Toniolo and Linder, Equation (3a)

Average Error: 1.0 → 0.0

Time: 18.6s

Precision: binary64

Cost: 39424

Math

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

↓

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

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
    (pow
     (+ 1.0 (pow (* (* 2.0 (/ l Om)) (hypot (sin kx) (sin ky))) 2.0))
     -0.5)))))

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 + pow((1.0 + pow(((2.0 * (l / Om)) * hypot(sin(kx), sin(ky))), 2.0)), -0.5))));
}

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.pow((1.0 + Math.pow(((2.0 * (l / Om)) * Math.hypot(Math.sin(kx), Math.sin(ky))), 2.0)), -0.5))));
}

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.pow((1.0 + math.pow(((2.0 * (l / Om)) * math.hypot(math.sin(kx), math.sin(ky))), 2.0)), -0.5))))

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 + (Float64(1.0 + (Float64(Float64(2.0 * Float64(l / Om)) * hypot(sin(kx), sin(ky))) ^ 2.0)) ^ -0.5))))
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 + ((1.0 + (((2.0 * (l / Om)) * hypot(sin(kx), sin(ky))) ^ 2.0)) ^ -0.5))));
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[Power[N[(1.0 + N[Power[N[(N[(2.0 * N[(l / Om), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 1.0

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

2. Applied egg-rr 0.0

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

3. Final simplification 0.0

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

Alternatives

Alternative 1
Error	1.1
Cost	39624

\[\begin{array}{l} t_0 := \sqrt{0.5 \cdot \left(1 + {\left(1 + {\left(\left(2 \cdot \frac{\ell}{Om}\right) \cdot \sin kx\right)}^{2}\right)}^{-0.5}\right)}\\ \mathbf{if}\;\sin kx \leq -2 \cdot 10^{-24}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\sin kx \leq 3.5 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, \left(\ell \cdot \sin ky\right) \cdot \frac{2}{Om}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	3.8
Cost	20360

\[\begin{array}{l} t_0 := \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, \left(\ell \cdot \sin ky\right) \cdot \frac{2}{Om}\right)}\right)}\\ \mathbf{if}\;kx \leq -7.720151037051786 \cdot 10^{+134}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;kx \leq -2.476298399290351 \cdot 10^{-24}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + {\left(1 + 4 \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(kx \cdot kx\right)\right)\right)}^{-0.5}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	13.4
Cost	14608

\[\begin{array}{l} t_0 := \sqrt{0.5 \cdot \left(1 + {\left(1 + 4 \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(kx \cdot kx\right)\right)\right)}^{-0.5}\right)}\\ \mathbf{if}\;\ell \leq -7.657038531261075 \cdot 10^{+65}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;\ell \leq -7.948271989027809 \cdot 10^{-114}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 3.8102769140165084 \cdot 10^{-251}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 3.0390924099459512 \cdot 10^{+115}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{Om}{\ell} \cdot \frac{-0.25}{ky}}\\ \end{array} \]

Alternative 4
Error	14.6
Cost	6992

\[\begin{array}{l} \mathbf{if}\;Om \leq -4.5934798741974154 \cdot 10^{+84}:\\ \;\;\;\;1\\ \mathbf{elif}\;Om \leq -1.2402019339174103 \cdot 10^{+20}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om \leq -1.275300790021598 \cdot 10^{-48}:\\ \;\;\;\;1\\ \mathbf{elif}\;Om \leq 8.13354883724954 \cdot 10^{-52}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5
Error	24.0
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022316 
(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))))))))))