Toniolo and Linder, Equation (3a)

Average Error: 1.0 → 0.3

Time: 17.2s

Precision: binary64

Cost: 52932

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;{\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \leq 10^{+260}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{\sqrt{1 + {\left(\frac{\ell + \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

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
 (if (<= (pow (/ (* 2.0 l) Om) 2.0) 1e+260)
   (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)))))))))
   (sqrt 0.5)))

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 tmp;
	if (pow(((2.0 * l) / Om), 2.0) <= 1e+260) {
		tmp = 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)))))))));
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

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) :: tmp
    if ((((2.0d0 * l) / om) ** 2.0d0) <= 1d+260) then
        tmp = sqrt((0.5d0 + (0.5d0 * (1.0d0 / sqrt((1.0d0 + ((((l + l) / om) ** 2.0d0) * ((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))))))))
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
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 tmp;
	if (Math.pow(((2.0 * l) / Om), 2.0) <= 1e+260) {
		tmp = 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)))))))));
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

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):
	tmp = 0
	if math.pow(((2.0 * l) / Om), 2.0) <= 1e+260:
		tmp = 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)))))))))
	else:
		tmp = math.sqrt(0.5)
	return tmp

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)
	tmp = 0.0
	if ((Float64(Float64(2.0 * l) / Om) ^ 2.0) <= 1e+260)
		tmp = 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)))))))));
	else
		tmp = sqrt(0.5);
	end
	return tmp
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_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if ((((2.0 * l) / Om) ^ 2.0) <= 1e+260)
		tmp = sqrt((0.5 + (0.5 * (1.0 / sqrt((1.0 + ((((l + l) / Om) ^ 2.0) * ((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))))))));
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
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_] := If[LessEqual[N[Power[N[(N[(2.0 * l), $MachinePrecision] / Om), $MachinePrecision], 2.0], $MachinePrecision], 1e+260], 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], N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (/.f64 (*.f64 2 l) Om) 2) < 1.00000000000000007e260

   1. Initial program 0.0

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

   2. Simplified 0.0

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

      [Start] 0.0	\[ \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-37 [=>] 0.0	\[ \sqrt{\color{blue}{1 \cdot \frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}}} \]
      metadata-eval [=>] 0.0	\[ \sqrt{1 \cdot \color{blue}{0.5} + \frac{1}{2} \cdot \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}} \]
      metadata-eval [=>] 0.0	\[ \sqrt{\color{blue}{0.5} + \frac{1}{2} \cdot \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}} \]
      metadata-eval [=>] 0.0	\[ \sqrt{0.5 + \color{blue}{0.5} \cdot \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.0	\[ \sqrt{0.5 + 0.5 \cdot \frac{1}{\sqrt{1 + {\left(\frac{\color{blue}{\ell \cdot 2}}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}} \]
      metadata-eval [<=] 0.0	\[ \sqrt{0.5 + 0.5 \cdot \frac{1}{\sqrt{1 + {\left(\frac{\ell \cdot \color{blue}{\left(1 + 1\right)}}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}} \]
      rational_best_oopsla_all_46_json_45_simplify-23 [<=] 0.0	\[ \sqrt{0.5 + 0.5 \cdot \frac{1}{\sqrt{1 + {\left(\frac{\color{blue}{1 \cdot \ell + \ell \cdot 1}}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [<=] 0.0	\[ \sqrt{0.5 + 0.5 \cdot \frac{1}{\sqrt{1 + {\left(\frac{\color{blue}{\ell \cdot 1} + \ell \cdot 1}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}} \]
      rational_best_oopsla_all_46_json_45_simplify-52 [=>] 0.0	\[ \sqrt{0.5 + 0.5 \cdot \frac{1}{\sqrt{1 + {\left(\frac{\color{blue}{\ell} + \ell \cdot 1}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}} \]
      rational_best_oopsla_all_46_json_45_simplify-52 [=>] 0.0	\[ \sqrt{0.5 + 0.5 \cdot \frac{1}{\sqrt{1 + {\left(\frac{\ell + \color{blue}{\ell}}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}} \]

   if 1.00000000000000007e260 < (pow.f64 (/.f64 (*.f64 2 l) Om) 2)

   1. Initial program 3.2

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

   2. Simplified 3.2

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

      [Start] 3.2	\[ \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-37 [=>] 3.2	\[ \sqrt{\color{blue}{1 \cdot \frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}}} \]
      metadata-eval [=>] 3.2	\[ \sqrt{1 \cdot \color{blue}{0.5} + \frac{1}{2} \cdot \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}} \]
      metadata-eval [=>] 3.2	\[ \sqrt{\color{blue}{0.5} + \frac{1}{2} \cdot \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}} \]
      metadata-eval [=>] 3.2	\[ \sqrt{0.5 + \color{blue}{0.5} \cdot \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 3.2	\[ \sqrt{0.5 + 0.5 \cdot \frac{1}{\sqrt{1 + {\left(\frac{\color{blue}{\ell \cdot 2}}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}} \]
      metadata-eval [<=] 3.2	\[ \sqrt{0.5 + 0.5 \cdot \frac{1}{\sqrt{1 + {\left(\frac{\ell \cdot \color{blue}{\left(1 + 1\right)}}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}} \]
      rational_best_oopsla_all_46_json_45_simplify-23 [<=] 3.2	\[ \sqrt{0.5 + 0.5 \cdot \frac{1}{\sqrt{1 + {\left(\frac{\color{blue}{1 \cdot \ell + \ell \cdot 1}}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [<=] 3.2	\[ \sqrt{0.5 + 0.5 \cdot \frac{1}{\sqrt{1 + {\left(\frac{\color{blue}{\ell \cdot 1} + \ell \cdot 1}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}} \]
      rational_best_oopsla_all_46_json_45_simplify-52 [=>] 3.2	\[ \sqrt{0.5 + 0.5 \cdot \frac{1}{\sqrt{1 + {\left(\frac{\color{blue}{\ell} + \ell \cdot 1}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}} \]
      rational_best_oopsla_all_46_json_45_simplify-52 [=>] 3.2	\[ \sqrt{0.5 + 0.5 \cdot \frac{1}{\sqrt{1 + {\left(\frac{\ell + \color{blue}{\ell}}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}} \]

   3. Taylor expanded in l around inf 3.6

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

   4. Taylor expanded in l around inf 0.9

      \[\leadsto \color{blue}{\sqrt{0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \leq 10^{+260}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{\sqrt{1 + {\left(\frac{\ell + \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternatives

Alternative 1
Error	8.1
Cost	40080

\[\begin{array}{l} t_0 := {\sin ky}^{2}\\ \mathbf{if}\;\ell \leq -5.5 \cdot 10^{+150}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-112}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{1 + 2 \cdot \frac{{\ell}^{2} \cdot t_0}{{Om}^{2}}}}\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{-181}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 3.7 \cdot 10^{+144}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{\sqrt{1 + 4 \cdot \frac{t_0 \cdot {\ell}^{2}}{{Om}^{2}}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 2
Error	7.3
Cost	39812

\[\begin{array}{l} \mathbf{if}\;ky \leq 6.5 \cdot 10^{+61}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{\sqrt{1 + {\left(\frac{\ell + \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {ky}^{2}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{\sqrt{1 + 4 \cdot \frac{{\sin ky}^{2} \cdot {\ell}^{2}}{{Om}^{2}}}}}\\ \end{array} \]

Alternative 3
Error	8.2
Cost	33680

\[\begin{array}{l} t_0 := \sqrt{0.5 + 0.5 \cdot \frac{1}{1 + 2 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}}}\\ \mathbf{if}\;\ell \leq -1.55 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-119}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{-181}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 4.9 \cdot 10^{+144}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 4
Error	14.0
Cost	6728

\[\begin{array}{l} \mathbf{if}\;Om \leq -2 \cdot 10^{-40}:\\ \;\;\;\;1\\ \mathbf{elif}\;Om \leq 150000:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5
Error	27.9
Cost	6464

\[\sqrt{0.5} \]

Reproduce

herbie shell --seed 2023090 
(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))))))))))