Migdal et al, Equation (51)

Average Error: 0.5 → 0.4

Time: 10.7s

Precision: binary64

Cost: 32896

Math

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

↓

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

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 (* (* 2.0 PI) n)))
   (/ (* (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 = (2.0 * ((double) M_PI)) * n;
	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 = (2.0 * Math.PI) * n;
	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 = (2.0 * math.pi) * n
	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(2.0 * pi) * n)
	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 = (2.0 * pi) * n;
	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[(2.0 * Pi), $MachinePrecision] * n), $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 0.5

   \[\leadsto \color{blue}{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(0.5 - \frac{k}{2}\right)}}{\sqrt{k}}} \]
   Proof
   (/.f64 (pow.f64 (*.f64 (PI.f64) (*.f64 2 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 (PI.f64) 2) n)) (-.f64 1/2 (/.f64 k 2))) (sqrt.f64 k)): 8 points increase in error, 0 points decrease in error
   (/.f64 (pow.f64 (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 (PI.f64))) n) (-.f64 1/2 (/.f64 k 2))) (sqrt.f64 k)): 1 points increase in error, 8 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)): 9 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, 9 points decrease in error
   (/.f64 (Rewrite=> sqr-pow_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)): 0 points increase in error, 0 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)): 9 points increase in error, 0 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, 1 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)))): 1 points increase in error, 8 points decrease in error

3. Applied egg-rr 0.4

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

4. Simplified 0.4

   \[\leadsto \frac{\color{blue}{\sqrt{\left(2 \cdot \pi\right) \cdot n} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(k \cdot -0.5\right)}}}{\sqrt{k}} \]
   Proof
   (/.f64 (*.f64 (sqrt.f64 (*.f64 (*.f64 2 (PI.f64)) n)) (pow.f64 (*.f64 (*.f64 2 (PI.f64)) n) (*.f64 k -1/2))) (sqrt.f64 k)): 0 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (sqrt.f64 (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 (PI.f64) 2)) n)) (pow.f64 (*.f64 (*.f64 2 (PI.f64)) n) (*.f64 k -1/2))) (sqrt.f64 k)): 5 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (sqrt.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (PI.f64) (*.f64 2 n)))) (pow.f64 (*.f64 (*.f64 2 (PI.f64)) n) (*.f64 k -1/2))) (sqrt.f64 k)): 0 points increase in error, 5 points decrease in error
   (/.f64 (*.f64 (sqrt.f64 (*.f64 (PI.f64) (*.f64 2 n))) (pow.f64 (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 (PI.f64) 2)) n) (*.f64 k -1/2))) (sqrt.f64 k)): 0 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (sqrt.f64 (*.f64 (PI.f64) (*.f64 2 n))) (pow.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (PI.f64) (*.f64 2 n))) (*.f64 k -1/2))) (sqrt.f64 k)): 0 points increase in error, 0 points decrease in error

5. Final simplification 0.4

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

Alternatives

Alternative 1
Error	0.5
Cost	19968

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

Alternative 2
Error	0.7
Cost	19908

\[\begin{array}{l} \mathbf{if}\;k \leq 2.8 \cdot 10^{-43}:\\ \;\;\;\;{\left(\frac{k}{\pi}\right)}^{-0.5} \cdot \sqrt{2 \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]

Alternative 3
Error	0.5
Cost	19904

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

Alternative 4
Error	22.2
Cost	19648

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

Alternative 5
Error	22.2
Cost	19584

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

Alternative 6
Error	22.2
Cost	19584

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

Alternative 7
Error	32.3
Cost	13248

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

Alternative 8
Error	32.3
Cost	13248

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

Alternative 9
Error	32.3
Cost	13248

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

Alternative 10
Error	32.9
Cost	13184

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

Alternative 11
Error	33.0
Cost	13184

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

Reproduce

herbie shell --seed 2022343 
(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))))