Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.5% → 98.4%

Time: 9.5s

Alternatives: 5

Speedup: 0.8×

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: 98.5% 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: 98.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} ky_m = \left|ky\right| \\ kx_m = \left|kx\right| \\ Om_m = \left|Om\right| \\ l_m = \left|\ell\right| \\ [l_m, Om_m, kx_m, ky_m] = \mathsf{sort}([l_m, Om_m, kx_m, ky_m])\\ \\ \begin{array}{l} t_0 := \frac{2 \cdot l\_m}{Om\_m}\\ \mathbf{if}\;{t\_0}^{2} \cdot \left({\sin kx\_m}^{2} + {\sin ky\_m}^{2}\right) \leq 1:\\ \;\;\;\;\sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.5}{\sin ky\_m \cdot t\_0} + 0.5}\\ \end{array} \end{array} \]

FPCore

ky_m = (fabs.f64 ky)
kx_m = (fabs.f64 kx)
Om_m = (fabs.f64 Om)
l_m = (fabs.f64 l)
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l_m Om_m kx_m ky_m)
 :precision binary64
 (let* ((t_0 (/ (* 2.0 l_m) Om_m)))
   (if (<= (* (pow t_0 2.0) (+ (pow (sin kx_m) 2.0) (pow (sin ky_m) 2.0))) 1.0)
     (sqrt 1.0)
     (sqrt (+ (/ 0.5 (* (sin ky_m) t_0)) 0.5)))))

C

ky_m = fabs(ky);
kx_m = fabs(kx);
Om_m = fabs(Om);
l_m = fabs(l);
assert(l_m < Om_m && Om_m < kx_m && kx_m < ky_m);
double code(double l_m, double Om_m, double kx_m, double ky_m) {
	double t_0 = (2.0 * l_m) / Om_m;
	double tmp;
	if ((pow(t_0, 2.0) * (pow(sin(kx_m), 2.0) + pow(sin(ky_m), 2.0))) <= 1.0) {
		tmp = sqrt(1.0);
	} else {
		tmp = sqrt(((0.5 / (sin(ky_m) * t_0)) + 0.5));
	}
	return tmp;
}

Fortran

ky_m = abs(ky)
kx_m = abs(kx)
Om_m = abs(om)
l_m = abs(l)
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
real(8) function code(l_m, om_m, kx_m, ky_m)
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om_m
    real(8), intent (in) :: kx_m
    real(8), intent (in) :: ky_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (2.0d0 * l_m) / om_m
    if (((t_0 ** 2.0d0) * ((sin(kx_m) ** 2.0d0) + (sin(ky_m) ** 2.0d0))) <= 1.0d0) then
        tmp = sqrt(1.0d0)
    else
        tmp = sqrt(((0.5d0 / (sin(ky_m) * t_0)) + 0.5d0))
    end if
    code = tmp
end function

Java

ky_m = Math.abs(ky);
kx_m = Math.abs(kx);
Om_m = Math.abs(Om);
l_m = Math.abs(l);
assert l_m < Om_m && Om_m < kx_m && kx_m < ky_m;
public static double code(double l_m, double Om_m, double kx_m, double ky_m) {
	double t_0 = (2.0 * l_m) / Om_m;
	double tmp;
	if ((Math.pow(t_0, 2.0) * (Math.pow(Math.sin(kx_m), 2.0) + Math.pow(Math.sin(ky_m), 2.0))) <= 1.0) {
		tmp = Math.sqrt(1.0);
	} else {
		tmp = Math.sqrt(((0.5 / (Math.sin(ky_m) * t_0)) + 0.5));
	}
	return tmp;
}

Python

ky_m = math.fabs(ky)
kx_m = math.fabs(kx)
Om_m = math.fabs(Om)
l_m = math.fabs(l)
[l_m, Om_m, kx_m, ky_m] = sort([l_m, Om_m, kx_m, ky_m])
def code(l_m, Om_m, kx_m, ky_m):
	t_0 = (2.0 * l_m) / Om_m
	tmp = 0
	if (math.pow(t_0, 2.0) * (math.pow(math.sin(kx_m), 2.0) + math.pow(math.sin(ky_m), 2.0))) <= 1.0:
		tmp = math.sqrt(1.0)
	else:
		tmp = math.sqrt(((0.5 / (math.sin(ky_m) * t_0)) + 0.5))
	return tmp

Julia

ky_m = abs(ky)
kx_m = abs(kx)
Om_m = abs(Om)
l_m = abs(l)
l_m, Om_m, kx_m, ky_m = sort([l_m, Om_m, kx_m, ky_m])
function code(l_m, Om_m, kx_m, ky_m)
	t_0 = Float64(Float64(2.0 * l_m) / Om_m)
	tmp = 0.0
	if (Float64((t_0 ^ 2.0) * Float64((sin(kx_m) ^ 2.0) + (sin(ky_m) ^ 2.0))) <= 1.0)
		tmp = sqrt(1.0);
	else
		tmp = sqrt(Float64(Float64(0.5 / Float64(sin(ky_m) * t_0)) + 0.5));
	end
	return tmp
end

MATLAB

ky_m = abs(ky);
kx_m = abs(kx);
Om_m = abs(Om);
l_m = abs(l);
l_m, Om_m, kx_m, ky_m = num2cell(sort([l_m, Om_m, kx_m, ky_m])){:}
function tmp_2 = code(l_m, Om_m, kx_m, ky_m)
	t_0 = (2.0 * l_m) / Om_m;
	tmp = 0.0;
	if (((t_0 ^ 2.0) * ((sin(kx_m) ^ 2.0) + (sin(ky_m) ^ 2.0))) <= 1.0)
		tmp = sqrt(1.0);
	else
		tmp = sqrt(((0.5 / (sin(ky_m) * t_0)) + 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

ky_m = N[Abs[ky], $MachinePrecision]
kx_m = N[Abs[kx], $MachinePrecision]
Om_m = N[Abs[Om], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l$95$m_, Om$95$m_, kx$95$m_, ky$95$m_] := Block[{t$95$0 = N[(N[(2.0 * l$95$m), $MachinePrecision] / Om$95$m), $MachinePrecision]}, If[LessEqual[N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[(N[Power[N[Sin[kx$95$m], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0], N[Sqrt[1.0], $MachinePrecision], N[Sqrt[N[(N[(0.5 / N[(N[Sin[ky$95$m], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0

      \[\leadsto \sqrt{\color{blue}{1}} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.0%

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

   if 1 < (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))

   1. Initial program 95.9%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 97.7

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

   5. Applied rewrites 97.7%

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

   6. Taylor expanded in kx around 0

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

   8. Applied rewrites 83.5%

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

      1. lift-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

   10. Applied rewrites 83.5%

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

Alternative 2: 98.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} ky_m = \left|ky\right| \\ kx_m = \left|kx\right| \\ Om_m = \left|Om\right| \\ l_m = \left|\ell\right| \\ [l_m, Om_m, kx_m, ky_m] = \mathsf{sort}([l_m, Om_m, kx_m, ky_m])\\ \\ \sqrt{{2}^{-1} \cdot \left(1 + {\left(\sqrt{1 + {\left(\frac{2 \cdot l\_m}{Om\_m}\right)}^{2} \cdot \left({\sin kx\_m}^{2} + {\sin ky\_m}^{2}\right)}\right)}^{-1}\right)} \end{array} \]

FPCore

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

C

ky_m = fabs(ky);
kx_m = fabs(kx);
Om_m = fabs(Om);
l_m = fabs(l);
assert(l_m < Om_m && Om_m < kx_m && kx_m < ky_m);
double code(double l_m, double Om_m, double kx_m, double ky_m) {
	return sqrt((pow(2.0, -1.0) * (1.0 + pow(sqrt((1.0 + (pow(((2.0 * l_m) / Om_m), 2.0) * (pow(sin(kx_m), 2.0) + pow(sin(ky_m), 2.0))))), -1.0))));
}

Fortran

ky_m = abs(ky)
kx_m = abs(kx)
Om_m = abs(om)
l_m = abs(l)
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
real(8) function code(l_m, om_m, kx_m, ky_m)
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om_m
    real(8), intent (in) :: kx_m
    real(8), intent (in) :: ky_m
    code = sqrt(((2.0d0 ** (-1.0d0)) * (1.0d0 + (sqrt((1.0d0 + ((((2.0d0 * l_m) / om_m) ** 2.0d0) * ((sin(kx_m) ** 2.0d0) + (sin(ky_m) ** 2.0d0))))) ** (-1.0d0)))))
end function

Java

ky_m = Math.abs(ky);
kx_m = Math.abs(kx);
Om_m = Math.abs(Om);
l_m = Math.abs(l);
assert l_m < Om_m && Om_m < kx_m && kx_m < ky_m;
public static double code(double l_m, double Om_m, double kx_m, double ky_m) {
	return Math.sqrt((Math.pow(2.0, -1.0) * (1.0 + Math.pow(Math.sqrt((1.0 + (Math.pow(((2.0 * l_m) / Om_m), 2.0) * (Math.pow(Math.sin(kx_m), 2.0) + Math.pow(Math.sin(ky_m), 2.0))))), -1.0))));
}

Python

ky_m = math.fabs(ky)
kx_m = math.fabs(kx)
Om_m = math.fabs(Om)
l_m = math.fabs(l)
[l_m, Om_m, kx_m, ky_m] = sort([l_m, Om_m, kx_m, ky_m])
def code(l_m, Om_m, kx_m, ky_m):
	return math.sqrt((math.pow(2.0, -1.0) * (1.0 + math.pow(math.sqrt((1.0 + (math.pow(((2.0 * l_m) / Om_m), 2.0) * (math.pow(math.sin(kx_m), 2.0) + math.pow(math.sin(ky_m), 2.0))))), -1.0))))

Julia

ky_m = abs(ky)
kx_m = abs(kx)
Om_m = abs(Om)
l_m = abs(l)
l_m, Om_m, kx_m, ky_m = sort([l_m, Om_m, kx_m, ky_m])
function code(l_m, Om_m, kx_m, ky_m)
	return sqrt(Float64((2.0 ^ -1.0) * Float64(1.0 + (sqrt(Float64(1.0 + Float64((Float64(Float64(2.0 * l_m) / Om_m) ^ 2.0) * Float64((sin(kx_m) ^ 2.0) + (sin(ky_m) ^ 2.0))))) ^ -1.0))))
end

MATLAB

ky_m = abs(ky);
kx_m = abs(kx);
Om_m = abs(Om);
l_m = abs(l);
l_m, Om_m, kx_m, ky_m = num2cell(sort([l_m, Om_m, kx_m, ky_m])){:}
function tmp = code(l_m, Om_m, kx_m, ky_m)
	tmp = sqrt(((2.0 ^ -1.0) * (1.0 + (sqrt((1.0 + ((((2.0 * l_m) / Om_m) ^ 2.0) * ((sin(kx_m) ^ 2.0) + (sin(ky_m) ^ 2.0))))) ^ -1.0))));
end

Wolfram

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

Derivation

1. Initial program 98.0%

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

3. Final simplification 98.0%

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

Alternative 3: 98.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0

      \[\leadsto \sqrt{\color{blue}{1}} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.0%

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

   if 1 < (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))

   1. Initial program 95.9%

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

   3. Taylor expanded in l around inf

      \[\leadsto \sqrt{\color{blue}{\frac{1}{2}}} \]
   4. Step-by-step derivation

   5. Applied rewrites 97.6%

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

Alternative 4: 98.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} ky_m = \left|ky\right| \\ kx_m = \left|kx\right| \\ Om_m = \left|Om\right| \\ l_m = \left|\ell\right| \\ [l_m, Om_m, kx_m, ky_m] = \mathsf{sort}([l_m, Om_m, kx_m, ky_m])\\ \\ \sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left({\sin ky\_m}^{2}, {\left(\frac{2 \cdot l\_m}{Om\_m}\right)}^{2}, 1\right)}} + 0.5} \end{array} \]

FPCore

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

C

ky_m = fabs(ky);
kx_m = fabs(kx);
Om_m = fabs(Om);
l_m = fabs(l);
assert(l_m < Om_m && Om_m < kx_m && kx_m < ky_m);
double code(double l_m, double Om_m, double kx_m, double ky_m) {
	return sqrt(((0.5 / sqrt(fma(pow(sin(ky_m), 2.0), pow(((2.0 * l_m) / Om_m), 2.0), 1.0))) + 0.5));
}

Julia

ky_m = abs(ky)
kx_m = abs(kx)
Om_m = abs(Om)
l_m = abs(l)
l_m, Om_m, kx_m, ky_m = sort([l_m, Om_m, kx_m, ky_m])
function code(l_m, Om_m, kx_m, ky_m)
	return sqrt(Float64(Float64(0.5 / sqrt(fma((sin(ky_m) ^ 2.0), (Float64(Float64(2.0 * l_m) / Om_m) ^ 2.0), 1.0))) + 0.5))
end

Wolfram

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

Derivation

1. Initial program 98.0%

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

3. Taylor expanded in kx around inf

   \[\leadsto \sqrt{\color{blue}{\frac{1}{2} \cdot \left(1 + \sqrt{\frac{1}{1 + 4 \cdot \frac{{\ell}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}{{Om}^{2}}}}\right)}} \]
4. Step-by-step derivation

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{1 + 4 \cdot \frac{{\ell}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}{{Om}^{2}}}}, \frac{1}{2}, \frac{1}{2}\right)}} \]

5. Applied rewrites 86.9%

   \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot 4, \frac{\mathsf{fma}\left(\sin ky, \sin ky, {\sin kx}^{2}\right)}{Om \cdot Om}, 1\right)}}, 0.5, 0.5\right)}} \]

6. Taylor expanded in kx around 0

   \[\leadsto \sqrt{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot 4, \frac{{\sin ky}^{2}}{Om \cdot Om}, 1\right)}}, \frac{1}{2}, \frac{1}{2}\right)} \]
7. Step-by-step derivation

8. Applied rewrites 75.5%

   \[\leadsto \sqrt{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(\left(\ell \cdot \ell\right) \cdot 4, \frac{{\sin ky}^{2}}{Om \cdot Om}, 1\right)}}, 0.5, 0.5\right)} \]
9. Step-by-step derivation

10. Applied rewrites 86.0%

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

Alternative 5: 56.3% accurate, 52.8× speedup

Math

\[\begin{array}{l} ky_m = \left|ky\right| \\ kx_m = \left|kx\right| \\ Om_m = \left|Om\right| \\ l_m = \left|\ell\right| \\ [l_m, Om_m, kx_m, ky_m] = \mathsf{sort}([l_m, Om_m, kx_m, ky_m])\\ \\ \sqrt{0.5} \end{array} \]

FPCore

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

C

ky_m = fabs(ky);
kx_m = fabs(kx);
Om_m = fabs(Om);
l_m = fabs(l);
assert(l_m < Om_m && Om_m < kx_m && kx_m < ky_m);
double code(double l_m, double Om_m, double kx_m, double ky_m) {
	return sqrt(0.5);
}

Fortran

ky_m = abs(ky)
kx_m = abs(kx)
Om_m = abs(om)
l_m = abs(l)
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
real(8) function code(l_m, om_m, kx_m, ky_m)
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om_m
    real(8), intent (in) :: kx_m
    real(8), intent (in) :: ky_m
    code = sqrt(0.5d0)
end function

Java

ky_m = Math.abs(ky);
kx_m = Math.abs(kx);
Om_m = Math.abs(Om);
l_m = Math.abs(l);
assert l_m < Om_m && Om_m < kx_m && kx_m < ky_m;
public static double code(double l_m, double Om_m, double kx_m, double ky_m) {
	return Math.sqrt(0.5);
}

Python

ky_m = math.fabs(ky)
kx_m = math.fabs(kx)
Om_m = math.fabs(Om)
l_m = math.fabs(l)
[l_m, Om_m, kx_m, ky_m] = sort([l_m, Om_m, kx_m, ky_m])
def code(l_m, Om_m, kx_m, ky_m):
	return math.sqrt(0.5)

Julia

ky_m = abs(ky)
kx_m = abs(kx)
Om_m = abs(Om)
l_m = abs(l)
l_m, Om_m, kx_m, ky_m = sort([l_m, Om_m, kx_m, ky_m])
function code(l_m, Om_m, kx_m, ky_m)
	return sqrt(0.5)
end

MATLAB

ky_m = abs(ky);
kx_m = abs(kx);
Om_m = abs(Om);
l_m = abs(l);
l_m, Om_m, kx_m, ky_m = num2cell(sort([l_m, Om_m, kx_m, ky_m])){:}
function tmp = code(l_m, Om_m, kx_m, ky_m)
	tmp = sqrt(0.5);
end

Wolfram

ky_m = N[Abs[ky], $MachinePrecision]
kx_m = N[Abs[kx], $MachinePrecision]
Om_m = N[Abs[Om], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l$95$m_, Om$95$m_, kx$95$m_, ky$95$m_] := N[Sqrt[0.5], $MachinePrecision]

Derivation

1. Initial program 98.0%

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

3. Taylor expanded in l around inf

   \[\leadsto \sqrt{\color{blue}{\frac{1}{2}}} \]
4. Step-by-step derivation

5. Applied rewrites 56.6%

   \[\leadsto \sqrt{\color{blue}{0.5}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024322 
(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))))))))))