Toniolo and Linder, Equation (3a)

Average Error: 1.0 → 0.0

Time: 21.3s

Precision: binary64

Cost: 45760

Math

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

↓

\[\sqrt{0.5 + 0.5 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{hypot}\left(1, \frac{2 \cdot \ell}{Om} \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}\right)\right)} \]

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
    (expm1
     (log1p
      (/ 1.0 (hypot 1.0 (* (/ (* 2.0 l) Om) (hypot (sin kx) (sin ky)))))))))))

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 * expm1(log1p((1.0 / hypot(1.0, (((2.0 * l) / Om) * hypot(sin(kx), sin(ky))))))))));
}

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.expm1(Math.log1p((1.0 / Math.hypot(1.0, (((2.0 * l) / Om) * Math.hypot(Math.sin(kx), Math.sin(ky))))))))));
}

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.expm1(math.log1p((1.0 / math.hypot(1.0, (((2.0 * l) / Om) * math.hypot(math.sin(kx), math.sin(ky))))))))))

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 * expm1(log1p(Float64(1.0 / hypot(1.0, Float64(Float64(Float64(2.0 * l) / Om) * hypot(sin(kx), sin(ky))))))))))
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[(Exp[N[Log[1 + N[(1.0 / N[Sqrt[1.0 ^ 2 + N[(N[(N[(2.0 * l), $MachinePrecision] / Om), $MachinePrecision] * N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $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

   \[\leadsto \sqrt{0.5 + 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{hypot}\left(1, \frac{2 \cdot \ell}{Om} \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}\right)\right)}} \]

4. Final simplification 0.0

   \[\leadsto \sqrt{0.5 + 0.5 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{hypot}\left(1, \frac{2 \cdot \ell}{Om} \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}\right)\right)} \]

Alternatives

Alternative 1
Error	4.7
Cost	39945

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -5 \cdot 10^{-76} \lor \neg \left(\sin ky \leq 2 \cdot 10^{-287}\right):\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{\left(2 \cdot \ell\right) \cdot \sin ky}{Om}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{\sqrt{1 + 4 \cdot \frac{{\sin kx}^{2}}{\frac{Om \cdot Om}{\ell \cdot \ell}}}}}\\ \end{array} \]

Alternative 2
Error	0.0
Cost	32832

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

Alternative 3
Error	3.2
Cost	20356

\[\begin{array}{l} \mathbf{if}\;\frac{2 \cdot \ell}{Om} \leq 5 \cdot 10^{+57}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{\left(2 \cdot \ell\right) \cdot \sin ky}{Om}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 4
Error	8.2
Cost	14409

\[\begin{array}{l} \mathbf{if}\;kx \leq -9.6 \cdot 10^{+117} \lor \neg \left(kx \leq -2.05 \cdot 10^{-162}\right):\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{2 \cdot \left(\ell \cdot ky\right)}{Om}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{\sqrt{1 + 4 \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(kx \cdot kx\right)\right)}}}\\ \end{array} \]

Alternative 5
Error	9.3
Cost	13961

\[\begin{array}{l} \mathbf{if}\;\ell \leq -0.0024 \lor \neg \left(\ell \leq 2.4 \cdot 10^{-170}\right):\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{2 \cdot \left(\ell \cdot ky\right)}{Om}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6
Error	11.6
Cost	8272

\[\begin{array}{l} t_0 := \sqrt{0.5 + 0.5 \cdot \frac{1}{1 + 2 \cdot \frac{\ell \cdot \ell}{\frac{Om \cdot Om}{kx \cdot kx}}}}\\ \mathbf{if}\;\ell \leq -5.2 \cdot 10^{+157}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;\ell \leq -33000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 7.8 \cdot 10^{-100}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 2.2 \cdot 10^{+139}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 7
Error	9.4
Cost	8272

\[\begin{array}{l} t_0 := \sqrt{0.5 + 0.5 \cdot \frac{1}{1 + \frac{2 \cdot \left(\ell \cdot \ell\right)}{\frac{Om}{kx} \cdot \frac{Om}{kx}}}}\\ \mathbf{if}\;\ell \leq -5.5 \cdot 10^{+160}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;\ell \leq -8.8 \cdot 10^{-75}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 3.25 \cdot 10^{-105}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 2.1 \cdot 10^{+139}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 8
Error	14.1
Cost	7632

\[\begin{array}{l} \mathbf{if}\;Om \leq -40000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;Om \leq 9 \cdot 10^{-44}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om \leq 1.7 \cdot 10^{+65}:\\ \;\;\;\;1\\ \mathbf{elif}\;Om \leq 3.2 \cdot 10^{+83}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \left(\frac{\frac{Om}{\ell}}{ky} \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9
Error	13.8
Cost	6728

\[\begin{array}{l} \mathbf{if}\;Om \leq -34000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;Om \leq 4.1 \cdot 10^{-44}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10
Error	23.6
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023039 
(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))))))))))