Migdal et al, Equation (51)

Average Error: 0.5 → 0.4

Time: 9.7s

Precision: binary64

Cost: 32896

Math

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

↓

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

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))))
   (/ (* (pow t_0 (* k -0.5)) (sqrt t_0)) (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 (pow(t_0, (k * -0.5)) * sqrt(t_0)) / 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.pow(t_0, (k * -0.5)) * Math.sqrt(t_0)) / 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.pow(t_0, (k * -0.5)) * math.sqrt(t_0)) / 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(n * Float64(2.0 * pi))
	return Float64(Float64((t_0 ^ Float64(k * -0.5)) * sqrt(t_0)) / 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 = ((t_0 ^ (k * -0.5)) * sqrt(t_0)) / 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 * Pi), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Power[t$95$0, N[(k * -0.5), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t$95$0], $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

   \[\leadsto \color{blue}{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\mathsf{fma}\left(k, -0.5, 0.5\right)\right)}}{\sqrt{k}}} \]
   Proof
   (/.f64 (pow.f64 (*.f64 2 (*.f64 (PI.f64) n)) (fma.f64 k -1/2 1/2)) (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)) (fma.f64 k -1/2 1/2)) (sqrt.f64 k)): 0 points increase in error, 0 points decrease in error
   (/.f64 (pow.f64 (*.f64 (*.f64 2 (PI.f64)) n) (fma.f64 k (Rewrite<= metadata-eval (neg.f64 1/2)) 1/2)) (sqrt.f64 k)): 0 points increase in error, 0 points decrease in error
   (/.f64 (pow.f64 (*.f64 (*.f64 2 (PI.f64)) n) (fma.f64 k (neg.f64 (Rewrite<= metadata-eval (/.f64 1 2))) 1/2)) (sqrt.f64 k)): 0 points increase in error, 0 points decrease in error
   (/.f64 (pow.f64 (*.f64 (*.f64 2 (PI.f64)) n) (fma.f64 k (neg.f64 (/.f64 1 2)) (Rewrite<= metadata-eval (/.f64 1 2)))) (sqrt.f64 k)): 0 points increase in error, 0 points decrease in error
   (/.f64 (pow.f64 (*.f64 (*.f64 2 (PI.f64)) n) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 k (neg.f64 (/.f64 1 2))) (/.f64 1 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<= *-commutative_binary64 (*.f64 (neg.f64 (/.f64 1 2)) k)) (/.f64 1 2))) (sqrt.f64 k)): 0 points increase in error, 0 points decrease in error
   (/.f64 (pow.f64 (*.f64 (*.f64 2 (PI.f64)) n) (+.f64 (*.f64 (neg.f64 (Rewrite=> metadata-eval 1/2)) k) (/.f64 1 2))) (sqrt.f64 k)): 0 points increase in error, 0 points decrease in error
   (/.f64 (pow.f64 (*.f64 (*.f64 2 (PI.f64)) n) (+.f64 (*.f64 (Rewrite=> metadata-eval -1/2) k) (/.f64 1 2))) (sqrt.f64 k)): 0 points increase in error, 0 points decrease in error
   (/.f64 (pow.f64 (*.f64 (*.f64 2 (PI.f64)) n) (+.f64 (*.f64 (Rewrite<= metadata-eval (/.f64 -1 2)) k) (/.f64 1 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<= associate-/r/_binary64 (/.f64 -1 (/.f64 2 k))) (/.f64 1 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<= associate-/l*_binary64 (/.f64 (*.f64 -1 k) 2)) (/.f64 1 2))) (sqrt.f64 k)): 0 points increase in error, 0 points decrease in error
   (/.f64 (pow.f64 (*.f64 (*.f64 2 (PI.f64)) n) (+.f64 (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 k)) 2) (/.f64 1 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<= distribute-neg-frac_binary64 (neg.f64 (/.f64 k 2))) (/.f64 1 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 (/.f64 1 2) (neg.f64 (/.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<= sub-neg_binary64 (-.f64 (/.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 (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)))): 23 points increase in error, 22 points decrease in error

3. Applied egg-rr 0.4

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

4. Final simplification 0.4

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

Alternatives

Alternative 1
Error	1.6
Cost	19908

\[\begin{array}{l} \mathbf{if}\;k \leq 1.3359675064954594 \cdot 10^{-102}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\pi}{k}} \cdot \sqrt{n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]

Alternative 2
Error	0.5
Cost	19904

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

Alternative 3
Error	3.9
Cost	19716

\[\begin{array}{l} \mathbf{if}\;k \leq 1.3359675064954594 \cdot 10^{-102}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\pi}{k}} \cdot \sqrt{n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(\left(\frac{\pi}{k} + 1\right) + -1\right)\right)}\\ \end{array} \]

Alternative 4
Error	13.1
Cost	13440

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

Alternative 5
Error	32.8
Cost	13184

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

Alternative 6
Error	32.8
Cost	13184

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

Reproduce

herbie shell --seed 2022306 
(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))))