Migdal et al, Equation (51)

Average Error: 0.5 → 0.4

Time: 1.4min

Precision: binary64

Cost: 32896

Math

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

↓

\[\begin{array}{l} t_0 := \left(n \cdot 2\right) \cdot \pi\\ \frac{\sqrt{t_0} \cdot {t_0}^{\left(k \cdot -0.5\right)}}{\sqrt{k}} \end{array} \]

FPCore

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

↓

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* (* n 2.0) PI)))
   (/ (* (sqrt t_0) (pow t_0 (* k -0.5))) (sqrt k))))

C

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

↓

double code(double k, double n) {
	double t_0 = (n * 2.0) * ((double) M_PI);
	return (sqrt(t_0) * pow(t_0, (k * -0.5))) / sqrt(k);
}

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));
}

↓

public static double code(double k, double n) {
	double t_0 = (n * 2.0) * Math.PI;
	return (Math.sqrt(t_0) * Math.pow(t_0, (k * -0.5))) / Math.sqrt(k);
}

Python

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

↓

def code(k, n):
	t_0 = (n * 2.0) * math.pi
	return (math.sqrt(t_0) * math.pow(t_0, (k * -0.5))) / math.sqrt(k)

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

↓

function code(k, n)
	t_0 = Float64(Float64(n * 2.0) * pi)
	return Float64(Float64(sqrt(t_0) * (t_0 ^ Float64(k * -0.5))) / sqrt(k))
end

MATLAB

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

↓

function tmp = code(k, n)
	t_0 = (n * 2.0) * pi;
	tmp = (sqrt(t_0) * (t_0 ^ (k * -0.5))) / sqrt(k);
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]

↓

code[k_, n_] := Block[{t$95$0 = N[(N[(n * 2.0), $MachinePrecision] * Pi), $MachinePrecision]}, N[(N[(N[Sqrt[t$95$0], $MachinePrecision] * N[Power[t$95$0, N[(k * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 0.5

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

2. Simplified 0.4

   \[\leadsto \color{blue}{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 - \frac{k}{2}\right)}}{\sqrt{k}}} \]
   Proof
   (/.f64 (pow.f64 (*.f64 2 (*.f64 (PI.f64) n)) (-.f64 1/2 (/.f64 k 2))) (sqrt.f64 k)): 0 points increase in error, 0 points decrease in error
   (/.f64 (pow.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 2 (PI.f64)) n)) (-.f64 1/2 (/.f64 k 2))) (sqrt.f64 k)): 0 points increase in error, 0 points decrease in error
   (/.f64 (pow.f64 (*.f64 (*.f64 2 (PI.f64)) n) (-.f64 (Rewrite<= metadata-eval (/.f64 1 2)) (/.f64 k 2))) (sqrt.f64 k)): 0 points increase in error, 0 points decrease in error
   (/.f64 (pow.f64 (*.f64 (*.f64 2 (PI.f64)) n) (Rewrite<= div-sub_binary64 (/.f64 (-.f64 1 k) 2))) (sqrt.f64 k)): 0 points increase in error, 0 points decrease in error
   (/.f64 (pow.f64 (*.f64 (*.f64 2 (PI.f64)) n) (Rewrite<= /-rgt-identity_binary64 (/.f64 (/.f64 (-.f64 1 k) 2) 1))) (sqrt.f64 k)): 0 points increase in error, 0 points decrease in error
   (/.f64 (pow.f64 (*.f64 (*.f64 2 (PI.f64)) n) (/.f64 (/.f64 (-.f64 1 k) 2) (Rewrite<= metadata-eval (/.f64 2 2)))) (sqrt.f64 k)): 0 points increase in error, 0 points decrease in error
   (/.f64 (pow.f64 (*.f64 (*.f64 2 (PI.f64)) n) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (/.f64 (-.f64 1 k) 2) 2) 2))) (sqrt.f64 k)): 0 points increase in error, 0 points decrease in error
   (/.f64 (pow.f64 (*.f64 (*.f64 2 (PI.f64)) n) (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (/.f64 (-.f64 1 k) 2) 2) 2))) (sqrt.f64 k)): 0 points increase in error, 0 points decrease in error
   (/.f64 (pow.f64 (*.f64 (*.f64 2 (PI.f64)) n) (Rewrite<= *-commutative_binary64 (*.f64 2 (/.f64 (/.f64 (-.f64 1 k) 2) 2)))) (sqrt.f64 k)): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= pow-sqr_binary64 (*.f64 (pow.f64 (*.f64 (*.f64 2 (PI.f64)) n) (/.f64 (/.f64 (-.f64 1 k) 2) 2)) (pow.f64 (*.f64 (*.f64 2 (PI.f64)) n) (/.f64 (/.f64 (-.f64 1 k) 2) 2)))) (sqrt.f64 k)): 52 points increase in error, 25 points decrease in error
   (/.f64 (Rewrite<= sqr-pow_binary64 (pow.f64 (*.f64 (*.f64 2 (PI.f64)) n) (/.f64 (-.f64 1 k) 2))) (sqrt.f64 k)): 25 points increase in error, 52 points decrease in error
   (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (pow.f64 (*.f64 (*.f64 2 (PI.f64)) n) (/.f64 (-.f64 1 k) 2)))) (sqrt.f64 k)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 1 (sqrt.f64 k)) (pow.f64 (*.f64 (*.f64 2 (PI.f64)) n) (/.f64 (-.f64 1 k) 2)))): 29 points increase in error, 19 points decrease in error

3. Applied egg-rr 0.4

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

4. Simplified 0.4

   \[\leadsto \frac{\color{blue}{\sqrt{\left(n \cdot 2\right) \cdot \pi} \cdot {\left(\left(n \cdot 2\right) \cdot \pi\right)}^{\left(k \cdot -0.5\right)}}}{\sqrt{k}} \]
   Proof
   (*.f64 (sqrt.f64 (*.f64 (*.f64 n 2) (PI.f64))) (pow.f64 (*.f64 (*.f64 n 2) (PI.f64)) (*.f64 k -1/2))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 (Rewrite<= associate-*r*_binary64 (*.f64 n (*.f64 2 (PI.f64))))) (pow.f64 (*.f64 (*.f64 n 2) (PI.f64)) (*.f64 k -1/2))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 (*.f64 n (*.f64 2 (PI.f64)))) (pow.f64 (Rewrite<= associate-*r*_binary64 (*.f64 n (*.f64 2 (PI.f64)))) (*.f64 k -1/2))): 0 points increase in error, 0 points decrease in error

5. Final simplification 0.4

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

Alternatives

Alternative 1
Error	0.5
Cost	32896

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

Alternative 2
Error	0.5
Cost	19968

\[\frac{{k}^{-0.5}}{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(-0.5 \cdot \left(1 - k\right)\right)}} \]

Alternative 3
Error	0.5
Cost	19908

\[\begin{array}{l} t_0 := n \cdot \left(2 \cdot \pi\right)\\ \mathbf{if}\;k \leq 5.3 \cdot 10^{-17}:\\ \;\;\;\;\frac{\sqrt{t_0}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{t_0}^{\left(1 - k\right)}}{k}}\\ \end{array} \]

Alternative 4
Error	0.4
Cost	19904

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

Alternative 5
Error	2.8
Cost	19716

\[\begin{array}{l} \mathbf{if}\;k \leq 2.9 \cdot 10^{-19}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{\frac{2}{\frac{k}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(\left(1 + \pi \cdot \frac{2}{k}\right) + -1\right)}\\ \end{array} \]

Alternative 6
Error	2.7
Cost	19716

\[\begin{array}{l} \mathbf{if}\;k \leq 2.05 \cdot 10^{-19}:\\ \;\;\;\;\frac{\sqrt{n \cdot 2}}{\sqrt{\frac{k}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(\left(1 + \pi \cdot \frac{2}{k}\right) + -1\right)}\\ \end{array} \]

Alternative 7
Error	2.7
Cost	19716

\[\begin{array}{l} \mathbf{if}\;k \leq 2.6 \cdot 10^{-19}:\\ \;\;\;\;\frac{\sqrt{n \cdot \left(2 \cdot \pi\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(\left(1 + \pi \cdot \frac{2}{k}\right) + -1\right)}\\ \end{array} \]

Alternative 8
Error	13.4
Cost	13440

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

Alternative 9
Error	32.6
Cost	13248

\[{\left(\frac{k}{n} \cdot \frac{0.5}{\pi}\right)}^{-0.5} \]

Alternative 10
Error	32.6
Cost	13248

\[{\left(\frac{k}{\frac{\pi}{\frac{0.5}{n}}}\right)}^{-0.5} \]

Alternative 11
Error	33.1
Cost	13184

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

Alternative 12
Error	33.1
Cost	13184

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

Alternative 13
Error	33.1
Cost	13184

\[\sqrt{\frac{\pi}{k \cdot \frac{0.5}{n}}} \]

Alternative 14
Error	33.1
Cost	13184

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

Reproduce

herbie shell --seed 2022325 
(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))))