Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.3% → 100.0%

Time: 21.8s

Precision: binary64

Cost: 45696

• Unsound rule application detected in e-graph. Results may not be sound.

Math

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

↓

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

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

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

   [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%

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

   [Start] 98.4	\[ \sqrt{0.5 + \frac{1}{\sqrt{1 + {\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot 0.5} \]
   inv-pow [=>] 98.4	\[ \sqrt{0.5 + \color{blue}{{\left(\sqrt{1 + {\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}\right)}^{-1}} \cdot 0.5} \]
   add-sqr-sqrt [=>] 98.4	\[ \sqrt{0.5 + {\color{blue}{\left(\sqrt{\sqrt{1 + {\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \sqrt{\sqrt{1 + {\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)}}^{-1} \cdot 0.5} \]
   unpow-prod-down [=>] 98.4	\[ \sqrt{0.5 + \color{blue}{\left({\left(\sqrt{\sqrt{1 + {\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)}^{-1} \cdot {\left(\sqrt{\sqrt{1 + {\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)}^{-1}\right)} \cdot 0.5} \]

4. Simplified 100.0%

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

   [Start] 100.0	\[ \sqrt{0.5 + \left({\left(\sqrt{\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}\right)}^{-1} \cdot {\left(\sqrt{\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}\right)}^{-1}\right) \cdot 0.5} \]
   pow-sqr [=>] 100.0	\[ \sqrt{0.5 + \color{blue}{{\left(\sqrt{\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}\right)}^{\left(2 \cdot -1\right)}} \cdot 0.5} \]
   *-commutative [=>] 100.0	\[ \sqrt{0.5 + {\left(\sqrt{\mathsf{hypot}\left(1, \color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \left(2 \cdot \frac{\ell}{Om}\right)}\right)}\right)}^{\left(2 \cdot -1\right)} \cdot 0.5} \]
   hypot-def [<=] 99.0	\[ \sqrt{0.5 + {\left(\sqrt{\mathsf{hypot}\left(1, \color{blue}{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}} \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}\right)}^{\left(2 \cdot -1\right)} \cdot 0.5} \]
   unpow2 [<=] 99.0	\[ \sqrt{0.5 + {\left(\sqrt{\mathsf{hypot}\left(1, \sqrt{\color{blue}{{\sin kx}^{2}} + \sin ky \cdot \sin ky} \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}\right)}^{\left(2 \cdot -1\right)} \cdot 0.5} \]
   unpow2 [<=] 99.0	\[ \sqrt{0.5 + {\left(\sqrt{\mathsf{hypot}\left(1, \sqrt{{\sin kx}^{2} + \color{blue}{{\sin ky}^{2}}} \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}\right)}^{\left(2 \cdot -1\right)} \cdot 0.5} \]
   +-commutative [<=] 99.0	\[ \sqrt{0.5 + {\left(\sqrt{\mathsf{hypot}\left(1, \sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}} \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}\right)}^{\left(2 \cdot -1\right)} \cdot 0.5} \]
   unpow2 [=>] 99.0	\[ \sqrt{0.5 + {\left(\sqrt{\mathsf{hypot}\left(1, \sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}} \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}\right)}^{\left(2 \cdot -1\right)} \cdot 0.5} \]
   unpow2 [=>] 99.0	\[ \sqrt{0.5 + {\left(\sqrt{\mathsf{hypot}\left(1, \sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}} \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}\right)}^{\left(2 \cdot -1\right)} \cdot 0.5} \]
   hypot-def [=>] 100.0	\[ \sqrt{0.5 + {\left(\sqrt{\mathsf{hypot}\left(1, \color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}\right)}^{\left(2 \cdot -1\right)} \cdot 0.5} \]
   metadata-eval [=>] 100.0	\[ \sqrt{0.5 + {\left(\sqrt{\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}\right)}^{\color{blue}{-2}} \cdot 0.5} \]

5. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	96.2%
Cost	32964

\[\begin{array}{l} \mathbf{if}\;ky \leq 2.65 \cdot 10^{-39}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{\sin kx}{\frac{Om}{\ell + \ell}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot {\left(\sqrt{\mathsf{hypot}\left(1, \frac{2}{\frac{Om}{\sin ky \cdot \ell}}\right)}\right)}^{-2}}\\ \end{array} \]

Alternative 2
Accuracy	96.2%
Cost	32964

\[\begin{array}{l} \mathbf{if}\;ky \leq 2 \cdot 10^{-50}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot {\left(\sqrt{\mathsf{hypot}\left(1, \frac{2 \cdot \ell}{\frac{Om}{\sin kx}}\right)}\right)}^{-2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot {\left(\sqrt{\mathsf{hypot}\left(1, \frac{2}{\frac{Om}{\sin ky \cdot \ell}}\right)}\right)}^{-2}}\\ \end{array} \]

Alternative 3
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 4
Accuracy	93.0%
Cost	26505

\[\begin{array}{l} \mathbf{if}\;Om \leq -4.2 \cdot 10^{-205} \lor \neg \left(Om \leq 2.2 \cdot 10^{-205}\right):\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{\sin kx}{\frac{Om}{\ell + \ell}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.25 \cdot \frac{Om}{\ell}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}\\ \end{array} \]

Alternative 5
Accuracy	93.4%
Cost	20681

\[\begin{array}{l} \mathbf{if}\;Om \leq -8.8 \cdot 10^{-204} \lor \neg \left(Om \leq 3.8 \cdot 10^{-205}\right):\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{\sin kx}{\frac{Om}{\ell + \ell}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{\sqrt{\mathsf{fma}\left(4, \frac{\left(\ell \cdot \frac{ky}{Om}\right) \cdot \left(ky \cdot \ell\right)}{Om}, 1\right)}}}\\ \end{array} \]

Alternative 6
Accuracy	93.2%
Cost	20233

\[\begin{array}{l} \mathbf{if}\;Om \leq -4.5 \cdot 10^{-205} \lor \neg \left(Om \leq 2.2 \cdot 10^{-205}\right):\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{\sin kx}{\frac{Om}{\ell + \ell}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{2 \cdot \frac{ky \cdot \ell}{Om} + 0.25 \cdot \frac{Om}{ky \cdot \ell}}}\\ \end{array} \]

Alternative 7
Accuracy	78.7%
Cost	6728

\[\begin{array}{l} \mathbf{if}\;\ell \leq -3.1 \cdot 10^{+51}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;\ell \leq 7 \cdot 10^{+82}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 8
Accuracy	55.8%
Cost	6464

\[\sqrt{0.5} \]

Reproduce

herbie shell --seed 2023160 
(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))))))))))