Toniolo and Linder, Equation (3a)

Average Error: 1.0 → 0.0

Time: 20.1s

Precision: binary64

Cost: 39744

Math

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

↓

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

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
   (*
    0.5
    (/
     1.0
     (sqrt
      (+
       1.0
       (+
        (+ 2.0 (pow (* (/ 2.0 Om) (* l (hypot (sin kx) (sin ky)))) 2.0))
        -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 + (0.5 * (1.0 / sqrt((1.0 + ((2.0 + pow(((2.0 / Om) * (l * hypot(sin(kx), sin(ky)))), 2.0)) + -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 + (0.5 * (1.0 / Math.sqrt((1.0 + ((2.0 + Math.pow(((2.0 / Om) * (l * Math.hypot(Math.sin(kx), Math.sin(ky)))), 2.0)) + -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 + (0.5 * (1.0 / math.sqrt((1.0 + ((2.0 + math.pow(((2.0 / Om) * (l * math.hypot(math.sin(kx), math.sin(ky)))), 2.0)) + -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(0.5 * Float64(1.0 / sqrt(Float64(1.0 + Float64(Float64(2.0 + (Float64(Float64(2.0 / Om) * Float64(l * hypot(sin(kx), sin(ky)))) ^ 2.0)) + -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 + (0.5 * (1.0 / sqrt((1.0 + ((2.0 + (((2.0 / Om) * (l * hypot(sin(kx), sin(ky)))) ^ 2.0)) + -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[(0.5 * N[(1.0 / N[Sqrt[N[(1.0 + N[(N[(2.0 + N[Power[N[(N[(2.0 / Om), $MachinePrecision] * N[(l * N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $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. Simplified 1.0

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

   [Start] 1.0	\[ \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
   distribute-lft-in [=>] 1.0	\[ \sqrt{\color{blue}{\frac{1}{2} \cdot 1 + \frac{1}{2} \cdot \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}}} \]
   metadata-eval [=>] 1.0	\[ \sqrt{\color{blue}{0.5} \cdot 1 + \frac{1}{2} \cdot \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}} \]
   metadata-eval [=>] 1.0	\[ \sqrt{\color{blue}{0.5} + \frac{1}{2} \cdot \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}} \]
   metadata-eval [=>] 1.0	\[ \sqrt{0.5 + \color{blue}{0.5} \cdot \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}} \]
   associate-/l* [=>] 1.0	\[ \sqrt{0.5 + 0.5 \cdot \frac{1}{\sqrt{1 + {\color{blue}{\left(\frac{2}{\frac{Om}{\ell}}\right)}}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}} \]

3. Applied egg-rr 0.0

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

4. Applied egg-rr 0.0

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

5. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.0
Cost	32832

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

Alternative 2
Error	2.9
Cost	27012

\[\begin{array}{l} \mathbf{if}\;ky \leq 1.1 \cdot 10^{-214}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{\sqrt{1 + \left(\left(2 + {\left(\frac{\sin kx \cdot \left(2 \cdot \ell\right)}{Om}\right)}^{2}\right) + -2\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \sin ky \cdot \frac{2 \cdot \ell}{Om}\right)}}\\ \end{array} \]

Alternative 3
Error	2.9
Cost	20228

\[\begin{array}{l} t_0 := \frac{2 \cdot \ell}{Om}\\ \mathbf{if}\;ky \leq 1.1 \cdot 10^{-214}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, \sin kx \cdot t_0\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \sin ky \cdot t_0\right)}}\\ \end{array} \]

Alternative 4
Error	4.0
Cost	19968

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

Alternative 5
Error	23.4
Cost	6993

\[\begin{array}{l} \mathbf{if}\;\ell \leq -7:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;\ell \leq 2.9 \cdot 10^{-96} \lor \neg \left(\ell \leq 5.4 \cdot 10^{-52}\right) \land \ell \leq 54000000000000:\\ \;\;\;\;1 + \frac{-0.5 \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(ky \cdot ky\right)\right)}{Om \cdot Om}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 6
Error	13.9
Cost	6992

\[\begin{array}{l} \mathbf{if}\;\ell \leq -3450000:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;\ell \leq 7.5 \cdot 10^{-94}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{-57}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+14}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 7
Error	42.3
Cost	960

\[1 + \frac{-0.5 \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(ky \cdot ky\right)\right)}{Om \cdot Om} \]

Reproduce

herbie shell --seed 2023053 
(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))))))))))