Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.5% → 100.0%

Time: 20.4s

Alternatives: 7

Speedup: 1.4×

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: 100.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \sqrt{0.5 + 0.5 \cdot {\left(1 + {\left(\frac{2}{\frac{Om}{\ell}} \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}^{2}\right)}^{-0.5}} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (sqrt
  (+
   0.5
   (*
    0.5
    (pow
     (+ 1.0 (pow (* (/ 2.0 (/ Om l)) (hypot (sin kx) (sin ky))) 2.0))
     -0.5)))))

C

double code(double l, double Om, double kx, double ky) {
	return sqrt((0.5 + (0.5 * pow((1.0 + pow(((2.0 / (Om / l)) * hypot(sin(kx), sin(ky))), 2.0)), -0.5))));
}

Java

public static double code(double l, double Om, double kx, double ky) {
	return Math.sqrt((0.5 + (0.5 * Math.pow((1.0 + Math.pow(((2.0 / (Om / l)) * Math.hypot(Math.sin(kx), Math.sin(ky))), 2.0)), -0.5))));
}

Python

def code(l, Om, kx, ky):
	return math.sqrt((0.5 + (0.5 * math.pow((1.0 + math.pow(((2.0 / (Om / l)) * math.hypot(math.sin(kx), math.sin(ky))), 2.0)), -0.5))))

Julia

function code(l, Om, kx, ky)
	return sqrt(Float64(0.5 + Float64(0.5 * (Float64(1.0 + (Float64(Float64(2.0 / Float64(Om / l)) * hypot(sin(kx), sin(ky))) ^ 2.0)) ^ -0.5))))
end

MATLAB

function tmp = code(l, Om, kx, ky)
	tmp = sqrt((0.5 + (0.5 * ((1.0 + (((2.0 / (Om / l)) * hypot(sin(kx), sin(ky))) ^ 2.0)) ^ -0.5))));
end

Wolfram

code[l_, Om_, kx_, ky_] := N[Sqrt[N[(0.5 + N[(0.5 * N[Power[N[(1.0 + N[Power[N[(N[(2.0 / N[(Om / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 97.6%

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

3. Simplified 97.6%

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

   1. inv-pow 97.6%

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

   2. sqrt-pow2 97.6%

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

6. Applied egg-rr 100.0%

   \[\leadsto \sqrt{0.5 + 0.5 \cdot \color{blue}{{\left(1 + {\left(\frac{2}{\frac{Om}{\ell}} \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}^{2}\right)}^{-0.5}}} \]
7. Add Preprocessing

Alternative 2: 100.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \sqrt{0.5 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{2}{\frac{Om}{\ell}} \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (sqrt
  (+
   0.5
   (*
    0.5
    (/ 1.0 (hypot 1.0 (* (/ 2.0 (/ Om l)) (hypot (sin kx) (sin ky)))))))))

C

double code(double l, double Om, double kx, double ky) {
	return sqrt((0.5 + (0.5 * (1.0 / hypot(1.0, ((2.0 / (Om / l)) * hypot(sin(kx), sin(ky))))))));
}

Java

public static double code(double l, double Om, double kx, double ky) {
	return Math.sqrt((0.5 + (0.5 * (1.0 / Math.hypot(1.0, ((2.0 / (Om / l)) * Math.hypot(Math.sin(kx), Math.sin(ky))))))));
}

Python

def code(l, Om, kx, ky):
	return math.sqrt((0.5 + (0.5 * (1.0 / math.hypot(1.0, ((2.0 / (Om / l)) * math.hypot(math.sin(kx), math.sin(ky))))))))

Julia

function code(l, Om, kx, ky)
	return sqrt(Float64(0.5 + Float64(0.5 * Float64(1.0 / hypot(1.0, Float64(Float64(2.0 / Float64(Om / l)) * hypot(sin(kx), sin(ky))))))))
end

MATLAB

function tmp = code(l, Om, kx, ky)
	tmp = sqrt((0.5 + (0.5 * (1.0 / hypot(1.0, ((2.0 / (Om / l)) * hypot(sin(kx), sin(ky))))))));
end

Wolfram

code[l_, Om_, kx_, ky_] := N[Sqrt[N[(0.5 + N[(0.5 * N[(1.0 / N[Sqrt[1.0 ^ 2 + N[(N[(2.0 / N[(Om / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 97.6%

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

3. Simplified 97.6%

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

   1. add-sqr-sqrt 97.6%

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

   2. hypot-1-def 97.6%

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

   3. sqrt-prod 97.6%

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

   4. sqrt-pow1 98.3%

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

   5. metadata-eval 98.3%

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

   6. pow1 98.3%

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

   7. clear-num 98.3%

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

   8. un-div-inv 98.3%

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

   9. unpow2 98.3%

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

   10. unpow2 98.3%

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

   11. hypot-define 100.0%

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

6. Applied egg-rr 100.0%

   \[\leadsto \sqrt{0.5 + 0.5 \cdot \frac{1}{\color{blue}{\mathsf{hypot}\left(1, \frac{2}{\frac{Om}{\ell}} \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}}} \]
7. Add Preprocessing

Alternative 3: 93.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \sqrt{0.5 + 0.5 \cdot {\left(1 + {\left(\frac{2}{\frac{Om}{\ell}} \cdot \sin ky\right)}^{2}\right)}^{-0.5}} \end{array} \]

FPCore

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

C

double code(double l, double Om, double kx, double ky) {
	return sqrt((0.5 + (0.5 * pow((1.0 + pow(((2.0 / (Om / l)) * sin(ky)), 2.0)), -0.5))));
}

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((0.5d0 + (0.5d0 * ((1.0d0 + (((2.0d0 / (om / l)) * sin(ky)) ** 2.0d0)) ** (-0.5d0)))))
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	return Math.sqrt((0.5 + (0.5 * Math.pow((1.0 + Math.pow(((2.0 / (Om / l)) * Math.sin(ky)), 2.0)), -0.5))));
}

Python

def code(l, Om, kx, ky):
	return math.sqrt((0.5 + (0.5 * math.pow((1.0 + math.pow(((2.0 / (Om / l)) * math.sin(ky)), 2.0)), -0.5))))

Julia

function code(l, Om, kx, ky)
	return sqrt(Float64(0.5 + Float64(0.5 * (Float64(1.0 + (Float64(Float64(2.0 / Float64(Om / l)) * sin(ky)) ^ 2.0)) ^ -0.5))))
end

MATLAB

function tmp = code(l, Om, kx, ky)
	tmp = sqrt((0.5 + (0.5 * ((1.0 + (((2.0 / (Om / l)) * sin(ky)) ^ 2.0)) ^ -0.5))));
end

Wolfram

code[l_, Om_, kx_, ky_] := N[Sqrt[N[(0.5 + N[(0.5 * N[Power[N[(1.0 + N[Power[N[(N[(2.0 / N[(Om / l), $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 97.6%

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

3. Simplified 97.6%

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

   1. inv-pow 97.6%

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

   2. sqrt-pow2 97.6%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in kx around 0 95.6%

   \[\leadsto \sqrt{0.5 + 0.5 \cdot {\left(1 + {\left(\frac{2}{\frac{Om}{\ell}} \cdot \color{blue}{\sin ky}\right)}^{2}\right)}^{-0.5}} \]
8. Add Preprocessing

Alternative 4: 93.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{2}{\frac{\frac{Om}{\ell}}{\sin ky}}\right)}\right)}^{1.5}} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (cbrt (pow (+ 0.5 (/ 0.5 (hypot 1.0 (/ 2.0 (/ (/ Om l) (sin ky)))))) 1.5)))

C

double code(double l, double Om, double kx, double ky) {
	return cbrt(pow((0.5 + (0.5 / hypot(1.0, (2.0 / ((Om / l) / sin(ky)))))), 1.5));
}

Java

public static double code(double l, double Om, double kx, double ky) {
	return Math.cbrt(Math.pow((0.5 + (0.5 / Math.hypot(1.0, (2.0 / ((Om / l) / Math.sin(ky)))))), 1.5));
}

Julia

function code(l, Om, kx, ky)
	return cbrt((Float64(0.5 + Float64(0.5 / hypot(1.0, Float64(2.0 / Float64(Float64(Om / l) / sin(ky)))))) ^ 1.5))
end

Wolfram

code[l_, Om_, kx_, ky_] := N[Power[N[Power[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + N[(2.0 / N[(N[(Om / l), $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]

Derivation

1. Initial program 97.6%

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

3. Simplified 97.6%

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

   1. add-sqr-sqrt 97.6%

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

   2. hypot-1-def 97.6%

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

   3. sqrt-prod 97.6%

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

   4. sqrt-pow1 98.3%

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

   5. metadata-eval 98.3%

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

   6. pow1 98.3%

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

   7. clear-num 98.3%

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

   8. un-div-inv 98.3%

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

   9. unpow2 98.3%

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

   10. unpow2 98.3%

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

   11. hypot-define 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in kx around 0 95.5%

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

   1. un-div-inv 95.5%

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

   2. associate-*l/ 95.5%

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

   3. associate-/l* 95.5%

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

9. Applied egg-rr 95.5%

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

    1. rem-log-exp 95.5%

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

    2. add-cbrt-cube 95.6%

       \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{0.5 + \log \left(e^{\frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \frac{\sin ky}{\frac{Om}{\ell}}\right)}}\right)} \cdot \sqrt{0.5 + \log \left(e^{\frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \frac{\sin ky}{\frac{Om}{\ell}}\right)}}\right)}\right) \cdot \sqrt{0.5 + \log \left(e^{\frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \frac{\sin ky}{\frac{Om}{\ell}}\right)}}\right)}}} \]

11. Applied egg-rr 95.5%

    \[\leadsto \color{blue}{\sqrt[3]{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{2}{\frac{\frac{Om}{\ell}}{\sin ky}}\right)}\right)}^{1.5}}} \]
12. Add Preprocessing

Alternative 5: 93.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \frac{\sin ky}{\frac{Om}{\ell}}\right)}} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 (* 2.0 (/ (sin ky) (/ Om l))))))))

C

double code(double l, double Om, double kx, double ky) {
	return sqrt((0.5 + (0.5 / hypot(1.0, (2.0 * (sin(ky) / (Om / l)))))));
}

Java

public static double code(double l, double Om, double kx, double ky) {
	return Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, (2.0 * (Math.sin(ky) / (Om / l)))))));
}

Python

def code(l, Om, kx, ky):
	return math.sqrt((0.5 + (0.5 / math.hypot(1.0, (2.0 * (math.sin(ky) / (Om / l)))))))

Julia

function code(l, Om, kx, ky)
	return sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, Float64(2.0 * Float64(sin(ky) / Float64(Om / l)))))))
end

MATLAB

function tmp = code(l, Om, kx, ky)
	tmp = sqrt((0.5 + (0.5 / hypot(1.0, (2.0 * (sin(ky) / (Om / l)))))));
end

Wolfram

code[l_, Om_, kx_, ky_] := N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + N[(2.0 * N[(N[Sin[ky], $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 97.6%

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

3. Simplified 97.6%

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

   1. add-sqr-sqrt 97.6%

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

   2. hypot-1-def 97.6%

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

   3. sqrt-prod 97.6%

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

   4. sqrt-pow1 98.3%

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

   5. metadata-eval 98.3%

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

   6. pow1 98.3%

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

   7. clear-num 98.3%

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

   8. un-div-inv 98.3%

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

   9. unpow2 98.3%

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

   10. unpow2 98.3%

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

   11. hypot-define 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in kx around 0 95.5%

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

   1. un-div-inv 95.5%

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

   2. associate-*l/ 95.5%

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

   3. associate-/l* 95.5%

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

9. Applied egg-rr 95.5%

   \[\leadsto \sqrt{0.5 + \color{blue}{\frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \frac{\sin ky}{\frac{Om}{\ell}}\right)}}} \]
10. Add Preprocessing

Alternative 6: 68.0% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq 7.6 \cdot 10^{-8}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky) :precision binary64 (if (<= Om 7.6e-8) (sqrt 0.5) 1.0))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 7.6e-8) {
		tmp = sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	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 (om <= 7.6d-8) then
        tmp = sqrt(0.5d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 7.6e-8) {
		tmp = Math.sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if Om <= 7.6e-8:
		tmp = math.sqrt(0.5)
	else:
		tmp = 1.0
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Om <= 7.6e-8)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (Om <= 7.6e-8)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[Om, 7.6e-8], N[Sqrt[0.5], $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if Om < 7.60000000000000056e-8

   1. Initial program 96.7%

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

   3. Simplified 96.7%

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

      1. add-sqr-sqrt 96.7%

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

      2. hypot-1-def 96.7%

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

      3. sqrt-prod 96.7%

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

      4. sqrt-pow1 97.6%

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

      5. metadata-eval 97.6%

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

      6. pow1 97.6%

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

      7. clear-num 97.6%

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

      8. un-div-inv 97.6%

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

      9. unpow2 97.6%

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

      10. unpow2 97.6%

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

      11. hypot-define 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in kx around 0 95.1%

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

      1. un-div-inv 95.1%

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

      2. associate-*l/ 95.1%

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

      3. associate-/l* 95.1%

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

   9. Applied egg-rr 95.1%

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

   10. Taylor expanded in Om around 0 63.0%

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

   if 7.60000000000000056e-8 < Om

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. hypot-1-def 100.0%

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

      3. sqrt-prod 100.0%

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

      4. sqrt-pow1 100.0%

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

      5. metadata-eval 100.0%

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

      6. pow1 100.0%

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

      7. clear-num 100.0%

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

      8. un-div-inv 100.0%

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

      9. unpow2 100.0%

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

      10. unpow2 100.0%

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

      11. hypot-define 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in kx around 0 96.7%

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

      1. un-div-inv 96.7%

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

      2. associate-*l/ 96.7%

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

      3. associate-/l* 96.7%

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

   9. Applied egg-rr 96.7%

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

   10. Taylor expanded in ky around 0 84.7%

       \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 62.8% accurate, 722.0× 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 97.6%

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

3. Simplified 97.6%

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

   1. add-sqr-sqrt 97.6%

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

   2. hypot-1-def 97.6%

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

   3. sqrt-prod 97.6%

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

   4. sqrt-pow1 98.3%

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

   5. metadata-eval 98.3%

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

   6. pow1 98.3%

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

   7. clear-num 98.3%

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

   8. un-div-inv 98.3%

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

   9. unpow2 98.3%

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

   10. unpow2 98.3%

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

   11. hypot-define 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in kx around 0 95.5%

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

   1. un-div-inv 95.5%

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

   2. associate-*l/ 95.5%

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

   3. associate-/l* 95.5%

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

9. Applied egg-rr 95.5%

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

10. Taylor expanded in ky around 0 64.1%

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

Reproduce

herbie shell --seed 2024076 -o generate:simplify
(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))))))))))