Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.6% → 99.0%

Time: 11.2s

Alternatives: 7

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: 98.6% 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: 99.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} ky_m = \left|ky\right| \\ kx_m = \left|kx\right| \\ [l, Om, kx_m, ky_m] = \mathsf{sort}([l, Om, kx_m, ky_m])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx\_m}^{2} + {\sin ky\_m}^{2}\right) \leq 10^{+28}:\\ \;\;\;\;\sqrt{{2}^{-1} \cdot \left(1 + {\left(\sqrt{\mathsf{fma}\left(\frac{\ell \cdot 2}{Om}, \frac{\left(1 - \cos \left(ky\_m \cdot 2\right)\right) \cdot \ell}{2 \cdot Om} \cdot 2, 1\right)}\right)}^{-1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

ky_m = (fabs.f64 ky)
kx_m = (fabs.f64 kx)
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l Om kx_m ky_m)
 :precision binary64
 (if (<=
      (*
       (pow (/ (* 2.0 l) Om) 2.0)
       (+ (pow (sin kx_m) 2.0) (pow (sin ky_m) 2.0)))
      1e+28)
   (sqrt
    (*
     (pow 2.0 -1.0)
     (+
      1.0
      (pow
       (sqrt
        (fma
         (/ (* l 2.0) Om)
         (* (/ (* (- 1.0 (cos (* ky_m 2.0))) l) (* 2.0 Om)) 2.0)
         1.0))
       -1.0))))
   (sqrt 0.5)))

C

ky_m = fabs(ky);
kx_m = fabs(kx);
assert(l < Om && Om < kx_m && kx_m < ky_m);
double code(double l, double Om, double kx_m, double ky_m) {
	double tmp;
	if ((pow(((2.0 * l) / Om), 2.0) * (pow(sin(kx_m), 2.0) + pow(sin(ky_m), 2.0))) <= 1e+28) {
		tmp = sqrt((pow(2.0, -1.0) * (1.0 + pow(sqrt(fma(((l * 2.0) / Om), ((((1.0 - cos((ky_m * 2.0))) * l) / (2.0 * Om)) * 2.0), 1.0)), -1.0))));
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Julia

ky_m = abs(ky)
kx_m = abs(kx)
l, Om, kx_m, ky_m = sort([l, Om, kx_m, ky_m])
function code(l, Om, kx_m, ky_m)
	tmp = 0.0
	if (Float64((Float64(Float64(2.0 * l) / Om) ^ 2.0) * Float64((sin(kx_m) ^ 2.0) + (sin(ky_m) ^ 2.0))) <= 1e+28)
		tmp = sqrt(Float64((2.0 ^ -1.0) * Float64(1.0 + (sqrt(fma(Float64(Float64(l * 2.0) / Om), Float64(Float64(Float64(Float64(1.0 - cos(Float64(ky_m * 2.0))) * l) / Float64(2.0 * Om)) * 2.0), 1.0)) ^ -1.0))));
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

Wolfram

ky_m = N[Abs[ky], $MachinePrecision]
kx_m = N[Abs[kx], $MachinePrecision]
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l_, Om_, kx$95$m_, ky$95$m_] := If[LessEqual[N[(N[Power[N[(N[(2.0 * l), $MachinePrecision] / Om), $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], 1e+28], N[Sqrt[N[(N[Power[2.0, -1.0], $MachinePrecision] * N[(1.0 + N[Power[N[Sqrt[N[(N[(N[(l * 2.0), $MachinePrecision] / Om), $MachinePrecision] * N[(N[(N[(N[(1.0 - N[Cos[N[(ky$95$m * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] / N[(2.0 * Om), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $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)))) < 9.99999999999999958e27

   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 ky around 0

      \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + \color{blue}{{ky}^{2} \cdot \left(1 + {ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right)\right)}\right)}}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + \color{blue}{\left(1 + {ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right)\right) \cdot {ky}^{2}}\right)}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + \color{blue}{\left(1 + {ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right)\right) \cdot {ky}^{2}}\right)}}\right)} \]

      3. +-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + \color{blue}{\left({ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right) + 1\right)} \cdot {ky}^{2}\right)}}\right)} \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + \left(\color{blue}{\left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right) \cdot {ky}^{2}} + 1\right) \cdot {ky}^{2}\right)}}\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + \color{blue}{\mathsf{fma}\left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}, {ky}^{2}, 1\right)} \cdot {ky}^{2}\right)}}\right)} \]

      6. sub-neg N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + \mathsf{fma}\left(\color{blue}{\frac{2}{45} \cdot {ky}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, {ky}^{2}, 1\right) \cdot {ky}^{2}\right)}}\right)} \]

      7. metadata-eval N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + \mathsf{fma}\left(\frac{2}{45} \cdot {ky}^{2} + \color{blue}{\frac{-1}{3}}, {ky}^{2}, 1\right) \cdot {ky}^{2}\right)}}\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{2}{45}, {ky}^{2}, \frac{-1}{3}\right)}, {ky}^{2}, 1\right) \cdot {ky}^{2}\right)}}\right)} \]

      9. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45}, \color{blue}{ky \cdot ky}, \frac{-1}{3}\right), {ky}^{2}, 1\right) \cdot {ky}^{2}\right)}}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45}, \color{blue}{ky \cdot ky}, \frac{-1}{3}\right), {ky}^{2}, 1\right) \cdot {ky}^{2}\right)}}\right)} \]

      11. unpow2 N/A

          \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45}, ky \cdot ky, \frac{-1}{3}\right), \color{blue}{ky \cdot ky}, 1\right) \cdot {ky}^{2}\right)}}\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45}, ky \cdot ky, \frac{-1}{3}\right), \color{blue}{ky \cdot ky}, 1\right) \cdot {ky}^{2}\right)}}\right)} \]

      13. unpow2 N/A

          \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45}, ky \cdot ky, \frac{-1}{3}\right), ky \cdot ky, 1\right) \cdot \color{blue}{\left(ky \cdot ky\right)}\right)}}\right)} \]

      14. lower-*.f64 62.1

          \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446, ky \cdot ky, -0.3333333333333333\right), ky \cdot ky, 1\right) \cdot \color{blue}{\left(ky \cdot ky\right)}\right)}}\right)} \]

   5. Applied rewrites 62.1%

      \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446, ky \cdot ky, -0.3333333333333333\right), ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}\right)}}\right)} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{\color{blue}{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45}, ky \cdot ky, \frac{-1}{3}\right), ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)\right)}}}\right)} \]

      2. +-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{\color{blue}{{\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45}, ky \cdot ky, \frac{-1}{3}\right), ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)\right) + 1}}}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{\color{blue}{{\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45}, ky \cdot ky, \frac{-1}{3}\right), ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)\right)} + 1}}\right)} \]

      4. lift-pow.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{\color{blue}{{\left(\frac{2 \cdot \ell}{Om}\right)}^{2}} \cdot \left({\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45}, ky \cdot ky, \frac{-1}{3}\right), ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)\right) + 1}}\right)} \]

      5. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{\color{blue}{\left(\frac{2 \cdot \ell}{Om} \cdot \frac{2 \cdot \ell}{Om}\right)} \cdot \left({\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45}, ky \cdot ky, \frac{-1}{3}\right), ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)\right) + 1}}\right)} \]

      6. associate-*l* N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{\color{blue}{\frac{2 \cdot \ell}{Om} \cdot \left(\frac{2 \cdot \ell}{Om} \cdot \left({\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45}, ky \cdot ky, \frac{-1}{3}\right), ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)\right)\right)} + 1}}\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{2 \cdot \ell}{Om}, \frac{2 \cdot \ell}{Om} \cdot \left({\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45}, ky \cdot ky, \frac{-1}{3}\right), ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)\right), 1\right)}}}\right)} \]

   7. Applied rewrites 64.6%

      \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{\ell \cdot 2}{Om}, \frac{\ell \cdot 2}{Om} \cdot \mathsf{fma}\left(\sin kx, \sin kx, \left(\mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446, ky \cdot ky, -0.3333333333333333\right), ky \cdot ky, 1\right) \cdot ky\right) \cdot ky\right), 1\right)}}}\right)} \]

   8. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 97.5

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

   10. Applied rewrites 97.5%

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

   12. Applied rewrites 97.3%

       \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{\mathsf{fma}\left(\frac{\ell \cdot 2}{Om}, \frac{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \ell}{2 \cdot Om} \cdot 2, 1\right)}}\right)} \]

   if 9.99999999999999958e27 < (*.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 96.0%

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
   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 100.0%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right) \leq 10^{+28}:\\ \;\;\;\;\sqrt{{2}^{-1} \cdot \left(1 + {\left(\sqrt{\mathsf{fma}\left(\frac{\ell \cdot 2}{Om}, \frac{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \ell}{2 \cdot Om} \cdot 2, 1\right)}\right)}^{-1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} ky_m = \left|ky\right| \\ kx_m = \left|kx\right| \\ [l, Om, kx_m, ky_m] = \mathsf{sort}([l, Om, kx_m, ky_m])\\ \\ \sqrt{{2}^{-1} \cdot \left(1 + {\left({\left({\left(\mathsf{fma}\left(\mathsf{fma}\left(\sin ky\_m, \sin ky\_m, {\sin kx\_m}^{2}\right), {\left(\frac{-2 \cdot \ell}{Om}\right)}^{2}, 1\right)\right)}^{0.25}\right)}^{2}\right)}^{-1}\right)} \end{array} \]

FPCore

ky_m = (fabs.f64 ky)
kx_m = (fabs.f64 kx)
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l Om kx_m ky_m)
 :precision binary64
 (sqrt
  (*
   (pow 2.0 -1.0)
   (+
    1.0
    (pow
     (pow
      (pow
       (fma
        (fma (sin ky_m) (sin ky_m) (pow (sin kx_m) 2.0))
        (pow (/ (* -2.0 l) Om) 2.0)
        1.0)
       0.25)
      2.0)
     -1.0)))))

C

ky_m = fabs(ky);
kx_m = fabs(kx);
assert(l < Om && Om < kx_m && kx_m < ky_m);
double code(double l, double Om, double kx_m, double ky_m) {
	return sqrt((pow(2.0, -1.0) * (1.0 + pow(pow(pow(fma(fma(sin(ky_m), sin(ky_m), pow(sin(kx_m), 2.0)), pow(((-2.0 * l) / Om), 2.0), 1.0), 0.25), 2.0), -1.0))));
}

Julia

ky_m = abs(ky)
kx_m = abs(kx)
l, Om, kx_m, ky_m = sort([l, Om, kx_m, ky_m])
function code(l, Om, kx_m, ky_m)
	return sqrt(Float64((2.0 ^ -1.0) * Float64(1.0 + (((fma(fma(sin(ky_m), sin(ky_m), (sin(kx_m) ^ 2.0)), (Float64(Float64(-2.0 * l) / Om) ^ 2.0), 1.0) ^ 0.25) ^ 2.0) ^ -1.0))))
end

Wolfram

ky_m = N[Abs[ky], $MachinePrecision]
kx_m = N[Abs[kx], $MachinePrecision]
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l_, Om_, kx$95$m_, ky$95$m_] := N[Sqrt[N[(N[Power[2.0, -1.0], $MachinePrecision] * N[(1.0 + N[Power[N[Power[N[Power[N[(N[(N[Sin[ky$95$m], $MachinePrecision] * N[Sin[ky$95$m], $MachinePrecision] + N[Power[N[Sin[kx$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(-2.0 * l), $MachinePrecision] / Om), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision], 0.25], $MachinePrecision], 2.0], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 98.4%

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

   1. lift-sqrt.f64 N/A

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

   2. pow1/2 N/A

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

   3. metadata-eval N/A

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

   4. lift-/.f64 N/A

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

   5. sqr-pow N/A

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

   6. pow2 N/A

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

   7. lower-pow.f64 N/A

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

4. Applied rewrites 98.4%

   \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\color{blue}{{\left({\left(\mathsf{fma}\left(\mathsf{fma}\left(\sin ky, \sin ky, {\sin kx}^{2}\right), {\left(\frac{-2 \cdot \ell}{Om}\right)}^{2}, 1\right)\right)}^{0.25}\right)}^{2}}}\right)} \]

5. Final simplification 98.4%

   \[\leadsto \sqrt{{2}^{-1} \cdot \left(1 + {\left({\left({\left(\mathsf{fma}\left(\mathsf{fma}\left(\sin ky, \sin ky, {\sin kx}^{2}\right), {\left(\frac{-2 \cdot \ell}{Om}\right)}^{2}, 1\right)\right)}^{0.25}\right)}^{2}\right)}^{-1}\right)} \]
6. Add Preprocessing

Alternative 3: 98.6% accurate, 0.8× speedup

Math

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

FPCore

ky_m = (fabs.f64 ky)
kx_m = (fabs.f64 kx)
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l Om kx_m ky_m)
 :precision binary64
 (sqrt
  (*
   (pow 2.0 -1.0)
   (+
    1.0
    (pow
     (sqrt
      (+
       1.0
       (*
        (pow (/ (* 2.0 l) Om) 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);
assert(l < Om && Om < kx_m && kx_m < ky_m);
double code(double l, double Om, double kx_m, double ky_m) {
	return sqrt((pow(2.0, -1.0) * (1.0 + pow(sqrt((1.0 + (pow(((2.0 * l) / Om), 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)
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
real(8) function code(l, om, kx_m, ky_m)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    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) / om) ** 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);
assert l < Om && Om < kx_m && kx_m < ky_m;
public static double code(double l, double Om, 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) / Om), 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)
[l, Om, kx_m, ky_m] = sort([l, Om, kx_m, ky_m])
def code(l, Om, 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) / Om), 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)
l, Om, kx_m, ky_m = sort([l, Om, kx_m, ky_m])
function code(l, Om, kx_m, ky_m)
	return sqrt(Float64((2.0 ^ -1.0) * Float64(1.0 + (sqrt(Float64(1.0 + Float64((Float64(Float64(2.0 * l) / Om) ^ 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);
l, Om, kx_m, ky_m = num2cell(sort([l, Om, kx_m, ky_m])){:}
function tmp = code(l, Om, kx_m, ky_m)
	tmp = sqrt(((2.0 ^ -1.0) * (1.0 + (sqrt((1.0 + ((((2.0 * l) / Om) ^ 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]
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l_, Om_, 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), $MachinePrecision] / Om), $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 98.4%

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

3. Final simplification 98.4%

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

Alternative 4: 98.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} ky_m = \left|ky\right| \\ kx_m = \left|kx\right| \\ [l, Om, kx_m, ky_m] = \mathsf{sort}([l, Om, kx_m, ky_m])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx\_m}^{2} + {\sin ky\_m}^{2}\right) \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.5}{\sin ky\_m \cdot \frac{\ell \cdot 2}{Om}} + 0.5}\\ \end{array} \end{array} \]

FPCore

ky_m = (fabs.f64 ky)
kx_m = (fabs.f64 kx)
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l Om kx_m ky_m)
 :precision binary64
 (if (<=
      (*
       (pow (/ (* 2.0 l) Om) 2.0)
       (+ (pow (sin kx_m) 2.0) (pow (sin ky_m) 2.0)))
      5e-9)
   (sqrt 1.0)
   (sqrt (+ (/ 0.5 (* (sin ky_m) (/ (* l 2.0) Om))) 0.5))))

C

ky_m = fabs(ky);
kx_m = fabs(kx);
assert(l < Om && Om < kx_m && kx_m < ky_m);
double code(double l, double Om, double kx_m, double ky_m) {
	double tmp;
	if ((pow(((2.0 * l) / Om), 2.0) * (pow(sin(kx_m), 2.0) + pow(sin(ky_m), 2.0))) <= 5e-9) {
		tmp = sqrt(1.0);
	} else {
		tmp = sqrt(((0.5 / (sin(ky_m) * ((l * 2.0) / Om))) + 0.5));
	}
	return tmp;
}

Fortran

ky_m = abs(ky)
kx_m = abs(kx)
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
real(8) function code(l, om, kx_m, ky_m)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx_m
    real(8), intent (in) :: ky_m
    real(8) :: tmp
    if (((((2.0d0 * l) / om) ** 2.0d0) * ((sin(kx_m) ** 2.0d0) + (sin(ky_m) ** 2.0d0))) <= 5d-9) then
        tmp = sqrt(1.0d0)
    else
        tmp = sqrt(((0.5d0 / (sin(ky_m) * ((l * 2.0d0) / om))) + 0.5d0))
    end if
    code = tmp
end function

Java

ky_m = Math.abs(ky);
kx_m = Math.abs(kx);
assert l < Om && Om < kx_m && kx_m < ky_m;
public static double code(double l, double Om, double kx_m, double ky_m) {
	double tmp;
	if ((Math.pow(((2.0 * l) / Om), 2.0) * (Math.pow(Math.sin(kx_m), 2.0) + Math.pow(Math.sin(ky_m), 2.0))) <= 5e-9) {
		tmp = Math.sqrt(1.0);
	} else {
		tmp = Math.sqrt(((0.5 / (Math.sin(ky_m) * ((l * 2.0) / Om))) + 0.5));
	}
	return tmp;
}

Python

ky_m = math.fabs(ky)
kx_m = math.fabs(kx)
[l, Om, kx_m, ky_m] = sort([l, Om, kx_m, ky_m])
def code(l, Om, kx_m, ky_m):
	tmp = 0
	if (math.pow(((2.0 * l) / Om), 2.0) * (math.pow(math.sin(kx_m), 2.0) + math.pow(math.sin(ky_m), 2.0))) <= 5e-9:
		tmp = math.sqrt(1.0)
	else:
		tmp = math.sqrt(((0.5 / (math.sin(ky_m) * ((l * 2.0) / Om))) + 0.5))
	return tmp

Julia

ky_m = abs(ky)
kx_m = abs(kx)
l, Om, kx_m, ky_m = sort([l, Om, kx_m, ky_m])
function code(l, Om, kx_m, ky_m)
	tmp = 0.0
	if (Float64((Float64(Float64(2.0 * l) / Om) ^ 2.0) * Float64((sin(kx_m) ^ 2.0) + (sin(ky_m) ^ 2.0))) <= 5e-9)
		tmp = sqrt(1.0);
	else
		tmp = sqrt(Float64(Float64(0.5 / Float64(sin(ky_m) * Float64(Float64(l * 2.0) / Om))) + 0.5));
	end
	return tmp
end

MATLAB

ky_m = abs(ky);
kx_m = abs(kx);
l, Om, kx_m, ky_m = num2cell(sort([l, Om, kx_m, ky_m])){:}
function tmp_2 = code(l, Om, kx_m, ky_m)
	tmp = 0.0;
	if (((((2.0 * l) / Om) ^ 2.0) * ((sin(kx_m) ^ 2.0) + (sin(ky_m) ^ 2.0))) <= 5e-9)
		tmp = sqrt(1.0);
	else
		tmp = sqrt(((0.5 / (sin(ky_m) * ((l * 2.0) / Om))) + 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

ky_m = N[Abs[ky], $MachinePrecision]
kx_m = N[Abs[kx], $MachinePrecision]
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l_, Om_, kx$95$m_, ky$95$m_] := If[LessEqual[N[(N[Power[N[(N[(2.0 * l), $MachinePrecision] / Om), $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], 5e-9], N[Sqrt[1.0], $MachinePrecision], N[Sqrt[N[(N[(0.5 / N[(N[Sin[ky$95$m], $MachinePrecision] * N[(N[(l * 2.0), $MachinePrecision] / Om), $MachinePrecision]), $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)))) < 5.0000000000000001e-9

   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 99.8%

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

   if 5.0000000000000001e-9 < (*.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 96.2%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. metadata-eval N/A

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

      4. lift-/.f64 N/A

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

      5. sqr-pow N/A

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

      6. pow2 N/A

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

      7. lower-pow.f64 N/A

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

   4. Applied rewrites 96.2%

      \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\color{blue}{{\left({\left(\mathsf{fma}\left(\mathsf{fma}\left(\sin ky, \sin ky, {\sin kx}^{2}\right), {\left(\frac{-2 \cdot \ell}{Om}\right)}^{2}, 1\right)\right)}^{0.25}\right)}^{2}}}\right)} \]

   5. Taylor expanded in l around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 97.8

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

   7. Applied rewrites 97.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

   9. Applied rewrites 97.8%

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

   10. Taylor expanded in kx around 0

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

   12. Applied rewrites 82.0%

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

Alternative 5: 98.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} ky_m = \left|ky\right| \\ kx_m = \left|kx\right| \\ [l, Om, kx_m, ky_m] = \mathsf{sort}([l, Om, kx_m, ky_m])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx\_m}^{2} + {\sin ky\_m}^{2}\right) \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

ky_m = (fabs.f64 ky)
kx_m = (fabs.f64 kx)
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l Om kx_m ky_m)
 :precision binary64
 (if (<=
      (*
       (pow (/ (* 2.0 l) Om) 2.0)
       (+ (pow (sin kx_m) 2.0) (pow (sin ky_m) 2.0)))
      5e-9)
   (sqrt 1.0)
   (sqrt 0.5)))

C

ky_m = fabs(ky);
kx_m = fabs(kx);
assert(l < Om && Om < kx_m && kx_m < ky_m);
double code(double l, double Om, double kx_m, double ky_m) {
	double tmp;
	if ((pow(((2.0 * l) / Om), 2.0) * (pow(sin(kx_m), 2.0) + pow(sin(ky_m), 2.0))) <= 5e-9) {
		tmp = sqrt(1.0);
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

ky_m = abs(ky)
kx_m = abs(kx)
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
real(8) function code(l, om, kx_m, ky_m)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx_m
    real(8), intent (in) :: ky_m
    real(8) :: tmp
    if (((((2.0d0 * l) / om) ** 2.0d0) * ((sin(kx_m) ** 2.0d0) + (sin(ky_m) ** 2.0d0))) <= 5d-9) 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);
assert l < Om && Om < kx_m && kx_m < ky_m;
public static double code(double l, double Om, double kx_m, double ky_m) {
	double tmp;
	if ((Math.pow(((2.0 * l) / Om), 2.0) * (Math.pow(Math.sin(kx_m), 2.0) + Math.pow(Math.sin(ky_m), 2.0))) <= 5e-9) {
		tmp = Math.sqrt(1.0);
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

ky_m = math.fabs(ky)
kx_m = math.fabs(kx)
[l, Om, kx_m, ky_m] = sort([l, Om, kx_m, ky_m])
def code(l, Om, kx_m, ky_m):
	tmp = 0
	if (math.pow(((2.0 * l) / Om), 2.0) * (math.pow(math.sin(kx_m), 2.0) + math.pow(math.sin(ky_m), 2.0))) <= 5e-9:
		tmp = math.sqrt(1.0)
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

ky_m = abs(ky)
kx_m = abs(kx)
l, Om, kx_m, ky_m = sort([l, Om, kx_m, ky_m])
function code(l, Om, kx_m, ky_m)
	tmp = 0.0
	if (Float64((Float64(Float64(2.0 * l) / Om) ^ 2.0) * Float64((sin(kx_m) ^ 2.0) + (sin(ky_m) ^ 2.0))) <= 5e-9)
		tmp = sqrt(1.0);
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

ky_m = abs(ky);
kx_m = abs(kx);
l, Om, kx_m, ky_m = num2cell(sort([l, Om, kx_m, ky_m])){:}
function tmp_2 = code(l, Om, kx_m, ky_m)
	tmp = 0.0;
	if (((((2.0 * l) / Om) ^ 2.0) * ((sin(kx_m) ^ 2.0) + (sin(ky_m) ^ 2.0))) <= 5e-9)
		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]
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l_, Om_, kx$95$m_, ky$95$m_] := If[LessEqual[N[(N[Power[N[(N[(2.0 * l), $MachinePrecision] / Om), $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], 5e-9], 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)))) < 5.0000000000000001e-9

   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 99.8%

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

   if 5.0000000000000001e-9 < (*.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 96.2%

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

   3. Taylor expanded in l around inf

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

   5. Applied rewrites 97.4%

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

Alternative 6: 98.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in ky around 0

   \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + \color{blue}{{ky}^{2} \cdot \left(1 + {ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right)\right)}\right)}}\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + \color{blue}{\left(1 + {ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right)\right) \cdot {ky}^{2}}\right)}}\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + \color{blue}{\left(1 + {ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right)\right) \cdot {ky}^{2}}\right)}}\right)} \]

   3. +-commutative N/A

      \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + \color{blue}{\left({ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right) + 1\right)} \cdot {ky}^{2}\right)}}\right)} \]

   4. *-commutative N/A

      \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + \left(\color{blue}{\left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right) \cdot {ky}^{2}} + 1\right) \cdot {ky}^{2}\right)}}\right)} \]

   5. lower-fma.f64 N/A

      \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + \color{blue}{\mathsf{fma}\left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}, {ky}^{2}, 1\right)} \cdot {ky}^{2}\right)}}\right)} \]

   6. sub-neg N/A

      \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + \mathsf{fma}\left(\color{blue}{\frac{2}{45} \cdot {ky}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, {ky}^{2}, 1\right) \cdot {ky}^{2}\right)}}\right)} \]

   7. metadata-eval N/A

      \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + \mathsf{fma}\left(\frac{2}{45} \cdot {ky}^{2} + \color{blue}{\frac{-1}{3}}, {ky}^{2}, 1\right) \cdot {ky}^{2}\right)}}\right)} \]

   8. lower-fma.f64 N/A

      \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{2}{45}, {ky}^{2}, \frac{-1}{3}\right)}, {ky}^{2}, 1\right) \cdot {ky}^{2}\right)}}\right)} \]

   9. unpow2 N/A

      \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45}, \color{blue}{ky \cdot ky}, \frac{-1}{3}\right), {ky}^{2}, 1\right) \cdot {ky}^{2}\right)}}\right)} \]

   10. lower-*.f64 N/A

       \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45}, \color{blue}{ky \cdot ky}, \frac{-1}{3}\right), {ky}^{2}, 1\right) \cdot {ky}^{2}\right)}}\right)} \]

   11. unpow2 N/A

       \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45}, ky \cdot ky, \frac{-1}{3}\right), \color{blue}{ky \cdot ky}, 1\right) \cdot {ky}^{2}\right)}}\right)} \]

   12. lower-*.f64 N/A

       \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45}, ky \cdot ky, \frac{-1}{3}\right), \color{blue}{ky \cdot ky}, 1\right) \cdot {ky}^{2}\right)}}\right)} \]

   13. unpow2 N/A

       \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45}, ky \cdot ky, \frac{-1}{3}\right), ky \cdot ky, 1\right) \cdot \color{blue}{\left(ky \cdot ky\right)}\right)}}\right)} \]

   14. lower-*.f64 75.2

       \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446, ky \cdot ky, -0.3333333333333333\right), ky \cdot ky, 1\right) \cdot \color{blue}{\left(ky \cdot ky\right)}\right)}}\right)} \]

5. Applied rewrites 75.2%

   \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446, ky \cdot ky, -0.3333333333333333\right), ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}\right)}}\right)} \]
6. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{\color{blue}{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45}, ky \cdot ky, \frac{-1}{3}\right), ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)\right)}}}\right)} \]

   2. +-commutative N/A

      \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{\color{blue}{{\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45}, ky \cdot ky, \frac{-1}{3}\right), ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)\right) + 1}}}\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{\color{blue}{{\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45}, ky \cdot ky, \frac{-1}{3}\right), ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)\right)} + 1}}\right)} \]

   4. lift-pow.f64 N/A

      \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{\color{blue}{{\left(\frac{2 \cdot \ell}{Om}\right)}^{2}} \cdot \left({\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45}, ky \cdot ky, \frac{-1}{3}\right), ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)\right) + 1}}\right)} \]

   5. unpow2 N/A

      \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{\color{blue}{\left(\frac{2 \cdot \ell}{Om} \cdot \frac{2 \cdot \ell}{Om}\right)} \cdot \left({\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45}, ky \cdot ky, \frac{-1}{3}\right), ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)\right) + 1}}\right)} \]

   6. associate-*l* N/A

      \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{\color{blue}{\frac{2 \cdot \ell}{Om} \cdot \left(\frac{2 \cdot \ell}{Om} \cdot \left({\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45}, ky \cdot ky, \frac{-1}{3}\right), ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)\right)\right)} + 1}}\right)} \]

   7. lower-fma.f64 N/A

      \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{2 \cdot \ell}{Om}, \frac{2 \cdot \ell}{Om} \cdot \left({\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45}, ky \cdot ky, \frac{-1}{3}\right), ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)\right), 1\right)}}}\right)} \]

7. Applied rewrites 76.9%

   \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{\ell \cdot 2}{Om}, \frac{\ell \cdot 2}{Om} \cdot \mathsf{fma}\left(\sin kx, \sin kx, \left(\mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446, ky \cdot ky, -0.3333333333333333\right), ky \cdot ky, 1\right) \cdot ky\right) \cdot ky\right), 1\right)}}}\right)} \]

8. Taylor expanded in kx around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-sin.f64 90.0

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

10. Applied rewrites 90.0%

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

11. Final simplification 90.0%

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

Alternative 7: 56.2% accurate, 52.8× speedup

Math

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

FPCore

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

C

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

Fortran

ky_m = abs(ky)
kx_m = abs(kx)
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
real(8) function code(l, om, kx_m, ky_m)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    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);
assert l < Om && Om < kx_m && kx_m < ky_m;
public static double code(double l, double Om, double kx_m, double ky_m) {
	return Math.sqrt(0.5);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.4%

   \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
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 51.7%

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

Reproduce

herbie shell --seed 2024305 
(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))))))))))