PI.f64 N/A

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

      19. lower--.f64 88.4

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

   3. Applied rewrites 88.4%

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

   4. Taylor expanded in a around inf

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

      1. lower-/.f64 64.5

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

   6. Applied rewrites 64.5%

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

   7. Taylor expanded in a around inf

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

      1. lower-*.f64 93.8

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

   9. Applied rewrites 93.8%

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

   10. Taylor expanded in a around 0

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

   12. Applied rewrites 63.1%

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

Alternative 4: 92.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -2.05 \cdot 10^{+68}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{b} \cdot 0.5}{a}}{a}\\ \mathbf{elif}\;a \leq -4.5 \cdot 10^{-72}:\\ \;\;\;\;\left(\frac{0.5}{a} \cdot \frac{\pi}{b - a}\right) \cdot \frac{-1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{b \cdot \left(b \cdot a\right)}{\pi}} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -2.05e+68)
   (/ (/ (* (/ PI b) 0.5) a) a)
   (if (<= a -4.5e-72)
     (* (* (/ 0.5 a) (/ PI (- b a))) (/ -1.0 b))
     (* (/ 1.0 (/ (* b (* b a)) PI)) 0.5))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -2.05e+68) {
		tmp = (((((double) M_PI) / b) * 0.5) / a) / a;
	} else if (a <= -4.5e-72) {
		tmp = ((0.5 / a) * (((double) M_PI) / (b - a))) * (-1.0 / b);
	} else {
		tmp = (1.0 / ((b * (b * a)) / ((double) M_PI))) * 0.5;
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -2.05e+68) {
		tmp = (((Math.PI / b) * 0.5) / a) / a;
	} else if (a <= -4.5e-72) {
		tmp = ((0.5 / a) * (Math.PI / (b - a))) * (-1.0 / b);
	} else {
		tmp = (1.0 / ((b * (b * a)) / Math.PI)) * 0.5;
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -2.05e+68:
		tmp = (((math.pi / b) * 0.5) / a) / a
	elif a <= -4.5e-72:
		tmp = ((0.5 / a) * (math.pi / (b - a))) * (-1.0 / b)
	else:
		tmp = (1.0 / ((b * (b * a)) / math.pi)) * 0.5
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -2.05e+68)
		tmp = Float64(Float64(Float64(Float64(pi / b) * 0.5) / a) / a);
	elseif (a <= -4.5e-72)
		tmp = Float64(Float64(Float64(0.5 / a) * Float64(pi / Float64(b - a))) * Float64(-1.0 / b));
	else
		tmp = Float64(Float64(1.0 / Float64(Float64(b * Float64(b * a)) / pi)) * 0.5);
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.05e+68)
		tmp = (((pi / b) * 0.5) / a) / a;
	elseif (a <= -4.5e-72)
		tmp = ((0.5 / a) * (pi / (b - a))) * (-1.0 / b);
	else
		tmp = (1.0 / ((b * (b * a)) / pi)) * 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, -2.05e+68], N[(N[(N[(N[(Pi / b), $MachinePrecision] * 0.5), $MachinePrecision] / a), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[a, -4.5e-72], N[(N[(N[(0.5 / a), $MachinePrecision] * N[(Pi / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[(b * N[(b * a), $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.05e68

   1. Initial program 78.8%

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

   2. Taylor expanded in 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 49.4

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

   4. Applied rewrites 49.4%

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

   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 56.6

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

   7. Applied rewrites 56.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 62.7

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

      6. lift-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-PI.f64 62.7

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

   9. Applied rewrites 62.7%

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

   if -2.05e68 < a < -4.5e-72

   1. Initial program 78.8%

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

      1. 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. difference-of-squares N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lift-PI.f64 N/A

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

      19. lower--.f64 88.4

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

   3. Applied rewrites 88.4%

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

   4. Taylor expanded in a around inf

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

      1. lower-/.f64 64.5

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

   6. Applied rewrites 64.5%

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

   7. Taylor expanded in a around inf

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

      1. lower-/.f64 61.4

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

   9. Applied rewrites 61.4%

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

   if -4.5e-72 < a

   1. Initial program 78.8%

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

   2. Taylor expanded in 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 57.3

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

   4. Applied rewrites 57.3%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. division-flip N/A

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

      4. lower-special-/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-special-/ N/A

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

      7. lower-/.f64 N/A

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

      8. lift-PI.f64 57.3

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

   6. Applied rewrites 57.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 63.0

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

   8. Applied rewrites 63.0%

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

Alternative 5: 89.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -4.5 \cdot 10^{-72}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{b} \cdot 0.5}{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{b \cdot \left(b \cdot a\right)}{\pi}} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -4.5e-72)
   (/ (/ (* (/ PI b) 0.5) a) a)
   (* (/ 1.0 (/ (* b (* b a)) PI)) 0.5)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -4.5e-72) {
		tmp = (((((double) M_PI) / b) * 0.5) / a) / a;
	} else {
		tmp = (1.0 / ((b * (b * a)) / ((double) M_PI))) * 0.5;
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -4.5e-72) {
		tmp = (((Math.PI / b) * 0.5) / a) / a;
	} else {
		tmp = (1.0 / ((b * (b * a)) / Math.PI)) * 0.5;
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -4.5e-72:
		tmp = (((math.pi / b) * 0.5) / a) / a
	else:
		tmp = (1.0 / ((b * (b * a)) / math.pi)) * 0.5
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -4.5e-72)
		tmp = Float64(Float64(Float64(Float64(pi / b) * 0.5) / a) / a);
	else
		tmp = Float64(Float64(1.0 / Float64(Float64(b * Float64(b * a)) / pi)) * 0.5);
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -4.5e-72)
		tmp = (((pi / b) * 0.5) / a) / a;
	else
		tmp = (1.0 / ((b * (b * a)) / pi)) * 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, -4.5e-72], N[(N[(N[(N[(Pi / b), $MachinePrecision] * 0.5), $MachinePrecision] / a), $MachinePrecision] / a), $MachinePrecision], N[(N[(1.0 / N[(N[(b * N[(b * a), $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.5e-72

   1. Initial program 78.8%

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

   2. Taylor expanded in 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 49.4

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

   4. Applied rewrites 49.4%

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

   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 56.6

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

   7. Applied rewrites 56.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 62.7

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

      6. lift-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-PI.f64 62.7

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

   9. Applied rewrites 62.7%

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

   if -4.5e-72 < a

   1. Initial program 78.8%

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

   2. Taylor expanded in 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 57.3

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

   4. Applied rewrites 57.3%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. division-flip N/A

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

      4. lower-special-/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-special-/ N/A

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

      7. lower-/.f64 N/A

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

      8. lift-PI.f64 57.3

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

   6. Applied rewrites 57.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 63.0

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

   8. Applied rewrites 63.0%

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

Alternative 6: 84.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -4.5 \cdot 10^{-72}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{b} \cdot 0.5}{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{b \cdot b} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -4.5e-72)
   (/ (/ (* (/ PI b) 0.5) a) a)
   (* (/ (/ PI a) (* b b)) 0.5)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -4.5e-72) {
		tmp = (((((double) M_PI) / b) * 0.5) / a) / a;
	} else {
		tmp = ((((double) M_PI) / a) / (b * b)) * 0.5;
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -4.5e-72) {
		tmp = (((Math.PI / b) * 0.5) / a) / a;
	} else {
		tmp = ((Math.PI / a) / (b * b)) * 0.5;
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -4.5e-72:
		tmp = (((math.pi / b) * 0.5) / a) / a
	else:
		tmp = ((math.pi / a) / (b * b)) * 0.5
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -4.5e-72)
		tmp = Float64(Float64(Float64(Float64(pi / b) * 0.5) / a) / a);
	else
		tmp = Float64(Float64(Float64(pi / a) / Float64(b * b)) * 0.5);
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -4.5e-72)
		tmp = (((pi / b) * 0.5) / a) / a;
	else
		tmp = ((pi / a) / (b * b)) * 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, -4.5e-72], N[(N[(N[(N[(Pi / b), $MachinePrecision] * 0.5), $MachinePrecision] / a), $MachinePrecision] / a), $MachinePrecision], N[(N[(N[(Pi / a), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.5e-72

   1. Initial program 78.8%

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

   2. Taylor expanded in 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 49.4

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

   4. Applied rewrites 49.4%

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

   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 56.6

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

   7. Applied rewrites 56.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 62.7

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

      6. lift-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-PI.f64 62.7

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

   9. Applied rewrites 62.7%

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

   if -4.5e-72 < a

   1. Initial program 78.8%

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

   2. Taylor expanded in 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 57.3

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

   4. Applied rewrites 57.3%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. division-flip N/A

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

      4. lower-special-/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-special-/ N/A

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

      7. lower-/.f64 N/A

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

      8. lift-PI.f64 57.3

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

   6. Applied rewrites 57.3%

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

      1. lift-/.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-special-/ N/A

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

      7. pow2 N/A

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

      8. *-commutative N/A

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

      9. lower-special-/ N/A

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

      10. division-flip N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-PI.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 57.2

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

   8. Applied rewrites 57.2%

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

Alternative 7: 78.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -4.5 \cdot 10^{-72}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{b}}{a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{b \cdot b} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -4.5e-72)
   (/ (* 0.5 (/ PI b)) (* a a))
   (* (/ (/ PI a) (* b b)) 0.5)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -4.5e-72) {
		tmp = (0.5 * (((double) M_PI) / b)) / (a * a);
	} else {
		tmp = ((((double) M_PI) / a) / (b * b)) * 0.5;
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -4.5e-72) {
		tmp = (0.5 * (Math.PI / b)) / (a * a);
	} else {
		tmp = ((Math.PI / a) / (b * b)) * 0.5;
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -4.5e-72:
		tmp = (0.5 * (math.pi / b)) / (a * a)
	else:
		tmp = ((math.pi / a) / (b * b)) * 0.5
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -4.5e-72)
		tmp = Float64(Float64(0.5 * Float64(pi / b)) / Float64(a * a));
	else
		tmp = Float64(Float64(Float64(pi / a) / Float64(b * b)) * 0.5);
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -4.5e-72)
		tmp = (0.5 * (pi / b)) / (a * a);
	else
		tmp = ((pi / a) / (b * b)) * 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, -4.5e-72], N[(N[(0.5 * N[(Pi / b), $MachinePrecision]), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi / a), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.5e-72

   1. Initial program 78.8%

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

   2. Taylor expanded in 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 49.4

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

   4. Applied rewrites 49.4%

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

   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 56.6

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

   7. Applied rewrites 56.6%

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

   if -4.5e-72 < a

   1. Initial program 78.8%

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

   2. Taylor expanded in 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 57.3

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

   4. Applied rewrites 57.3%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. division-flip N/A

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

      4. lower-special-/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-special-/ N/A

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

      7. lower-/.f64 N/A

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

      8. lift-PI.f64 57.3

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

   6. Applied rewrites 57.3%

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

      1. lift-/.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-special-/ N/A

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

      7. pow2 N/A

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

      8. *-commutative N/A

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

      9. lower-special-/ N/A

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

      10. division-flip N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-PI.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 57.2

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

   8. Applied rewrites 57.2%

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

Alternative 8: 78.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -4.5 \cdot 10^{-72}:\\ \;\;\;\;\frac{\pi}{\left(a \cdot a\right) \cdot b} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{b \cdot b} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -4.5e-72)
   (* (/ PI (* (* a a) b)) 0.5)
   (* (/ (/ PI a) (* b b)) 0.5)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -4.5e-72) {
		tmp = (((double) M_PI) / ((a * a) * b)) * 0.5;
	} else {
		tmp = ((((double) M_PI) / a) / (b * b)) * 0.5;
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -4.5e-72) {
		tmp = (Math.PI / ((a * a) * b)) * 0.5;
	} else {
		tmp = ((Math.PI / a) / (b * b)) * 0.5;
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -4.5e-72:
		tmp = (math.pi / ((a * a) * b)) * 0.5
	else:
		tmp = ((math.pi / a) / (b * b)) * 0.5
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -4.5e-72)
		tmp = Float64(Float64(pi / Float64(Float64(a * a) * b)) * 0.5);
	else
		tmp = Float64(Float64(Float64(pi / a) / Float64(b * b)) * 0.5);
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -4.5e-72)
		tmp = (pi / ((a * a) * b)) * 0.5;
	else
		tmp = ((pi / a) / (b * b)) * 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, -4.5e-72], N[(N[(Pi / N[(N[(a * a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(Pi / a), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.5e-72

   1. Initial program 78.8%

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

   2. Taylor expanded in 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 56.7

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

   4. Applied rewrites 56.7%

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

   if -4.5e-72 < a

   1. Initial program 78.8%

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

   2. Taylor expanded in 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 57.3

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

   4. Applied rewrites 57.3%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. division-flip N/A

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

      4. lower-special-/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-special-/ N/A

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

      7. lower-/.f64 N/A

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

      8. lift-PI.f64 57.3

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

   6. Applied rewrites 57.3%

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

      1. lift-/.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-special-/ N/A

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

      7. pow2 N/A

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

      8. *-commutative N/A

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

      9. lower-special-/ N/A

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

      10. division-flip N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-PI.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 57.2

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

   8. Applied rewrites 57.2%

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

Alternative 9: 78.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -4.5 \cdot 10^{-72}:\\ \;\;\;\;\frac{\pi}{\left(a \cdot a\right) \cdot b} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -4.5e-72)
   (* (/ 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 <= -4.5e-72) {
		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 <= -4.5e-72) {
		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 <= -4.5e-72:
		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 <= -4.5e-72)
		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 <= -4.5e-72)
		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, -4.5e-72], 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 < -4.5e-72

   1. Initial program 78.8%

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

   2. Taylor expanded in 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 56.7

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

   4. Applied rewrites 56.7%

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

   if -4.5e-72 < a

   1. Initial program 78.8%

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

   2. Taylor expanded in 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 57.3

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

   4. Applied rewrites 57.3%

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

Alternative 10: 56.7% 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 78.8%

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

2. Taylor expanded in 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 56.7

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

4. Applied rewrites 56.7%

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

Reproduce

herbie shell --seed 2025132 
(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))))