Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.5% → 99.3%

Time: 13.4s

Alternatives: 7

Speedup: 1.0×

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: 99.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} \leq 2 \cdot 10^{+25}:\\ \;\;\;\;\sqrt{\left(\frac{1}{\sqrt{\mathsf{fma}\left(4 \cdot \frac{\ell}{Om}, \left(0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \left(0.5 - \cos \left(kx + kx\right) \cdot 0.5\right)\right)\right) \cdot \frac{\ell}{Om}, 1\right)}} + 1\right) \cdot \frac{1}{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<=
      (* (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)) (pow (/ (* l 2.0) Om) 2.0))
      2e+25)
   (sqrt
    (*
     (+
      (/
       1.0
       (sqrt
        (fma
         (* 4.0 (/ l Om))
         (*
          (- 0.5 (- (* (cos (+ ky ky)) 0.5) (- 0.5 (* (cos (+ kx kx)) 0.5))))
          (/ l Om))
         1.0)))
      1.0)
     (/ 1.0 2.0)))
   (sqrt 0.5)))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)) * pow(((l * 2.0) / Om), 2.0)) <= 2e+25) {
		tmp = sqrt((((1.0 / sqrt(fma((4.0 * (l / Om)), ((0.5 - ((cos((ky + ky)) * 0.5) - (0.5 - (cos((kx + kx)) * 0.5)))) * (l / Om)), 1.0))) + 1.0) * (1.0 / 2.0)));
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Float64(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)) * (Float64(Float64(l * 2.0) / Om) ^ 2.0)) <= 2e+25)
		tmp = sqrt(Float64(Float64(Float64(1.0 / sqrt(fma(Float64(4.0 * Float64(l / Om)), Float64(Float64(0.5 - Float64(Float64(cos(Float64(ky + ky)) * 0.5) - Float64(0.5 - Float64(cos(Float64(kx + kx)) * 0.5)))) * Float64(l / Om)), 1.0))) + 1.0) * Float64(1.0 / 2.0)));
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

Wolfram

code[l_, Om_, 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 * 2.0), $MachinePrecision] / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 2e+25], N[Sqrt[N[(N[(N[(1.0 / N[Sqrt[N[(N[(4.0 * N[(l / Om), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 - N[(N[(N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] - N[(0.5 - N[(N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))) < 2.00000000000000018e25

   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 98.9%

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

   if 2.00000000000000018e25 < (*.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 96.3%

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
   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 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} \leq 2 \cdot 10^{+25}:\\ \;\;\;\;\sqrt{\left(\frac{1}{\sqrt{\mathsf{fma}\left(4 \cdot \frac{\ell}{Om}, \left(0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \left(0.5 - \cos \left(kx + kx\right) \cdot 0.5\right)\right)\right) \cdot \frac{\ell}{Om}, 1\right)}} + 1\right) \cdot \frac{1}{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[l_, Om_, 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 * 2.0), $MachinePrecision] / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 2.0], 1.0, N[Sqrt[N[(-0.25 * N[(Om / N[(N[Sin[ky], $MachinePrecision] * l), $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)))) < 2

   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 0.4%

      \[\leadsto \sqrt{\color{blue}{\frac{0.25 \cdot \frac{1}{\mathsf{fma}\left(0.5 - \left(0.5 \cdot \cos \left(ky + ky\right) - \left(0.5 - 0.5 \cdot \cos \left(kx + kx\right)\right)\right), \frac{\left(\ell \cdot \ell\right) \cdot 4}{Om \cdot Om}, 1\right)} - 0.25}{\frac{0.5}{\sqrt{\mathsf{fma}\left(0.5 - \left(0.5 \cdot \cos \left(ky + ky\right) - \left(0.5 - 0.5 \cdot \cos \left(kx + kx\right)\right)\right), \frac{\left(\ell \cdot \ell\right) \cdot 4}{Om \cdot Om}, 1\right)}} - 0.5}}} \]

   5. Taylor expanded in Om around inf

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

   7. Applied rewrites 98.6%

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

   8. Taylor expanded in Om around inf

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

   10. Applied rewrites 98.6%

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

   if 2 < (*.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 96.5%

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

   3. Taylor expanded in kx around 0

      \[\leadsto \sqrt{\color{blue}{\frac{1}{2} \cdot \left(1 + \sqrt{\frac{1}{1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}}}\right)}} \]
   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 67.5%

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

   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 86.1%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-0.25, \frac{Om}{\sin ky \cdot \ell}, 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} \leq 0.1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{Om}{\ell} \cdot 0.25, \frac{\sqrt{0.5}}{ky}, 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<=
      (* (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)) (pow (/ (* l 2.0) Om) 2.0))
      0.1)
   1.0
   (sqrt (fma (* (/ Om l) 0.25) (/ (sqrt 0.5) ky) 0.5))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)) * pow(((l * 2.0) / Om), 2.0)) <= 0.1) {
		tmp = 1.0;
	} else {
		tmp = sqrt(fma(((Om / l) * 0.25), (sqrt(0.5) / ky), 0.5));
	}
	return tmp;
}

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Float64(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)) * (Float64(Float64(l * 2.0) / Om) ^ 2.0)) <= 0.1)
		tmp = 1.0;
	else
		tmp = sqrt(fma(Float64(Float64(Om / l) * 0.25), Float64(sqrt(0.5) / ky), 0.5));
	end
	return tmp
end

Wolfram

code[l_, Om_, 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 * 2.0), $MachinePrecision] / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 0.1], 1.0, N[Sqrt[N[(N[(N[(Om / l), $MachinePrecision] * 0.25), $MachinePrecision] * N[(N[Sqrt[0.5], $MachinePrecision] / ky), $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)))) < 0.10000000000000001

   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 0.4%

      \[\leadsto \sqrt{\color{blue}{\frac{0.25 \cdot \frac{1}{\mathsf{fma}\left(0.5 - \left(0.5 \cdot \cos \left(ky + ky\right) - \left(0.5 - 0.5 \cdot \cos \left(kx + kx\right)\right)\right), \frac{\left(\ell \cdot \ell\right) \cdot 4}{Om \cdot Om}, 1\right)} - 0.25}{\frac{0.5}{\sqrt{\mathsf{fma}\left(0.5 - \left(0.5 \cdot \cos \left(ky + ky\right) - \left(0.5 - 0.5 \cdot \cos \left(kx + kx\right)\right)\right), \frac{\left(\ell \cdot \ell\right) \cdot 4}{Om \cdot Om}, 1\right)}} - 0.5}}} \]

   5. Taylor expanded in Om around inf

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

   7. Applied rewrites 99.2%

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

   8. Taylor expanded in Om around inf

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

   10. Applied rewrites 99.2%

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

   if 0.10000000000000001 < (*.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 96.6%

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

   4. Applied rewrites 72.9%

      \[\leadsto \sqrt{\color{blue}{\frac{0.25 \cdot \frac{1}{\mathsf{fma}\left(0.5 - \left(0.5 \cdot \cos \left(ky + ky\right) - \left(0.5 - 0.5 \cdot \cos \left(kx + kx\right)\right)\right), \frac{\left(\ell \cdot \ell\right) \cdot 4}{Om \cdot Om}, 1\right)} - 0.25}{\frac{0.5}{\sqrt{\mathsf{fma}\left(0.5 - \left(0.5 \cdot \cos \left(ky + ky\right) - \left(0.5 - 0.5 \cdot \cos \left(kx + kx\right)\right)\right), \frac{\left(\ell \cdot \ell\right) \cdot 4}{Om \cdot Om}, 1\right)}} - 0.5}}} \]

   5. Taylor expanded in Om around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 82.7%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(0.25 \cdot \frac{Om}{\ell}, \sqrt{\frac{1}{1 - 0.5 \cdot \left(\cos \left(2 \cdot kx\right) + \cos \left(2 \cdot ky\right)\right)}}, 0.5\right)}} \]
   8. Step-by-step derivation

   9. Applied rewrites 62.2%

      \[\leadsto \sqrt{\mathsf{fma}\left(0.25 \cdot \frac{Om}{\ell}, \sqrt{\frac{1}{1 - 0.5 \cdot \left(\cos \left(ky + ky\right) + \cos \left(ky + ky\right)\right)}}, 0.5\right)} \]

   10. Taylor expanded in ky around 0

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

   12. Applied rewrites 85.4%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} \leq 0.1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{Om}{\ell} \cdot 0.25, \frac{\sqrt{0.5}}{ky}, 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double l, double Om, double kx, double ky) {
	return sqrt((((1.0 / sqrt((((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)) * pow(((l * 2.0) / Om), 2.0)) + 1.0))) + 1.0) * (1.0 / 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 / sqrt(((((sin(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)) * (((l * 2.0d0) / om) ** 2.0d0)) + 1.0d0))) + 1.0d0) * (1.0d0 / 2.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

3. Final simplification 98.4%

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

Alternative 5: 91.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} \leq 0.1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{Om}{ky \cdot \ell}, 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<=
      (* (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)) (pow (/ (* l 2.0) Om) 2.0))
      0.1)
   1.0
   (sqrt (fma 0.25 (/ Om (* ky l)) 0.5))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)) * pow(((l * 2.0) / Om), 2.0)) <= 0.1) {
		tmp = 1.0;
	} else {
		tmp = sqrt(fma(0.25, (Om / (ky * l)), 0.5));
	}
	return tmp;
}

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Float64(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)) * (Float64(Float64(l * 2.0) / Om) ^ 2.0)) <= 0.1)
		tmp = 1.0;
	else
		tmp = sqrt(fma(0.25, Float64(Om / Float64(ky * l)), 0.5));
	end
	return tmp
end

Wolfram

code[l_, Om_, 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 * 2.0), $MachinePrecision] / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 0.1], 1.0, N[Sqrt[N[(0.25 * N[(Om / N[(ky * l), $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)))) < 0.10000000000000001

   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 0.4%

      \[\leadsto \sqrt{\color{blue}{\frac{0.25 \cdot \frac{1}{\mathsf{fma}\left(0.5 - \left(0.5 \cdot \cos \left(ky + ky\right) - \left(0.5 - 0.5 \cdot \cos \left(kx + kx\right)\right)\right), \frac{\left(\ell \cdot \ell\right) \cdot 4}{Om \cdot Om}, 1\right)} - 0.25}{\frac{0.5}{\sqrt{\mathsf{fma}\left(0.5 - \left(0.5 \cdot \cos \left(ky + ky\right) - \left(0.5 - 0.5 \cdot \cos \left(kx + kx\right)\right)\right), \frac{\left(\ell \cdot \ell\right) \cdot 4}{Om \cdot Om}, 1\right)}} - 0.5}}} \]

   5. Taylor expanded in Om around inf

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

   7. Applied rewrites 99.2%

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

   8. Taylor expanded in Om around inf

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

   10. Applied rewrites 99.2%

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

   if 0.10000000000000001 < (*.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 96.6%

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

   3. Taylor expanded in kx around 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 66.9%

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

   6. Taylor expanded in Om around 0

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

   8. Applied rewrites 85.2%

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

   9. Taylor expanded in ky around 0

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

   11. Applied rewrites 85.3%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} \leq 0.1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{Om}{ky \cdot \ell}, 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} \leq 3.8:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<=
      (* (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)) (pow (/ (* l 2.0) Om) 2.0))
      3.8)
   1.0
   (sqrt 0.5)))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)) * pow(((l * 2.0) / Om), 2.0)) <= 3.8) {
		tmp = 1.0;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

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
    real(8) :: tmp
    if ((((sin(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)) * (((l * 2.0d0) / om) ** 2.0d0)) <= 3.8d0) then
        tmp = 1.0d0
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)) * Math.pow(((l * 2.0) / Om), 2.0)) <= 3.8) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if ((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)) * math.pow(((l * 2.0) / Om), 2.0)) <= 3.8:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Float64(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)) * (Float64(Float64(l * 2.0) / Om) ^ 2.0)) <= 3.8)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if ((((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)) * (((l * 2.0) / Om) ^ 2.0)) <= 3.8)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, 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 * 2.0), $MachinePrecision] / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.8], 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)))) < 3.7999999999999998

   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 0.4%

      \[\leadsto \sqrt{\color{blue}{\frac{0.25 \cdot \frac{1}{\mathsf{fma}\left(0.5 - \left(0.5 \cdot \cos \left(ky + ky\right) - \left(0.5 - 0.5 \cdot \cos \left(kx + kx\right)\right)\right), \frac{\left(\ell \cdot \ell\right) \cdot 4}{Om \cdot Om}, 1\right)} - 0.25}{\frac{0.5}{\sqrt{\mathsf{fma}\left(0.5 - \left(0.5 \cdot \cos \left(ky + ky\right) - \left(0.5 - 0.5 \cdot \cos \left(kx + kx\right)\right)\right), \frac{\left(\ell \cdot \ell\right) \cdot 4}{Om \cdot Om}, 1\right)}} - 0.5}}} \]

   5. Taylor expanded in Om around inf

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

   7. Applied rewrites 98.6%

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

   8. Taylor expanded in Om around inf

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

   10. Applied rewrites 98.6%

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

   if 3.7999999999999998 < (*.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 96.5%

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

   3. Taylor expanded in Om around 0

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

   5. Applied rewrites 97.3%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} \leq 3.8:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.4% accurate, 581.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (l Om kx ky) :precision binary64 1.0)

C

double code(double l, double Om, double kx, double ky) {
	return 1.0;
}

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 = 1.0d0
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	return 1.0;
}

Python

def code(l, Om, kx, ky):
	return 1.0

Julia

function code(l, Om, kx, ky)
	return 1.0
end

MATLAB

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

Wolfram

code[l_, Om_, kx_, ky_] := 1.0

Derivation

1. Initial program 98.4%

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

4. Applied rewrites 33.8%

   \[\leadsto \sqrt{\color{blue}{\frac{0.25 \cdot \frac{1}{\mathsf{fma}\left(0.5 - \left(0.5 \cdot \cos \left(ky + ky\right) - \left(0.5 - 0.5 \cdot \cos \left(kx + kx\right)\right)\right), \frac{\left(\ell \cdot \ell\right) \cdot 4}{Om \cdot Om}, 1\right)} - 0.25}{\frac{0.5}{\sqrt{\mathsf{fma}\left(0.5 - \left(0.5 \cdot \cos \left(ky + ky\right) - \left(0.5 - 0.5 \cdot \cos \left(kx + kx\right)\right)\right), \frac{\left(\ell \cdot \ell\right) \cdot 4}{Om \cdot Om}, 1\right)}} - 0.5}}} \]

5. Taylor expanded in Om around inf

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

7. Applied rewrites 62.7%

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

8. Taylor expanded in Om around inf

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

10. Applied rewrites 62.7%

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

Reproduce

herbie shell --seed 2024240 
(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))))))))))