Toniolo and Linder, Equation (3a)

Average Error: 1.2 → 0.0

Time: 16.7s

Precision: binary64

Cost: 32832

Math

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

↓

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

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 (hypot 1.0 (* (* 2.0 l) (/ (hypot (sin kx) (sin ky)) Om)))))))

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 / hypot(1.0, ((2.0 * l) * (hypot(sin(kx), sin(ky)) / Om))))));
}

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

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.hypot(1.0, ((2.0 * l) * (math.hypot(math.sin(kx), math.sin(ky)) / Om))))))

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

Derivation

1. Initial program 1.2

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

2. Simplified 1.2

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

   [Start] 1.2	\[ \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
   distribute-lft-in [=>] 1.2	\[ \sqrt{\color{blue}{\frac{1}{2} \cdot 1 + \frac{1}{2} \cdot \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}}} \]
   metadata-eval [=>] 1.2	\[ \sqrt{\color{blue}{0.5} \cdot 1 + \frac{1}{2} \cdot \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}} \]
   metadata-eval [=>] 1.2	\[ \sqrt{\color{blue}{0.5} + \frac{1}{2} \cdot \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}} \]
   metadata-eval [=>] 1.2	\[ \sqrt{0.5 + \color{blue}{0.5} \cdot \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}} \]
   associate-/l* [=>] 1.2	\[ \sqrt{0.5 + 0.5 \cdot \frac{1}{\sqrt{1 + {\color{blue}{\left(\frac{2}{\frac{Om}{\ell}}\right)}}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}} \]

3. Applied egg-rr 0.0

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

4. Applied egg-rr 0.0

   \[\leadsto \sqrt{0.5 + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{0.5}{\mathsf{hypot}\left(1, \left(2 \cdot \ell\right) \cdot \frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{Om}\right)}\right)} - 1\right)}} \]

5. Simplified 0.0

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

   [Start] 0.0	\[ \sqrt{0.5 + \left(e^{\mathsf{log1p}\left(\frac{0.5}{\mathsf{hypot}\left(1, \left(2 \cdot \ell\right) \cdot \frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{Om}\right)}\right)} - 1\right)} \]
   expm1-def [=>] 0.0	\[ \sqrt{0.5 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.5}{\mathsf{hypot}\left(1, \left(2 \cdot \ell\right) \cdot \frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{Om}\right)}\right)\right)}} \]
   expm1-log1p [=>] 0.0	\[ \sqrt{0.5 + \color{blue}{\frac{0.5}{\mathsf{hypot}\left(1, \left(2 \cdot \ell\right) \cdot \frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{Om}\right)}}} \]

6. Final simplification 0.0

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

Alternatives

Alternative 1
Error	4.3
Cost	33033

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq 0.59 \lor \neg \left(\sin ky \leq 0.68\right):\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \left(2 \cdot \ell\right) \cdot \frac{\sin kx}{Om}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{Om}{\ell \cdot \sin ky} \cdot -0.25}\\ \end{array} \]

Alternative 2
Error	1.3
Cost	33033

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -0.16 \lor \neg \left(\sin ky \leq 10^{-161}\right):\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{\ell}{0.5} \cdot \frac{\sin ky}{Om}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \left(2 \cdot \ell\right) \cdot \frac{\sin kx}{Om}\right)}}\\ \end{array} \]

Alternative 3
Error	13.9
Cost	8536

\[\begin{array}{l} t_0 := \sqrt{0.5 + 0.5 \cdot \frac{1}{1 + 2 \cdot \frac{kx \cdot kx}{\frac{\frac{Om}{\ell}}{\frac{\ell}{Om}}}}}\\ \mathbf{if}\;Om \leq -6.1 \cdot 10^{+201}:\\ \;\;\;\;1\\ \mathbf{elif}\;Om \leq -5.7 \cdot 10^{+125}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;Om \leq -1.9 \cdot 10^{-36}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{1 + \frac{2 \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(ky \cdot ky\right)\right)}{Om \cdot Om}}}\\ \mathbf{elif}\;Om \leq -1.6 \cdot 10^{-213}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;Om \leq 5.6 \cdot 10^{-120}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om \leq 2.9 \cdot 10^{-62}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{1 + 2 \cdot \frac{\left(kx \cdot kx\right) \cdot \left(\ell \cdot \ell\right)}{Om \cdot Om}}}\\ \mathbf{elif}\;Om \leq 5 \cdot 10^{+85}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4
Error	14.0
Cost	8272

\[\begin{array}{l} \mathbf{if}\;Om \leq -6.1 \cdot 10^{+201}:\\ \;\;\;\;1\\ \mathbf{elif}\;Om \leq -1 \cdot 10^{-213}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{1 + 2 \cdot \frac{kx \cdot kx}{\frac{\frac{Om}{\ell}}{\frac{\ell}{Om}}}}}\\ \mathbf{elif}\;Om \leq 4.4 \cdot 10^{-121}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om \leq 2.9 \cdot 10^{-62}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{1 + 2 \cdot \frac{\left(kx \cdot kx\right) \cdot \left(\ell \cdot \ell\right)}{Om \cdot Om}}}\\ \mathbf{elif}\;Om \leq 3.6 \cdot 10^{+85}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5
Error	14.4
Cost	8008

\[\begin{array}{l} \mathbf{if}\;Om \leq -6.1 \cdot 10^{+201}:\\ \;\;\;\;1\\ \mathbf{elif}\;Om \leq -1.35 \cdot 10^{-213}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{1 + 2 \cdot \frac{kx \cdot kx}{\frac{\frac{Om}{\ell}}{\frac{\ell}{Om}}}}}\\ \mathbf{elif}\;Om \leq 3.5 \cdot 10^{+85}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6
Error	15.0
Cost	6728

\[\begin{array}{l} \mathbf{if}\;Om \leq -1.2 \cdot 10^{+44}:\\ \;\;\;\;1\\ \mathbf{elif}\;Om \leq 4 \cdot 10^{+85}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7
Error	23.9
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023027 
(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))))))))))