Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.4% → 100.0%

Time: 15.0s

Alternatives: 5

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: 98.4% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

   1. expm1-log1p-u 98.8%

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

   2. expm1-udef 98.8%

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

5. Applied egg-rr 100.0%

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

   1. expm1-def 100.0%

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

   2. expm1-log1p 100.0%

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

   3. *-commutative 100.0%

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

7. Simplified 100.0%

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

8. Final simplification 100.0%

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

Alternative 2: 93.6% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

4. Taylor expanded in kx around 0 76.6%

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

   1. associate-*r/ 76.6%

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

   2. associate-*r* 76.6%

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

   3. unpow2 76.6%

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

   4. unpow2 76.6%

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

6. Simplified 76.6%

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

   1. add-sqr-sqrt 76.6%

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

   2. hypot-1-def 76.6%

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

   3. sqrt-div 76.6%

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

   4. *-commutative 76.6%

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

   5. sqrt-prod 78.2%

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

   6. unpow2 78.2%

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

   7. sqrt-prod 40.8%

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

   8. add-sqr-sqrt 85.3%

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

   9. *-commutative 85.3%

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

   10. sqrt-prod 85.3%

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

   11. sqrt-prod 44.9%

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

   12. add-sqr-sqrt 89.0%

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

   13. metadata-eval 89.0%

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

   14. sqrt-prod 48.9%

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

   15. add-sqr-sqrt 94.0%

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

8. Applied egg-rr 94.0%

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

   1. un-div-inv 94.0%

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

   2. associate-/l* 94.0%

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

10. Applied egg-rr 94.0%

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

11. Final simplification 94.0%

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

Alternative 3: 68.8% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq 1.15 \cdot 10^{-123}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om \leq 2.1 \cdot 10^{+65}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{1 + 2 \cdot \frac{ky \cdot ky}{\frac{Om \cdot Om}{\ell \cdot \ell}}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= Om 1.15e-123)
   (sqrt 0.5)
   (if (<= Om 2.1e+65)
     (sqrt
      (+
       0.5
       (* 0.5 (/ 1.0 (+ 1.0 (* 2.0 (/ (* ky ky) (/ (* Om Om) (* l l)))))))))
     1.0)))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 1.15e-123) {
		tmp = sqrt(0.5);
	} else if (Om <= 2.1e+65) {
		tmp = sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * ((ky * ky) / ((Om * Om) / (l * l)))))))));
	} 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 <= 1.15d-123) then
        tmp = sqrt(0.5d0)
    else if (om <= 2.1d+65) then
        tmp = sqrt((0.5d0 + (0.5d0 * (1.0d0 / (1.0d0 + (2.0d0 * ((ky * ky) / ((om * om) / (l * l)))))))))
    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 <= 1.15e-123) {
		tmp = Math.sqrt(0.5);
	} else if (Om <= 2.1e+65) {
		tmp = Math.sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * ((ky * ky) / ((Om * Om) / (l * l)))))))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if Om <= 1.15e-123:
		tmp = math.sqrt(0.5)
	elif Om <= 2.1e+65:
		tmp = math.sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * ((ky * ky) / ((Om * Om) / (l * l)))))))))
	else:
		tmp = 1.0
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Om <= 1.15e-123)
		tmp = sqrt(0.5);
	elseif (Om <= 2.1e+65)
		tmp = sqrt(Float64(0.5 + Float64(0.5 * Float64(1.0 / Float64(1.0 + Float64(2.0 * Float64(Float64(ky * ky) / Float64(Float64(Om * Om) / Float64(l * l)))))))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (Om <= 1.15e-123)
		tmp = sqrt(0.5);
	elseif (Om <= 2.1e+65)
		tmp = sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * ((ky * ky) / ((Om * Om) / (l * l)))))))));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[Om, 1.15e-123], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[Om, 2.1e+65], N[Sqrt[N[(0.5 + N[(0.5 * N[(1.0 / N[(1.0 + N[(2.0 * N[(N[(ky * ky), $MachinePrecision] / N[(N[(Om * Om), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if Om < 1.14999999999999993e-123

   1. Initial program 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in Om around 0 49.0%

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

      1. associate-*r* 49.0%

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

      2. unpow2 49.0%

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

      3. unpow2 49.0%

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

      4. hypot-def 50.3%

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

      5. *-commutative 50.3%

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

   6. Simplified 50.3%

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

   7. Taylor expanded in l around inf 58.7%

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

   if 1.14999999999999993e-123 < Om < 2.09999999999999991e65

   1. Initial program 97.8%

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

   3. Simplified 97.8%

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

   4. Taylor expanded in kx around 0 88.9%

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

      1. associate-*r/ 88.9%

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

      2. associate-*r* 88.9%

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

      3. unpow2 88.9%

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

      4. unpow2 88.9%

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

   6. Simplified 88.9%

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

   7. Taylor expanded in ky around 0 85.5%

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

      1. associate-/l* 82.8%

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

      2. unpow2 82.8%

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

      3. unpow2 82.8%

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

      4. unpow2 82.8%

         \[\leadsto \sqrt{0.5 + 0.5 \cdot \frac{1}{1 + 2 \cdot \frac{ky \cdot ky}{\frac{Om \cdot Om}{\color{blue}{\ell \cdot \ell}}}}} \]

   9. Simplified 82.8%

      \[\leadsto \sqrt{0.5 + 0.5 \cdot \frac{1}{\color{blue}{1 + 2 \cdot \frac{ky \cdot ky}{\frac{Om \cdot Om}{\ell \cdot \ell}}}}} \]

   if 2.09999999999999991e65 < Om

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in kx around 0 92.5%

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

      1. associate-*r/ 92.5%

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

      2. associate-*r* 92.5%

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

      3. unpow2 92.5%

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

      4. unpow2 92.5%

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

   6. Simplified 92.5%

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

      1. add-sqr-sqrt 92.5%

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

      2. hypot-1-def 92.5%

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

      3. sqrt-div 92.5%

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

      4. *-commutative 92.5%

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

      5. sqrt-prod 92.5%

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

      6. unpow2 92.5%

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

      7. sqrt-prod 49.4%

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

      8. add-sqr-sqrt 92.5%

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

      9. *-commutative 92.5%

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

      10. sqrt-prod 92.5%

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

      11. sqrt-prod 45.9%

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

      12. add-sqr-sqrt 96.9%

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

      13. metadata-eval 96.9%

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

      14. sqrt-prod 98.4%

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

      15. add-sqr-sqrt 98.4%

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

   8. Applied egg-rr 98.4%

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

   9. Taylor expanded in ky around 0 90.6%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 1.15 \cdot 10^{-123}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om \leq 2.1 \cdot 10^{+65}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{1 + 2 \cdot \frac{ky \cdot ky}{\frac{Om \cdot Om}{\ell \cdot \ell}}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4: 67.3% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq 2 \cdot 10^{-21}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky) :precision binary64 (if (<= Om 2e-21) (sqrt 0.5) 1.0))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 2e-21) {
		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 <= 2d-21) 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 <= 2e-21) {
		tmp = Math.sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if Om <= 2e-21:
		tmp = math.sqrt(0.5)
	else:
		tmp = 1.0
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Om <= 2e-21)
		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 <= 2e-21)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[Om, 2e-21], N[Sqrt[0.5], $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if Om < 1.99999999999999982e-21

   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%

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

   4. Taylor expanded in Om around 0 51.8%

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

      1. associate-*r* 51.8%

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

      2. unpow2 51.8%

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

      3. unpow2 51.8%

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

      4. hypot-def 52.9%

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

      5. *-commutative 52.9%

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

   6. Simplified 52.9%

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

   7. Taylor expanded in l around inf 60.8%

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

   if 1.99999999999999982e-21 < Om

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in kx around 0 91.6%

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

      1. associate-*r/ 91.6%

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

      2. associate-*r* 91.6%

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

      3. unpow2 91.6%

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

      4. unpow2 91.6%

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

   6. Simplified 91.6%

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

      1. add-sqr-sqrt 91.6%

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

      2. hypot-1-def 91.6%

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

      3. sqrt-div 91.6%

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

      4. *-commutative 91.6%

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

      5. sqrt-prod 91.6%

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

      6. unpow2 91.6%

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

      7. sqrt-prod 43.9%

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

      8. add-sqr-sqrt 93.1%

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

      9. *-commutative 93.1%

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

      10. sqrt-prod 93.1%

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

      11. sqrt-prod 42.7%

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

      12. add-sqr-sqrt 96.4%

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

      13. metadata-eval 96.4%

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

      14. sqrt-prod 97.6%

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

      15. add-sqr-sqrt 97.6%

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

   8. Applied egg-rr 97.6%

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

   9. Taylor expanded in ky around 0 82.1%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 2 \cdot 10^{-21}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5: 56.5% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{0.5} \end{array} \]

FPCore

(FPCore (l Om kx ky) :precision binary64 (sqrt 0.5))

C

double code(double l, double Om, double kx, double ky) {
	return sqrt(0.5);
}

Fortran

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    code = sqrt(0.5d0)
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	return Math.sqrt(0.5);
}

Python

def code(l, Om, kx, ky):
	return math.sqrt(0.5)

Julia

function code(l, Om, kx, ky)
	return sqrt(0.5)
end

MATLAB

function tmp = code(l, Om, kx, ky)
	tmp = sqrt(0.5);
end

Wolfram

code[l_, Om_, kx_, ky_] := N[Sqrt[0.5], $MachinePrecision]

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

4. Taylor expanded in Om around 0 44.3%

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

   1. associate-*r* 44.3%

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

   2. unpow2 44.3%

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

   3. unpow2 44.3%

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

   4. hypot-def 45.2%

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

   5. *-commutative 45.2%

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

6. Simplified 45.2%

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

7. Taylor expanded in l around inf 54.6%

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

8. Final simplification 54.6%

   \[\leadsto \sqrt{0.5} \]

Reproduce

herbie shell --seed 2023279 
(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))))))))))