Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.3% → 99.2%

Time: 13.5s

Alternatives: 6

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: 99.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} Om_m = \left|Om\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} t_0 := \frac{l\_m \cdot 2}{Om\_m}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+106}:\\ \;\;\;\;e^{\left(\log 0.5 + \mathsf{log1p}\left({\left(\mathsf{fma}\left(\mathsf{fma}\left(\sin ky, \sin ky, {\sin kx}^{2}\right), {t\_0}^{2}, 1\right)\right)}^{-0.5}\right)\right) \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

Om_m = (fabs.f64 Om)
l_m = (fabs.f64 l)
(FPCore (l_m Om_m kx ky)
 :precision binary64
 (let* ((t_0 (/ (* l_m 2.0) Om_m)))
   (if (<= t_0 2e+106)
     (exp
      (*
       (+
        (log 0.5)
        (log1p
         (pow
          (fma (fma (sin ky) (sin ky) (pow (sin kx) 2.0)) (pow t_0 2.0) 1.0)
          -0.5)))
       0.5))
     (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 t_0 = (l_m * 2.0) / Om_m;
	double tmp;
	if (t_0 <= 2e+106) {
		tmp = exp(((log(0.5) + log1p(pow(fma(fma(sin(ky), sin(ky), pow(sin(kx), 2.0)), pow(t_0, 2.0), 1.0), -0.5))) * 0.5));
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Julia

Om_m = abs(Om)
l_m = abs(l)
function code(l_m, Om_m, kx, ky)
	t_0 = Float64(Float64(l_m * 2.0) / Om_m)
	tmp = 0.0
	if (t_0 <= 2e+106)
		tmp = exp(Float64(Float64(log(0.5) + log1p((fma(fma(sin(ky), sin(ky), (sin(kx) ^ 2.0)), (t_0 ^ 2.0), 1.0) ^ -0.5))) * 0.5));
	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_] := Block[{t$95$0 = N[(N[(l$95$m * 2.0), $MachinePrecision] / Om$95$m), $MachinePrecision]}, If[LessEqual[t$95$0, 2e+106], N[Exp[N[(N[(N[Log[0.5], $MachinePrecision] + N[Log[1 + N[Power[N[(N[(N[Sin[ky], $MachinePrecision] * N[Sin[ky], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[t$95$0, 2.0], $MachinePrecision] + 1.0), $MachinePrecision], -0.5], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) l) Om) < 2.00000000000000018e106

   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 rewrites 98.6%

      \[\leadsto \color{blue}{e^{\left(\mathsf{log1p}\left({\left(\mathsf{fma}\left(\mathsf{fma}\left(\sin ky, \sin ky, {\sin kx}^{2}\right), {\left(\frac{\ell \cdot 2}{Om}\right)}^{2}, 1\right)\right)}^{-0.5}\right) + \log 0.5\right) \cdot 0.5}} \]

   if 2.00000000000000018e106 < (/.f64 (*.f64 #s(literal 2 binary64) l) Om)

   1. Initial program 95.3%

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

   3. Taylor expanded in Om around 0

      \[\leadsto \sqrt{\color{blue}{\frac{1}{2}}} \]
   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 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\ell \cdot 2}{Om} \leq 2 \cdot 10^{+106}:\\ \;\;\;\;e^{\left(\log 0.5 + \mathsf{log1p}\left({\left(\mathsf{fma}\left(\mathsf{fma}\left(\sin ky, \sin ky, {\sin kx}^{2}\right), {\left(\frac{\ell \cdot 2}{Om}\right)}^{2}, 1\right)\right)}^{-0.5}\right)\right) \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} Om_m = \left|Om\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} t_0 := \frac{l\_m \cdot 2}{Om\_m}\\ \mathbf{if}\;\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {t\_0}^{2} \leq 1000000000:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(t\_0, \frac{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot l\_m}{Om\_m \cdot 2} \cdot 2, 1\right)}}, 0.5, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \left(\frac{l\_m}{Om\_m} \cdot 2\right)} + 1\right) \cdot \frac{1}{2}}\\ \end{array} \end{array} \]

FPCore

Om_m = (fabs.f64 Om)
l_m = (fabs.f64 l)
(FPCore (l_m Om_m kx ky)
 :precision binary64
 (let* ((t_0 (/ (* l_m 2.0) Om_m)))
   (if (<=
        (* (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)) (pow t_0 2.0))
        1000000000.0)
     (sqrt
      (fma
       (sqrt
        (/
         1.0
         (fma
          t_0
          (* (/ (* (- 1.0 (cos (* ky 2.0))) l_m) (* Om_m 2.0)) 2.0)
          1.0)))
       0.5
       0.5))
     (sqrt
      (*
       (+ (/ 1.0 (* (hypot (sin kx) (sin ky)) (* (/ l_m Om_m) 2.0))) 1.0)
       (/ 1.0 2.0))))))

C

Om_m = fabs(Om);
l_m = fabs(l);
double code(double l_m, double Om_m, double kx, double ky) {
	double t_0 = (l_m * 2.0) / Om_m;
	double tmp;
	if (((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)) * pow(t_0, 2.0)) <= 1000000000.0) {
		tmp = sqrt(fma(sqrt((1.0 / fma(t_0, ((((1.0 - cos((ky * 2.0))) * l_m) / (Om_m * 2.0)) * 2.0), 1.0))), 0.5, 0.5));
	} else {
		tmp = sqrt((((1.0 / (hypot(sin(kx), sin(ky)) * ((l_m / Om_m) * 2.0))) + 1.0) * (1.0 / 2.0)));
	}
	return tmp;
}

Julia

Om_m = abs(Om)
l_m = abs(l)
function code(l_m, Om_m, kx, ky)
	t_0 = Float64(Float64(l_m * 2.0) / Om_m)
	tmp = 0.0
	if (Float64(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)) * (t_0 ^ 2.0)) <= 1000000000.0)
		tmp = sqrt(fma(sqrt(Float64(1.0 / fma(t_0, Float64(Float64(Float64(Float64(1.0 - cos(Float64(ky * 2.0))) * l_m) / Float64(Om_m * 2.0)) * 2.0), 1.0))), 0.5, 0.5));
	else
		tmp = sqrt(Float64(Float64(Float64(1.0 / Float64(hypot(sin(kx), sin(ky)) * Float64(Float64(l_m / Om_m) * 2.0))) + 1.0) * Float64(1.0 / 2.0)));
	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_] := Block[{t$95$0 = N[(N[(l$95$m * 2.0), $MachinePrecision] / Om$95$m), $MachinePrecision]}, If[LessEqual[N[(N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision], 1000000000.0], N[Sqrt[N[(N[Sqrt[N[(1.0 / N[(t$95$0 * N[(N[(N[(N[(1.0 - N[Cos[N[(ky * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * l$95$m), $MachinePrecision] / N[(Om$95$m * 2.0), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(1.0 / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] * N[(N[(l$95$m / Om$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $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)))) < 1e9

   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 inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 83.9%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in kx around 0

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

   10. Applied rewrites 99.2%

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

   12. Applied rewrites 99.2%

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

   if 1e9 < (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))

   1. Initial program 95.0%

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

   3. Taylor expanded in Om around 0

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

      1. 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. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 99.5

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

   5. Applied rewrites 99.5%

      \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\color{blue}{\left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} \leq 1000000000:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(\frac{\ell \cdot 2}{Om}, \frac{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \ell}{Om \cdot 2} \cdot 2, 1\right)}}, 0.5, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \left(\frac{\ell}{Om} \cdot 2\right)} + 1\right) \cdot \frac{1}{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} Om_m = \left|Om\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} t_0 := \frac{l\_m \cdot 2}{Om\_m}\\ \mathbf{if}\;\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {t\_0}^{2} \leq 2 \cdot 10^{+28}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(t\_0, \frac{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot l\_m}{Om\_m \cdot 2} \cdot 2, 1\right)}}, 0.5, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

Om_m = (fabs.f64 Om)
l_m = (fabs.f64 l)
(FPCore (l_m Om_m kx ky)
 :precision binary64
 (let* ((t_0 (/ (* l_m 2.0) Om_m)))
   (if (<= (* (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)) (pow t_0 2.0)) 2e+28)
     (sqrt
      (fma
       (sqrt
        (/
         1.0
         (fma
          t_0
          (* (/ (* (- 1.0 (cos (* ky 2.0))) l_m) (* Om_m 2.0)) 2.0)
          1.0)))
       0.5
       0.5))
     (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 t_0 = (l_m * 2.0) / Om_m;
	double tmp;
	if (((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)) * pow(t_0, 2.0)) <= 2e+28) {
		tmp = sqrt(fma(sqrt((1.0 / fma(t_0, ((((1.0 - cos((ky * 2.0))) * l_m) / (Om_m * 2.0)) * 2.0), 1.0))), 0.5, 0.5));
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Julia

Om_m = abs(Om)
l_m = abs(l)
function code(l_m, Om_m, kx, ky)
	t_0 = Float64(Float64(l_m * 2.0) / Om_m)
	tmp = 0.0
	if (Float64(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)) * (t_0 ^ 2.0)) <= 2e+28)
		tmp = sqrt(fma(sqrt(Float64(1.0 / fma(t_0, Float64(Float64(Float64(Float64(1.0 - cos(Float64(ky * 2.0))) * l_m) / Float64(Om_m * 2.0)) * 2.0), 1.0))), 0.5, 0.5));
	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_] := Block[{t$95$0 = N[(N[(l$95$m * 2.0), $MachinePrecision] / Om$95$m), $MachinePrecision]}, If[LessEqual[N[(N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision], 2e+28], N[Sqrt[N[(N[Sqrt[N[(1.0 / N[(t$95$0 * N[(N[(N[(N[(1.0 - N[Cos[N[(ky * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * l$95$m), $MachinePrecision] / N[(Om$95$m * 2.0), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $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)))) < 1.99999999999999992e28

   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 inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 83.0%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in kx around 0

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

   10. Applied rewrites 98.4%

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

   12. Applied rewrites 98.4%

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

   if 1.99999999999999992e28 < (*.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 94.8%

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

   3. Taylor expanded in Om around 0

      \[\leadsto \sqrt{\color{blue}{\frac{1}{2}}} \]
   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 99.0%

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

Alternative 4: 98.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} Om_m = \left|Om\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{l\_m \cdot 2}{Om\_m}\right)}^{2} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

Om_m = (fabs.f64 Om)
l_m = (fabs.f64 l)
(FPCore (l_m Om_m kx ky)
 :precision binary64
 (if (<=
      (*
       (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0))
       (pow (/ (* l_m 2.0) Om_m) 2.0))
      5e-6)
   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(sin(ky), 2.0) + pow(sin(kx), 2.0)) * pow(((l_m * 2.0) / Om_m), 2.0)) <= 5e-6) {
		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 ((((sin(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)) * (((l_m * 2.0d0) / om_m) ** 2.0d0)) <= 5d-6) 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(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)) * Math.pow(((l_m * 2.0) / Om_m), 2.0)) <= 5e-6) {
		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(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)) * math.pow(((l_m * 2.0) / Om_m), 2.0)) <= 5e-6:
		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((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)) * (Float64(Float64(l_m * 2.0) / Om_m) ^ 2.0)) <= 5e-6)
		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 ((((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)) * (((l_m * 2.0) / Om_m) ^ 2.0)) <= 5e-6)
		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[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(l$95$m * 2.0), $MachinePrecision] / Om$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 5e-6], 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)))) < 5.00000000000000041e-6

   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 Om around 0

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

      1. 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. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 3.2

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

   5. Applied rewrites 3.2%

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

   7. Applied rewrites 2.9%

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

   9. Applied rewrites 2.9%

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

   10. Taylor expanded in Om around inf

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

   12. Applied rewrites 99.4%

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

   if 5.00000000000000041e-6 < (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))

   1. Initial program 95.1%

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

   3. Taylor expanded in Om around 0

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

   5. Applied rewrites 98.1%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} Om_m = \left|Om\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} t_0 := \frac{l\_m \cdot 2}{Om\_m}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+106}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(t\_0, \mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right) \cdot t\_0, 1\right)}}, 0.5, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

Om_m = (fabs.f64 Om)
l_m = (fabs.f64 l)
(FPCore (l_m Om_m kx ky)
 :precision binary64
 (let* ((t_0 (/ (* l_m 2.0) Om_m)))
   (if (<= t_0 2e+106)
     (sqrt
      (fma
       (sqrt
        (/
         1.0
         (fma t_0 (* (fma (sin kx) (sin kx) (pow (sin ky) 2.0)) t_0) 1.0)))
       0.5
       0.5))
     (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 t_0 = (l_m * 2.0) / Om_m;
	double tmp;
	if (t_0 <= 2e+106) {
		tmp = sqrt(fma(sqrt((1.0 / fma(t_0, (fma(sin(kx), sin(kx), pow(sin(ky), 2.0)) * t_0), 1.0))), 0.5, 0.5));
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Julia

Om_m = abs(Om)
l_m = abs(l)
function code(l_m, Om_m, kx, ky)
	t_0 = Float64(Float64(l_m * 2.0) / Om_m)
	tmp = 0.0
	if (t_0 <= 2e+106)
		tmp = sqrt(fma(sqrt(Float64(1.0 / fma(t_0, Float64(fma(sin(kx), sin(kx), (sin(ky) ^ 2.0)) * t_0), 1.0))), 0.5, 0.5));
	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_] := Block[{t$95$0 = N[(N[(l$95$m * 2.0), $MachinePrecision] / Om$95$m), $MachinePrecision]}, If[LessEqual[t$95$0, 2e+106], N[Sqrt[N[(N[Sqrt[N[(1.0 / N[(t$95$0 * N[(N[(N[Sin[kx], $MachinePrecision] * N[Sin[kx], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]], $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) l) Om) < 2.00000000000000018e106

   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

   3. Taylor expanded in kx around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 81.4%

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

   7. Applied rewrites 98.8%

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

   if 2.00000000000000018e106 < (/.f64 (*.f64 #s(literal 2 binary64) l) Om)

   1. Initial program 95.3%

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

   3. Taylor expanded in Om around 0

      \[\leadsto \sqrt{\color{blue}{\frac{1}{2}}} \]
   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 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\ell \cdot 2}{Om} \leq 2 \cdot 10^{+106}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(\frac{\ell \cdot 2}{Om}, \mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right) \cdot \frac{\ell \cdot 2}{Om}, 1\right)}}, 0.5, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.2% 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 98.0%

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

3. Taylor expanded in Om around 0

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

   1. 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. unpow2 N/A

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

   6. unpow2 N/A

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

   7. lower-hypot.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. lower-sin.f64 41.3

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

5. Applied rewrites 41.3%

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

7. Applied rewrites 35.3%

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

9. Applied rewrites 35.3%

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

10. Taylor expanded in Om around inf

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

12. Applied rewrites 67.7%

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

Reproduce

herbie shell --seed 2024276 
(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))))))))))