Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.3% → 100.0%

Time: 17.1s

Alternatives: 6

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: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ {\left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \left(\frac{\ell}{Om} \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right)}\right)}^{1.5}\right)}^{0.3333333333333333} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (pow
  (pow
   (+ 0.5 (/ 0.5 (hypot 1.0 (* 2.0 (* (/ l Om) (hypot (sin kx) (sin ky)))))))
   1.5)
  0.3333333333333333))

C

double code(double l, double Om, double kx, double ky) {
	return pow(pow((0.5 + (0.5 / hypot(1.0, (2.0 * ((l / Om) * hypot(sin(kx), sin(ky))))))), 1.5), 0.3333333333333333);
}

Java

public static double code(double l, double Om, double kx, double ky) {
	return Math.pow(Math.pow((0.5 + (0.5 / Math.hypot(1.0, (2.0 * ((l / Om) * Math.hypot(Math.sin(kx), Math.sin(ky))))))), 1.5), 0.3333333333333333);
}

Python

def code(l, Om, kx, ky):
	return math.pow(math.pow((0.5 + (0.5 / math.hypot(1.0, (2.0 * ((l / Om) * math.hypot(math.sin(kx), math.sin(ky))))))), 1.5), 0.3333333333333333)

Julia

function code(l, Om, kx, ky)
	return (Float64(0.5 + Float64(0.5 / hypot(1.0, Float64(2.0 * Float64(Float64(l / Om) * hypot(sin(kx), sin(ky))))))) ^ 1.5) ^ 0.3333333333333333
end

MATLAB

function tmp = code(l, Om, kx, ky)
	tmp = ((0.5 + (0.5 / hypot(1.0, (2.0 * ((l / Om) * hypot(sin(kx), sin(ky))))))) ^ 1.5) ^ 0.3333333333333333;
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[(l / Om), $MachinePrecision] * N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 0.3333333333333333], $MachinePrecision]

Derivation

1. Initial program 99.2%

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

3. Simplified 99.2%

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

   1. expm1-log1p-u 99.2%

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

   2. expm1-udef 99.2%

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

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%

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

   4. associate-*r/ 100.0%

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

8. Simplified 100.0%

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

   1. add-cbrt-cube 99.9%

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

   2. pow1/3 100.0%

      \[\leadsto \color{blue}{{\left(\left(\sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \frac{2 \cdot \ell}{Om}\right)} \cdot 0.5} \cdot \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \frac{2 \cdot \ell}{Om}\right)} \cdot 0.5}\right) \cdot \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \frac{2 \cdot \ell}{Om}\right)} \cdot 0.5}\right)}^{0.3333333333333333}} \]

10. Applied egg-rr 100.0%

    \[\leadsto \color{blue}{{\left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \left(\frac{\ell}{Om} \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right)}\right)}^{1.5}\right)}^{0.3333333333333333}} \]

11. Final simplification 100.0%

    \[\leadsto {\left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \left(\frac{\ell}{Om} \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right)}\right)}^{1.5}\right)}^{0.3333333333333333} \]
12. 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 99.2%

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

3. Simplified 99.2%

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

   1. expm1-log1p-u 99.2%

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

   2. expm1-udef 99.2%

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

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%

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

   4. associate-*r/ 100.0%

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

8. Simplified 100.0%

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

   1. expm1-log1p-u 99.3%

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

   2. expm1-udef 99.3%

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

10. Applied egg-rr 99.3%

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

    1. expm1-def 99.3%

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

    2. expm1-log1p 100.0%

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

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: 93.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 3.8 \cdot 10^{+208}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \frac{\ell}{\frac{Om}{\sin kx}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= l 3.8e+208)
   (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 (* 2.0 (/ l (/ Om (sin kx))))))))
   (sqrt 0.5)))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 3.8e+208) {
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (2.0 * (l / (Om / sin(kx))))))));
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 3.8e+208) {
		tmp = Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, (2.0 * (l / (Om / Math.sin(kx))))))));
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if l <= 3.8e+208:
		tmp = math.sqrt((0.5 + (0.5 / math.hypot(1.0, (2.0 * (l / (Om / math.sin(kx))))))))
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (l <= 3.8e+208)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, Float64(2.0 * Float64(l / Float64(Om / sin(kx))))))));
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (l <= 3.8e+208)
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (2.0 * (l / (Om / sin(kx))))))));
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[l, 3.8e+208], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + N[(2.0 * N[(l / N[(Om / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 3.8000000000000002e208

   1. Initial program 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in ky around 0 84.3%

      \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\sqrt{1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin kx}^{2}}{{Om}^{2}}}}} \cdot 0.5} \]
   6. Step-by-step derivation

      1. associate-/l* 84.3%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + 4 \cdot \color{blue}{\frac{{\ell}^{2}}{\frac{{Om}^{2}}{{\sin kx}^{2}}}}}} \cdot 0.5} \]

      2. associate-/r/ 84.3%

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

      3. associate-*l* 84.3%

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

      4. metadata-eval 84.3%

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

      5. unpow2 84.3%

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

      6. unpow2 84.3%

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

      7. times-frac 92.6%

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

      8. swap-sqr 92.6%

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

      9. unpow2 92.6%

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

      10. swap-sqr 95.1%

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

      11. hypot-1-def 95.1%

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

      12. *-commutative 95.1%

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

      13. associate-*r/ 95.1%

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

   7. Simplified 95.1%

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

      1. expm1-log1p-u 94.5%

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

      2. expm1-udef 94.5%

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

      3. *-commutative 94.5%

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

      4. div-inv 94.5%

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

      5. *-commutative 94.5%

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

      6. associate-*r/ 94.5%

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

      7. associate-*l* 94.5%

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

   9. Applied egg-rr 94.5%

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

       1. expm1-def 94.5%

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

       2. expm1-log1p 95.1%

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

       3. associate-*l/ 95.1%

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

       4. associate-/l* 95.1%

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

   11. Simplified 95.1%

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

   if 3.8000000000000002e208 < l

   1. Initial program 96.6%

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

   3. Simplified 96.6%

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

   5. Taylor expanded in ky around 0 69.0%

      \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\sqrt{1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin kx}^{2}}{{Om}^{2}}}}} \cdot 0.5} \]
   6. Step-by-step derivation

      1. associate-/l* 69.0%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + 4 \cdot \color{blue}{\frac{{\ell}^{2}}{\frac{{Om}^{2}}{{\sin kx}^{2}}}}}} \cdot 0.5} \]

      2. associate-/r/ 69.0%

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

      3. associate-*l* 69.0%

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

      4. metadata-eval 69.0%

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

      5. unpow2 69.0%

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

      6. unpow2 69.0%

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

      7. times-frac 84.1%

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

      8. swap-sqr 84.1%

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

      9. unpow2 84.1%

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

      10. swap-sqr 89.0%

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

      11. hypot-1-def 89.0%

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

      12. *-commutative 89.0%

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

      13. associate-*r/ 89.0%

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

   7. Simplified 89.0%

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

   8. Taylor expanded in l around inf 94.5%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.8 \cdot 10^{+208}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \frac{\ell}{\frac{Om}{\sin kx}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.8% 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 99.2%

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

3. Simplified 99.2%

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

   1. expm1-log1p-u 99.2%

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

   2. expm1-udef 99.2%

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

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%

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

   4. associate-*r/ 100.0%

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

8. Simplified 100.0%

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

   1. expm1-log1p-u 99.3%

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

   2. expm1-udef 99.3%

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

10. Applied egg-rr 99.3%

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

    1. expm1-def 99.3%

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

    2. expm1-log1p 100.0%

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

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 92.0%

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

14. Final simplification 92.0%

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

Alternative 5: 67.5% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq 1.1 \cdot 10^{-14}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om \leq 6200000000:\\ \;\;\;\;1\\ \mathbf{elif}\;Om \leq 2 \cdot 10^{+45}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= Om 1.1e-14)
   (sqrt 0.5)
   (if (<= Om 6200000000.0) 1.0 (if (<= Om 2e+45) (sqrt 0.5) 1.0))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 1.1e-14) {
		tmp = sqrt(0.5);
	} else if (Om <= 6200000000.0) {
		tmp = 1.0;
	} else if (Om <= 2e+45) {
		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 <= 1.1d-14) then
        tmp = sqrt(0.5d0)
    else if (om <= 6200000000.0d0) then
        tmp = 1.0d0
    else if (om <= 2d+45) 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 <= 1.1e-14) {
		tmp = Math.sqrt(0.5);
	} else if (Om <= 6200000000.0) {
		tmp = 1.0;
	} else if (Om <= 2e+45) {
		tmp = Math.sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if Om <= 1.1e-14:
		tmp = math.sqrt(0.5)
	elif Om <= 6200000000.0:
		tmp = 1.0
	elif Om <= 2e+45:
		tmp = math.sqrt(0.5)
	else:
		tmp = 1.0
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Om <= 1.1e-14)
		tmp = sqrt(0.5);
	elseif (Om <= 6200000000.0)
		tmp = 1.0;
	elseif (Om <= 2e+45)
		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 <= 1.1e-14)
		tmp = sqrt(0.5);
	elseif (Om <= 6200000000.0)
		tmp = 1.0;
	elseif (Om <= 2e+45)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[Om, 1.1e-14], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[Om, 6200000000.0], 1.0, If[LessEqual[Om, 2e+45], N[Sqrt[0.5], $MachinePrecision], 1.0]]]

Derivation

1. Split input into 2 regimes

2. if Om < 1.1e-14 or 6.2e9 < Om < 1.9999999999999999e45

   1. Initial program 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in ky around 0 83.0%

      \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\sqrt{1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin kx}^{2}}{{Om}^{2}}}}} \cdot 0.5} \]
   6. Step-by-step derivation

      1. associate-/l* 83.0%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + 4 \cdot \color{blue}{\frac{{\ell}^{2}}{\frac{{Om}^{2}}{{\sin kx}^{2}}}}}} \cdot 0.5} \]

      2. associate-/r/ 83.0%

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

      3. associate-*l* 83.0%

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

      4. metadata-eval 83.0%

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

      5. unpow2 83.0%

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

      6. unpow2 83.0%

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

      7. times-frac 89.9%

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

      8. swap-sqr 89.9%

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

      9. unpow2 89.9%

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

      10. swap-sqr 93.6%

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

      11. hypot-1-def 93.6%

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

      12. *-commutative 93.6%

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

      13. associate-*r/ 93.6%

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

   7. Simplified 93.6%

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

   8. Taylor expanded in l around inf 66.0%

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

   if 1.1e-14 < Om < 6.2e9 or 1.9999999999999999e45 < 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%

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

      4. associate-*r/ 100.0%

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

   8. Simplified 100.0%

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

      1. expm1-log1p-u 99.8%

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

      2. expm1-udef 99.8%

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

   10. Applied egg-rr 99.8%

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

       1. expm1-def 99.8%

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

       2. expm1-log1p 100.0%

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

   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 l around 0 87.9%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 1.1 \cdot 10^{-14}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om \leq 6200000000:\\ \;\;\;\;1\\ \mathbf{elif}\;Om \leq 2 \cdot 10^{+45}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 61.6% 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 99.2%

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

3. Simplified 99.2%

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

   1. expm1-log1p-u 99.2%

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

   2. expm1-udef 99.2%

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

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%

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

   4. associate-*r/ 100.0%

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

8. Simplified 100.0%

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

   1. expm1-log1p-u 99.3%

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

   2. expm1-udef 99.3%

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

10. Applied egg-rr 99.3%

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

    1. expm1-def 99.3%

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

    2. expm1-log1p 100.0%

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

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 l around 0 61.7%

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

14. Final simplification 61.7%

    \[\leadsto 1 \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024010 
(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))))))))))