Toniolo and Linder, Equation (3a)

Average Error: 1.1 → 1.1

Time: 19.4s

Precision: binary64

Cost: 46080

Math

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

↓

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

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
   (+
    1.0
    (/
     1.0
     (sqrt
      (+
       1.0
       (*
        (pow (* l (/ 2.0 Om)) 2.0)
        (+ (pow (sin kx) 2.0) (pow (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 * (1.0 + (1.0 / sqrt((1.0 + (pow((l * (2.0 / Om)), 2.0) * (pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))))))));
}

Fortran

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    code = sqrt(((1.0d0 / 2.0d0) * (1.0d0 + (1.0d0 / sqrt((1.0d0 + ((((2.0d0 * l) / om) ** 2.0d0) * ((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))))))))
end function

↓

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    code = sqrt((0.5d0 * (1.0d0 + (1.0d0 / sqrt((1.0d0 + (((l * (2.0d0 / om)) ** 2.0d0) * ((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))))))))
end function

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 * (1.0 + (1.0 / Math.sqrt((1.0 + (Math.pow((l * (2.0 / Om)), 2.0) * (Math.pow(Math.sin(kx), 2.0) + Math.pow(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 * (1.0 + (1.0 / math.sqrt((1.0 + (math.pow((l * (2.0 / Om)), 2.0) * (math.pow(math.sin(kx), 2.0) + math.pow(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(1.0 + Float64(1.0 / sqrt(Float64(1.0 + Float64((Float64(l * Float64(2.0 / Om)) ^ 2.0) * Float64((sin(kx) ^ 2.0) + (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 * (1.0 + (1.0 / sqrt((1.0 + (((l * (2.0 / Om)) ^ 2.0) * ((sin(kx) ^ 2.0) + (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[(1.0 + N[(1.0 / N[Sqrt[N[(1.0 + N[(N[Power[N[(l * N[(2.0 / Om), $MachinePrecision]), $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]

Derivation

1. Initial program 1.1

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

2. Simplified 1.1

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

   [Start] 1.1	\[ \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
   metadata-eval [=>] 1.1	\[ \sqrt{\color{blue}{0.5} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
   rational.json-simplify-49 [=>] 1.1	\[ \sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{1 + {\color{blue}{\left(\ell \cdot \frac{2}{Om}\right)}}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]

3. Final simplification 1.1

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

Alternatives

Alternative 1
Error	6.7
Cost	39944

\[\begin{array}{l} \mathbf{if}\;kx \leq -7.8 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{1}{-1 + \left(1 - \left(-\sqrt{1 + 4 \cdot \frac{{\left(\sin kx \cdot \ell\right)}^{2}}{{Om}^{2}}}\right)\right)}\right)}\\ \mathbf{elif}\;kx \leq 1.7 \cdot 10^{-116}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\ell \cdot \frac{2}{Om}\right)}^{2} \cdot \left({kx}^{2} + {\sin ky}^{2}\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{1 + 4 \cdot \frac{{\left(\sin ky \cdot \ell\right)}^{2}}{{Om}^{2}}}}\right)}\\ \end{array} \]

Alternative 2
Error	6.7
Cost	33616

\[\begin{array}{l} t_0 := \sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{1 + 4 \cdot \frac{{\left(\sin ky \cdot \ell\right)}^{2}}{{Om}^{2}}}}\right)}\\ \mathbf{if}\;Om \leq -3.7 \cdot 10^{+197}:\\ \;\;\;\;1\\ \mathbf{elif}\;Om \leq -2.8 \cdot 10^{+58}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;Om \leq -1.7 \cdot 10^{-162}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{1 + 4 \cdot \frac{{\left(\sin kx \cdot \ell\right)}^{2}}{{Om}^{2}}}}\right)}\\ \mathbf{elif}\;Om \leq 8.2 \cdot 10^{-152}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	6.8
Cost	33484

\[\begin{array}{l} t_0 := \sqrt{0.5 \cdot \left(1 + \frac{1}{1 + \frac{2 \cdot {\left(\ell \cdot \sin ky\right)}^{2}}{{Om}^{2}}}\right)}\\ \mathbf{if}\;Om \leq -3.7 \cdot 10^{+197}:\\ \;\;\;\;1\\ \mathbf{elif}\;Om \leq -2 \cdot 10^{+31}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;Om \leq -1.6 \cdot 10^{-162}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{1 + 4 \cdot \frac{{\left(\sin kx \cdot \ell\right)}^{2}}{{Om}^{2}}}}\right)}\\ \mathbf{elif}\;Om \leq 4.6 \cdot 10^{-153}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	6.8
Cost	27216

\[\begin{array}{l} t_0 := \sqrt{0.5 \cdot \left(1 + \frac{1}{1 + \frac{2 \cdot {\left(\ell \cdot \sin ky\right)}^{2}}{{Om}^{2}}}\right)}\\ \mathbf{if}\;Om \leq -3.7 \cdot 10^{+197}:\\ \;\;\;\;1\\ \mathbf{elif}\;Om \leq -5 \cdot 10^{+27}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;Om \leq -3.8 \cdot 10^{-162}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{1}{1 + 2 \cdot \frac{{\left(\ell \cdot \sin kx\right)}^{2}}{{Om}^{2}}}\right)}\\ \mathbf{elif}\;Om \leq 4.6 \cdot 10^{-153}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	7.4
Cost	26952

\[\begin{array}{l} t_0 := \sqrt{0.5 \cdot \left(1 + \frac{1}{1 + 2 \cdot \frac{{\left(\ell \cdot \sin kx\right)}^{2}}{{Om}^{2}}}\right)}\\ \mathbf{if}\;Om \leq -2 \cdot 10^{-162}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;Om \leq 1.45 \cdot 10^{-194}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	14.1
Cost	6728

\[\begin{array}{l} \mathbf{if}\;Om \leq -6.8 \cdot 10^{-135}:\\ \;\;\;\;1\\ \mathbf{elif}\;Om \leq 880000000:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7
Error	28.6
Cost	6464

\[\sqrt{0.5} \]

Reproduce

herbie shell --seed 2023068 
(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))))))))))