Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 4.3s

Alternatives: 7

Speedup: 1.1×

• could not determine a ground truth

  1. k = 2.650753454383447e+73
  2. n = -3.5488537072875985e+294

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in k around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 99.4

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

4. Applied rewrites 99.4%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*l* N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lift-PI.f64 N/A

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

   16. lift-sqrt.f64 99.5

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

6. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. *-lft-identity 99.5

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 99.5

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

8. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. count-2-rev N/A

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

   4. lower-+.f64 99.5

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

10. Applied rewrites 99.5%

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

Alternative 2: 49.3% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in k around 0

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

   1. sqrt-unprod N/A

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lift-PI.f64 49.4

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

4. Applied rewrites 49.4%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. sqrt-prod N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. sqrt-unprod N/A

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

   7. lower-*.f64 N/A

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

   8. sqrt-unprod N/A

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

   9. lift-*.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-sqrt.f64 49.2

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

6. Applied rewrites 49.2%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. sqrt-prod N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lower-sqrt.f64 49.3

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

8. Applied rewrites 49.3%

   \[\leadsto \frac{1}{\sqrt{k}} \cdot \left(\left(\sqrt{\pi} \cdot \sqrt{2}\right) \cdot \sqrt{\color{blue}{n}}\right) \]
9. Add Preprocessing

Alternative 3: 49.2% accurate, 2.8× speedup

Math

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

FPCore

(FPCore (k n)
 :precision binary64
 (* (/ 1.0 (sqrt k)) (* (sqrt (+ PI PI)) (sqrt n))))

C

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

Java

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

Python

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

Julia

function code(k, n)
	return Float64(Float64(1.0 / sqrt(k)) * Float64(sqrt(Float64(pi + pi)) * sqrt(n)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in k around 0

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

   1. sqrt-unprod N/A

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lift-PI.f64 49.4

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

4. Applied rewrites 49.4%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. sqrt-prod N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. sqrt-unprod N/A

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

   7. lower-*.f64 N/A

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

   8. sqrt-unprod N/A

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

   9. lift-*.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-sqrt.f64 49.2

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

6. Applied rewrites 49.2%

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

   1. lift-PI.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. count-2-rev N/A

      \[\leadsto \frac{1}{\sqrt{k}} \cdot \left(\sqrt{\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)} \cdot \sqrt{n}\right) \]

   5. lower-+.f64 N/A

      \[\leadsto \frac{1}{\sqrt{k}} \cdot \left(\sqrt{\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)} \cdot \sqrt{n}\right) \]

   6. lift-PI.f64 N/A

      \[\leadsto \frac{1}{\sqrt{k}} \cdot \left(\sqrt{\pi + \mathsf{PI}\left(\right)} \cdot \sqrt{n}\right) \]

   7. lift-PI.f64 49.2

      \[\leadsto \frac{1}{\sqrt{k}} \cdot \left(\sqrt{\pi + \pi} \cdot \sqrt{n}\right) \]

8. Applied rewrites 49.2%

   \[\leadsto \frac{1}{\sqrt{k}} \cdot \left(\sqrt{\pi + \pi} \cdot \sqrt{n}\right) \]
9. Add Preprocessing

Alternative 4: 49.4% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

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

3. Applied rewrites 99.5%

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

4. Taylor expanded in k around 0

   \[\leadsto \frac{\color{blue}{\sqrt{n \cdot \mathsf{PI}\left(\right)} \cdot \sqrt{2}}}{\sqrt{k}} \]
5. 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. lift-*.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. sqr-pow N/A

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

   6. lift-*.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. sqrt-unprod N/A

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

   13. *-commutative N/A

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

   14. *-commutative N/A

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

6. Applied rewrites 49.4%

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

Alternative 5: 38.0% accurate, 4.8× speedup

Math

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

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(Float64(Float64(pi * 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[(N[(N[(Pi * n), $MachinePrecision] / k), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}} \cdot \sqrt{2}} \]
3. 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 38.0

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

4. Applied rewrites 38.0%

   \[\leadsto \color{blue}{\sqrt{\frac{\pi \cdot n}{k} \cdot 2}} \]
5. Add Preprocessing

Alternative 6: 37.9% 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 99.4%

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

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}} \cdot \sqrt{2}} \]
3. 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 38.0

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

4. Applied rewrites 38.0%

   \[\leadsto \color{blue}{\sqrt{\frac{\pi \cdot n}{k} \cdot 2}} \]
5. 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 37.9

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

6. Applied rewrites 37.9%

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

Alternative 7: 38.0% 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 99.4%

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

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}} \cdot \sqrt{2}} \]
3. 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 38.0

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

4. Applied rewrites 38.0%

   \[\leadsto \color{blue}{\sqrt{\frac{\pi \cdot n}{k} \cdot 2}} \]
5. 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 38.0

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

6. Applied rewrites 38.0%

   \[\leadsto \sqrt{\left(\pi \cdot \frac{n}{k}\right) \cdot 2} \]
7. 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 38.0

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

8. Applied rewrites 38.0%

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

Reproduce

herbie shell --seed 2025097 
(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))))