NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.8% → 99.7%

Time: 11.6s

Alternatives: 10

Speedup: 2.0×

Specification

Math

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

FPCore

(FPCore (a b)
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 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), $MachinePrecision] * N[(1 / N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1 / a), $MachinePrecision] - N[(1 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 78.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b)
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 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), $MachinePrecision] * N[(1 / N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1 / a), $MachinePrecision] - N[(1 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.7% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (a b)
  :precision binary64
  (/ (* (/ PI (+ b a)) 1/2) (* b a)))

C

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

Java

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

Python

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

Julia

function code(a, b)
	return Float64(Float64(Float64(pi / Float64(b + a)) * 0.5) / Float64(b * a))
end

MATLAB

function tmp = code(a, b)
	tmp = ((pi / (b + a)) * 0.5) / (b * a);
end

Wolfram

code[a_, b_] := N[(N[(N[(Pi / N[(b + a), $MachinePrecision]), $MachinePrecision] * 1/2), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.8%

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

3. Applied rewrites 99.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. associate-/l* N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-/r* N/A

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

   8. lift-*.f64 N/A

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

   9. times-frac N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. associate-/l* N/A

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

   13. lower-*.f64 N/A

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

5. Applied rewrites 81.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/r* N/A

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

   9. associate-*r/ N/A

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

   10. *-inverses N/A

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

   11. *-rgt-identity N/A

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. times-frac N/A

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

   15. associate-*l/ N/A

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

   16. lower-/.f64 N/A

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

7. Applied rewrites 99.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lift-/.f64 N/A

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

   5. associate-*r/ N/A

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

   6. lower-/.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 99.7%

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 99.7%

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 99.7%

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

9. Applied rewrites 99.7%

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

Alternative 2: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\mathsf{max}\left(a, b\right) \leq 5000000000000000079514455548799590234180404281972640694890663778873919386085190530406734992928407552:\\ \;\;\;\;\frac{\pi \cdot \frac{1}{2}}{\left(\left(\mathsf{min}\left(a, b\right) + \mathsf{max}\left(a, b\right)\right) \cdot \mathsf{max}\left(a, b\right)\right) \cdot \mathsf{min}\left(a, b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{\mathsf{min}\left(a, b\right) \cdot \mathsf{max}\left(a, b\right)} \cdot \frac{\frac{1}{2}}{\mathsf{max}\left(a, b\right)}\\ \end{array} \]

FPCore

(FPCore (a b)
  :precision binary64
  (if (<=
     (fmax a b)
     5000000000000000079514455548799590234180404281972640694890663778873919386085190530406734992928407552)
  (/
   (* PI 1/2)
   (* (* (+ (fmin a b) (fmax a b)) (fmax a b)) (fmin a b)))
  (* (/ PI (* (fmin a b) (fmax a b))) (/ 1/2 (fmax a b)))))

C

double code(double a, double b) {
	double tmp;
	if (fmax(a, b) <= 5e+99) {
		tmp = (((double) M_PI) * 0.5) / (((fmin(a, b) + fmax(a, b)) * fmax(a, b)) * fmin(a, b));
	} else {
		tmp = (((double) M_PI) / (fmin(a, b) * fmax(a, b))) * (0.5 / fmax(a, b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (fmax(a, b) <= 5e+99) {
		tmp = (Math.PI * 0.5) / (((fmin(a, b) + fmax(a, b)) * fmax(a, b)) * fmin(a, b));
	} else {
		tmp = (Math.PI / (fmin(a, b) * fmax(a, b))) * (0.5 / fmax(a, b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if fmax(a, b) <= 5e+99:
		tmp = (math.pi * 0.5) / (((fmin(a, b) + fmax(a, b)) * fmax(a, b)) * fmin(a, b))
	else:
		tmp = (math.pi / (fmin(a, b) * fmax(a, b))) * (0.5 / fmax(a, b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (fmax(a, b) <= 5e+99)
		tmp = Float64(Float64(pi * 0.5) / Float64(Float64(Float64(fmin(a, b) + fmax(a, b)) * fmax(a, b)) * fmin(a, b)));
	else
		tmp = Float64(Float64(pi / Float64(fmin(a, b) * fmax(a, b))) * Float64(0.5 / fmax(a, b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (max(a, b) <= 5e+99)
		tmp = (pi * 0.5) / (((min(a, b) + max(a, b)) * max(a, b)) * min(a, b));
	else
		tmp = (pi / (min(a, b) * max(a, b))) * (0.5 / max(a, b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[Max[a, b], $MachinePrecision], 5000000000000000079514455548799590234180404281972640694890663778873919386085190530406734992928407552], N[(N[(Pi * 1/2), $MachinePrecision] / N[(N[(N[(N[Min[a, b], $MachinePrecision] + N[Max[a, b], $MachinePrecision]), $MachinePrecision] * N[Max[a, b], $MachinePrecision]), $MachinePrecision] * N[Min[a, b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / N[(N[Min[a, b], $MachinePrecision] * N[Max[a, b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1/2 / N[Max[a, b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.0000000000000001e99

   1. Initial program 78.8%

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

   3. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r* N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 81.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. frac-times N/A

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

      10. *-inverses N/A

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

      11. lift-/.f64 N/A

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

      12. *-lft-identity 99.1%

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

   7. Applied rewrites 93.2%

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

   if 5.0000000000000001e99 < b

   1. Initial program 78.8%

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

   3. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r* N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 81.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. associate-*r/ N/A

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

      10. *-inverses N/A

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

      11. *-rgt-identity N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. times-frac N/A

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

      16. lower-*.f64 N/A

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 62.4%

       \[\leadsto \frac{\pi}{a \cdot b} \cdot \frac{\frac{1}{2}}{\color{blue}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.6% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (a b)
  :precision binary64
  (/ (* PI (/ 1/2 (* a b))) (+ a b)))

C

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

Java

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

Python

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

Julia

function code(a, b)
	return Float64(Float64(pi * Float64(0.5 / Float64(a * b))) / Float64(a + b))
end

MATLAB

function tmp = code(a, b)
	tmp = (pi * (0.5 / (a * b))) / (a + b);
end

Wolfram

code[a_, b_] := N[(N[(Pi * N[(1/2 / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a + b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.8%

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

3. Applied rewrites 99.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. associate-/l* N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-/r* N/A

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

   8. lift-*.f64 N/A

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

   9. times-frac N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. associate-/l* N/A

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

   13. lower-*.f64 N/A

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

5. Applied rewrites 81.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/r* N/A

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

   9. associate-*r/ N/A

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

   10. *-inverses N/A

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

   11. *-rgt-identity N/A

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. times-frac N/A

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

   15. associate-*l/ N/A

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

   16. lower-/.f64 N/A

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

7. Applied rewrites 99.6%

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

Alternative 4: 99.4% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (a b)
  :precision binary64
  (* (/ PI (* a b)) (/ 1/2 (+ a b))))

C

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

Java

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

Python

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

Julia

function code(a, b)
	return Float64(Float64(pi / Float64(a * b)) * Float64(0.5 / Float64(a + b)))
end

MATLAB

function tmp = code(a, b)
	tmp = (pi / (a * b)) * (0.5 / (a + b));
end

Wolfram

code[a_, b_] := N[(N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(1/2 / N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.8%

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

3. Applied rewrites 99.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. associate-/l* N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-/r* N/A

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

   8. lift-*.f64 N/A

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

   9. times-frac N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. associate-/l* N/A

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

   13. lower-*.f64 N/A

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

5. Applied rewrites 81.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/r* N/A

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

   9. associate-*r/ N/A

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

   10. *-inverses N/A

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

   11. *-rgt-identity N/A

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. times-frac N/A

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

   16. lower-*.f64 N/A

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

7. Applied rewrites 99.6%

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

Alternative 5: 99.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(a, b\right) + \mathsf{max}\left(a, b\right)\\ \mathbf{if}\;\mathsf{max}\left(a, b\right) \leq 1999999999999999849735523237985764085089341739669676922878451944450588399951586053206326987525635307503060116827311064565678080:\\ \;\;\;\;\frac{\pi \cdot \frac{1}{2}}{\left(t\_0 \cdot \mathsf{max}\left(a, b\right)\right) \cdot \mathsf{min}\left(a, b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{2} \cdot \pi}{\left(t\_0 \cdot \mathsf{min}\left(a, b\right)\right) \cdot \left(-\mathsf{max}\left(a, b\right)\right)}\\ \end{array} \]

FPCore

(FPCore (a b)
  :precision binary64
  (let* ((t_0 (+ (fmin a b) (fmax a b))))
  (if (<=
       (fmax a b)
       1999999999999999849735523237985764085089341739669676922878451944450588399951586053206326987525635307503060116827311064565678080)
    (/ (* PI 1/2) (* (* t_0 (fmax a b)) (fmin a b)))
    (/ (* -1/2 PI) (* (* t_0 (fmin a b)) (- (fmax a b)))))))

C

double code(double a, double b) {
	double t_0 = fmin(a, b) + fmax(a, b);
	double tmp;
	if (fmax(a, b) <= 2e+126) {
		tmp = (((double) M_PI) * 0.5) / ((t_0 * fmax(a, b)) * fmin(a, b));
	} else {
		tmp = (-0.5 * ((double) M_PI)) / ((t_0 * fmin(a, b)) * -fmax(a, b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = fmin(a, b) + fmax(a, b);
	double tmp;
	if (fmax(a, b) <= 2e+126) {
		tmp = (Math.PI * 0.5) / ((t_0 * fmax(a, b)) * fmin(a, b));
	} else {
		tmp = (-0.5 * Math.PI) / ((t_0 * fmin(a, b)) * -fmax(a, b));
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = fmin(a, b) + fmax(a, b)
	tmp = 0
	if fmax(a, b) <= 2e+126:
		tmp = (math.pi * 0.5) / ((t_0 * fmax(a, b)) * fmin(a, b))
	else:
		tmp = (-0.5 * math.pi) / ((t_0 * fmin(a, b)) * -fmax(a, b))
	return tmp

Julia

function code(a, b)
	t_0 = Float64(fmin(a, b) + fmax(a, b))
	tmp = 0.0
	if (fmax(a, b) <= 2e+126)
		tmp = Float64(Float64(pi * 0.5) / Float64(Float64(t_0 * fmax(a, b)) * fmin(a, b)));
	else
		tmp = Float64(Float64(-0.5 * pi) / Float64(Float64(t_0 * fmin(a, b)) * Float64(-fmax(a, b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = min(a, b) + max(a, b);
	tmp = 0.0;
	if (max(a, b) <= 2e+126)
		tmp = (pi * 0.5) / ((t_0 * max(a, b)) * min(a, b));
	else
		tmp = (-0.5 * pi) / ((t_0 * min(a, b)) * -max(a, b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[Min[a, b], $MachinePrecision] + N[Max[a, b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Max[a, b], $MachinePrecision], 1999999999999999849735523237985764085089341739669676922878451944450588399951586053206326987525635307503060116827311064565678080], N[(N[(Pi * 1/2), $MachinePrecision] / N[(N[(t$95$0 * N[Max[a, b], $MachinePrecision]), $MachinePrecision] * N[Min[a, b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1/2 * Pi), $MachinePrecision] / N[(N[(t$95$0 * N[Min[a, b], $MachinePrecision]), $MachinePrecision] * (-N[Max[a, b], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.9999999999999998e126

   1. Initial program 78.8%

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

   3. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r* N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 81.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. frac-times N/A

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

      10. *-inverses N/A

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

      11. lift-/.f64 N/A

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

      12. *-lft-identity 99.1%

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

   7. Applied rewrites 93.2%

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

   if 1.9999999999999998e126 < b

   1. Initial program 78.8%

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

   3. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r* N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 81.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. associate-*r/ N/A

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

      10. *-inverses N/A

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

      11. frac-2neg N/A

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

      12. *-rgt-identity N/A

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

      13. lower-/.f64 N/A

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

   7. Applied rewrites 93.1%

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

Alternative 6: 99.1% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (a b)
  :precision binary64
  (* (/ (* PI 1/2) (* (+ a b) (* a b))) 1))

C

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

Java

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

Python

def code(a, b):
	return ((math.pi * 0.5) / ((a + b) * (a * b))) * 1.0

Julia

function code(a, b)
	return Float64(Float64(Float64(pi * 0.5) / Float64(Float64(a + b) * Float64(a * b))) * 1.0)
end

MATLAB

function tmp = code(a, b)
	tmp = ((pi * 0.5) / ((a + b) * (a * b))) * 1.0;
end

Wolfram

code[a_, b_] := N[(N[(N[(Pi * 1/2), $MachinePrecision] / N[(N[(a + b), $MachinePrecision] * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1), $MachinePrecision]

Derivation

1. Initial program 78.8%

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

3. Applied rewrites 99.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lift-/.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. frac-times N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/r* N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. times-frac N/A

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

   12. lower-*.f64 N/A

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

5. Applied rewrites 99.1%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 99.1%

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

Alternative 7: 99.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\mathsf{max}\left(a, b\right) \leq 5000000000000000298915391230258075925874645126169045354368179749161004102875565468155280170533300701722840996122161770682942226432:\\ \;\;\;\;\frac{\pi \cdot \frac{1}{2}}{\left(\left(\mathsf{min}\left(a, b\right) + \mathsf{max}\left(a, b\right)\right) \cdot \mathsf{max}\left(a, b\right)\right) \cdot \mathsf{min}\left(a, b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2}}{\left(\left(\mathsf{max}\left(a, b\right) + \mathsf{min}\left(a, b\right)\right) \cdot \mathsf{min}\left(a, b\right)\right) \cdot \mathsf{max}\left(a, b\right)} \cdot \pi\\ \end{array} \]

FPCore

(FPCore (a b)
  :precision binary64
  (if (<=
     (fmax a b)
     5000000000000000298915391230258075925874645126169045354368179749161004102875565468155280170533300701722840996122161770682942226432)
  (/
   (* PI 1/2)
   (* (* (+ (fmin a b) (fmax a b)) (fmax a b)) (fmin a b)))
  (*
   (/ 1/2 (* (* (+ (fmax a b) (fmin a b)) (fmin a b)) (fmax a b)))
   PI)))

C

double code(double a, double b) {
	double tmp;
	if (fmax(a, b) <= 5e+129) {
		tmp = (((double) M_PI) * 0.5) / (((fmin(a, b) + fmax(a, b)) * fmax(a, b)) * fmin(a, b));
	} else {
		tmp = (0.5 / (((fmax(a, b) + fmin(a, b)) * fmin(a, b)) * fmax(a, b))) * ((double) M_PI);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (fmax(a, b) <= 5e+129) {
		tmp = (Math.PI * 0.5) / (((fmin(a, b) + fmax(a, b)) * fmax(a, b)) * fmin(a, b));
	} else {
		tmp = (0.5 / (((fmax(a, b) + fmin(a, b)) * fmin(a, b)) * fmax(a, b))) * Math.PI;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if fmax(a, b) <= 5e+129:
		tmp = (math.pi * 0.5) / (((fmin(a, b) + fmax(a, b)) * fmax(a, b)) * fmin(a, b))
	else:
		tmp = (0.5 / (((fmax(a, b) + fmin(a, b)) * fmin(a, b)) * fmax(a, b))) * math.pi
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (fmax(a, b) <= 5e+129)
		tmp = Float64(Float64(pi * 0.5) / Float64(Float64(Float64(fmin(a, b) + fmax(a, b)) * fmax(a, b)) * fmin(a, b)));
	else
		tmp = Float64(Float64(0.5 / Float64(Float64(Float64(fmax(a, b) + fmin(a, b)) * fmin(a, b)) * fmax(a, b))) * pi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (max(a, b) <= 5e+129)
		tmp = (pi * 0.5) / (((min(a, b) + max(a, b)) * max(a, b)) * min(a, b));
	else
		tmp = (0.5 / (((max(a, b) + min(a, b)) * min(a, b)) * max(a, b))) * pi;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[Max[a, b], $MachinePrecision], 5000000000000000298915391230258075925874645126169045354368179749161004102875565468155280170533300701722840996122161770682942226432], N[(N[(Pi * 1/2), $MachinePrecision] / N[(N[(N[(N[Min[a, b], $MachinePrecision] + N[Max[a, b], $MachinePrecision]), $MachinePrecision] * N[Max[a, b], $MachinePrecision]), $MachinePrecision] * N[Min[a, b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1/2 / N[(N[(N[(N[Max[a, b], $MachinePrecision] + N[Min[a, b], $MachinePrecision]), $MachinePrecision] * N[Min[a, b], $MachinePrecision]), $MachinePrecision] * N[Max[a, b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.0000000000000003e129

   1. Initial program 78.8%

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

   3. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r* N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 81.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. frac-times N/A

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

      10. *-inverses N/A

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

      11. lift-/.f64 N/A

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

      12. *-lft-identity 99.1%

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

   7. Applied rewrites 93.2%

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

   if 5.0000000000000003e129 < b

   1. Initial program 78.8%

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

   3. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r* N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 81.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. associate-*r/ N/A

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

      10. *-inverses N/A

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

      11. *-rgt-identity N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   7. Applied rewrites 93.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 93.0%

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 93.0%

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

   9. Applied rewrites 93.0%

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

Alternative 8: 99.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\mathsf{min}\left(a, b\right) \leq -200000000000000000:\\ \;\;\;\;\frac{\frac{1}{2}}{\left(\left(\mathsf{min}\left(a, b\right) + \mathsf{max}\left(a, b\right)\right) \cdot \mathsf{max}\left(a, b\right)\right) \cdot \mathsf{min}\left(a, b\right)} \cdot \pi\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2}}{\left(\left(\mathsf{max}\left(a, b\right) + \mathsf{min}\left(a, b\right)\right) \cdot \mathsf{min}\left(a, b\right)\right) \cdot \mathsf{max}\left(a, b\right)} \cdot \pi\\ \end{array} \]

FPCore

(FPCore (a b)
  :precision binary64
  (if (<= (fmin a b) -200000000000000000)
  (*
   (/ 1/2 (* (* (+ (fmin a b) (fmax a b)) (fmax a b)) (fmin a b)))
   PI)
  (*
   (/ 1/2 (* (* (+ (fmax a b) (fmin a b)) (fmin a b)) (fmax a b)))
   PI)))

C

double code(double a, double b) {
	double tmp;
	if (fmin(a, b) <= -2e+17) {
		tmp = (0.5 / (((fmin(a, b) + fmax(a, b)) * fmax(a, b)) * fmin(a, b))) * ((double) M_PI);
	} else {
		tmp = (0.5 / (((fmax(a, b) + fmin(a, b)) * fmin(a, b)) * fmax(a, b))) * ((double) M_PI);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (fmin(a, b) <= -2e+17) {
		tmp = (0.5 / (((fmin(a, b) + fmax(a, b)) * fmax(a, b)) * fmin(a, b))) * Math.PI;
	} else {
		tmp = (0.5 / (((fmax(a, b) + fmin(a, b)) * fmin(a, b)) * fmax(a, b))) * Math.PI;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if fmin(a, b) <= -2e+17:
		tmp = (0.5 / (((fmin(a, b) + fmax(a, b)) * fmax(a, b)) * fmin(a, b))) * math.pi
	else:
		tmp = (0.5 / (((fmax(a, b) + fmin(a, b)) * fmin(a, b)) * fmax(a, b))) * math.pi
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (fmin(a, b) <= -2e+17)
		tmp = Float64(Float64(0.5 / Float64(Float64(Float64(fmin(a, b) + fmax(a, b)) * fmax(a, b)) * fmin(a, b))) * pi);
	else
		tmp = Float64(Float64(0.5 / Float64(Float64(Float64(fmax(a, b) + fmin(a, b)) * fmin(a, b)) * fmax(a, b))) * pi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (min(a, b) <= -2e+17)
		tmp = (0.5 / (((min(a, b) + max(a, b)) * max(a, b)) * min(a, b))) * pi;
	else
		tmp = (0.5 / (((max(a, b) + min(a, b)) * min(a, b)) * max(a, b))) * pi;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[Min[a, b], $MachinePrecision], -200000000000000000], N[(N[(1/2 / N[(N[(N[(N[Min[a, b], $MachinePrecision] + N[Max[a, b], $MachinePrecision]), $MachinePrecision] * N[Max[a, b], $MachinePrecision]), $MachinePrecision] * N[Min[a, b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision], N[(N[(1/2 / N[(N[(N[(N[Max[a, b], $MachinePrecision] + N[Min[a, b], $MachinePrecision]), $MachinePrecision] * N[Min[a, b], $MachinePrecision]), $MachinePrecision] * N[Max[a, b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2e17

   1. Initial program 78.8%

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

   3. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r* N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 81.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. associate-*r/ N/A

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

      10. *-inverses N/A

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

      11. *-rgt-identity N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   7. Applied rewrites 93.1%

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

   if -2e17 < a

   1. Initial program 78.8%

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

   3. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r* N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 81.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. associate-*r/ N/A

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

      10. *-inverses N/A

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

      11. *-rgt-identity N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   7. Applied rewrites 93.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 93.0%

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 93.0%

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

   9. Applied rewrites 93.0%

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

Alternative 9: 92.8% accurate, 0.2× speedup

Math

\[\frac{\frac{1}{2}}{\left(\left(\mathsf{min}\left(a, b\right) + \mathsf{max}\left(a, b\right)\right) \cdot \mathsf{max}\left(a, b\right)\right) \cdot \mathsf{min}\left(a, b\right)} \cdot \pi \]

FPCore

(FPCore (a b)
  :precision binary64
  (* (/ 1/2 (* (* (+ (fmin a b) (fmax a b)) (fmax a b)) (fmin a b))) PI))

C

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

Java

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

Python

def code(a, b):
	return (0.5 / (((fmin(a, b) + fmax(a, b)) * fmax(a, b)) * fmin(a, b))) * math.pi

Julia

function code(a, b)
	return Float64(Float64(0.5 / Float64(Float64(Float64(fmin(a, b) + fmax(a, b)) * fmax(a, b)) * fmin(a, b))) * pi)
end

MATLAB

function tmp = code(a, b)
	tmp = (0.5 / (((min(a, b) + max(a, b)) * max(a, b)) * min(a, b))) * pi;
end

Wolfram

code[a_, b_] := N[(N[(1/2 / N[(N[(N[(N[Min[a, b], $MachinePrecision] + N[Max[a, b], $MachinePrecision]), $MachinePrecision] * N[Max[a, b], $MachinePrecision]), $MachinePrecision] * N[Min[a, b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]

Derivation

1. Initial program 78.8%

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

3. Applied rewrites 99.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. associate-/l* N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-/r* N/A

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

   8. lift-*.f64 N/A

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

   9. times-frac N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. associate-/l* N/A

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

   13. lower-*.f64 N/A

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

5. Applied rewrites 81.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/r* N/A

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

   9. associate-*r/ N/A

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

   10. *-inverses N/A

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

   11. *-rgt-identity N/A

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

   12. lift-*.f64 N/A

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

   13. associate-/l* N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

7. Applied rewrites 93.1%

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

Alternative 10: 56.4% accurate, 2.6× speedup

Math

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

FPCore

(FPCore (a b)
  :precision binary64
  (* (/ 1/2 (* (* b b) a)) PI))

C

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

Java

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

Python

def code(a, b):
	return (0.5 / ((b * b) * a)) * math.pi

Julia

function code(a, b)
	return Float64(Float64(0.5 / Float64(Float64(b * b) * a)) * pi)
end

MATLAB

function tmp = code(a, b)
	tmp = (0.5 / ((b * b) * a)) * pi;
end

Wolfram

code[a_, b_] := N[(N[(1/2 / N[(N[(b * b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]

Derivation

1. Initial program 78.8%

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

3. Applied rewrites 99.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. associate-/l* N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-/r* N/A

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

   8. lift-*.f64 N/A

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

   9. times-frac N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. associate-/l* N/A

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

   13. lower-*.f64 N/A

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

5. Applied rewrites 81.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/r* N/A

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

   9. associate-*r/ N/A

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

   10. *-inverses N/A

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

   11. *-rgt-identity N/A

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

   12. lift-*.f64 N/A

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

   13. associate-/l* N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

7. Applied rewrites 93.1%

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

8. Taylor expanded in a around 0

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

10. Applied rewrites 56.4%

    \[\leadsto \frac{\frac{1}{2}}{\left(\color{blue}{b} \cdot b\right) \cdot a} \cdot \pi \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025271 -o generate:evaluate
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))