Toniolo and Linder, Equation (3a)

Average Accuracy: 98.4% → 100.0%

Time: 18.1s

Precision: binary64

Cost: 39488

Math

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

↓

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

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

Derivation

1. Initial program 98.4%

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

2. Simplified 98.4%

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

   [Start] 98.4	\[ \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
   distribute-rgt-in [=>] 98.4	\[ \sqrt{\color{blue}{1 \cdot \frac{1}{2} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}}} \]
   metadata-eval [=>] 98.4	\[ \sqrt{1 \cdot \color{blue}{0.5} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}} \]
   metadata-eval [=>] 98.4	\[ \sqrt{\color{blue}{0.5} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}} \]
   associate-/l* [=>] 98.4	\[ \sqrt{0.5 + \frac{1}{\sqrt{1 + {\color{blue}{\left(\frac{2}{\frac{Om}{\ell}}\right)}}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}} \]
   metadata-eval [=>] 98.4	\[ \sqrt{0.5 + \frac{1}{\sqrt{1 + {\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \color{blue}{0.5}} \]

3. Applied egg-rr 100.0%

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

   [Start] 98.4	\[ \sqrt{0.5 + \frac{1}{\sqrt{1 + {\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot 0.5} \]
   pow1 [=>] 98.4	\[ \sqrt{0.5 + \frac{1}{\sqrt{1 + \color{blue}{{\left({\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)\right)}^{1}}}} \cdot 0.5} \]
   add-sqr-sqrt [=>] 98.4	\[ \sqrt{0.5 + \frac{1}{\sqrt{1 + {\color{blue}{\left(\sqrt{{\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)} \cdot \sqrt{{\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}\right)}}^{1}}} \cdot 0.5} \]
   pow2 [=>] 98.4	\[ \sqrt{0.5 + \frac{1}{\sqrt{1 + {\color{blue}{\left({\left(\sqrt{{\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}\right)}^{2}\right)}}^{1}}} \cdot 0.5} \]
   sqrt-prod [=>] 98.4	\[ \sqrt{0.5 + \frac{1}{\sqrt{1 + {\left({\color{blue}{\left(\sqrt{{\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2}} \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}}^{2}\right)}^{1}}} \cdot 0.5} \]
   unpow2 [=>] 98.4	\[ \sqrt{0.5 + \frac{1}{\sqrt{1 + {\left({\left(\sqrt{\color{blue}{\frac{2}{\frac{Om}{\ell}} \cdot \frac{2}{\frac{Om}{\ell}}}} \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}^{2}\right)}^{1}}} \cdot 0.5} \]
   sqrt-prod [=>] 56.4	\[ \sqrt{0.5 + \frac{1}{\sqrt{1 + {\left({\left(\color{blue}{\left(\sqrt{\frac{2}{\frac{Om}{\ell}}} \cdot \sqrt{\frac{2}{\frac{Om}{\ell}}}\right)} \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}^{2}\right)}^{1}}} \cdot 0.5} \]
   add-sqr-sqrt [<=] 98.8	\[ \sqrt{0.5 + \frac{1}{\sqrt{1 + {\left({\left(\color{blue}{\frac{2}{\frac{Om}{\ell}}} \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}^{2}\right)}^{1}}} \cdot 0.5} \]
   div-inv [=>] 98.8	\[ \sqrt{0.5 + \frac{1}{\sqrt{1 + {\left({\left(\color{blue}{\left(2 \cdot \frac{1}{\frac{Om}{\ell}}\right)} \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}^{2}\right)}^{1}}} \cdot 0.5} \]
   clear-num [<=] 98.8	\[ \sqrt{0.5 + \frac{1}{\sqrt{1 + {\left({\left(\left(2 \cdot \color{blue}{\frac{\ell}{Om}}\right) \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}^{2}\right)}^{1}}} \cdot 0.5} \]
   unpow2 [=>] 98.8	\[ \sqrt{0.5 + \frac{1}{\sqrt{1 + {\left({\left(\left(2 \cdot \frac{\ell}{Om}\right) \cdot \sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}\right)}^{2}\right)}^{1}}} \cdot 0.5} \]
   unpow2 [=>] 98.8	\[ \sqrt{0.5 + \frac{1}{\sqrt{1 + {\left({\left(\left(2 \cdot \frac{\ell}{Om}\right) \cdot \sqrt{\sin kx \cdot \sin kx + \color{blue}{\sin ky \cdot \sin ky}}\right)}^{2}\right)}^{1}}} \cdot 0.5} \]
   hypot-def [=>] 100.0	\[ \sqrt{0.5 + \frac{1}{\sqrt{1 + {\left({\left(\left(2 \cdot \frac{\ell}{Om}\right) \cdot \color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\right)}^{2}\right)}^{1}}} \cdot 0.5} \]

4. Simplified 100.0%

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

   [Start] 100.0	\[ \sqrt{0.5 + \frac{1}{\sqrt{1 + {\left({\left(\left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}^{2}\right)}^{1}}} \cdot 0.5} \]
   unpow1 [=>] 100.0	\[ \sqrt{0.5 + \frac{1}{\sqrt{1 + \color{blue}{{\left(\left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}^{2}}}} \cdot 0.5} \]
   *-commutative [=>] 100.0	\[ \sqrt{0.5 + \frac{1}{\sqrt{1 + {\color{blue}{\left(\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}}^{2}}} \cdot 0.5} \]
   hypot-def [<=] 98.8	\[ \sqrt{0.5 + \frac{1}{\sqrt{1 + {\left(\color{blue}{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}} \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}^{2}}} \cdot 0.5} \]
   unpow2 [<=] 98.8	\[ \sqrt{0.5 + \frac{1}{\sqrt{1 + {\left(\sqrt{\color{blue}{{\sin kx}^{2}} + \sin ky \cdot \sin ky} \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}^{2}}} \cdot 0.5} \]
   unpow2 [<=] 98.8	\[ \sqrt{0.5 + \frac{1}{\sqrt{1 + {\left(\sqrt{{\sin kx}^{2} + \color{blue}{{\sin ky}^{2}}} \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}^{2}}} \cdot 0.5} \]
   +-commutative [<=] 98.8	\[ \sqrt{0.5 + \frac{1}{\sqrt{1 + {\left(\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}} \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}^{2}}} \cdot 0.5} \]
   unpow2 [=>] 98.8	\[ \sqrt{0.5 + \frac{1}{\sqrt{1 + {\left(\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}} \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}^{2}}} \cdot 0.5} \]
   unpow2 [=>] 98.8	\[ \sqrt{0.5 + \frac{1}{\sqrt{1 + {\left(\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}} \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}^{2}}} \cdot 0.5} \]
   hypot-def [=>] 100.0	\[ \sqrt{0.5 + \frac{1}{\sqrt{1 + {\left(\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}^{2}}} \cdot 0.5} \]

5. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	96.2%
Cost	33544

\[\begin{array}{l} t_0 := \frac{2 \cdot \ell}{Om}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+51}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;t_0 \leq 5000000:\\ \;\;\;\;\sqrt{0.5 + \sqrt{\frac{0.25}{1 + {\left(\frac{\ell}{Om} \cdot \left(\sin kx \cdot 2\right)\right)}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(0.5 + \frac{Om \cdot \frac{0.25}{\ell}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)}^{1.5}\right)}^{0.3333333333333333}\\ \end{array} \]

Alternative 2
Accuracy	100.0%
Cost	32960

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

Alternative 3
Accuracy	91.9%
Cost	26628

\[\begin{array}{l} t_0 := \frac{\ell}{Om} \cdot \left(\sin kx \cdot 2\right)\\ \mathbf{if}\;kx \leq 7.6 \cdot 10^{-267}:\\ \;\;\;\;\sqrt{0.5 + \sqrt{\frac{0.25}{1 + {t_0}^{2}}}}\\ \mathbf{elif}\;kx \leq 2.35 \cdot 10^{-198}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, t_0\right)}}\\ \end{array} \]

Alternative 4
Accuracy	91.9%
Cost	20233

\[\begin{array}{l} \mathbf{if}\;kx \leq 10^{-267} \lor \neg \left(kx \leq 2.3 \cdot 10^{-198}\right):\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{\ell}{Om} \cdot \left(\sin kx \cdot 2\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 5
Accuracy	79.0%
Cost	6728

\[\begin{array}{l} \mathbf{if}\;Om \leq -5 \cdot 10^{-21}:\\ \;\;\;\;1\\ \mathbf{elif}\;Om \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6
Accuracy	55.8%
Cost	6464

\[\sqrt{0.5} \]

Reproduce

herbie shell --seed 2023146 
(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))))))))))