Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.3% → 98.1%

Time: 9.9s

Alternatives: 7

Speedup: 1.1×

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: 98.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} Om_m = \left|Om\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{2 \cdot l\_m}{Om\_m} \leq 2 \cdot 10^{+60}:\\ \;\;\;\;\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + \frac{l\_m}{Om\_m} \cdot \left(\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right) \cdot \left(\frac{l\_m}{Om\_m} \cdot 4\right)\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

Om_m = (fabs.f64 Om)
l_m = (fabs.f64 l)
(FPCore (l_m Om_m kx ky)
 :precision binary64
 (if (<= (/ (* 2.0 l_m) Om_m) 2e+60)
   (sqrt
    (*
     (/ 1.0 2.0)
     (+
      1.0
      (/
       1.0
       (sqrt
        (+
         1.0
         (*
          (/ l_m Om_m)
          (*
           (fma (- 1.0 (cos (+ ky ky))) 0.5 (+ 0.5 (* -0.5 (cos (+ kx kx)))))
           (* (/ l_m Om_m) 4.0)))))))))
   (sqrt 0.5)))

C

Om_m = fabs(Om);
l_m = fabs(l);
double code(double l_m, double Om_m, double kx, double ky) {
	double tmp;
	if (((2.0 * l_m) / Om_m) <= 2e+60) {
		tmp = sqrt(((1.0 / 2.0) * (1.0 + (1.0 / sqrt((1.0 + ((l_m / Om_m) * (fma((1.0 - cos((ky + ky))), 0.5, (0.5 + (-0.5 * cos((kx + kx))))) * ((l_m / Om_m) * 4.0)))))))));
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Julia

Om_m = abs(Om)
l_m = abs(l)
function code(l_m, Om_m, kx, ky)
	tmp = 0.0
	if (Float64(Float64(2.0 * l_m) / Om_m) <= 2e+60)
		tmp = sqrt(Float64(Float64(1.0 / 2.0) * Float64(1.0 + Float64(1.0 / sqrt(Float64(1.0 + Float64(Float64(l_m / Om_m) * Float64(fma(Float64(1.0 - cos(Float64(ky + ky))), 0.5, Float64(0.5 + Float64(-0.5 * cos(Float64(kx + kx))))) * Float64(Float64(l_m / Om_m) * 4.0)))))))));
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

Wolfram

Om_m = N[Abs[Om], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[l$95$m_, Om$95$m_, kx_, ky_] := If[LessEqual[N[(N[(2.0 * l$95$m), $MachinePrecision] / Om$95$m), $MachinePrecision], 2e+60], N[Sqrt[N[(N[(1.0 / 2.0), $MachinePrecision] * N[(1.0 + N[(1.0 / N[Sqrt[N[(1.0 + N[(N[(l$95$m / Om$95$m), $MachinePrecision] * N[(N[(N[(1.0 - N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(0.5 + N[(-0.5 * N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l$95$m / Om$95$m), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) l) Om) < 1.9999999999999999e60

   1. Initial program 98.6%

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

   4. Applied egg-rr 95.8%

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

   if 1.9999999999999999e60 < (/.f64 (*.f64 #s(literal 2 binary64) l) Om)

   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 kx around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 79.5

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

   5. Simplified 79.5%

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

   6. Taylor expanded in l around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 91.5

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

   8. Simplified 91.5%

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

   9. Taylor expanded in Om around 0

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

       1. lower-sqrt.f64 100.0

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

   11. Simplified 100.0%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 \cdot \ell}{Om} \leq 2 \cdot 10^{+60}:\\ \;\;\;\;\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + \frac{\ell}{Om} \cdot \left(\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right) \cdot \left(\frac{\ell}{Om} \cdot 4\right)\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} Om_m = \left|Om\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{2 \cdot l\_m}{Om\_m}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right) \leq 0.02:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{Om\_m}{l\_m \cdot \sin ky}, 0.5\right)}\\ \end{array} \end{array} \]

FPCore

Om_m = (fabs.f64 Om)
l_m = (fabs.f64 l)
(FPCore (l_m Om_m kx ky)
 :precision binary64
 (if (<=
      (*
       (pow (/ (* 2.0 l_m) Om_m) 2.0)
       (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))
      0.02)
   1.0
   (sqrt (fma 0.25 (/ Om_m (* l_m (sin ky))) 0.5))))

C

Om_m = fabs(Om);
l_m = fabs(l);
double code(double l_m, double Om_m, double kx, double ky) {
	double tmp;
	if ((pow(((2.0 * l_m) / Om_m), 2.0) * (pow(sin(kx), 2.0) + pow(sin(ky), 2.0))) <= 0.02) {
		tmp = 1.0;
	} else {
		tmp = sqrt(fma(0.25, (Om_m / (l_m * sin(ky))), 0.5));
	}
	return tmp;
}

Julia

Om_m = abs(Om)
l_m = abs(l)
function code(l_m, Om_m, kx, ky)
	tmp = 0.0
	if (Float64((Float64(Float64(2.0 * l_m) / Om_m) ^ 2.0) * Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))) <= 0.02)
		tmp = 1.0;
	else
		tmp = sqrt(fma(0.25, Float64(Om_m / Float64(l_m * sin(ky))), 0.5));
	end
	return tmp
end

Wolfram

Om_m = N[Abs[Om], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[l$95$m_, Om$95$m_, kx_, ky_] := 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], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.02], 1.0, N[Sqrt[N[(0.25 * N[(Om$95$m / N[(l$95$m * N[Sin[ky], $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)))) < 0.0200000000000000004

   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

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

   5. Simplified 99.3%

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \color{blue}{2}} \]

      3. metadata-eval N/A

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

      4. metadata-eval 99.3

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

   7. Applied egg-rr 99.3%

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

   if 0.0200000000000000004 < (*.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 97.5%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 75.4

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

   5. Simplified 75.4%

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

   6. Taylor expanded in l around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 86.3

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

   8. Simplified 86.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-sqrt.f64 86.3

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

      9. lift-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. distribute-lft-in N/A

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

   10. Applied egg-rr 86.3%

       \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(0.25, \frac{Om}{\ell \cdot \sin ky}, 0.5\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 91.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} Om_m = \left|Om\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{2 \cdot l\_m}{Om\_m}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right) \leq 0.02:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-0.25, \frac{Om\_m}{l\_m \cdot \sin ky}, 0.5\right)}\\ \end{array} \end{array} \]

FPCore

Om_m = (fabs.f64 Om)
l_m = (fabs.f64 l)
(FPCore (l_m Om_m kx ky)
 :precision binary64
 (if (<=
      (*
       (pow (/ (* 2.0 l_m) Om_m) 2.0)
       (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))
      0.02)
   1.0
   (sqrt (fma -0.25 (/ Om_m (* l_m (sin ky))) 0.5))))

C

Om_m = fabs(Om);
l_m = fabs(l);
double code(double l_m, double Om_m, double kx, double ky) {
	double tmp;
	if ((pow(((2.0 * l_m) / Om_m), 2.0) * (pow(sin(kx), 2.0) + pow(sin(ky), 2.0))) <= 0.02) {
		tmp = 1.0;
	} else {
		tmp = sqrt(fma(-0.25, (Om_m / (l_m * sin(ky))), 0.5));
	}
	return tmp;
}

Julia

Om_m = abs(Om)
l_m = abs(l)
function code(l_m, Om_m, kx, ky)
	tmp = 0.0
	if (Float64((Float64(Float64(2.0 * l_m) / Om_m) ^ 2.0) * Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))) <= 0.02)
		tmp = 1.0;
	else
		tmp = sqrt(fma(-0.25, Float64(Om_m / Float64(l_m * sin(ky))), 0.5));
	end
	return tmp
end

Wolfram

Om_m = N[Abs[Om], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[l$95$m_, Om$95$m_, kx_, ky_] := 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], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.02], 1.0, N[Sqrt[N[(-0.25 * N[(Om$95$m / N[(l$95$m * N[Sin[ky], $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)))) < 0.0200000000000000004

   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

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

   5. Simplified 99.3%

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \color{blue}{2}} \]

      3. metadata-eval N/A

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

      4. metadata-eval 99.3

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

   7. Applied egg-rr 99.3%

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

   if 0.0200000000000000004 < (*.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 97.5%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 75.4

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

   5. Simplified 75.4%

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

      1. metadata-eval 75.4

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

   7. Applied egg-rr 75.4%

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

   8. Taylor expanded in l around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 86.2

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

   10. Simplified 86.2%

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

       1. lift-*.f64 N/A

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

       2. lift-sin.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. lift-+.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-sqrt.f64 86.2

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

       8. lift-*.f64 N/A

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

       9. lift-+.f64 N/A

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

       10. +-commutative N/A

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

       11. distribute-lft-in N/A

           \[\leadsto \sqrt{\color{blue}{\frac{1}{2} \cdot \frac{\frac{-1}{2} \cdot Om}{\ell \cdot \sin ky} + \frac{1}{2} \cdot 1}} \]

       12. lift-/.f64 N/A

           \[\leadsto \sqrt{\frac{1}{2} \cdot \color{blue}{\frac{\frac{-1}{2} \cdot Om}{\ell \cdot \sin ky}} + \frac{1}{2} \cdot 1} \]

       13. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{1}{2} \cdot \frac{\color{blue}{\frac{-1}{2} \cdot Om}}{\ell \cdot \sin ky} + \frac{1}{2} \cdot 1} \]

       14. associate-/l* N/A

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

       15. associate-*r* N/A

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

       16. metadata-eval N/A

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

   12. Applied egg-rr 86.2%

       \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(-0.25, \frac{Om}{\ell \cdot \sin ky}, 0.5\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 98.3% accurate, 1.0× speedup

Math

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

FPCore

Om_m = (fabs.f64 Om)
l_m = (fabs.f64 l)
(FPCore (l_m Om_m kx ky)
 :precision binary64
 (sqrt
  (*
   (/ 1.0 2.0)
   (+
    1.0
    (/
     1.0
     (sqrt
      (+
       1.0
       (*
        (pow (/ (* 2.0 l_m) Om_m) 2.0)
        (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))))))

C

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

Fortran

Om_m = abs(om)
l_m = abs(l)
real(8) function code(l_m, om_m, kx, ky)
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om_m
    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_m) / om_m) ** 2.0d0) * ((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))))))))
end function

Java

Om_m = Math.abs(Om);
l_m = Math.abs(l);
public static double code(double l_m, double Om_m, double kx, double ky) {
	return Math.sqrt(((1.0 / 2.0) * (1.0 + (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), 2.0)))))))));
}

Python

Om_m = math.fabs(Om)
l_m = math.fabs(l)
def code(l_m, Om_m, kx, ky):
	return math.sqrt(((1.0 / 2.0) * (1.0 + (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), 2.0)))))))))

Julia

Om_m = abs(Om)
l_m = abs(l)
function code(l_m, Om_m, 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_m) / Om_m) ^ 2.0) * Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))))))))
end

MATLAB

Om_m = abs(Om);
l_m = abs(l);
function tmp = code(l_m, Om_m, kx, ky)
	tmp = sqrt(((1.0 / 2.0) * (1.0 + (1.0 / sqrt((1.0 + ((((2.0 * l_m) / Om_m) ^ 2.0) * ((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))))))));
end

Wolfram

Om_m = N[Abs[Om], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[l$95$m_, Om$95$m_, 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$95$m), $MachinePrecision] / Om$95$m), $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]

Derivation

1. Initial program 98.8%

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

Alternative 5: 98.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} Om_m = \left|Om\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{2 \cdot l\_m}{Om\_m}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right) \leq 0.02:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

Om_m = (fabs.f64 Om)
l_m = (fabs.f64 l)
(FPCore (l_m Om_m kx ky)
 :precision binary64
 (if (<=
      (*
       (pow (/ (* 2.0 l_m) Om_m) 2.0)
       (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))
      0.02)
   1.0
   (sqrt 0.5)))

C

Om_m = fabs(Om);
l_m = fabs(l);
double code(double l_m, double Om_m, double kx, double ky) {
	double tmp;
	if ((pow(((2.0 * l_m) / Om_m), 2.0) * (pow(sin(kx), 2.0) + pow(sin(ky), 2.0))) <= 0.02) {
		tmp = 1.0;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

Om_m = abs(om)
l_m = abs(l)
real(8) function code(l_m, om_m, kx, ky)
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om_m
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: tmp
    if (((((2.0d0 * l_m) / om_m) ** 2.0d0) * ((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0))) <= 0.02d0) then
        tmp = 1.0d0
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

Om_m = Math.abs(Om);
l_m = Math.abs(l);
public static double code(double l_m, double Om_m, double kx, double ky) {
	double tmp;
	if ((Math.pow(((2.0 * l_m) / Om_m), 2.0) * (Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0))) <= 0.02) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

Om_m = math.fabs(Om)
l_m = math.fabs(l)
def code(l_m, Om_m, kx, ky):
	tmp = 0
	if (math.pow(((2.0 * l_m) / Om_m), 2.0) * (math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0))) <= 0.02:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

Om_m = abs(Om)
l_m = abs(l)
function code(l_m, Om_m, kx, ky)
	tmp = 0.0
	if (Float64((Float64(Float64(2.0 * l_m) / Om_m) ^ 2.0) * Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))) <= 0.02)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

Om_m = abs(Om);
l_m = abs(l);
function tmp_2 = code(l_m, Om_m, kx, ky)
	tmp = 0.0;
	if (((((2.0 * l_m) / Om_m) ^ 2.0) * ((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))) <= 0.02)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

Om_m = N[Abs[Om], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[l$95$m_, Om$95$m_, kx_, ky_] := 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], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.02], 1.0, 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)))) < 0.0200000000000000004

   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

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

   5. Simplified 99.3%

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \color{blue}{2}} \]

      3. metadata-eval N/A

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

      4. metadata-eval 99.3

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

   7. Applied egg-rr 99.3%

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

   if 0.0200000000000000004 < (*.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 97.5%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 75.4

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

   5. Simplified 75.4%

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

   6. Taylor expanded in l around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 86.3

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

   8. Simplified 86.3%

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

   9. Taylor expanded in Om around 0

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

       1. lower-sqrt.f64 98.4

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

   11. Simplified 98.4%

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

Alternative 6: 95.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} Om_m = \left|Om\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{2 \cdot l\_m}{Om\_m} \leq 2 \cdot 10^{+60}:\\ \;\;\;\;\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + \frac{l\_m}{Om\_m} \cdot \frac{2 \cdot \left(l\_m \cdot \left(1 - \cos \left(ky \cdot -2\right)\right)\right)}{Om\_m}}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

Om_m = (fabs.f64 Om)
l_m = (fabs.f64 l)
(FPCore (l_m Om_m kx ky)
 :precision binary64
 (if (<= (/ (* 2.0 l_m) Om_m) 2e+60)
   (sqrt
    (*
     (/ 1.0 2.0)
     (+
      1.0
      (/
       1.0
       (sqrt
        (+
         1.0
         (*
          (/ l_m Om_m)
          (/ (* 2.0 (* l_m (- 1.0 (cos (* ky -2.0))))) Om_m))))))))
   (sqrt 0.5)))

C

Om_m = fabs(Om);
l_m = fabs(l);
double code(double l_m, double Om_m, double kx, double ky) {
	double tmp;
	if (((2.0 * l_m) / Om_m) <= 2e+60) {
		tmp = sqrt(((1.0 / 2.0) * (1.0 + (1.0 / sqrt((1.0 + ((l_m / Om_m) * ((2.0 * (l_m * (1.0 - cos((ky * -2.0))))) / Om_m))))))));
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

Om_m = abs(om)
l_m = abs(l)
real(8) function code(l_m, om_m, kx, ky)
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om_m
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: tmp
    if (((2.0d0 * l_m) / om_m) <= 2d+60) then
        tmp = sqrt(((1.0d0 / 2.0d0) * (1.0d0 + (1.0d0 / sqrt((1.0d0 + ((l_m / om_m) * ((2.0d0 * (l_m * (1.0d0 - cos((ky * (-2.0d0)))))) / om_m))))))))
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

Om_m = Math.abs(Om);
l_m = Math.abs(l);
public static double code(double l_m, double Om_m, double kx, double ky) {
	double tmp;
	if (((2.0 * l_m) / Om_m) <= 2e+60) {
		tmp = Math.sqrt(((1.0 / 2.0) * (1.0 + (1.0 / Math.sqrt((1.0 + ((l_m / Om_m) * ((2.0 * (l_m * (1.0 - Math.cos((ky * -2.0))))) / Om_m))))))));
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

Om_m = math.fabs(Om)
l_m = math.fabs(l)
def code(l_m, Om_m, kx, ky):
	tmp = 0
	if ((2.0 * l_m) / Om_m) <= 2e+60:
		tmp = math.sqrt(((1.0 / 2.0) * (1.0 + (1.0 / math.sqrt((1.0 + ((l_m / Om_m) * ((2.0 * (l_m * (1.0 - math.cos((ky * -2.0))))) / Om_m))))))))
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

Om_m = abs(Om)
l_m = abs(l)
function code(l_m, Om_m, kx, ky)
	tmp = 0.0
	if (Float64(Float64(2.0 * l_m) / Om_m) <= 2e+60)
		tmp = sqrt(Float64(Float64(1.0 / 2.0) * Float64(1.0 + Float64(1.0 / sqrt(Float64(1.0 + Float64(Float64(l_m / Om_m) * Float64(Float64(2.0 * Float64(l_m * Float64(1.0 - cos(Float64(ky * -2.0))))) / Om_m))))))));
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

Om_m = abs(Om);
l_m = abs(l);
function tmp_2 = code(l_m, Om_m, kx, ky)
	tmp = 0.0;
	if (((2.0 * l_m) / Om_m) <= 2e+60)
		tmp = sqrt(((1.0 / 2.0) * (1.0 + (1.0 / sqrt((1.0 + ((l_m / Om_m) * ((2.0 * (l_m * (1.0 - cos((ky * -2.0))))) / Om_m))))))));
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

Om_m = N[Abs[Om], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[l$95$m_, Om$95$m_, kx_, ky_] := If[LessEqual[N[(N[(2.0 * l$95$m), $MachinePrecision] / Om$95$m), $MachinePrecision], 2e+60], N[Sqrt[N[(N[(1.0 / 2.0), $MachinePrecision] * N[(1.0 + N[(1.0 / N[Sqrt[N[(1.0 + N[(N[(l$95$m / Om$95$m), $MachinePrecision] * N[(N[(2.0 * N[(l$95$m * N[(1.0 - N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) l) Om) < 1.9999999999999999e60

   1. Initial program 98.6%

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

   4. Applied egg-rr 95.8%

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

   5. Taylor expanded in kx around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft-neg-in N/A

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

      7. lower--.f64 N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 87.1

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

   7. Simplified 87.1%

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

   if 1.9999999999999999e60 < (/.f64 (*.f64 #s(literal 2 binary64) l) Om)

   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 kx around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 79.5

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

   5. Simplified 79.5%

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

   6. Taylor expanded in l around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 91.5

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

   8. Simplified 91.5%

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

   9. Taylor expanded in Om around 0

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

       1. lower-sqrt.f64 100.0

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

   11. Simplified 100.0%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 \cdot \ell}{Om} \leq 2 \cdot 10^{+60}:\\ \;\;\;\;\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + \frac{\ell}{Om} \cdot \frac{2 \cdot \left(\ell \cdot \left(1 - \cos \left(ky \cdot -2\right)\right)\right)}{Om}}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.8% accurate, 581.0× speedup

Math

\[\begin{array}{l} Om_m = \left|Om\right| \\ l_m = \left|\ell\right| \\ 1 \end{array} \]

FPCore

Om_m = (fabs.f64 Om)
l_m = (fabs.f64 l)
(FPCore (l_m Om_m kx ky) :precision binary64 1.0)

C

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

Fortran

Om_m = abs(om)
l_m = abs(l)
real(8) function code(l_m, om_m, kx, ky)
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om_m
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    code = 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

Om_m = abs(Om);
l_m = abs(l);
function tmp = code(l_m, Om_m, kx, ky)
	tmp = 1.0;
end

Wolfram

Om_m = N[Abs[Om], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[l$95$m_, Om$95$m_, kx_, ky_] := 1.0

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in l around 0

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

5. Simplified 61.5%

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

   1. metadata-eval N/A

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

   2. metadata-eval N/A

      \[\leadsto \sqrt{\frac{1}{2} \cdot \color{blue}{2}} \]

   3. metadata-eval N/A

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

   4. metadata-eval 61.5

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

7. Applied egg-rr 61.5%

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

Reproduce

herbie shell --seed 2024219 
(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))))))))))