NMSE Section 6.1 mentioned, B

Percentage Accurate: 77.9% → 99.0%

Time: 10.7s

Alternatives: 7

Speedup: 2.4×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 10^{+46}:\\ \;\;\;\;\frac{\pi}{a \cdot \left(\left(b + a\right) \cdot \left(b \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(a \cdot \left(b + a\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 1e+46)
   (/ PI (* a (* (+ b a) (* b 2.0))))
   (* PI (/ 0.5 (* b (* a (+ b a)))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 1e+46) {
		tmp = ((double) M_PI) / (a * ((b + a) * (b * 2.0)));
	} else {
		tmp = ((double) M_PI) * (0.5 / (b * (a * (b + a))));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 1e+46) {
		tmp = Math.PI / (a * ((b + a) * (b * 2.0)));
	} else {
		tmp = Math.PI * (0.5 / (b * (a * (b + a))));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 1e+46:
		tmp = math.pi / (a * ((b + a) * (b * 2.0)))
	else:
		tmp = math.pi * (0.5 / (b * (a * (b + a))))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 1e+46)
		tmp = Float64(pi / Float64(a * Float64(Float64(b + a) * Float64(b * 2.0))));
	else
		tmp = Float64(pi * Float64(0.5 / Float64(b * Float64(a * Float64(b + a)))));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1e+46)
		tmp = pi / (a * ((b + a) * (b * 2.0)));
	else
		tmp = pi * (0.5 / (b * (a * (b + a))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[b, 1e+46], N[(Pi / N[(a * N[(N[(b + a), $MachinePrecision] * N[(b * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(0.5 / N[(b * N[(a * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 9.9999999999999999e45

   1. Initial program 83.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. associate-*r/ N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. difference-of-squares N/A

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

      11. *-commutative N/A

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

      12. *-rgt-identity N/A

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

      13. *-lft-identity N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

   4. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. *-rgt-identity N/A

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

      10. *-lft-identity N/A

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

      11. *-commutative N/A

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

      12. frac-sub N/A

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

      13. lift-+.f64 N/A

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

      14. lift--.f64 N/A

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

      15. difference-of-squares N/A

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

      16. associate-*l/ N/A

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

   6. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.7

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. metadata-eval N/A

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

      13. div-inv N/A

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

      14. lower-*.f64 N/A

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

      15. div-inv N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 99.7

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

   8. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 99.7

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

   10. Applied rewrites 99.7%

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

   if 9.9999999999999999e45 < b

   1. Initial program 67.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. associate-*r/ N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. difference-of-squares N/A

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

      11. *-commutative N/A

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

      12. *-rgt-identity N/A

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

      13. *-lft-identity N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

   4. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. *-rgt-identity N/A

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

      10. *-lft-identity N/A

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

      11. *-commutative N/A

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

      12. frac-sub N/A

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

      13. lift-+.f64 N/A

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

      14. lift--.f64 N/A

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

      15. difference-of-squares N/A

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

      16. associate-*l/ N/A

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

   6. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.8

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. metadata-eval N/A

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

      13. div-inv N/A

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

      14. lower-*.f64 N/A

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

      15. div-inv N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 99.8

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

   8. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. *-lft-identity N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. frac-times N/A

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

      13. lift-/.f64 N/A

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

      14. div-inv N/A

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

      15. metadata-eval N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*l* N/A

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

      20. lower-*.f64 N/A

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

   10. Applied rewrites 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 10^{+46}:\\ \;\;\;\;\frac{\pi}{a \cdot \left(\left(b + a\right) \cdot \left(b \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(a \cdot \left(b + a\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1.5 \cdot 10^{+92}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(a \cdot \left(b + a\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -1.5e+92)
   (/ (* PI 0.5) (* a (* b a)))
   (* PI (/ 0.5 (* b (* a (+ b a)))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -1.5e+92) {
		tmp = (((double) M_PI) * 0.5) / (a * (b * a));
	} else {
		tmp = ((double) M_PI) * (0.5 / (b * (a * (b + a))));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -1.5e+92) {
		tmp = (Math.PI * 0.5) / (a * (b * a));
	} else {
		tmp = Math.PI * (0.5 / (b * (a * (b + a))));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -1.5e+92:
		tmp = (math.pi * 0.5) / (a * (b * a))
	else:
		tmp = math.pi * (0.5 / (b * (a * (b + a))))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -1.5e+92)
		tmp = Float64(Float64(pi * 0.5) / Float64(a * Float64(b * a)));
	else
		tmp = Float64(pi * Float64(0.5 / Float64(b * Float64(a * Float64(b + a)))));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.5e+92)
		tmp = (pi * 0.5) / (a * (b * a));
	else
		tmp = pi * (0.5 / (b * (a * (b + a))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -1.5e+92], N[(N[(Pi * 0.5), $MachinePrecision] / N[(a * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(0.5 / N[(b * N[(a * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.50000000000000007e92

   1. Initial program 64.5%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 98.4

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

   5. Applied rewrites 98.4%

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

   if -1.50000000000000007e92 < a

   1. Initial program 84.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. associate-*r/ N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. difference-of-squares N/A

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

      11. *-commutative N/A

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

      12. *-rgt-identity N/A

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

      13. *-lft-identity N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

   4. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. *-rgt-identity N/A

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

      10. *-lft-identity N/A

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

      11. *-commutative N/A

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

      12. frac-sub N/A

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

      13. lift-+.f64 N/A

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

      14. lift--.f64 N/A

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

      15. difference-of-squares N/A

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

      16. associate-*l/ N/A

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

   6. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.2

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. metadata-eval N/A

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

      13. div-inv N/A

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

      14. lower-*.f64 N/A

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

      15. div-inv N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 99.2

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

   8. Applied rewrites 99.2%

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

      1. lift-/.f64 N/A

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

      2. *-lft-identity N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. frac-times N/A

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

      13. lift-/.f64 N/A

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

      14. div-inv N/A

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

      15. metadata-eval N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*l* N/A

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

      20. lower-*.f64 N/A

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

   10. Applied rewrites 99.1%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.5 \cdot 10^{+92}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(a \cdot \left(b + a\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024225 
(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))))