Toniolo and Linder, Equation (3a)

Average Accuracy: 98.4% → 100.0%

Time: 20.9s

Precision: binary64

Cost: 45824

Math

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

↓

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

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

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

   [Start] 98.4	\[ \sqrt{0.5 + \frac{1}{\sqrt{1 + {\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot 0.5} \]
   add-sqr-sqrt [=>] 98.4	\[ \sqrt{0.5 + \frac{1}{\color{blue}{\sqrt{\sqrt{1 + {\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \sqrt{\sqrt{1 + {\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}}} \cdot 0.5} \]
   pow2 [=>] 98.4	\[ \sqrt{0.5 + \frac{1}{\color{blue}{{\left(\sqrt{\sqrt{1 + {\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)}^{2}}} \cdot 0.5} \]

4. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	95.7%
Cost	32964

\[\begin{array}{l} \mathbf{if}\;kx \leq -2 \cdot 10^{-84}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot {\left(\sqrt{\mathsf{hypot}\left(1, 2 \cdot \frac{\sin kx}{\frac{Om}{\ell}}\right)}\right)}^{-2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{2}{\frac{Om}{\ell \cdot \sin ky}}\right)}}\\ \end{array} \]

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	95.7%
Cost	20228

\[\begin{array}{l} \mathbf{if}\;kx \leq -1.5 \cdot 10^{-85}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{\sin kx}{Om} \cdot \left(\ell \cdot 2\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{2}{\frac{Om}{\ell \cdot \sin ky}}\right)}}\\ \end{array} \]

Alternative 4
Accuracy	92.2%
Cost	20100

\[\begin{array}{l} \mathbf{if}\;\ell \leq 10^{+119}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{\sin kx}{Om} \cdot \left(\ell \cdot 2\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 5
Accuracy	80.4%
Cost	14408

\[\begin{array}{l} \mathbf{if}\;Om \leq -8.2 \cdot 10^{+20}:\\ \;\;\;\;1\\ \mathbf{elif}\;Om \leq -5.3 \cdot 10^{-150}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{\sqrt{1 + \frac{4 \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(kx \cdot kx\right)\right)}{Om \cdot Om}}}}\\ \mathbf{elif}\;Om \leq 2.5 \cdot 10^{-161}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om \leq 1.55 \cdot 10^{-134}:\\ \;\;\;\;1\\ \mathbf{elif}\;Om \leq 48000000:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{1 + 2 \cdot \frac{\ell \cdot \ell}{\frac{Om \cdot Om}{kx \cdot kx}}}}\\ \mathbf{elif}\;Om \leq 2 \cdot 10^{+14}:\\ \;\;\;\;1\\ \mathbf{elif}\;Om \leq 5 \cdot 10^{+41}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6
Accuracy	80.0%
Cost	8404

\[\begin{array}{l} t_0 := \sqrt{0.5 + 0.5 \cdot \frac{1}{1 + 2 \cdot \frac{\ell \cdot \ell}{\frac{Om \cdot Om}{kx \cdot kx}}}}\\ \mathbf{if}\;Om \leq -4 \cdot 10^{+21}:\\ \;\;\;\;1\\ \mathbf{elif}\;Om \leq -8.6 \cdot 10^{-134}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;Om \leq 2.6 \cdot 10^{-161}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om \leq 1.55 \cdot 10^{-134}:\\ \;\;\;\;1\\ \mathbf{elif}\;Om \leq 70000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;Om \leq 1.9 \cdot 10^{+16}:\\ \;\;\;\;1\\ \mathbf{elif}\;Om \leq 3.8 \cdot 10^{+46}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7
Accuracy	76.7%
Cost	7256

\[\begin{array}{l} \mathbf{if}\;Om \leq -1.25 \cdot 10^{+17}:\\ \;\;\;\;1\\ \mathbf{elif}\;Om \leq -1.56 \cdot 10^{-22}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om \leq -5 \cdot 10^{-96}:\\ \;\;\;\;1\\ \mathbf{elif}\;Om \leq 2.6 \cdot 10^{-161}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om \leq 1.6 \cdot 10^{-134}:\\ \;\;\;\;1\\ \mathbf{elif}\;Om \leq 3.7 \cdot 10^{+46}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8
Accuracy	56.9%
Cost	6464

\[\sqrt{0.5} \]

Reproduce

herbie shell --seed 2023151 
(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))))))))))