Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.3% → 100.0%

Time: 14.1s

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, 1.4× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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. add-sqr-sqrt 98.8%

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

   2. hypot-1-def 98.8%

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

   3. sqrt-prod 98.8%

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

   4. unpow2 98.8%

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

   5. sqrt-prod 52.8%

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

   6. add-sqr-sqrt 98.9%

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

   7. associate-/r/ 98.9%

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

   8. *-commutative 98.9%

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

   9. unpow2 98.9%

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

   10. unpow2 98.9%

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

   11. hypot-def 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 93.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.1 \cdot 10^{+231}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \frac{\sin kx}{\frac{Om}{\ell}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= l 1.1e+231)
   (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 (* 2.0 (/ (sin kx) (/ Om l)))))))
   (sqrt 0.5)))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 1.1e+231) {
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (2.0 * (sin(kx) / (Om / l)))))));
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 1.1e+231) {
		tmp = Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, (2.0 * (Math.sin(kx) / (Om / l)))))));
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if l <= 1.1e+231:
		tmp = math.sqrt((0.5 + (0.5 / math.hypot(1.0, (2.0 * (math.sin(kx) / (Om / l)))))))
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (l <= 1.1e+231)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, Float64(2.0 * Float64(sin(kx) / Float64(Om / l)))))));
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (l <= 1.1e+231)
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (2.0 * (sin(kx) / (Om / l)))))));
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[l, 1.1e+231], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + N[(2.0 * N[(N[Sin[kx], $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.09999999999999996e231

   1. Initial program 98.7%

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

   3. Simplified 98.7%

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

   4. Taylor expanded in ky around 0 84.6%

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

      1. associate-/l* 82.7%

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

      2. associate-/r/ 84.2%

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

      3. associate-*l* 84.2%

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

      4. *-commutative 84.2%

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

      5. unpow2 84.2%

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

      6. unpow2 84.2%

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

      7. times-frac 90.9%

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

      8. metadata-eval 90.9%

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

      9. swap-sqr 90.9%

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

      10. associate-*l/ 90.9%

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

      11. associate-*r/ 90.9%

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

      12. associate-*l/ 90.9%

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

      13. associate-*r/ 90.9%

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

      14. unpow2 90.9%

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

      15. swap-sqr 95.7%

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

      16. *-commutative 95.7%

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

      17. *-commutative 95.7%

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

   6. Simplified 95.7%

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

      1. expm1-log1p-u 95.1%

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

      2. expm1-udef 95.1%

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

      3. un-div-inv 95.1%

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

      4. *-commutative 95.1%

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

      5. clear-num 95.1%

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

      6. un-div-inv 95.1%

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

   8. Applied egg-rr 95.1%

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

      1. expm1-def 95.1%

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

      2. expm1-log1p 95.7%

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

   10. Simplified 95.7%

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

   if 1.09999999999999996e231 < l

   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 Om around 0 94.1%

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

      1. unpow2 94.1%

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

      2. unpow2 94.1%

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

      3. hypot-def 94.1%

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

   6. Simplified 94.1%

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

   7. Taylor expanded in l around inf 95.3%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.1 \cdot 10^{+231}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \frac{\sin kx}{\frac{Om}{\ell}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 3: 93.8% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[l_, Om_, kx_, ky_] := N[Sqrt[N[(0.5 + N[(0.5 * N[(1.0 / N[Sqrt[1.0 ^ 2 + N[(N[(l * N[(2.0 / Om), $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $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. add-sqr-sqrt 98.8%

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

   2. hypot-1-def 98.8%

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

   3. sqrt-prod 98.8%

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

   4. unpow2 98.8%

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

   5. sqrt-prod 52.8%

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

   6. add-sqr-sqrt 98.9%

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

   7. associate-/r/ 98.9%

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

   8. *-commutative 98.9%

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

   9. unpow2 98.9%

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

   10. unpow2 98.9%

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

   11. hypot-def 100.0%

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

5. Applied egg-rr 100.0%

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

6. Taylor expanded in kx around 0 91.1%

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

7. Final simplification 91.1%

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

Alternative 4: 84.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.6 \cdot 10^{+233}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \frac{\ell \cdot kx}{Om}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= l 1.6e+233)
   (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 (* 2.0 (/ (* l kx) Om))))))
   (sqrt 0.5)))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 1.6e+233) {
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (2.0 * ((l * kx) / Om))))));
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 1.6e+233) {
		tmp = Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, (2.0 * ((l * kx) / Om))))));
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if l <= 1.6e+233:
		tmp = math.sqrt((0.5 + (0.5 / math.hypot(1.0, (2.0 * ((l * kx) / Om))))))
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (l <= 1.6e+233)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, Float64(2.0 * Float64(Float64(l * kx) / Om))))));
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (l <= 1.6e+233)
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (2.0 * ((l * kx) / Om))))));
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[l, 1.6e+233], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + N[(2.0 * N[(N[(l * kx), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.60000000000000009e233

   1. Initial program 98.7%

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

   3. Simplified 98.7%

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

   4. Taylor expanded in ky around 0 84.6%

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

      1. associate-/l* 82.7%

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

      2. associate-/r/ 84.2%

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

      3. associate-*l* 84.2%

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

      4. *-commutative 84.2%

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

      5. unpow2 84.2%

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

      6. unpow2 84.2%

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

      7. times-frac 90.9%

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

      8. metadata-eval 90.9%

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

      9. swap-sqr 90.9%

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

      10. associate-*l/ 90.9%

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

      11. associate-*r/ 90.9%

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

      12. associate-*l/ 90.9%

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

      13. associate-*r/ 90.9%

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

      14. unpow2 90.9%

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

      15. swap-sqr 95.7%

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

      16. *-commutative 95.7%

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

      17. *-commutative 95.7%

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

   6. Simplified 95.7%

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

      1. expm1-log1p-u 95.1%

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

      2. expm1-udef 95.1%

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

      3. un-div-inv 95.1%

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

      4. *-commutative 95.1%

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

      5. clear-num 95.1%

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

      6. un-div-inv 95.1%

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

   8. Applied egg-rr 95.1%

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

      1. expm1-def 95.1%

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

      2. expm1-log1p 95.7%

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

   10. Simplified 95.7%

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

   11. Taylor expanded in kx around 0 88.4%

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

   if 1.60000000000000009e233 < l

   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 Om around 0 94.1%

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

      1. unpow2 94.1%

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

      2. unpow2 94.1%

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

      3. hypot-def 94.1%

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

   6. Simplified 94.1%

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

   7. Taylor expanded in l around inf 95.3%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.6 \cdot 10^{+233}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \frac{\ell \cdot kx}{Om}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 5: 67.8% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq 1.6 \cdot 10^{-22}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky) :precision binary64 (if (<= Om 1.6e-22) (sqrt 0.5) 1.0))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 1.6e-22) {
		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 <= 1.6d-22) 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 <= 1.6e-22) {
		tmp = Math.sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if Om <= 1.6e-22:
		tmp = math.sqrt(0.5)
	else:
		tmp = 1.0
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Om <= 1.6e-22)
		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 <= 1.6e-22)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[Om, 1.6e-22], N[Sqrt[0.5], $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if Om < 1.59999999999999994e-22

   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%

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

   4. Taylor expanded in Om around 0 54.8%

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

      1. unpow2 54.8%

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

      2. unpow2 54.8%

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

      3. hypot-def 56.3%

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

   6. Simplified 56.3%

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

   7. Taylor expanded in l around inf 63.6%

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

   if 1.59999999999999994e-22 < 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 ky around 0 86.2%

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

      1. associate-/l* 84.5%

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

      2. associate-/r/ 86.2%

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

      3. associate-*l* 86.2%

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

      4. *-commutative 86.2%

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

      5. unpow2 86.2%

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

      6. unpow2 86.2%

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

      7. times-frac 96.6%

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

      8. metadata-eval 96.6%

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

      9. swap-sqr 96.6%

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

      10. associate-*l/ 96.6%

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

      11. associate-*r/ 96.6%

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

      12. associate-*l/ 96.6%

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

      13. associate-*r/ 96.6%

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

      14. unpow2 96.6%

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

      15. swap-sqr 98.6%

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

      16. *-commutative 98.6%

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

      17. *-commutative 98.6%

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

   6. Simplified 98.6%

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

      1. expm1-log1p-u 98.3%

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

      2. expm1-udef 98.3%

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

      3. un-div-inv 98.3%

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

      4. *-commutative 98.3%

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

      5. clear-num 98.3%

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

      6. un-div-inv 98.3%

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

   8. Applied egg-rr 98.3%

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

      1. expm1-def 98.3%

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

      2. expm1-log1p 98.6%

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

   10. Simplified 98.6%

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

   11. Taylor expanded in kx around 0 87.6%

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

   12. Taylor expanded in kx around 0 80.7%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 1.6 \cdot 10^{-22}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6: 62.3% 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 98.8%

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

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 ky around 0 82.5%

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

   1. associate-/l* 80.8%

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

   2. associate-/r/ 82.1%

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

   3. associate-*l* 82.1%

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

   4. *-commutative 82.1%

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

   5. unpow2 82.1%

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

   6. unpow2 82.1%

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

   7. times-frac 89.7%

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

   8. metadata-eval 89.7%

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

   9. swap-sqr 89.7%

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

   10. associate-*l/ 89.7%

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

   11. associate-*r/ 89.7%

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

   12. associate-*l/ 89.7%

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

   13. associate-*r/ 89.7%

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

   14. unpow2 89.7%

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

   15. swap-sqr 95.1%

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

   16. *-commutative 95.1%

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

   17. *-commutative 95.1%

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

6. Simplified 95.1%

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

   1. expm1-log1p-u 94.4%

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

   2. expm1-udef 94.4%

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

   3. un-div-inv 94.4%

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

   4. *-commutative 94.4%

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

   5. clear-num 94.4%

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

   6. un-div-inv 94.4%

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

8. Applied egg-rr 94.4%

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

   1. expm1-def 94.4%

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

   2. expm1-log1p 95.1%

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

10. Simplified 95.1%

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

11. Taylor expanded in kx around 0 88.2%

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

12. Taylor expanded in kx around 0 60.8%

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

13. Final simplification 60.8%

    \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023306 
(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))))))))))