Toniolo and Linder, Equation (3a)

Average Error: 0.9 → 0.9

Time: 21.0s

Precision: binary64

Cost: 46080

Math

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

↓

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

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
    (/
     1.0
     (sqrt
      (+
       1.0
       (*
        (pow (/ (+ l l) 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 + (0.5 * (1.0 / sqrt((1.0 + (pow(((l + l) / 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 + (0.5d0 * (1.0d0 / sqrt((1.0d0 + ((((l + l) / 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 + (0.5 * (1.0 / Math.sqrt((1.0 + (Math.pow(((l + l) / 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 + (0.5 * (1.0 / math.sqrt((1.0 + (math.pow(((l + l) / 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(0.5 * Float64(1.0 / sqrt(Float64(1.0 + Float64((Float64(Float64(l + l) / 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 + (0.5 * (1.0 / sqrt((1.0 + ((((l + l) / 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[(0.5 * N[(1.0 / N[Sqrt[N[(1.0 + N[(N[Power[N[(N[(l + 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]

Derivation

1. Initial program 0.9

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

2. Simplified 0.9

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

   [Start] 0.9	\[ \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
   rational_best_45_simplify-83 [=>] 0.9	\[ \sqrt{\color{blue}{1 \cdot \frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}}} \]
   metadata-eval [=>] 0.9	\[ \sqrt{1 \cdot \color{blue}{0.5} + \frac{1}{2} \cdot \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}} \]
   metadata-eval [=>] 0.9	\[ \sqrt{\color{blue}{0.5} + \frac{1}{2} \cdot \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}} \]
   metadata-eval [=>] 0.9	\[ \sqrt{0.5 + \color{blue}{0.5} \cdot \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}} \]
   rational_best_45_simplify-91 [=>] 0.9	\[ \sqrt{0.5 + 0.5 \cdot \frac{1}{\sqrt{1 + {\left(\frac{\color{blue}{\ell \cdot 2}}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}} \]
   metadata-eval [<=] 0.9	\[ \sqrt{0.5 + 0.5 \cdot \frac{1}{\sqrt{1 + {\left(\frac{\ell \cdot \color{blue}{\left(1 + 1\right)}}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}} \]
   rational_best_45_simplify-71 [<=] 0.9	\[ \sqrt{0.5 + 0.5 \cdot \frac{1}{\sqrt{1 + {\left(\frac{\color{blue}{1 \cdot \ell + \ell \cdot 1}}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}} \]
   rational_best_45_simplify-91 [<=] 0.9	\[ \sqrt{0.5 + 0.5 \cdot \frac{1}{\sqrt{1 + {\left(\frac{\color{blue}{\ell \cdot 1} + \ell \cdot 1}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}} \]
   rational_best_45_simplify-8 [=>] 0.9	\[ \sqrt{0.5 + 0.5 \cdot \frac{1}{\sqrt{1 + {\left(\frac{\color{blue}{\ell} + \ell \cdot 1}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}} \]
   rational_best_45_simplify-8 [=>] 0.9	\[ \sqrt{0.5 + 0.5 \cdot \frac{1}{\sqrt{1 + {\left(\frac{\ell + \color{blue}{\ell}}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}} \]

3. Final simplification 0.9

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

Alternatives

Alternative 1
Error	8.0
Cost	40080

\[\begin{array}{l} t_0 := \sqrt{0.5 + 0.5 \cdot \frac{1}{\sqrt{1 + 4 \cdot \frac{{\sin kx}^{2} \cdot {\ell}^{2}}{{Om}^{2}}}}}\\ \mathbf{if}\;\ell \leq -1.7 \cdot 10^{+127}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;\ell \leq -3.4 \cdot 10^{-117}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{-170}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{+152}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 2
Error	13.4
Cost	33548

\[\begin{array}{l} t_0 := \sin kx \cdot \ell\\ \mathbf{if}\;\ell \leq -1.2 \cdot 10^{+49}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{-170}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{+36}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{\sqrt{1 + 4 \cdot \frac{{\ell}^{2} \cdot {kx}^{2}}{{Om}^{2}}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{-0.25 \cdot \frac{Om}{t_0} + -2 \cdot \frac{t_0}{Om}}}\\ \end{array} \]

Alternative 3
Error	13.4
Cost	27148

\[\begin{array}{l} t_0 := \sin kx \cdot \ell\\ \mathbf{if}\;\ell \leq -3 \cdot 10^{+48}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{-170}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 2.05 \cdot 10^{+36}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{1 + 2 \cdot \frac{{\ell}^{2} \cdot {kx}^{2}}{{Om}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{-0.25 \cdot \frac{Om}{t_0} + -2 \cdot \frac{t_0}{Om}}}\\ \end{array} \]

Alternative 4
Error	15.1
Cost	20808

\[\begin{array}{l} t_0 := \sin kx \cdot \ell\\ \mathbf{if}\;\ell \leq -1.02 \cdot 10^{+48}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;\ell \leq 1.02 \cdot 10^{-137}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{-0.25 \cdot \frac{Om}{t_0} + -2 \cdot \frac{t_0}{Om}}}\\ \end{array} \]

Alternative 5
Error	14.9
Cost	6728

\[\begin{array}{l} \mathbf{if}\;\ell \leq -4.9 \cdot 10^{+49}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;\ell \leq 1.02 \cdot 10^{-137}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 6
Error	27.9
Cost	6464

\[\sqrt{0.5} \]

Reproduce

herbie shell --seed 2023098 
(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))))))))))