Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.4%

Time: 10.7s

Alternatives: 10

Speedup: 1.1×

• Could not converge on a ground truth

  1. k = 6.379554807829026e+255
  2. n = -6.0608172485600066e+84

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: 99.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(\mathsf{fma}\left(k, -0.5, 0.5\right)\right)}}{\sqrt{k}} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (/ (pow (* n (* 2.0 PI)) (fma k -0.5 0.5)) (sqrt k)))

C

double code(double k, double n) {
	return pow((n * (2.0 * ((double) M_PI))), fma(k, -0.5, 0.5)) / sqrt(k);
}

Julia

function code(k, n)
	return Float64((Float64(n * Float64(2.0 * pi)) ^ fma(k, -0.5, 0.5)) / sqrt(k))
end

Wolfram

code[k_, n_] := N[(N[Power[N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision], N[(k * -0.5 + 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. Add Preprocessing
3. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. div-sub N/A

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

   5. metadata-eval N/A

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

   6. sub-neg N/A

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

   7. +-commutative N/A

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

   8. unpow-prod-up N/A

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

   9. lower-*.f64 N/A

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

4. Applied rewrites 99.3%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. un-div-inv N/A

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

   5. lower-/.f64 99.4

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

6. Applied rewrites 99.5%

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

Alternative 2: 61.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(2 \cdot \pi\right)\\ \mathbf{if}\;\frac{1}{\sqrt{k}} \cdot {t\_0}^{\left(\frac{1 - k}{2}\right)} \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \pi} \cdot \sqrt{n \cdot \sqrt{\frac{1}{k} \cdot \frac{1}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{t\_0}}{\sqrt{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* n (* 2.0 PI))))
   (if (<= (* (/ 1.0 (sqrt k)) (pow t_0 (/ (- 1.0 k) 2.0))) 0.0)
     (* (sqrt (* 2.0 PI)) (sqrt (* n (sqrt (* (/ 1.0 k) (/ 1.0 k))))))
     (/ (sqrt t_0) (sqrt k)))))

C

double code(double k, double n) {
	double t_0 = n * (2.0 * ((double) M_PI));
	double tmp;
	if (((1.0 / sqrt(k)) * pow(t_0, ((1.0 - k) / 2.0))) <= 0.0) {
		tmp = sqrt((2.0 * ((double) M_PI))) * sqrt((n * sqrt(((1.0 / k) * (1.0 / k)))));
	} else {
		tmp = sqrt(t_0) / sqrt(k);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = n * (2.0 * Math.PI);
	double tmp;
	if (((1.0 / Math.sqrt(k)) * Math.pow(t_0, ((1.0 - k) / 2.0))) <= 0.0) {
		tmp = Math.sqrt((2.0 * Math.PI)) * Math.sqrt((n * Math.sqrt(((1.0 / k) * (1.0 / k)))));
	} else {
		tmp = Math.sqrt(t_0) / Math.sqrt(k);
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = n * (2.0 * math.pi)
	tmp = 0
	if ((1.0 / math.sqrt(k)) * math.pow(t_0, ((1.0 - k) / 2.0))) <= 0.0:
		tmp = math.sqrt((2.0 * math.pi)) * math.sqrt((n * math.sqrt(((1.0 / k) * (1.0 / k)))))
	else:
		tmp = math.sqrt(t_0) / math.sqrt(k)
	return tmp

Julia

function code(k, n)
	t_0 = Float64(n * Float64(2.0 * pi))
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(k)) * (t_0 ^ Float64(Float64(1.0 - k) / 2.0))) <= 0.0)
		tmp = Float64(sqrt(Float64(2.0 * pi)) * sqrt(Float64(n * sqrt(Float64(Float64(1.0 / k) * Float64(1.0 / k))))));
	else
		tmp = Float64(sqrt(t_0) / sqrt(k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	t_0 = n * (2.0 * pi);
	tmp = 0.0;
	if (((1.0 / sqrt(k)) * (t_0 ^ ((1.0 - k) / 2.0))) <= 0.0)
		tmp = sqrt((2.0 * pi)) * sqrt((n * sqrt(((1.0 / k) * (1.0 / k)))));
	else
		tmp = sqrt(t_0) / sqrt(k);
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / N[Sqrt[k], $MachinePrecision]), $MachinePrecision] * N[Power[t$95$0, N[(N[(1.0 - k), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[Sqrt[N[(2.0 * Pi), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(n * N[Sqrt[N[(N[(1.0 / k), $MachinePrecision] * N[(1.0 / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $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)))) < 0.0

   1. Initial program 100.0%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   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. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 3.2

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

   5. Applied rewrites 3.2%

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

   7. Applied rewrites 3.2%

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

   9. Applied rewrites 3.2%

      \[\leadsto \sqrt{2 \cdot \pi} \cdot \color{blue}{\sqrt{n \cdot \frac{1}{k}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 48.9%

       \[\leadsto \sqrt{2 \cdot \pi} \cdot \sqrt{n \cdot \sqrt{\frac{1}{k} \cdot \frac{1}{k}}} \]

   if 0.0 < (*.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 99.2%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   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. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 49.6

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

   5. Applied rewrites 49.6%

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

   7. Applied rewrites 65.4%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{k}} \cdot {\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(\frac{1 - k}{2}\right)} \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \pi} \cdot \sqrt{n \cdot \sqrt{\frac{1}{k} \cdot \frac{1}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{n \cdot \left(2 \cdot \pi\right)}}{\sqrt{k}}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024229 
(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))))