Toniolo and Linder, Equation (3a)

Average Error: 1.1 → 0.0

Time: 15.3s

Precision: binary64

Cost: 45696

Math

\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]

↓

\[\sqrt{0.5 \cdot \left(1 + \sqrt{{\left(\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right)}^{-2}}\right)} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (sqrt
  (*
   (/ 1.0 2.0)
   (+
    1.0
    (/
     1.0
     (sqrt
      (+
       1.0
       (*
        (pow (/ (* 2.0 l) Om) 2.0)
        (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))))))

↓

(FPCore (l Om kx ky)
 :precision binary64
 (sqrt
  (*
   0.5
   (+
    1.0
    (sqrt
     (pow (hypot 1.0 (* (* 2.0 (/ l Om)) (hypot (sin kx) (sin ky)))) -2.0))))))

C

double code(double l, double Om, double kx, double ky) {
	return sqrt(((1.0 / 2.0) * (1.0 + (1.0 / sqrt((1.0 + (pow(((2.0 * l) / Om), 2.0) * (pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))))))));
}

↓

double code(double l, double Om, double kx, double ky) {
	return sqrt((0.5 * (1.0 + sqrt(pow(hypot(1.0, ((2.0 * (l / Om)) * hypot(sin(kx), sin(ky)))), -2.0)))));
}

Java

public static double code(double l, double Om, double kx, double ky) {
	return Math.sqrt(((1.0 / 2.0) * (1.0 + (1.0 / Math.sqrt((1.0 + (Math.pow(((2.0 * l) / Om), 2.0) * (Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))))))));
}

↓

public static double code(double l, double Om, double kx, double ky) {
	return Math.sqrt((0.5 * (1.0 + Math.sqrt(Math.pow(Math.hypot(1.0, ((2.0 * (l / Om)) * Math.hypot(Math.sin(kx), Math.sin(ky)))), -2.0)))));
}

Python

def code(l, Om, kx, ky):
	return math.sqrt(((1.0 / 2.0) * (1.0 + (1.0 / math.sqrt((1.0 + (math.pow(((2.0 * l) / Om), 2.0) * (math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))))))))

↓

def code(l, Om, kx, ky):
	return math.sqrt((0.5 * (1.0 + math.sqrt(math.pow(math.hypot(1.0, ((2.0 * (l / Om)) * math.hypot(math.sin(kx), math.sin(ky)))), -2.0)))))

Julia

function code(l, Om, kx, ky)
	return sqrt(Float64(Float64(1.0 / 2.0) * Float64(1.0 + Float64(1.0 / sqrt(Float64(1.0 + Float64((Float64(Float64(2.0 * l) / Om) ^ 2.0) * Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))))))))
end

↓

function code(l, Om, kx, ky)
	return sqrt(Float64(0.5 * Float64(1.0 + sqrt((hypot(1.0, Float64(Float64(2.0 * Float64(l / Om)) * hypot(sin(kx), sin(ky)))) ^ -2.0)))))
end

MATLAB

function tmp = code(l, Om, kx, ky)
	tmp = sqrt(((1.0 / 2.0) * (1.0 + (1.0 / sqrt((1.0 + ((((2.0 * l) / Om) ^ 2.0) * ((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))))))));
end

↓

function tmp = code(l, Om, kx, ky)
	tmp = sqrt((0.5 * (1.0 + sqrt((hypot(1.0, ((2.0 * (l / Om)) * hypot(sin(kx), sin(ky)))) ^ -2.0)))));
end

Wolfram

code[l_, Om_, kx_, ky_] := N[Sqrt[N[(N[(1.0 / 2.0), $MachinePrecision] * N[(1.0 + N[(1.0 / N[Sqrt[N[(1.0 + N[(N[Power[N[(N[(2.0 * l), $MachinePrecision] / Om), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

↓

code[l_, Om_, kx_, ky_] := N[Sqrt[N[(0.5 * N[(1.0 + N[Sqrt[N[Power[N[Sqrt[1.0 ^ 2 + N[(N[(2.0 * N[(l / Om), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision], -2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 1.1

   \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]

2. Applied egg-rr 0.0

   \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \color{blue}{\sqrt{{\left(\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right)}^{-2}}}\right)} \]

3. Final simplification 0.0

   \[\leadsto \sqrt{0.5 \cdot \left(1 + \sqrt{{\left(\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right)}^{-2}}\right)} \]

Alternatives

Alternative 1
Error	4.2
Cost	39944

\[\begin{array}{l} t_0 := 0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{\ell}{Om} \cdot \left(2 \cdot \sin kx\right)\right)}\\ \mathbf{if}\;\sin kx \leq -2 \cdot 10^{-202}:\\ \;\;\;\;\sqrt[3]{{t_0}^{1.5}}\\ \mathbf{elif}\;\sin kx \leq 1.5 \cdot 10^{-283}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{1 + \frac{4}{Om \cdot Om} \cdot \left({\sin ky}^{2} \cdot \left(\ell \cdot \ell\right)\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;{t_0}^{0.5}\\ \end{array} \]

Alternative 2
Error	4.2
Cost	39944

\[\begin{array}{l} \mathbf{if}\;\sin kx \leq -2 \cdot 10^{-202}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \sqrt{{\left(\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \sin kx\right)\right)}^{-2}}\right)}\\ \mathbf{elif}\;\sin kx \leq 1.5 \cdot 10^{-283}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{1 + \frac{4}{Om \cdot Om} \cdot \left({\sin ky}^{2} \cdot \left(\ell \cdot \ell\right)\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{\ell}{Om} \cdot \left(2 \cdot \sin kx\right)\right)}\right)}^{0.5}\\ \end{array} \]

Alternative 3
Error	3.9
Cost	20032

\[{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{\ell}{Om} \cdot \left(2 \cdot \sin kx\right)\right)}\right)}^{0.5} \]

Alternative 4
Error	9.3
Cost	13828

\[\begin{array}{l} \mathbf{if}\;Om \leq 2.5415763215822063 \cdot 10^{+100}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, \frac{2 \cdot \left(\ell \cdot kx\right)}{Om}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5
Error	14.5
Cost	13640

\[\begin{array}{l} \mathbf{if}\;Om \leq -1.3286209987640612 \cdot 10^{-34}:\\ \;\;\;\;1\\ \mathbf{elif}\;Om \leq -1.3823469613072965 \cdot 10^{-70}:\\ \;\;\;\;\sqrt{0.5 + 0.25 \cdot \frac{Om}{\ell \cdot \sin kx}}\\ \mathbf{elif}\;Om \leq -2.40475969879833 \cdot 10^{-154}:\\ \;\;\;\;1\\ \mathbf{elif}\;Om \leq 8.880699496075667 \cdot 10^{-23}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6
Error	14.2
Cost	6992

\[\begin{array}{l} \mathbf{if}\;Om \leq -1.3286209987640612 \cdot 10^{-34}:\\ \;\;\;\;1\\ \mathbf{elif}\;Om \leq -1.3823469613072965 \cdot 10^{-70}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om \leq -2.40475969879833 \cdot 10^{-154}:\\ \;\;\;\;1\\ \mathbf{elif}\;Om \leq 8.880699496075667 \cdot 10^{-23}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7
Error	27.6
Cost	6464

\[\sqrt{0.5} \]

Reproduce

herbie shell --seed 2022312 
(FPCore (l Om kx ky)
  :name "Toniolo and Linder, Equation (3a)"
  :precision binary64
  (sqrt (* (/ 1.0 2.0) (+ 1.0 (/ 1.0 (sqrt (+ 1.0 (* (pow (/ (* 2.0 l) Om) 2.0) (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))))))