Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.3% → 98.6%

Time: 10.8s

Alternatives: 3

Speedup: 1.0×

Specification

Math

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

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))))))))))

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)))))))));
}

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

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)))))))));
}

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)))))))))

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

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

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]

Initial Program: 98.3% accurate, 1.0× speedup

Math

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

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))))))))))

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)))))))));
}

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

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)))))))));
}

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)))))))))

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

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

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]

Alternative 1: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ Om_m = \left|Om\right| \\ ky_m = \left|ky\right| \\ [l_m, Om_m, kx, ky_m] = \mathsf{sort}([l_m, Om_m, kx, ky_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{2 \cdot l\_m}{Om\_m} \leq 2 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{\sqrt{1 + {\left(2 \cdot \frac{l\_m}{Om\_m}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky\_m}^{2}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.25 \cdot \frac{\frac{Om\_m}{ky\_m}}{l\_m}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
Om_m = (fabs.f64 Om)
ky_m = (fabs.f64 ky)
NOTE: l_m, Om_m, kx, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l_m Om_m kx ky_m)
 :precision binary64
 (if (<= (/ (* 2.0 l_m) Om_m) 2e+151)
   (sqrt
    (+
     0.5
     (*
      0.5
      (/
       1.0
       (sqrt
        (+
         1.0
         (*
          (pow (* 2.0 (/ l_m Om_m)) 2.0)
          (+ (pow (sin kx) 2.0) (pow (sin ky_m) 2.0)))))))))
   (sqrt (+ 0.5 (* 0.25 (/ (/ Om_m ky_m) l_m))))))

C

l_m = fabs(l);
Om_m = fabs(Om);
ky_m = fabs(ky);
assert(l_m < Om_m && Om_m < kx && kx < ky_m);
double code(double l_m, double Om_m, double kx, double ky_m) {
	double tmp;
	if (((2.0 * l_m) / Om_m) <= 2e+151) {
		tmp = sqrt((0.5 + (0.5 * (1.0 / sqrt((1.0 + (pow((2.0 * (l_m / Om_m)), 2.0) * (pow(sin(kx), 2.0) + pow(sin(ky_m), 2.0)))))))));
	} else {
		tmp = sqrt((0.5 + (0.25 * ((Om_m / ky_m) / l_m))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
Om_m = abs(om)
ky_m = abs(ky)
NOTE: l_m, Om_m, kx, and ky_m should be sorted in increasing order before calling this function.
real(8) function code(l_m, om_m, kx, ky_m)
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om_m
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky_m
    real(8) :: tmp
    if (((2.0d0 * l_m) / om_m) <= 2d+151) then
        tmp = sqrt((0.5d0 + (0.5d0 * (1.0d0 / sqrt((1.0d0 + (((2.0d0 * (l_m / om_m)) ** 2.0d0) * ((sin(kx) ** 2.0d0) + (sin(ky_m) ** 2.0d0)))))))))
    else
        tmp = sqrt((0.5d0 + (0.25d0 * ((om_m / ky_m) / l_m))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
Om_m = Math.abs(Om);
ky_m = Math.abs(ky);
assert l_m < Om_m && Om_m < kx && kx < ky_m;
public static double code(double l_m, double Om_m, double kx, double ky_m) {
	double tmp;
	if (((2.0 * l_m) / Om_m) <= 2e+151) {
		tmp = Math.sqrt((0.5 + (0.5 * (1.0 / Math.sqrt((1.0 + (Math.pow((2.0 * (l_m / Om_m)), 2.0) * (Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky_m), 2.0)))))))));
	} else {
		tmp = Math.sqrt((0.5 + (0.25 * ((Om_m / ky_m) / l_m))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
Om_m = math.fabs(Om)
ky_m = math.fabs(ky)
[l_m, Om_m, kx, ky_m] = sort([l_m, Om_m, kx, ky_m])
def code(l_m, Om_m, kx, ky_m):
	tmp = 0
	if ((2.0 * l_m) / Om_m) <= 2e+151:
		tmp = math.sqrt((0.5 + (0.5 * (1.0 / math.sqrt((1.0 + (math.pow((2.0 * (l_m / Om_m)), 2.0) * (math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky_m), 2.0)))))))))
	else:
		tmp = math.sqrt((0.5 + (0.25 * ((Om_m / ky_m) / l_m))))
	return tmp

Julia

l_m = abs(l)
Om_m = abs(Om)
ky_m = abs(ky)
l_m, Om_m, kx, ky_m = sort([l_m, Om_m, kx, ky_m])
function code(l_m, Om_m, kx, ky_m)
	tmp = 0.0
	if (Float64(Float64(2.0 * l_m) / Om_m) <= 2e+151)
		tmp = sqrt(Float64(0.5 + Float64(0.5 * Float64(1.0 / sqrt(Float64(1.0 + Float64((Float64(2.0 * Float64(l_m / Om_m)) ^ 2.0) * Float64((sin(kx) ^ 2.0) + (sin(ky_m) ^ 2.0)))))))));
	else
		tmp = sqrt(Float64(0.5 + Float64(0.25 * Float64(Float64(Om_m / ky_m) / l_m))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
Om_m = abs(Om);
ky_m = abs(ky);
l_m, Om_m, kx, ky_m = num2cell(sort([l_m, Om_m, kx, ky_m])){:}
function tmp_2 = code(l_m, Om_m, kx, ky_m)
	tmp = 0.0;
	if (((2.0 * l_m) / Om_m) <= 2e+151)
		tmp = sqrt((0.5 + (0.5 * (1.0 / sqrt((1.0 + (((2.0 * (l_m / Om_m)) ^ 2.0) * ((sin(kx) ^ 2.0) + (sin(ky_m) ^ 2.0)))))))));
	else
		tmp = sqrt((0.5 + (0.25 * ((Om_m / ky_m) / l_m))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
Om_m = N[Abs[Om], $MachinePrecision]
ky_m = N[Abs[ky], $MachinePrecision]
NOTE: l_m, Om_m, kx, and ky_m should be sorted in increasing order before calling this function.
code[l$95$m_, Om$95$m_, kx_, ky$95$m_] := If[LessEqual[N[(N[(2.0 * l$95$m), $MachinePrecision] / Om$95$m), $MachinePrecision], 2e+151], N[Sqrt[N[(0.5 + N[(0.5 * N[(1.0 / N[Sqrt[N[(1.0 + N[(N[Power[N[(2.0 * N[(l$95$m / Om$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 + N[(0.25 * N[(N[(Om$95$m / ky$95$m), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) l) Om) < 2.00000000000000003e151

   1. Initial program 98.7%

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

   3. Simplified 98.7%

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

   if 2.00000000000000003e151 < (/.f64 (*.f64 #s(literal 2 binary64) l) Om)

   1. Initial program 90.2%

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

   3. Simplified 90.2%

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

   5. Taylor expanded in l around inf 90.8%

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

      1. unpow2 90.8%

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

      2. unpow2 90.8%

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

      3. hypot-undefine 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in kx around 0 95.5%

      \[\leadsto \color{blue}{\sqrt{0.5 + 0.25 \cdot \frac{Om}{\ell \cdot \sin ky}}} \]
   9. Step-by-step derivation

      1. +-commutative 95.5%

         \[\leadsto \sqrt{\color{blue}{0.25 \cdot \frac{Om}{\ell \cdot \sin ky} + 0.5}} \]

   10. Simplified 95.5%

       \[\leadsto \color{blue}{\sqrt{0.25 \cdot \frac{Om}{\ell \cdot \sin ky} + 0.5}} \]

   11. Taylor expanded in ky around 0 95.5%

       \[\leadsto \sqrt{\color{blue}{0.25 \cdot \frac{Om}{ky \cdot \ell}} + 0.5} \]
   12. Step-by-step derivation

       1. associate-/r* 95.7%

          \[\leadsto \sqrt{0.25 \cdot \color{blue}{\frac{\frac{Om}{ky}}{\ell}} + 0.5} \]

   13. Simplified 95.7%

       \[\leadsto \sqrt{\color{blue}{0.25 \cdot \frac{\frac{Om}{ky}}{\ell}} + 0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 \cdot \ell}{Om} \leq 2 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{\sqrt{1 + {\left(2 \cdot \frac{\ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.25 \cdot \frac{\frac{Om}{ky}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.5% accurate, 6.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ Om_m = \left|Om\right| \\ ky_m = \left|ky\right| \\ [l_m, Om_m, kx, ky_m] = \mathsf{sort}([l_m, Om_m, kx, ky_m])\\ \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 1.15 \cdot 10^{-37}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
Om_m = (fabs.f64 Om)
ky_m = (fabs.f64 ky)
NOTE: l_m, Om_m, kx, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l_m Om_m kx ky_m)
 :precision binary64
 (if (<= l_m 1.15e-37) 1.0 (sqrt 0.5)))

C

l_m = fabs(l);
Om_m = fabs(Om);
ky_m = fabs(ky);
assert(l_m < Om_m && Om_m < kx && kx < ky_m);
double code(double l_m, double Om_m, double kx, double ky_m) {
	double tmp;
	if (l_m <= 1.15e-37) {
		tmp = 1.0;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

l_m = abs(l)
Om_m = abs(om)
ky_m = abs(ky)
NOTE: l_m, Om_m, kx, and ky_m should be sorted in increasing order before calling this function.
real(8) function code(l_m, om_m, kx, ky_m)
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om_m
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky_m
    real(8) :: tmp
    if (l_m <= 1.15d-37) then
        tmp = 1.0d0
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
Om_m = Math.abs(Om);
ky_m = Math.abs(ky);
assert l_m < Om_m && Om_m < kx && kx < ky_m;
public static double code(double l_m, double Om_m, double kx, double ky_m) {
	double tmp;
	if (l_m <= 1.15e-37) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
Om_m = math.fabs(Om)
ky_m = math.fabs(ky)
[l_m, Om_m, kx, ky_m] = sort([l_m, Om_m, kx, ky_m])
def code(l_m, Om_m, kx, ky_m):
	tmp = 0
	if l_m <= 1.15e-37:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

l_m = abs(l)
Om_m = abs(Om)
ky_m = abs(ky)
l_m, Om_m, kx, ky_m = sort([l_m, Om_m, kx, ky_m])
function code(l_m, Om_m, kx, ky_m)
	tmp = 0.0
	if (l_m <= 1.15e-37)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

l_m = abs(l);
Om_m = abs(Om);
ky_m = abs(ky);
l_m, Om_m, kx, ky_m = num2cell(sort([l_m, Om_m, kx, ky_m])){:}
function tmp_2 = code(l_m, Om_m, kx, ky_m)
	tmp = 0.0;
	if (l_m <= 1.15e-37)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
Om_m = N[Abs[Om], $MachinePrecision]
ky_m = N[Abs[ky], $MachinePrecision]
NOTE: l_m, Om_m, kx, and ky_m should be sorted in increasing order before calling this function.
code[l$95$m_, Om$95$m_, kx_, ky$95$m_] := If[LessEqual[l$95$m, 1.15e-37], 1.0, N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.15e-37

   1. Initial program 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in l around 0 71.2%

      \[\leadsto \sqrt{0.5 + 0.5 \cdot \frac{1}{\color{blue}{1}}} \]
   6. Step-by-step derivation

      1. metadata-eval 71.2%

         \[\leadsto \sqrt{0.5 + 0.5 \cdot \color{blue}{1}} \]

      2. metadata-eval 71.2%

         \[\leadsto \sqrt{0.5 + \color{blue}{0.5}} \]

      3. metadata-eval 71.2%

         \[\leadsto \sqrt{\color{blue}{1}} \]

      4. metadata-eval 71.2%

         \[\leadsto \color{blue}{1} \]

   7. Applied egg-rr 71.2%

      \[\leadsto \color{blue}{1} \]

   if 1.15e-37 < l

   1. Initial program 95.7%

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

   3. Simplified 95.7%

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

   5. Taylor expanded in l around inf 70.2%

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

      1. unpow2 70.2%

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

      2. unpow2 70.2%

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

      3. hypot-undefine 73.0%

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

   7. Simplified 73.0%

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

   8. Taylor expanded in l around inf 77.5%

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

Alternative 3: 62.3% accurate, 722.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ Om_m = \left|Om\right| \\ ky_m = \left|ky\right| \\ [l_m, Om_m, kx, ky_m] = \mathsf{sort}([l_m, Om_m, kx, ky_m])\\ \\ 1 \end{array} \]

FPCore

l_m = (fabs.f64 l)
Om_m = (fabs.f64 Om)
ky_m = (fabs.f64 ky)
NOTE: l_m, Om_m, kx, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l_m Om_m kx ky_m) :precision binary64 1.0)

C

l_m = fabs(l);
Om_m = fabs(Om);
ky_m = fabs(ky);
assert(l_m < Om_m && Om_m < kx && kx < ky_m);
double code(double l_m, double Om_m, double kx, double ky_m) {
	return 1.0;
}

Fortran

l_m = abs(l)
Om_m = abs(om)
ky_m = abs(ky)
NOTE: l_m, Om_m, kx, and ky_m should be sorted in increasing order before calling this function.
real(8) function code(l_m, om_m, kx, ky_m)
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om_m
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky_m
    code = 1.0d0
end function

Java

l_m = Math.abs(l);
Om_m = Math.abs(Om);
ky_m = Math.abs(ky);
assert l_m < Om_m && Om_m < kx && kx < ky_m;
public static double code(double l_m, double Om_m, double kx, double ky_m) {
	return 1.0;
}

Python

l_m = math.fabs(l)
Om_m = math.fabs(Om)
ky_m = math.fabs(ky)
[l_m, Om_m, kx, ky_m] = sort([l_m, Om_m, kx, ky_m])
def code(l_m, Om_m, kx, ky_m):
	return 1.0

Julia

l_m = abs(l)
Om_m = abs(Om)
ky_m = abs(ky)
l_m, Om_m, kx, ky_m = sort([l_m, Om_m, kx, ky_m])
function code(l_m, Om_m, kx, ky_m)
	return 1.0
end

MATLAB

l_m = abs(l);
Om_m = abs(Om);
ky_m = abs(ky);
l_m, Om_m, kx, ky_m = num2cell(sort([l_m, Om_m, kx, ky_m])){:}
function tmp = code(l_m, Om_m, kx, ky_m)
	tmp = 1.0;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
Om_m = N[Abs[Om], $MachinePrecision]
ky_m = N[Abs[ky], $MachinePrecision]
NOTE: l_m, Om_m, kx, and ky_m should be sorted in increasing order before calling this function.
code[l$95$m_, Om$95$m_, kx_, ky$95$m_] := 1.0

Derivation

1. Initial program 97.6%

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

3. Simplified 97.6%

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

5. Taylor expanded in l around 0 63.1%

   \[\leadsto \sqrt{0.5 + 0.5 \cdot \frac{1}{\color{blue}{1}}} \]
6. Step-by-step derivation

   1. metadata-eval 63.1%

      \[\leadsto \sqrt{0.5 + 0.5 \cdot \color{blue}{1}} \]

   2. metadata-eval 63.1%

      \[\leadsto \sqrt{0.5 + \color{blue}{0.5}} \]

   3. metadata-eval 63.1%

      \[\leadsto \sqrt{\color{blue}{1}} \]

   4. metadata-eval 63.1%

      \[\leadsto \color{blue}{1} \]

7. Applied egg-rr 63.1%

   \[\leadsto \color{blue}{1} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024144 
(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))))))))))