Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.5% → 100.0%

Time: 21.5s

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, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

3. Simplified 98.4%

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

   1. inv-pow 98.4%

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

   2. add-sqr-sqrt 98.4%

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

   3. unpow-prod-down 98.4%

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

6. Applied egg-rr 100.0%

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

   1. pow-sqr 100.0%

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

8. Simplified 100.0%

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

   1. sqrt-pow2 100.0%

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

   2. metadata-eval 100.0%

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

   3. inv-pow 100.0%

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

   4. expm1-log1p-u 100.0%

      \[\leadsto \sqrt{0.5 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{hypot}\left(1, \frac{\ell}{Om} \cdot \left(2 \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right)}\right)\right)} \cdot 0.5} \]

   5. expm1-udef 100.0%

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

   6. log1p-udef 100.0%

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

   7. +-commutative 100.0%

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

   8. add-exp-log 100.0%

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

   9. +-commutative 100.0%

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

   10. *-commutative 100.0%

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

   11. associate-*l* 100.0%

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

10. Applied egg-rr 100.0%

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

    1. associate--l+ 100.0%

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

    2. *-commutative 100.0%

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

12. Simplified 100.0%

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

13. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

function tmp = code(l, Om, kx, ky)
	tmp = sqrt((0.5 + (0.5 / hypot(1.0, (2.0 * ((l / Om) * hypot(sin(kx), sin(ky))))))));
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[(l / Om), $MachinePrecision] * N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 98.4%

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

3. Simplified 98.4%

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

   1. expm1-log1p-u 98.4%

      \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 + {\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}\right)\right)}} \cdot 0.5} \]

   2. expm1-udef 98.4%

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

6. Applied egg-rr 100.0%

   \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right)} - 1}} \cdot 0.5} \]
7. Step-by-step derivation

   1. expm1-def 100.0%

      \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right)\right)}} \cdot 0.5} \]

   2. expm1-log1p 100.0%

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

   3. *-commutative 100.0%

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

   4. hypot-def 98.6%

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

   5. unpow2 98.6%

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

   6. unpow2 98.6%

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

   7. associate-*l* 98.6%

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

   8. unpow2 98.6%

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

   9. unpow2 98.6%

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

   10. hypot-def 100.0%

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

8. Simplified 100.0%

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

   1. expm1-log1p-u 99.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \frac{\ell}{Om} \cdot \left(2 \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right)} \cdot 0.5}\right)\right)} \]

   2. expm1-udef 99.3%

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

   3. associate-*l/ 99.3%

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

   4. metadata-eval 99.3%

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

   5. *-commutative 99.3%

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

   6. associate-*l* 99.3%

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

10. Applied egg-rr 99.3%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \left(\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \frac{\ell}{Om}\right)\right)}}\right)} - 1} \]
11. Step-by-step derivation

    1. expm1-def 99.3%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \left(\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \frac{\ell}{Om}\right)\right)}}\right)\right)} \]

    2. expm1-log1p 100.0%

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

    3. *-commutative 100.0%

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

12. Simplified 100.0%

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

13. Final simplification 100.0%

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

Alternative 3: 93.6% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

3. Simplified 98.4%

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

   1. inv-pow 98.4%

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

   2. add-sqr-sqrt 98.4%

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

   3. unpow-prod-down 98.4%

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

6. Applied egg-rr 100.0%

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

   1. pow-sqr 100.0%

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

8. Simplified 100.0%

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

   1. sqrt-pow2 100.0%

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

   2. metadata-eval 100.0%

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

   3. inv-pow 100.0%

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

   4. expm1-log1p-u 100.0%

      \[\leadsto \sqrt{0.5 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{hypot}\left(1, \frac{\ell}{Om} \cdot \left(2 \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right)}\right)\right)} \cdot 0.5} \]

   5. expm1-udef 100.0%

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

   6. log1p-udef 100.0%

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

   7. +-commutative 100.0%

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

   8. add-exp-log 100.0%

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

   9. +-commutative 100.0%

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

   10. *-commutative 100.0%

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

   11. associate-*l* 100.0%

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

10. Applied egg-rr 100.0%

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

    1. associate--l+ 100.0%

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

    2. *-commutative 100.0%

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

12. Simplified 100.0%

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

13. Taylor expanded in kx around 0 91.7%

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

14. Final simplification 91.7%

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

Alternative 4: 93.6% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

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 * ((l * Math.sin(ky)) / Om))))));
}

Python

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

Julia

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

MATLAB

function tmp = code(l, Om, kx, ky)
	tmp = sqrt((0.5 + (0.5 / hypot(1.0, (2.0 * ((l * sin(ky)) / Om))))));
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[(l * N[Sin[ky], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 98.4%

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

3. Simplified 98.4%

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

   1. expm1-log1p-u 98.4%

      \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 + {\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}\right)\right)}} \cdot 0.5} \]

   2. expm1-udef 98.4%

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

6. Applied egg-rr 100.0%

   \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right)} - 1}} \cdot 0.5} \]
7. Step-by-step derivation

   1. expm1-def 100.0%

      \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right)\right)}} \cdot 0.5} \]

   2. expm1-log1p 100.0%

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

   3. *-commutative 100.0%

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

   4. hypot-def 98.6%

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

   5. unpow2 98.6%

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

   6. unpow2 98.6%

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

   7. associate-*l* 98.6%

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

   8. unpow2 98.6%

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

   9. unpow2 98.6%

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

   10. hypot-def 100.0%

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

8. Simplified 100.0%

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

   1. expm1-log1p-u 99.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \frac{\ell}{Om} \cdot \left(2 \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right)} \cdot 0.5}\right)\right)} \]

   2. expm1-udef 99.3%

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

   3. associate-*l/ 99.3%

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

   4. metadata-eval 99.3%

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

   5. *-commutative 99.3%

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

   6. associate-*l* 99.3%

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

10. Applied egg-rr 99.3%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \left(\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \frac{\ell}{Om}\right)\right)}}\right)} - 1} \]
11. Step-by-step derivation

    1. expm1-def 99.3%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \left(\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \frac{\ell}{Om}\right)\right)}}\right)\right)} \]

    2. expm1-log1p 100.0%

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

    3. *-commutative 100.0%

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

12. Simplified 100.0%

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

13. Taylor expanded in kx around 0 91.7%

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

14. Final simplification 91.7%

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

Alternative 5: 85.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq 2.2 \cdot 10^{+97}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \frac{\ell \cdot ky}{Om}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= Om 2.2e+97)
   (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 (* 2.0 (/ (* l ky) Om))))))
   1.0))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 2.2e+97) {
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (2.0 * ((l * ky) / Om))))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 2.2e+97) {
		tmp = Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, (2.0 * ((l * ky) / Om))))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if Om <= 2.2e+97:
		tmp = math.sqrt((0.5 + (0.5 / math.hypot(1.0, (2.0 * ((l * ky) / Om))))))
	else:
		tmp = 1.0
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Om <= 2.2e+97)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, Float64(2.0 * Float64(Float64(l * ky) / Om))))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (Om <= 2.2e+97)
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (2.0 * ((l * ky) / Om))))));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[Om, 2.2e+97], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + N[(2.0 * N[(N[(l * ky), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if Om < 2.2000000000000001e97

   1. Initial program 98.1%

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

   3. Simplified 98.1%

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

      1. expm1-log1p-u 98.1%

         \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 + {\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}\right)\right)}} \cdot 0.5} \]

      2. expm1-udef 98.1%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right)} - 1}} \cdot 0.5} \]
   7. Step-by-step derivation

      1. expm1-def 100.0%

         \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right)\right)}} \cdot 0.5} \]

      2. expm1-log1p 100.0%

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

      3. *-commutative 100.0%

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

      4. hypot-def 98.3%

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

      5. unpow2 98.3%

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

      6. unpow2 98.3%

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

      7. associate-*l* 98.3%

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

      8. unpow2 98.3%

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

      9. unpow2 98.3%

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

      10. hypot-def 100.0%

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

   8. Simplified 100.0%

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

      1. expm1-log1p-u 99.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \frac{\ell}{Om} \cdot \left(2 \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right)} \cdot 0.5}\right)\right)} \]

      2. expm1-udef 99.2%

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

      3. associate-*l/ 99.2%

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

      4. metadata-eval 99.2%

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

      5. *-commutative 99.2%

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

      6. associate-*l* 99.2%

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

   10. Applied egg-rr 99.2%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \left(\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \frac{\ell}{Om}\right)\right)}}\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 99.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \left(\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \frac{\ell}{Om}\right)\right)}}\right)\right)} \]

       2. expm1-log1p 100.0%

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

       3. *-commutative 100.0%

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

   12. Simplified 100.0%

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

   13. Taylor expanded in kx around 0 91.0%

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

   14. Taylor expanded in ky around 0 82.7%

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

       1. *-commutative 82.7%

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

   16. Simplified 82.7%

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

   if 2.2000000000000001e97 < 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%

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

      1. expm1-log1p-u 100.0%

         \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 + {\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}\right)\right)}} \cdot 0.5} \]

      2. expm1-udef 100.0%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right)} - 1}} \cdot 0.5} \]
   7. Step-by-step derivation

      1. expm1-def 100.0%

         \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right)\right)}} \cdot 0.5} \]

      2. expm1-log1p 100.0%

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

      3. *-commutative 100.0%

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

      4. hypot-def 100.0%

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

      5. unpow2 100.0%

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

      6. unpow2 100.0%

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

      7. associate-*l* 100.0%

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

      8. unpow2 100.0%

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

      9. unpow2 100.0%

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

      10. hypot-def 100.0%

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

   8. Simplified 100.0%

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

      1. expm1-log1p-u 99.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \frac{\ell}{Om} \cdot \left(2 \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right)} \cdot 0.5}\right)\right)} \]

      2. expm1-udef 99.8%

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

      3. associate-*l/ 99.8%

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

      4. metadata-eval 99.8%

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

      5. *-commutative 99.8%

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

      6. associate-*l* 99.8%

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

   10. Applied egg-rr 99.8%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \left(\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \frac{\ell}{Om}\right)\right)}}\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 99.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \left(\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \frac{\ell}{Om}\right)\right)}}\right)\right)} \]

       2. expm1-log1p 100.0%

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

       3. *-commutative 100.0%

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

   12. Simplified 100.0%

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

       1. add-cbrt-cube 100.0%

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

       2. add-sqr-sqrt 100.0%

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

       3. pow1 100.0%

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

       4. pow1/2 100.0%

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

       5. pow-prod-up 100.0%

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

   14. Applied egg-rr 100.0%

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

   15. Taylor expanded in l around 0 90.0%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 2.2 \cdot 10^{+97}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \frac{\ell \cdot ky}{Om}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.1% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq 5 \cdot 10^{-106}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om \leq 4 \cdot 10^{-83}:\\ \;\;\;\;1\\ \mathbf{elif}\;Om \leq 5 \cdot 10^{-43}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om \leq 10^{-19}:\\ \;\;\;\;1\\ \mathbf{elif}\;Om \leq 7.5 \cdot 10^{+28}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= Om 5e-106)
   (sqrt 0.5)
   (if (<= Om 4e-83)
     1.0
     (if (<= Om 5e-43)
       (sqrt 0.5)
       (if (<= Om 1e-19) 1.0 (if (<= Om 7.5e+28) (sqrt 0.5) 1.0))))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 5e-106) {
		tmp = sqrt(0.5);
	} else if (Om <= 4e-83) {
		tmp = 1.0;
	} else if (Om <= 5e-43) {
		tmp = sqrt(0.5);
	} else if (Om <= 1e-19) {
		tmp = 1.0;
	} else if (Om <= 7.5e+28) {
		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 <= 5d-106) then
        tmp = sqrt(0.5d0)
    else if (om <= 4d-83) then
        tmp = 1.0d0
    else if (om <= 5d-43) then
        tmp = sqrt(0.5d0)
    else if (om <= 1d-19) then
        tmp = 1.0d0
    else if (om <= 7.5d+28) 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 <= 5e-106) {
		tmp = Math.sqrt(0.5);
	} else if (Om <= 4e-83) {
		tmp = 1.0;
	} else if (Om <= 5e-43) {
		tmp = Math.sqrt(0.5);
	} else if (Om <= 1e-19) {
		tmp = 1.0;
	} else if (Om <= 7.5e+28) {
		tmp = Math.sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if Om <= 5e-106:
		tmp = math.sqrt(0.5)
	elif Om <= 4e-83:
		tmp = 1.0
	elif Om <= 5e-43:
		tmp = math.sqrt(0.5)
	elif Om <= 1e-19:
		tmp = 1.0
	elif Om <= 7.5e+28:
		tmp = math.sqrt(0.5)
	else:
		tmp = 1.0
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Om <= 5e-106)
		tmp = sqrt(0.5);
	elseif (Om <= 4e-83)
		tmp = 1.0;
	elseif (Om <= 5e-43)
		tmp = sqrt(0.5);
	elseif (Om <= 1e-19)
		tmp = 1.0;
	elseif (Om <= 7.5e+28)
		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 <= 5e-106)
		tmp = sqrt(0.5);
	elseif (Om <= 4e-83)
		tmp = 1.0;
	elseif (Om <= 5e-43)
		tmp = sqrt(0.5);
	elseif (Om <= 1e-19)
		tmp = 1.0;
	elseif (Om <= 7.5e+28)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[Om, 5e-106], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[Om, 4e-83], 1.0, If[LessEqual[Om, 5e-43], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[Om, 1e-19], 1.0, If[LessEqual[Om, 7.5e+28], N[Sqrt[0.5], $MachinePrecision], 1.0]]]]]

Derivation

1. Split input into 2 regimes

2. if Om < 4.99999999999999983e-106 or 4.0000000000000001e-83 < Om < 5.00000000000000019e-43 or 9.9999999999999998e-20 < Om < 7.4999999999999998e28

   1. Initial program 98.4%

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

   3. Simplified 98.4%

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

   5. Taylor expanded in Om around 0 57.0%

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

      1. associate-*r* 57.0%

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

      2. *-commutative 57.0%

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

      3. associate-*l* 57.0%

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

      4. unpow2 57.0%

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

      5. unpow2 57.0%

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

      6. hypot-def 58.6%

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

   7. Simplified 58.6%

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

   8. Taylor expanded in l around inf 65.7%

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

   if 4.99999999999999983e-106 < Om < 4.0000000000000001e-83 or 5.00000000000000019e-43 < Om < 9.9999999999999998e-20 or 7.4999999999999998e28 < Om

   1. Initial program 98.5%

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

   3. Simplified 98.5%

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

      1. expm1-log1p-u 98.5%

         \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 + {\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}\right)\right)}} \cdot 0.5} \]

      2. expm1-udef 98.5%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right)} - 1}} \cdot 0.5} \]
   7. Step-by-step derivation

      1. expm1-def 100.0%

         \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right)\right)}} \cdot 0.5} \]

      2. expm1-log1p 100.0%

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

      3. *-commutative 100.0%

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

      4. hypot-def 98.8%

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

      5. unpow2 98.8%

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

      6. unpow2 98.8%

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

      7. associate-*l* 98.8%

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

      8. unpow2 98.8%

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

      9. unpow2 98.8%

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

      10. hypot-def 100.0%

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

   8. Simplified 100.0%

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

      1. expm1-log1p-u 99.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \frac{\ell}{Om} \cdot \left(2 \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right)} \cdot 0.5}\right)\right)} \]

      2. expm1-udef 99.8%

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

      3. associate-*l/ 99.8%

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

      4. metadata-eval 99.8%

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

      5. *-commutative 99.8%

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

      6. associate-*l* 99.8%

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

   10. Applied egg-rr 99.8%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \left(\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \frac{\ell}{Om}\right)\right)}}\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 99.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \left(\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \frac{\ell}{Om}\right)\right)}}\right)\right)} \]

       2. expm1-log1p 100.0%

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

       3. *-commutative 100.0%

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

   12. Simplified 100.0%

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

       1. add-cbrt-cube 100.0%

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

       2. add-sqr-sqrt 100.0%

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

       3. pow1 100.0%

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

       4. pow1/2 100.0%

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

       5. pow-prod-up 100.0%

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

   14. Applied egg-rr 100.0%

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

   15. Taylor expanded in l around 0 87.9%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 5 \cdot 10^{-106}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om \leq 4 \cdot 10^{-83}:\\ \;\;\;\;1\\ \mathbf{elif}\;Om \leq 5 \cdot 10^{-43}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om \leq 10^{-19}:\\ \;\;\;\;1\\ \mathbf{elif}\;Om \leq 7.5 \cdot 10^{+28}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.5% 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 98.4%

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

3. Simplified 98.4%

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

   1. expm1-log1p-u 98.4%

      \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 + {\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}\right)\right)}} \cdot 0.5} \]

   2. expm1-udef 98.4%

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

6. Applied egg-rr 100.0%

   \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right)} - 1}} \cdot 0.5} \]
7. Step-by-step derivation

   1. expm1-def 100.0%

      \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right)\right)}} \cdot 0.5} \]

   2. expm1-log1p 100.0%

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

   3. *-commutative 100.0%

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

   4. hypot-def 98.6%

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

   5. unpow2 98.6%

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

   6. unpow2 98.6%

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

   7. associate-*l* 98.6%

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

   8. unpow2 98.6%

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

   9. unpow2 98.6%

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

   10. hypot-def 100.0%

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

8. Simplified 100.0%

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

   1. expm1-log1p-u 99.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \frac{\ell}{Om} \cdot \left(2 \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right)} \cdot 0.5}\right)\right)} \]

   2. expm1-udef 99.3%

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

   3. associate-*l/ 99.3%

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

   4. metadata-eval 99.3%

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

   5. *-commutative 99.3%

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

   6. associate-*l* 99.3%

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

10. Applied egg-rr 99.3%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \left(\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \frac{\ell}{Om}\right)\right)}}\right)} - 1} \]
11. Step-by-step derivation

    1. expm1-def 99.3%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \left(\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \frac{\ell}{Om}\right)\right)}}\right)\right)} \]

    2. expm1-log1p 100.0%

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

    3. *-commutative 100.0%

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

12. Simplified 100.0%

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

    1. add-cbrt-cube 100.0%

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

    2. add-sqr-sqrt 100.0%

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

    3. pow1 100.0%

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

    4. pow1/2 100.0%

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

    5. pow-prod-up 100.0%

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

14. Applied egg-rr 100.0%

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

15. Taylor expanded in l around 0 61.8%

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

16. Final simplification 61.8%

    \[\leadsto 1 \]
17. Add Preprocessing

Reproduce

herbie shell --seed 2024029 
(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))))))))))