Toniolo and Linder, Equation (3a)

Average Error: 1.0 → 0.0

Time: 19.0s

Precision: binary64

Cost: 40256

\[ \begin{array}{c}[kx, ky] = \mathsf{sort}([kx, ky])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} t_0 := 2 \cdot \frac{\ell}{Om}\\ \sqrt{0.5 + 0.5 \cdot \frac{1}{\sqrt{1 + \left({\left(\sin kx \cdot t_0\right)}^{2} + \left(1 + \left({\left(t_0 \cdot \sin ky\right)}^{2} + -1\right)\right)\right)}}} \end{array} \]

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
 (let* ((t_0 (* 2.0 (/ l Om))))
   (sqrt
    (+
     0.5
     (*
      0.5
      (/
       1.0
       (sqrt
        (+
         1.0
         (+
          (pow (* (sin kx) t_0) 2.0)
          (+ 1.0 (+ (pow (* t_0 (sin ky)) 2.0) -1.0)))))))))))

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) {
	double t_0 = 2.0 * (l / Om);
	return sqrt((0.5 + (0.5 * (1.0 / sqrt((1.0 + (pow((sin(kx) * t_0), 2.0) + (1.0 + (pow((t_0 * sin(ky)), 2.0) + -1.0)))))))));
}

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
    real(8) :: t_0
    t_0 = 2.0d0 * (l / om)
    code = sqrt((0.5d0 + (0.5d0 * (1.0d0 / sqrt((1.0d0 + (((sin(kx) * t_0) ** 2.0d0) + (1.0d0 + (((t_0 * sin(ky)) ** 2.0d0) + (-1.0d0))))))))))
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) {
	double t_0 = 2.0 * (l / Om);
	return Math.sqrt((0.5 + (0.5 * (1.0 / Math.sqrt((1.0 + (Math.pow((Math.sin(kx) * t_0), 2.0) + (1.0 + (Math.pow((t_0 * Math.sin(ky)), 2.0) + -1.0)))))))));
}

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):
	t_0 = 2.0 * (l / Om)
	return math.sqrt((0.5 + (0.5 * (1.0 / math.sqrt((1.0 + (math.pow((math.sin(kx) * t_0), 2.0) + (1.0 + (math.pow((t_0 * math.sin(ky)), 2.0) + -1.0)))))))))

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)
	t_0 = Float64(2.0 * Float64(l / Om))
	return sqrt(Float64(0.5 + Float64(0.5 * Float64(1.0 / sqrt(Float64(1.0 + Float64((Float64(sin(kx) * t_0) ^ 2.0) + Float64(1.0 + Float64((Float64(t_0 * sin(ky)) ^ 2.0) + -1.0)))))))))
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)
	t_0 = 2.0 * (l / Om);
	tmp = sqrt((0.5 + (0.5 * (1.0 / sqrt((1.0 + (((sin(kx) * t_0) ^ 2.0) + (1.0 + (((t_0 * sin(ky)) ^ 2.0) + -1.0)))))))));
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_] := Block[{t$95$0 = N[(2.0 * N[(l / Om), $MachinePrecision]), $MachinePrecision]}, N[Sqrt[N[(0.5 + N[(0.5 * N[(1.0 / N[Sqrt[N[(1.0 + N[(N[Power[N[(N[Sin[kx], $MachinePrecision] * t$95$0), $MachinePrecision], 2.0], $MachinePrecision] + N[(1.0 + N[(N[Power[N[(t$95$0 * N[Sin[ky], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $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

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

4. Simplified 0.0

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

   [Start] 0.0	\[ \sqrt{0.5 + 0.5 \cdot \frac{1}{\sqrt{1 + \left({\left(\frac{2 \cdot \ell}{Om} \cdot \sin kx\right)}^{2} + \left(\left(1 + {\left(\sin ky \cdot \frac{2 \cdot \ell}{Om}\right)}^{2}\right) - 1\right)\right)}}} \]
   *-commutative [=>] 0.0	\[ \sqrt{0.5 + 0.5 \cdot \frac{1}{\sqrt{1 + \left({\color{blue}{\left(\sin kx \cdot \frac{2 \cdot \ell}{Om}\right)}}^{2} + \left(\left(1 + {\left(\sin ky \cdot \frac{2 \cdot \ell}{Om}\right)}^{2}\right) - 1\right)\right)}}} \]
   associate-*r/ [<=] 0.0	\[ \sqrt{0.5 + 0.5 \cdot \frac{1}{\sqrt{1 + \left({\left(\sin kx \cdot \color{blue}{\left(2 \cdot \frac{\ell}{Om}\right)}\right)}^{2} + \left(\left(1 + {\left(\sin ky \cdot \frac{2 \cdot \ell}{Om}\right)}^{2}\right) - 1\right)\right)}}} \]
   associate--l+ [=>] 0.0	\[ \sqrt{0.5 + 0.5 \cdot \frac{1}{\sqrt{1 + \left({\left(\sin kx \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}^{2} + \color{blue}{\left(1 + \left({\left(\sin ky \cdot \frac{2 \cdot \ell}{Om}\right)}^{2} - 1\right)\right)}\right)}}} \]
   sub-neg [=>] 0.0	\[ \sqrt{0.5 + 0.5 \cdot \frac{1}{\sqrt{1 + \left({\left(\sin kx \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}^{2} + \left(1 + \color{blue}{\left({\left(\sin ky \cdot \frac{2 \cdot \ell}{Om}\right)}^{2} + \left(-1\right)\right)}\right)\right)}}} \]
   associate-*r/ [<=] 0.0	\[ \sqrt{0.5 + 0.5 \cdot \frac{1}{\sqrt{1 + \left({\left(\sin kx \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}^{2} + \left(1 + \left({\left(\sin ky \cdot \color{blue}{\left(2 \cdot \frac{\ell}{Om}\right)}\right)}^{2} + \left(-1\right)\right)\right)\right)}}} \]
   metadata-eval [=>] 0.0	\[ \sqrt{0.5 + 0.5 \cdot \frac{1}{\sqrt{1 + \left({\left(\sin kx \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}^{2} + \left(1 + \left({\left(\sin ky \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}^{2} + \color{blue}{-1}\right)\right)\right)}}} \]

5. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.9
Cost	33161

\[\begin{array}{l} \mathbf{if}\;\sin kx \leq -2 \cdot 10^{-139} \lor \neg \left(\sin kx \leq 1.25 \cdot 10^{-108}\right):\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, \sin kx \cdot \frac{2 \cdot \ell}{Om}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \sin ky\right)}}\\ \end{array} \]

Alternative 2
Error	0.9
Cost	33160

\[\begin{array}{l} \mathbf{if}\;\sin kx \leq -1 \cdot 10^{-137}:\\ \;\;\;\;\sqrt[3]{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{\ell}{0.5} \cdot \frac{\sin kx}{Om}\right)}\right)}^{1.5}}\\ \mathbf{elif}\;\sin kx \leq 4 \cdot 10^{-109}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \sin ky\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, \sin kx \cdot \frac{2 \cdot \ell}{Om}\right)}}\\ \end{array} \]

Alternative 3
Error	0.0
Cost	32832

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

Alternative 4
Error	4.0
Cost	19968

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

Alternative 5
Error	15.7
Cost	7500

\[\begin{array}{l} \mathbf{if}\;\ell \leq -7.8 \cdot 10^{+68}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;\ell \leq 9.2 \cdot 10^{+55}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 8.8 \cdot 10^{+148}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{Om \cdot -0.5}{\ell \cdot ky}}\\ \mathbf{elif}\;\ell \leq 1.95 \cdot 10^{+217}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 6
Error	15.2
Cost	6992

\[\begin{array}{l} \mathbf{if}\;\ell \leq -5.3 \cdot 10^{+70}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;\ell \leq 1.65 \cdot 10^{+54}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 3.4 \cdot 10^{+149}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;\ell \leq 9.5 \cdot 10^{+216}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 7
Error	24.2
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022364 
(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))))))))))