Toniolo and Linder, Equation (3a)

Average Error: 0.9 → 0.0

Time: 42.0s

Precision: binary64

Cost: 40128

Math

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

↓

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

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
  (*
   (/ 1.0 2.0)
   (+
    1.0
    (/
     1.0
     (sqrt
      (+
       1.0
       (+
        (pow (* l (* (/ 2.0 Om) (sin ky))) 2.0)
        (pow (* l (* (/ 2.0 Om) (sin kx))) 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(((1.0 / 2.0) * (1.0 + (1.0 / sqrt((1.0 + (pow((l * ((2.0 / Om) * sin(ky))), 2.0) + pow((l * ((2.0 / Om) * sin(kx))), 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(((1.0d0 / 2.0d0) * (1.0d0 + (1.0d0 / sqrt((1.0d0 + (((l * ((2.0d0 / om) * sin(ky))) ** 2.0d0) + ((l * ((2.0d0 / om) * sin(kx))) ** 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(((1.0 / 2.0) * (1.0 + (1.0 / Math.sqrt((1.0 + (Math.pow((l * ((2.0 / Om) * Math.sin(ky))), 2.0) + Math.pow((l * ((2.0 / Om) * Math.sin(kx))), 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(((1.0 / 2.0) * (1.0 + (1.0 / math.sqrt((1.0 + (math.pow((l * ((2.0 / Om) * math.sin(ky))), 2.0) + math.pow((l * ((2.0 / Om) * math.sin(kx))), 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(Float64(1.0 / 2.0) * Float64(1.0 + Float64(1.0 / sqrt(Float64(1.0 + Float64((Float64(l * Float64(Float64(2.0 / Om) * sin(ky))) ^ 2.0) + (Float64(l * Float64(Float64(2.0 / Om) * sin(kx))) ^ 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(((1.0 / 2.0) * (1.0 + (1.0 / sqrt((1.0 + (((l * ((2.0 / Om) * sin(ky))) ^ 2.0) + ((l * ((2.0 / Om) * sin(kx))) ^ 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[(N[(1.0 / 2.0), $MachinePrecision] * N[(1.0 + N[(1.0 / N[Sqrt[N[(1.0 + N[(N[Power[N[(l * N[(N[(2.0 / Om), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(l * N[(N[(2.0 / Om), $MachinePrecision] * N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $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. Applied egg-rr 0.0

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

Alternatives

Alternative 1
Error	1.5
Cost	40068

\[\begin{array}{l} t_0 := \frac{\ell \cdot 2}{Om}\\ \mathbf{if}\;\sin kx \leq -0.1:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{\frac{\left(\left(2 - \cos \left(kx + kx\right)\right) - \cos \left(ky + ky\right)\right) \cdot {t_0}^{2} + 2}{2}}}}\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-180}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(\left(\sin ky \cdot \frac{\ell}{Om}\right) \cdot 2, 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(\sin kx \cdot t_0, 1\right)}}\\ \end{array} \]

Alternative 2
Error	0.2
Cost	34504

\[\begin{array}{l} t_0 := \frac{2 \cdot \ell}{Om}\\ t_1 := \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + \left({\left(\frac{2 \cdot \left(\ell \cdot ky\right)}{Om}\right)}^{2} + {\left(\ell \cdot \left(\frac{2}{Om} \cdot \sin kx\right)\right)}^{2}\right)}}\right)}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+94}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 500000000:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{\frac{\left(\left(2 - \cos \left(kx + kx\right)\right) - \cos \left(ky + ky\right)\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} + 2}{2}}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	3.3
Cost	33032

\[\begin{array}{l} t_0 := \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(\sin kx \cdot \frac{\ell \cdot 2}{Om}, 1\right)}}\\ \mathbf{if}\;\sin kx \leq -1 \cdot 10^{-151}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-255}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{2 \cdot \left(0.5 + {\left(\frac{\ell \cdot ky}{Om}\right)}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	1.5
Cost	20232

\[\begin{array}{l} t_0 := \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(\sin kx \cdot \frac{\ell \cdot 2}{Om}, 1\right)}}\\ \mathbf{if}\;kx \leq -2:\\ \;\;\;\;t_0\\ \mathbf{elif}\;kx \leq 1.46 \cdot 10^{-179}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(\left(\sin ky \cdot \frac{\ell}{Om}\right) \cdot 2, 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	8.0
Cost	14152

\[\begin{array}{l} t_0 := \sqrt{0.5 + \frac{0.5}{\mathsf{fma}\left(2, kx \cdot \left(\left(kx \cdot \ell\right) \cdot \frac{\frac{\ell}{Om}}{Om}\right), 1\right)}}\\ \mathbf{if}\;ky \leq -1 \cdot 10^{+204}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;ky \leq -5.5 \cdot 10^{-180}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{2 \cdot \left(0.5 + {\left(\frac{\ell \cdot ky}{Om}\right)}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	8.2
Cost	13960

\[\begin{array}{l} t_0 := \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(\frac{2 \cdot \left(\ell \cdot kx\right)}{Om}, 1\right)}}\\ \mathbf{if}\;ky \leq -6.2 \cdot 10^{+202}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;ky \leq -1.25 \cdot 10^{-179}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{2 \cdot \left(0.5 + {\left(\frac{\ell \cdot ky}{Om}\right)}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	9.3
Cost	13832

\[\begin{array}{l} t_0 := \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(\frac{2 \cdot \left(\ell \cdot kx\right)}{Om}, 1\right)}}\\ \mathbf{if}\;ky \leq -8 \cdot 10^{+82}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;ky \leq -8.8 \cdot 10^{-44}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{1 + \frac{\left(\left(\ell \cdot \ell\right) \cdot \left(ky \cdot ky\right)\right) \cdot 2}{Om \cdot Om}}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	16.4
Cost	7880

\[\begin{array}{l} \mathbf{if}\;Om \leq -1 \cdot 10^{+66}:\\ \;\;\;\;\sqrt{1}\\ \mathbf{elif}\;Om \leq 4.6 \cdot 10^{+52}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{1 + \frac{\left(\left(\ell \cdot \ell\right) \cdot \left(ky \cdot ky\right)\right) \cdot 2}{Om \cdot Om}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1}\\ \end{array} \]

Alternative 9
Error	24.2
Cost	6464

\[\sqrt{1} \]

Alternative 10
Error	42.8
Cost	960

\[1 + \frac{\left(\left(\ell \cdot \ell\right) \cdot \left(kx \cdot kx\right)\right) \cdot -0.5}{Om \cdot Om} \]

Reproduce

herbie shell --seed 2023010 
(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))))))))))