Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.5% → 98.5%

Time: 10.3s

Alternatives: 5

Speedup: N/A×

Specification

Math

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

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (sqrt
  (*
   (/ 1.0 2.0)
   (+
    1.0
    (/
     1.0
     (sqrt
      (+
       1.0
       (*
        (pow (/ (* 2.0 l) Om) 2.0)
        (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))))))

C

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

Fortran

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    code = sqrt(((1.0d0 / 2.0d0) * (1.0d0 + (1.0d0 / sqrt((1.0d0 + ((((2.0d0 * l) / om) ** 2.0d0) * ((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))))))))
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	return Math.sqrt(((1.0 / 2.0) * (1.0 + (1.0 / Math.sqrt((1.0 + (Math.pow(((2.0 * l) / Om), 2.0) * (Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))))))));
}

Python

def code(l, Om, kx, ky):
	return math.sqrt(((1.0 / 2.0) * (1.0 + (1.0 / math.sqrt((1.0 + (math.pow(((2.0 * l) / Om), 2.0) * (math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))))))))

Julia

function code(l, Om, kx, ky)
	return sqrt(Float64(Float64(1.0 / 2.0) * Float64(1.0 + Float64(1.0 / sqrt(Float64(1.0 + Float64((Float64(Float64(2.0 * l) / Om) ^ 2.0) * Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))))))))
end

MATLAB

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

Wolfram

code[l_, Om_, kx_, ky_] := N[Sqrt[N[(N[(1.0 / 2.0), $MachinePrecision] * N[(1.0 + N[(1.0 / N[Sqrt[N[(1.0 + N[(N[Power[N[(N[(2.0 * l), $MachinePrecision] / Om), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 98.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (sqrt
  (*
   (/ 1.0 2.0)
   (+
    1.0
    (/
     1.0
     (sqrt
      (+
       1.0
       (*
        (pow (/ (* 2.0 l) Om) 2.0)
        (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))))))

C

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

Fortran

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    code = sqrt(((1.0d0 / 2.0d0) * (1.0d0 + (1.0d0 / sqrt((1.0d0 + ((((2.0d0 * l) / om) ** 2.0d0) * ((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))))))))
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	return Math.sqrt(((1.0 / 2.0) * (1.0 + (1.0 / Math.sqrt((1.0 + (Math.pow(((2.0 * l) / Om), 2.0) * (Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))))))));
}

Python

def code(l, Om, kx, ky):
	return math.sqrt(((1.0 / 2.0) * (1.0 + (1.0 / math.sqrt((1.0 + (math.pow(((2.0 * l) / Om), 2.0) * (math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))))))))

Julia

function code(l, Om, kx, ky)
	return sqrt(Float64(Float64(1.0 / 2.0) * Float64(1.0 + Float64(1.0 / sqrt(Float64(1.0 + Float64((Float64(Float64(2.0 * l) / Om) ^ 2.0) * Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))))))))
end

MATLAB

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

Wolfram

code[l_, Om_, kx_, ky_] := N[Sqrt[N[(N[(1.0 / 2.0), $MachinePrecision] * N[(1.0 + N[(1.0 / N[Sqrt[N[(1.0 + N[(N[Power[N[(N[(2.0 * l), $MachinePrecision] / Om), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 98.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} kx_m = \left|kx\right| \\ [l, Om, kx_m, ky] = \mathsf{sort}([l, Om, kx_m, ky])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx\_m}^{2} + {\sin ky}^{2}\right) \leq 10^{+15}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(\frac{1 - \cos \left(2 \cdot ky\right)}{2 \cdot Om} \cdot \left(\frac{\ell}{Om} \cdot \ell\right), 4, 1\right)\right)}^{-1}}, 0.5, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

kx_m = (fabs.f64 kx)
NOTE: l, Om, kx_m, and ky should be sorted in increasing order before calling this function.
(FPCore (l Om kx_m ky)
 :precision binary64
 (if (<=
      (*
       (pow (/ (* 2.0 l) Om) 2.0)
       (+ (pow (sin kx_m) 2.0) (pow (sin ky) 2.0)))
      1e+15)
   (sqrt
    (fma
     (sqrt
      (pow
       (fma (* (/ (- 1.0 (cos (* 2.0 ky))) (* 2.0 Om)) (* (/ l Om) l)) 4.0 1.0)
       -1.0))
     0.5
     0.5))
   (sqrt 0.5)))

C

kx_m = fabs(kx);
assert(l < Om && Om < kx_m && kx_m < ky);
double code(double l, double Om, double kx_m, double ky) {
	double tmp;
	if ((pow(((2.0 * l) / Om), 2.0) * (pow(sin(kx_m), 2.0) + pow(sin(ky), 2.0))) <= 1e+15) {
		tmp = sqrt(fma(sqrt(pow(fma((((1.0 - cos((2.0 * ky))) / (2.0 * Om)) * ((l / Om) * l)), 4.0, 1.0), -1.0)), 0.5, 0.5));
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Julia

kx_m = abs(kx)
l, Om, kx_m, ky = sort([l, Om, kx_m, ky])
function code(l, Om, kx_m, ky)
	tmp = 0.0
	if (Float64((Float64(Float64(2.0 * l) / Om) ^ 2.0) * Float64((sin(kx_m) ^ 2.0) + (sin(ky) ^ 2.0))) <= 1e+15)
		tmp = sqrt(fma(sqrt((fma(Float64(Float64(Float64(1.0 - cos(Float64(2.0 * ky))) / Float64(2.0 * Om)) * Float64(Float64(l / Om) * l)), 4.0, 1.0) ^ -1.0)), 0.5, 0.5));
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

Wolfram

kx_m = N[Abs[kx], $MachinePrecision]
NOTE: l, Om, kx_m, and ky should be sorted in increasing order before calling this function.
code[l_, Om_, kx$95$m_, ky_] := If[LessEqual[N[(N[Power[N[(N[(2.0 * l), $MachinePrecision] / Om), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[N[Sin[kx$95$m], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+15], N[Sqrt[N[(N[Sqrt[N[Power[N[(N[(N[(N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * Om), $MachinePrecision]), $MachinePrecision] * N[(N[(l / Om), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * 4.0 + 1.0), $MachinePrecision], -1.0], $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)))) < 1e15

   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

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 92.7%

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

   7. Applied rewrites 97.9%

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

   9. Applied rewrites 97.6%

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

   if 1e15 < (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))

   1. Initial program 95.9%

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

   3. Taylor expanded in l around inf

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

   5. Applied rewrites 98.7%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right) \leq 10^{+15}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(\frac{1 - \cos \left(2 \cdot ky\right)}{2 \cdot Om} \cdot \left(\frac{\ell}{Om} \cdot \ell\right), 4, 1\right)\right)}^{-1}}, 0.5, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.2% accurate, 0.9× speedup

Math

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

FPCore

kx_m = (fabs.f64 kx)
NOTE: l, Om, kx_m, and ky should be sorted in increasing order before calling this function.
(FPCore (l Om kx_m ky)
 :precision binary64
 (if (<=
      (*
       (pow (/ (* 2.0 l) Om) 2.0)
       (+ (pow (sin kx_m) 2.0) (pow (sin ky) 2.0)))
      0.02)
   1.0
   (sqrt (fma (/ Om (* (sin ky) l)) -0.25 0.5))))

C

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

Julia

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

Wolfram

kx_m = N[Abs[kx], $MachinePrecision]
NOTE: l, Om, kx_m, and ky should be sorted in increasing order before calling this function.
code[l_, Om_, kx$95$m_, ky_] := If[LessEqual[N[(N[Power[N[(N[(2.0 * l), $MachinePrecision] / Om), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[N[Sin[kx$95$m], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.02], 1.0, N[Sqrt[N[(N[(Om / N[(N[Sin[ky], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * -0.25 + 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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{{\left({\left(\mathsf{fma}\left({\left(\mathsf{fma}\left(\mathsf{fma}\left(\sin ky, \sin ky, {\sin kx}^{2}\right), {\left(\frac{\ell \cdot 2}{Om}\right)}^{2}, 1\right)\right)}^{-0.5}, 0.5, 0.5\right)\right)}^{0.25}\right)}^{2}} \]

   5. Taylor expanded in kx around 0

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

      1. lower-pow.f64 N/A

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

      2. lower-sin.f64 99.8

         \[\leadsto {\left({\left(\mathsf{fma}\left({\left(\mathsf{fma}\left({\color{blue}{\sin ky}}^{2}, {\left(\frac{\ell \cdot 2}{Om}\right)}^{2}, 1\right)\right)}^{-0.5}, 0.5, 0.5\right)\right)}^{0.25}\right)}^{2} \]

   7. Applied rewrites 99.8%

      \[\leadsto {\left({\left(\mathsf{fma}\left({\left(\mathsf{fma}\left(\color{blue}{{\sin ky}^{2}}, {\left(\frac{\ell \cdot 2}{Om}\right)}^{2}, 1\right)\right)}^{-0.5}, 0.5, 0.5\right)\right)}^{0.25}\right)}^{2} \]

   8. Taylor expanded in l around 0

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

   10. Applied rewrites 98.9%

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

   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 96.0%

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

   3. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 77.3%

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

   6. Taylor expanded in l around -inf

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

   8. Applied rewrites 85.4%

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

Alternative 3: 98.3% accurate, 1.1× speedup

Math

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

FPCore

kx_m = (fabs.f64 kx)
NOTE: l, Om, kx_m, and ky should be sorted in increasing order before calling this function.
(FPCore (l Om kx_m ky)
 :precision binary64
 (if (<=
      (*
       (pow (/ (* 2.0 l) Om) 2.0)
       (+ (pow (sin kx_m) 2.0) (pow (sin ky) 2.0)))
      0.02)
   1.0
   (sqrt 0.5)))

C

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

Fortran

kx_m = abs(kx)
NOTE: l, Om, kx_m, and ky should be sorted in increasing order before calling this function.
real(8) function code(l, om, kx_m, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx_m
    real(8), intent (in) :: ky
    real(8) :: tmp
    if (((((2.0d0 * l) / om) ** 2.0d0) * ((sin(kx_m) ** 2.0d0) + (sin(ky) ** 2.0d0))) <= 0.02d0) then
        tmp = 1.0d0
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

kx_m = abs(kx);
l, Om, kx_m, ky = num2cell(sort([l, Om, kx_m, ky])){:}
function tmp_2 = code(l, Om, kx_m, ky)
	tmp = 0.0;
	if (((((2.0 * l) / Om) ^ 2.0) * ((sin(kx_m) ^ 2.0) + (sin(ky) ^ 2.0))) <= 0.02)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{{\left({\left(\mathsf{fma}\left({\left(\mathsf{fma}\left(\mathsf{fma}\left(\sin ky, \sin ky, {\sin kx}^{2}\right), {\left(\frac{\ell \cdot 2}{Om}\right)}^{2}, 1\right)\right)}^{-0.5}, 0.5, 0.5\right)\right)}^{0.25}\right)}^{2}} \]

   5. Taylor expanded in kx around 0

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

      1. lower-pow.f64 N/A

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

      2. lower-sin.f64 99.8

         \[\leadsto {\left({\left(\mathsf{fma}\left({\left(\mathsf{fma}\left({\color{blue}{\sin ky}}^{2}, {\left(\frac{\ell \cdot 2}{Om}\right)}^{2}, 1\right)\right)}^{-0.5}, 0.5, 0.5\right)\right)}^{0.25}\right)}^{2} \]

   7. Applied rewrites 99.8%

      \[\leadsto {\left({\left(\mathsf{fma}\left({\left(\mathsf{fma}\left(\color{blue}{{\sin ky}^{2}}, {\left(\frac{\ell \cdot 2}{Om}\right)}^{2}, 1\right)\right)}^{-0.5}, 0.5, 0.5\right)\right)}^{0.25}\right)}^{2} \]

   8. Taylor expanded in l around 0

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

   10. Applied rewrites 98.9%

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

   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 96.0%

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

   3. Taylor expanded in l around inf

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

   5. Applied rewrites 97.8%

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

Alternative 4: 93.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} kx_m = \left|kx\right| \\ [l, Om, kx_m, ky] = \mathsf{sort}([l, Om, kx_m, ky])\\ \\ \sqrt{\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(\frac{{\sin ky}^{2}}{Om} \cdot \left(\frac{\ell}{Om} \cdot \ell\right), 4, 1\right)\right)}^{-1}}, 0.5, 0.5\right)} \end{array} \]

FPCore

kx_m = (fabs.f64 kx)
NOTE: l, Om, kx_m, and ky should be sorted in increasing order before calling this function.
(FPCore (l Om kx_m ky)
 :precision binary64
 (sqrt
  (fma
   (sqrt (pow (fma (* (/ (pow (sin ky) 2.0) Om) (* (/ l Om) l)) 4.0 1.0) -1.0))
   0.5
   0.5)))

C

kx_m = fabs(kx);
assert(l < Om && Om < kx_m && kx_m < ky);
double code(double l, double Om, double kx_m, double ky) {
	return sqrt(fma(sqrt(pow(fma(((pow(sin(ky), 2.0) / Om) * ((l / Om) * l)), 4.0, 1.0), -1.0)), 0.5, 0.5));
}

Julia

kx_m = abs(kx)
l, Om, kx_m, ky = sort([l, Om, kx_m, ky])
function code(l, Om, kx_m, ky)
	return sqrt(fma(sqrt((fma(Float64(Float64((sin(ky) ^ 2.0) / Om) * Float64(Float64(l / Om) * l)), 4.0, 1.0) ^ -1.0)), 0.5, 0.5))
end

Wolfram

kx_m = N[Abs[kx], $MachinePrecision]
NOTE: l, Om, kx_m, and ky should be sorted in increasing order before calling this function.
code[l_, Om_, kx$95$m_, ky_] := N[Sqrt[N[(N[Sqrt[N[Power[N[(N[(N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] / Om), $MachinePrecision] * N[(N[(l / Om), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * 4.0 + 1.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 98.0%

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

3. Taylor expanded in kx around 0

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f64 N/A

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

5. Applied rewrites 85.9%

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

7. Applied rewrites 89.3%

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

8. Final simplification 89.3%

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

Alternative 5: 62.9% accurate, 581.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

4. Applied rewrites 97.3%

   \[\leadsto \color{blue}{{\left({\left(\mathsf{fma}\left({\left(\mathsf{fma}\left(\mathsf{fma}\left(\sin ky, \sin ky, {\sin kx}^{2}\right), {\left(\frac{\ell \cdot 2}{Om}\right)}^{2}, 1\right)\right)}^{-0.5}, 0.5, 0.5\right)\right)}^{0.25}\right)}^{2}} \]

5. Taylor expanded in kx around 0

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

   1. lower-pow.f64 N/A

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

   2. lower-sin.f64 88.3

      \[\leadsto {\left({\left(\mathsf{fma}\left({\left(\mathsf{fma}\left({\color{blue}{\sin ky}}^{2}, {\left(\frac{\ell \cdot 2}{Om}\right)}^{2}, 1\right)\right)}^{-0.5}, 0.5, 0.5\right)\right)}^{0.25}\right)}^{2} \]

7. Applied rewrites 88.3%

   \[\leadsto {\left({\left(\mathsf{fma}\left({\left(\mathsf{fma}\left(\color{blue}{{\sin ky}^{2}}, {\left(\frac{\ell \cdot 2}{Om}\right)}^{2}, 1\right)\right)}^{-0.5}, 0.5, 0.5\right)\right)}^{0.25}\right)}^{2} \]

8. Taylor expanded in l around 0

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

10. Applied rewrites 61.0%

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

Reproduce

herbie shell --seed 2024326 
(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))))))))))