Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 6.0s

Alternatives: 5

Speedup: 1.1×

• Could not converge on a ground truth

  1. k = 5.7678911092463516e+209
  2. n = -1.957753582132397e+119

Specification

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (* (/ 1.0 (sqrt k)) (pow (* (* 2.0 PI) n) (/ (- 1.0 k) 2.0))))

C

double code(double k, double n) {
	return (1.0 / sqrt(k)) * pow(((2.0 * ((double) M_PI)) * n), ((1.0 - k) / 2.0));
}

Java

public static double code(double k, double n) {
	return (1.0 / Math.sqrt(k)) * Math.pow(((2.0 * Math.PI) * n), ((1.0 - k) / 2.0));
}

Python

def code(k, n):
	return (1.0 / math.sqrt(k)) * math.pow(((2.0 * math.pi) * n), ((1.0 - k) / 2.0))

Julia

function code(k, n)
	return Float64(Float64(1.0 / sqrt(k)) * (Float64(Float64(2.0 * pi) * n) ^ Float64(Float64(1.0 - k) / 2.0)))
end

MATLAB

function tmp = code(k, n)
	tmp = (1.0 / sqrt(k)) * (((2.0 * pi) * n) ^ ((1.0 - k) / 2.0));
end

Wolfram

code[k_, n_] := N[(N[(1.0 / N[Sqrt[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 * Pi), $MachinePrecision] * n), $MachinePrecision], N[(N[(1.0 - k), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (* (/ 1.0 (sqrt k)) (pow (* (* 2.0 PI) n) (/ (- 1.0 k) 2.0))))

C

double code(double k, double n) {
	return (1.0 / sqrt(k)) * pow(((2.0 * ((double) M_PI)) * n), ((1.0 - k) / 2.0));
}

Java

public static double code(double k, double n) {
	return (1.0 / Math.sqrt(k)) * Math.pow(((2.0 * Math.PI) * n), ((1.0 - k) / 2.0));
}

Python

def code(k, n):
	return (1.0 / math.sqrt(k)) * math.pow(((2.0 * math.pi) * n), ((1.0 - k) / 2.0))

Julia

function code(k, n)
	return Float64(Float64(1.0 / sqrt(k)) * (Float64(Float64(2.0 * pi) * n) ^ Float64(Float64(1.0 - k) / 2.0)))
end

MATLAB

function tmp = code(k, n)
	tmp = (1.0 / sqrt(k)) * (((2.0 * pi) * n) ^ ((1.0 - k) / 2.0));
end

Wolfram

code[k_, n_] := N[(N[(1.0 / N[Sqrt[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 * Pi), $MachinePrecision] * n), $MachinePrecision], N[(N[(1.0 - k), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(n \cdot \left(\pi + \pi\right)\right)}^{\left(\mathsf{fma}\left(-0.5, k, 0.5\right)\right)}}{\sqrt{k}} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (/ (pow (* n (+ PI PI)) (fma -0.5 k 0.5)) (sqrt k)))

C

double code(double k, double n) {
	return pow((n * (((double) M_PI) + ((double) M_PI))), fma(-0.5, k, 0.5)) / sqrt(k);
}

Julia

function code(k, n)
	return Float64((Float64(n * Float64(pi + pi)) ^ fma(-0.5, k, 0.5)) / sqrt(k))
end

Wolfram

code[k_, n_] := N[(N[Power[N[(n * N[(Pi + Pi), $MachinePrecision]), $MachinePrecision], N[(-0.5 * k + 0.5), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}} \]

   2. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\sqrt{k}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

   3. lift-sqrt.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\sqrt{k}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

   4. lift-pow.f64 N/A

      \[\leadsto \frac{1}{\sqrt{k}} \cdot \color{blue}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\color{blue}{\left(\left(2 \cdot \pi\right) \cdot n\right)}}^{\left(\frac{1 - k}{2}\right)} \]

   6. lift-PI.f64 N/A

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right)} \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

   8. lift--.f64 N/A

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{\color{blue}{1 - k}}{2}\right)} \]

   9. lift-/.f64 N/A

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\color{blue}{\left(\frac{1 - k}{2}\right)}} \]

   10. associate-*l/ N/A

       \[\leadsto \color{blue}{\frac{1 \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}} \]

   11. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{1 \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}} \]

4. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\frac{1 \cdot {\left(\left(\pi \cdot 2\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}} \]

5. Taylor expanded in k around 0

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

   1. +-commutative N/A

      \[\leadsto \frac{1 \cdot {\left(\left(\pi \cdot 2\right) \cdot n\right)}^{\left(\frac{-1}{2} \cdot k + \color{blue}{\frac{1}{2}}\right)}}{\sqrt{k}} \]

   2. lower-fma.f64 99.3

      \[\leadsto \frac{1 \cdot {\left(\left(\pi \cdot 2\right) \cdot n\right)}^{\left(\mathsf{fma}\left(-0.5, \color{blue}{k}, 0.5\right)\right)}}{\sqrt{k}} \]

7. Applied rewrites 99.3%

   \[\leadsto \frac{1 \cdot {\left(\left(\pi \cdot 2\right) \cdot n\right)}^{\color{blue}{\left(\mathsf{fma}\left(-0.5, k, 0.5\right)\right)}}}{\sqrt{k}} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{1 \cdot {\left(\left(\pi \cdot 2\right) \cdot n\right)}^{\left(\mathsf{fma}\left(\frac{-1}{2}, k, \frac{1}{2}\right)\right)}}}{\sqrt{k}} \]

   2. *-lft-identity 99.3

      \[\leadsto \frac{\color{blue}{{\left(\left(\pi \cdot 2\right) \cdot n\right)}^{\left(\mathsf{fma}\left(-0.5, k, 0.5\right)\right)}}}{\sqrt{k}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{{\color{blue}{\left(\left(\pi \cdot 2\right) \cdot n\right)}}^{\left(\mathsf{fma}\left(\frac{-1}{2}, k, \frac{1}{2}\right)\right)}}{\sqrt{k}} \]

   4. *-commutative N/A

      \[\leadsto \frac{{\color{blue}{\left(n \cdot \left(\pi \cdot 2\right)\right)}}^{\left(\mathsf{fma}\left(\frac{-1}{2}, k, \frac{1}{2}\right)\right)}}{\sqrt{k}} \]

   5. lift-*.f64 99.3

      \[\leadsto \frac{{\color{blue}{\left(n \cdot \left(\pi \cdot 2\right)\right)}}^{\left(\mathsf{fma}\left(-0.5, k, 0.5\right)\right)}}{\sqrt{k}} \]

9. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\frac{{\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(\mathsf{fma}\left(-0.5, k, 0.5\right)\right)}}{\sqrt{k}}} \]
10. Step-by-step derivation

    1. lift-PI.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. *-commutative N/A

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

    4. count-2-rev N/A

       \[\leadsto \frac{{\left(n \cdot \color{blue}{\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right)}\right)}^{\left(\mathsf{fma}\left(\frac{-1}{2}, k, \frac{1}{2}\right)\right)}}{\sqrt{k}} \]

    5. lift-+.f64 N/A

       \[\leadsto \frac{{\left(n \cdot \color{blue}{\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right)}\right)}^{\left(\mathsf{fma}\left(\frac{-1}{2}, k, \frac{1}{2}\right)\right)}}{\sqrt{k}} \]

    6. lift-PI.f64 N/A

       \[\leadsto \frac{{\left(n \cdot \left(\color{blue}{\pi} + \mathsf{PI}\left(\right)\right)\right)}^{\left(\mathsf{fma}\left(\frac{-1}{2}, k, \frac{1}{2}\right)\right)}}{\sqrt{k}} \]

    7. lift-PI.f64 99.3

       \[\leadsto \frac{{\left(n \cdot \left(\pi + \color{blue}{\pi}\right)\right)}^{\left(\mathsf{fma}\left(-0.5, k, 0.5\right)\right)}}{\sqrt{k}} \]

11. Applied rewrites 99.3%

    \[\leadsto \frac{{\left(n \cdot \color{blue}{\left(\pi + \pi\right)}\right)}^{\left(\mathsf{fma}\left(-0.5, k, 0.5\right)\right)}}{\sqrt{k}} \]
12. Add Preprocessing

Alternative 2: 49.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \leq 2 \cdot 10^{+148}:\\ \;\;\;\;\sqrt{\left(n \cdot \frac{\pi}{k}\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot k\right) \cdot \left(\pi \cdot n\right)}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= (* (/ 1.0 (sqrt k)) (pow (* (* 2.0 PI) n) (/ (- 1.0 k) 2.0))) 2e+148)
   (sqrt (* (* n (/ PI k)) 2.0))
   (/ (sqrt (* (* 2.0 k) (* PI n))) k)))

C

double code(double k, double n) {
	double tmp;
	if (((1.0 / sqrt(k)) * pow(((2.0 * ((double) M_PI)) * n), ((1.0 - k) / 2.0))) <= 2e+148) {
		tmp = sqrt(((n * (((double) M_PI) / k)) * 2.0));
	} else {
		tmp = sqrt(((2.0 * k) * (((double) M_PI) * n))) / k;
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (((1.0 / Math.sqrt(k)) * Math.pow(((2.0 * Math.PI) * n), ((1.0 - k) / 2.0))) <= 2e+148) {
		tmp = Math.sqrt(((n * (Math.PI / k)) * 2.0));
	} else {
		tmp = Math.sqrt(((2.0 * k) * (Math.PI * n))) / k;
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if ((1.0 / math.sqrt(k)) * math.pow(((2.0 * math.pi) * n), ((1.0 - k) / 2.0))) <= 2e+148:
		tmp = math.sqrt(((n * (math.pi / k)) * 2.0))
	else:
		tmp = math.sqrt(((2.0 * k) * (math.pi * n))) / k
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(k)) * (Float64(Float64(2.0 * pi) * n) ^ Float64(Float64(1.0 - k) / 2.0))) <= 2e+148)
		tmp = sqrt(Float64(Float64(n * Float64(pi / k)) * 2.0));
	else
		tmp = Float64(sqrt(Float64(Float64(2.0 * k) * Float64(pi * n))) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (((1.0 / sqrt(k)) * (((2.0 * pi) * n) ^ ((1.0 - k) / 2.0))) <= 2e+148)
		tmp = sqrt(((n * (pi / k)) * 2.0));
	else
		tmp = sqrt(((2.0 * k) * (pi * n))) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 * Pi), $MachinePrecision] * n), $MachinePrecision], N[(N[(1.0 - k), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e+148], N[Sqrt[N[(N[(n * N[(Pi / k), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(2.0 * k), $MachinePrecision] * N[(Pi * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 k)) (pow.f64 (*.f64 (*.f64 #s(literal 2 binary64) (PI.f64)) n) (/.f64 (-.f64 #s(literal 1 binary64) k) #s(literal 2 binary64)))) < 2.0000000000000001e148

   1. Initial program 99.0%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}} \cdot \sqrt{2}} \]
   4. Step-by-step derivation

      1. sqrt-unprod N/A

         \[\leadsto \sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2} \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{\mathsf{PI}\left(\right) \cdot n}{k} \cdot 2} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{\mathsf{PI}\left(\right) \cdot n}{k} \cdot 2} \]

      7. lift-PI.f64 66.3

         \[\leadsto \sqrt{\frac{\pi \cdot n}{k} \cdot 2} \]

   5. Applied rewrites 66.3%

      \[\leadsto \color{blue}{\sqrt{\frac{\pi \cdot n}{k} \cdot 2}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{\pi \cdot n}{k} \cdot 2} \]

      2. lift-PI.f64 N/A

         \[\leadsto \sqrt{\frac{\mathsf{PI}\left(\right) \cdot n}{k} \cdot 2} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\mathsf{PI}\left(\right) \cdot n}{k} \cdot 2} \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2} \]

      5. associate-/l* N/A

         \[\leadsto \sqrt{\left(n \cdot \frac{\mathsf{PI}\left(\right)}{k}\right) \cdot 2} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(n \cdot \frac{\mathsf{PI}\left(\right)}{k}\right) \cdot 2} \]

      7. lower-/.f64 N/A

         \[\leadsto \sqrt{\left(n \cdot \frac{\mathsf{PI}\left(\right)}{k}\right) \cdot 2} \]

      8. lift-PI.f64 66.3

         \[\leadsto \sqrt{\left(n \cdot \frac{\pi}{k}\right) \cdot 2} \]

   7. Applied rewrites 66.3%

      \[\leadsto \sqrt{\left(n \cdot \frac{\pi}{k}\right) \cdot 2} \]

   if 2.0000000000000001e148 < (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 k)) (pow.f64 (*.f64 (*.f64 #s(literal 2 binary64) (PI.f64)) n) (/.f64 (-.f64 #s(literal 1 binary64) k) #s(literal 2 binary64))))

   1. Initial program 99.7%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot \left(\sqrt{{k}^{3} \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\log \left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right)\right) + \sqrt{k \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)} \cdot \sqrt{2}}{k}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot \left(\sqrt{{k}^{3} \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\log \left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right)\right) + \sqrt{k \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)} \cdot \sqrt{2}}{\color{blue}{k}} \]

   5. Applied rewrites 81.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\log \left(\left(\pi \cdot 2\right) \cdot n\right) \cdot \sqrt{2}\right) \cdot \sqrt{\left({k}^{3} \cdot n\right) \cdot \pi}, -0.5, \sqrt{\left(\left(\pi \cdot n\right) \cdot k\right) \cdot 2}\right)}{k}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{\sqrt{k \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)} \cdot \sqrt{2}}{k} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sqrt{k \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)} \cdot \sqrt{2}}{k} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{k \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)} \cdot \sqrt{2}}{k} \]

      3. lift-PI.f64 N/A

         \[\leadsto \frac{\sqrt{k \cdot \left(\pi \cdot n\right)} \cdot \sqrt{2}}{k} \]

      4. *-commutative N/A

         \[\leadsto \frac{\sqrt{\left(\pi \cdot n\right) \cdot k} \cdot \sqrt{2}}{k} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(\pi \cdot n\right) \cdot k} \cdot \sqrt{2}}{k} \]

      6. sqrt-prod N/A

         \[\leadsto \frac{\sqrt{\left(\left(\pi \cdot n\right) \cdot k\right) \cdot 2}}{k} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(\left(\pi \cdot n\right) \cdot k\right) \cdot 2}}{k} \]

      8. lift-sqrt.f64 38.1

         \[\leadsto \frac{\sqrt{\left(\left(\pi \cdot n\right) \cdot k\right) \cdot 2}}{k} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(\left(\pi \cdot n\right) \cdot k\right) \cdot 2}}{k} \]

      10. *-commutative N/A

          \[\leadsto \frac{\sqrt{2 \cdot \left(\left(\pi \cdot n\right) \cdot k\right)}}{k} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{2 \cdot \left(\left(\pi \cdot n\right) \cdot k\right)}}{k} \]

      12. *-commutative N/A

          \[\leadsto \frac{\sqrt{2 \cdot \left(k \cdot \left(\pi \cdot n\right)\right)}}{k} \]

      13. lift-PI.f64 N/A

          \[\leadsto \frac{\sqrt{2 \cdot \left(k \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}}{k} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{2 \cdot \left(k \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}}{k} \]

      15. *-commutative N/A

          \[\leadsto \frac{\sqrt{2 \cdot \left(k \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}}{k} \]

      16. associate-*r* N/A

          \[\leadsto \frac{\sqrt{\left(2 \cdot k\right) \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)}}{k} \]

   8. Applied rewrites 38.1%

      \[\leadsto \frac{\sqrt{\left(2 \cdot k\right) \cdot \left(\pi \cdot n\right)}}{k} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 49.4% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{n \cdot \left(\pi \cdot 2\right)}}{\sqrt{k}} \end{array} \]

FPCore

(FPCore (k n) :precision binary64 (/ (sqrt (* n (* PI 2.0))) (sqrt k)))

C

double code(double k, double n) {
	return sqrt((n * (((double) M_PI) * 2.0))) / sqrt(k);
}

Java

public static double code(double k, double n) {
	return Math.sqrt((n * (Math.PI * 2.0))) / Math.sqrt(k);
}

Python

def code(k, n):
	return math.sqrt((n * (math.pi * 2.0))) / math.sqrt(k)

Julia

function code(k, n)
	return Float64(sqrt(Float64(n * Float64(pi * 2.0))) / sqrt(k))
end

MATLAB

function tmp = code(k, n)
	tmp = sqrt((n * (pi * 2.0))) / sqrt(k);
end

Wolfram

code[k_, n_] := N[(N[Sqrt[N[(n * N[(Pi * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}} \]

   2. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\sqrt{k}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

   3. lift-sqrt.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\sqrt{k}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

   4. lift-pow.f64 N/A

      \[\leadsto \frac{1}{\sqrt{k}} \cdot \color{blue}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\color{blue}{\left(\left(2 \cdot \pi\right) \cdot n\right)}}^{\left(\frac{1 - k}{2}\right)} \]

   6. lift-PI.f64 N/A

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right)} \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

   8. lift--.f64 N/A

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{\color{blue}{1 - k}}{2}\right)} \]

   9. lift-/.f64 N/A

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\color{blue}{\left(\frac{1 - k}{2}\right)}} \]

   10. associate-*l/ N/A

       \[\leadsto \color{blue}{\frac{1 \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}} \]

   11. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{1 \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}} \]

4. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\frac{1 \cdot {\left(\left(\pi \cdot 2\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}} \]

5. Taylor expanded in k around 0

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

   1. *-lft-identity N/A

      \[\leadsto \frac{\color{blue}{\sqrt{n \cdot \mathsf{PI}\left(\right)}} \cdot \sqrt{2}}{\sqrt{k}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\sqrt{\color{blue}{n} \cdot \mathsf{PI}\left(\right)} \cdot \sqrt{2}}{\sqrt{k}} \]

   3. sqrt-unprod N/A

      \[\leadsto \frac{\sqrt{\left(n \cdot \mathsf{PI}\left(\right)\right) \cdot 2}}{\sqrt{k}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\sqrt{2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)}}{\sqrt{k}} \]

   5. *-commutative N/A

      \[\leadsto \frac{\sqrt{2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)}}{\sqrt{k}} \]

   6. associate-*l* N/A

      \[\leadsto \frac{\sqrt{\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n}}{\sqrt{k}} \]

   7. *-commutative N/A

      \[\leadsto \frac{\sqrt{\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot n}}{\sqrt{k}} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\sqrt{\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot n}}{\sqrt{k}} \]

   9. lift-PI.f64 N/A

      \[\leadsto \frac{\sqrt{\left(\pi \cdot 2\right) \cdot n}}{\sqrt{k}} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{\sqrt{\left(\pi \cdot 2\right) \cdot n}}{\sqrt{k}} \]

   11. lift-sqrt.f64 56.0

       \[\leadsto \frac{\sqrt{\left(\pi \cdot 2\right) \cdot n}}{\sqrt{k}} \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{\sqrt{\left(\pi \cdot 2\right) \cdot n}}{\sqrt{k}} \]

   13. *-commutative N/A

       \[\leadsto \frac{\sqrt{n \cdot \left(\pi \cdot 2\right)}}{\sqrt{k}} \]

   14. lower-*.f64 56.0

       \[\leadsto \frac{\sqrt{n \cdot \left(\pi \cdot 2\right)}}{\sqrt{k}} \]

7. Applied rewrites 56.0%

   \[\leadsto \frac{\color{blue}{\sqrt{n \cdot \left(\pi \cdot 2\right)}}}{\sqrt{k}} \]
8. Add Preprocessing

Alternative 4: 38.2% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left(n \cdot \frac{\pi}{k}\right) \cdot 2} \end{array} \]

FPCore

(FPCore (k n) :precision binary64 (sqrt (* (* n (/ PI k)) 2.0)))

C

double code(double k, double n) {
	return sqrt(((n * (((double) M_PI) / k)) * 2.0));
}

Java

public static double code(double k, double n) {
	return Math.sqrt(((n * (Math.PI / k)) * 2.0));
}

Python

def code(k, n):
	return math.sqrt(((n * (math.pi / k)) * 2.0))

Julia

function code(k, n)
	return sqrt(Float64(Float64(n * Float64(pi / k)) * 2.0))
end

MATLAB

function tmp = code(k, n)
	tmp = sqrt(((n * (pi / k)) * 2.0));
end

Wolfram

code[k_, n_] := N[Sqrt[N[(N[(n * N[(Pi / k), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}} \cdot \sqrt{2}} \]
4. Step-by-step derivation

   1. sqrt-unprod N/A

      \[\leadsto \sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2} \]

   2. lower-sqrt.f64 N/A

      \[\leadsto \sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2} \]

   3. lower-*.f64 N/A

      \[\leadsto \sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2} \]

   4. lower-/.f64 N/A

      \[\leadsto \sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2} \]

   5. *-commutative N/A

      \[\leadsto \sqrt{\frac{\mathsf{PI}\left(\right) \cdot n}{k} \cdot 2} \]

   6. lower-*.f64 N/A

      \[\leadsto \sqrt{\frac{\mathsf{PI}\left(\right) \cdot n}{k} \cdot 2} \]

   7. lift-PI.f64 44.0

      \[\leadsto \sqrt{\frac{\pi \cdot n}{k} \cdot 2} \]

5. Applied rewrites 44.0%

   \[\leadsto \color{blue}{\sqrt{\frac{\pi \cdot n}{k} \cdot 2}} \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \sqrt{\frac{\pi \cdot n}{k} \cdot 2} \]

   2. lift-PI.f64 N/A

      \[\leadsto \sqrt{\frac{\mathsf{PI}\left(\right) \cdot n}{k} \cdot 2} \]

   3. lift-*.f64 N/A

      \[\leadsto \sqrt{\frac{\mathsf{PI}\left(\right) \cdot n}{k} \cdot 2} \]

   4. *-commutative N/A

      \[\leadsto \sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2} \]

   5. associate-/l* N/A

      \[\leadsto \sqrt{\left(n \cdot \frac{\mathsf{PI}\left(\right)}{k}\right) \cdot 2} \]

   6. lower-*.f64 N/A

      \[\leadsto \sqrt{\left(n \cdot \frac{\mathsf{PI}\left(\right)}{k}\right) \cdot 2} \]

   7. lower-/.f64 N/A

      \[\leadsto \sqrt{\left(n \cdot \frac{\mathsf{PI}\left(\right)}{k}\right) \cdot 2} \]

   8. lift-PI.f64 44.1

      \[\leadsto \sqrt{\left(n \cdot \frac{\pi}{k}\right) \cdot 2} \]

7. Applied rewrites 44.1%

   \[\leadsto \sqrt{\left(n \cdot \frac{\pi}{k}\right) \cdot 2} \]
8. Add Preprocessing

Alternative 5: 38.2% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{\pi \cdot \left(\frac{n}{k} \cdot 2\right)} \end{array} \]

FPCore

(FPCore (k n) :precision binary64 (sqrt (* PI (* (/ n k) 2.0))))

C

double code(double k, double n) {
	return sqrt((((double) M_PI) * ((n / k) * 2.0)));
}

Java

public static double code(double k, double n) {
	return Math.sqrt((Math.PI * ((n / k) * 2.0)));
}

Python

def code(k, n):
	return math.sqrt((math.pi * ((n / k) * 2.0)))

Julia

function code(k, n)
	return sqrt(Float64(pi * Float64(Float64(n / k) * 2.0)))
end

MATLAB

function tmp = code(k, n)
	tmp = sqrt((pi * ((n / k) * 2.0)));
end

Wolfram

code[k_, n_] := N[Sqrt[N[(Pi * N[(N[(n / k), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}} \cdot \sqrt{2}} \]
4. Step-by-step derivation

   1. sqrt-unprod N/A

      \[\leadsto \sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2} \]

   2. lower-sqrt.f64 N/A

      \[\leadsto \sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2} \]

   3. lower-*.f64 N/A

      \[\leadsto \sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2} \]

   4. lower-/.f64 N/A

      \[\leadsto \sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2} \]

   5. *-commutative N/A

      \[\leadsto \sqrt{\frac{\mathsf{PI}\left(\right) \cdot n}{k} \cdot 2} \]

   6. lower-*.f64 N/A

      \[\leadsto \sqrt{\frac{\mathsf{PI}\left(\right) \cdot n}{k} \cdot 2} \]

   7. lift-PI.f64 44.0

      \[\leadsto \sqrt{\frac{\pi \cdot n}{k} \cdot 2} \]

5. Applied rewrites 44.0%

   \[\leadsto \color{blue}{\sqrt{\frac{\pi \cdot n}{k} \cdot 2}} \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \sqrt{\frac{\pi \cdot n}{k} \cdot 2} \]

   2. lift-PI.f64 N/A

      \[\leadsto \sqrt{\frac{\mathsf{PI}\left(\right) \cdot n}{k} \cdot 2} \]

   3. lift-*.f64 N/A

      \[\leadsto \sqrt{\frac{\mathsf{PI}\left(\right) \cdot n}{k} \cdot 2} \]

   4. associate-/l* N/A

      \[\leadsto \sqrt{\left(\mathsf{PI}\left(\right) \cdot \frac{n}{k}\right) \cdot 2} \]

   5. lower-*.f64 N/A

      \[\leadsto \sqrt{\left(\mathsf{PI}\left(\right) \cdot \frac{n}{k}\right) \cdot 2} \]

   6. lift-PI.f64 N/A

      \[\leadsto \sqrt{\left(\pi \cdot \frac{n}{k}\right) \cdot 2} \]

   7. lower-/.f64 44.1

      \[\leadsto \sqrt{\left(\pi \cdot \frac{n}{k}\right) \cdot 2} \]

7. Applied rewrites 44.1%

   \[\leadsto \sqrt{\left(\pi \cdot \frac{n}{k}\right) \cdot 2} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sqrt{\left(\pi \cdot \frac{n}{k}\right) \cdot 2} \]

   2. lift-PI.f64 N/A

      \[\leadsto \sqrt{\left(\mathsf{PI}\left(\right) \cdot \frac{n}{k}\right) \cdot 2} \]

   3. lift-*.f64 N/A

      \[\leadsto \sqrt{\left(\mathsf{PI}\left(\right) \cdot \frac{n}{k}\right) \cdot 2} \]

   4. associate-*l* N/A

      \[\leadsto \sqrt{\mathsf{PI}\left(\right) \cdot \left(\frac{n}{k} \cdot 2\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \sqrt{\mathsf{PI}\left(\right) \cdot \left(\frac{n}{k} \cdot 2\right)} \]

   6. lift-PI.f64 N/A

      \[\leadsto \sqrt{\pi \cdot \left(\frac{n}{k} \cdot 2\right)} \]

   7. lower-*.f64 44.1

      \[\leadsto \sqrt{\pi \cdot \left(\frac{n}{k} \cdot 2\right)} \]

9. Applied rewrites 44.1%

   \[\leadsto \sqrt{\pi \cdot \left(\frac{n}{k} \cdot 2\right)} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025072 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1.0 (sqrt k)) (pow (* (* 2.0 PI) n) (/ (- 1.0 k) 2.0))))