Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.1% → 100.0%

Time: 24.3s

Alternatives: 10

Speedup: 1.0×

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: 98.1% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. add-sqr-sqrt 100.0%

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

   2. pow2 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 94.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 78.0%

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

   1. associate-*r/ 78.0%

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

   2. associate-*l/ 77.5%

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

   3. unpow2 77.5%

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

   4. *-commutative 77.5%

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

   5. *-commutative 77.5%

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

   6. unpow2 77.5%

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

   7. unpow2 77.5%

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

   8. unswap-sqr 83.0%

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

   9. associate-/r* 83.0%

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

   10. metadata-eval 83.0%

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

   11. associate-*l/ 83.0%

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

   12. associate-*r/ 83.0%

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

   13. swap-sqr 92.4%

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

   14. associate-*r* 92.4%

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

   15. associate-*r* 92.4%

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

6. Simplified 92.4%

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

   1. expm1-log1p-u 91.9%

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

   2. expm1-udef 91.9%

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

   3. un-div-inv 91.9%

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

   4. div-inv 91.9%

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

   5. clear-num 91.9%

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

8. Applied egg-rr 91.9%

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

   1. expm1-def 91.9%

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

   2. expm1-log1p 92.4%

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

   3. associate-*r* 92.4%

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

   4. *-commutative 92.4%

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

10. Simplified 92.4%

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

    1. add-log-exp 92.4%

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

    2. associate-*l* 92.4%

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

12. Applied egg-rr 92.4%

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

13. Final simplification 92.4%

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

Alternative 3: 94.4% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 78.0%

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

   1. associate-*r/ 78.0%

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

   2. associate-*l/ 77.5%

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

   3. unpow2 77.5%

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

   4. *-commutative 77.5%

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

   5. *-commutative 77.5%

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

   6. unpow2 77.5%

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

   7. unpow2 77.5%

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

   8. unswap-sqr 83.0%

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

   9. associate-/r* 83.0%

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

   10. metadata-eval 83.0%

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

   11. associate-*l/ 83.0%

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

   12. associate-*r/ 83.0%

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

   13. swap-sqr 92.4%

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

   14. associate-*r* 92.4%

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

   15. associate-*r* 92.4%

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

6. Simplified 92.4%

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

   1. expm1-log1p-u 91.9%

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

   2. expm1-udef 91.9%

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

   3. un-div-inv 91.9%

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

   4. div-inv 91.9%

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

   5. clear-num 91.9%

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

8. Applied egg-rr 91.9%

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

   1. expm1-def 91.9%

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

   2. expm1-log1p 92.4%

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

   3. associate-*r* 92.4%

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

   4. *-commutative 92.4%

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

10. Simplified 92.4%

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

11. Final simplification 92.4%

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

Alternative 4: 63.1% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq 8500000000000:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \frac{\ell \cdot ky}{\frac{Om \cdot \frac{Om}{\ell}}{ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= Om 8500000000000.0)
   (sqrt 0.5)
   (+ 1.0 (* -0.5 (/ (* l ky) (/ (* Om (/ Om l)) ky))))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 8500000000000.0) {
		tmp = sqrt(0.5);
	} else {
		tmp = 1.0 + (-0.5 * ((l * ky) / ((Om * (Om / l)) / ky)));
	}
	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 <= 8500000000000.0d0) then
        tmp = sqrt(0.5d0)
    else
        tmp = 1.0d0 + ((-0.5d0) * ((l * ky) / ((om * (om / l)) / ky)))
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 8500000000000.0) {
		tmp = Math.sqrt(0.5);
	} else {
		tmp = 1.0 + (-0.5 * ((l * ky) / ((Om * (Om / l)) / ky)));
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if Om <= 8500000000000.0:
		tmp = math.sqrt(0.5)
	else:
		tmp = 1.0 + (-0.5 * ((l * ky) / ((Om * (Om / l)) / ky)))
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Om <= 8500000000000.0)
		tmp = sqrt(0.5);
	else
		tmp = Float64(1.0 + Float64(-0.5 * Float64(Float64(l * ky) / Float64(Float64(Om * Float64(Om / l)) / ky))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (Om <= 8500000000000.0)
		tmp = sqrt(0.5);
	else
		tmp = 1.0 + (-0.5 * ((l * ky) / ((Om * (Om / l)) / ky)));
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[Om, 8500000000000.0], N[Sqrt[0.5], $MachinePrecision], N[(1.0 + N[(-0.5 * N[(N[(l * ky), $MachinePrecision] / N[(N[(Om * N[(Om / l), $MachinePrecision]), $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Om < 8.5e12

   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 74.8%

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

      1. associate-*r/ 74.8%

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

      2. associate-*l/ 74.2%

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

      3. unpow2 74.2%

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

      4. *-commutative 74.2%

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

      5. *-commutative 74.2%

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

      6. unpow2 74.2%

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

      7. unpow2 74.2%

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

      8. unswap-sqr 79.0%

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

      9. associate-/r* 79.0%

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

      10. metadata-eval 79.0%

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

      11. associate-*l/ 79.0%

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

      12. associate-*r/ 79.0%

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

      13. swap-sqr 92.9%

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

      14. associate-*r* 92.9%

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

      15. associate-*r* 92.9%

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

   6. Simplified 92.9%

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

   7. Taylor expanded in l around inf 60.3%

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

   if 8.5e12 < Om

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in kx around 0 83.4%

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

      1. associate-*r/ 83.4%

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

      2. associate-*l/ 83.4%

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

      3. unpow2 83.4%

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

      4. *-commutative 83.4%

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

      5. *-commutative 83.4%

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

      6. unpow2 83.4%

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

      7. unpow2 83.4%

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

      8. unswap-sqr 89.8%

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

      9. associate-/r* 89.8%

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

      10. metadata-eval 89.8%

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

      11. associate-*l/ 89.8%

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

      12. associate-*r/ 89.8%

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

      13. swap-sqr 93.1%

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

      14. associate-*r* 93.1%

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

      15. associate-*r* 93.1%

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

   6. Simplified 93.1%

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

   7. Taylor expanded in ky around 0 55.3%

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

      1. unpow2 55.3%

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

      2. times-frac 55.3%

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

      3. unpow2 55.3%

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

      4. unpow2 55.3%

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

   9. Simplified 55.3%

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

       1. *-commutative 55.3%

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

       2. associate-/l* 59.6%

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

       3. associate-/l* 66.3%

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

       4. frac-times 71.7%

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

   11. Applied egg-rr 71.7%

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

       1. associate-*r/ 75.0%

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

   13. Applied egg-rr 75.0%

       \[\leadsto 1 + -0.5 \cdot \frac{\ell \cdot ky}{\color{blue}{\frac{\frac{Om}{\ell} \cdot Om}{ky}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 8500000000000:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \frac{\ell \cdot ky}{\frac{Om \cdot \frac{Om}{\ell}}{ky}}\\ \end{array} \]

Alternative 5: 66.9% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky) :precision binary64 (if (<= Om 2e-19) (sqrt 0.5) 1.0))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 2e-19) {
		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-19) 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-19) {
		tmp = Math.sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if Om <= 2e-19:
		tmp = math.sqrt(0.5)
	else:
		tmp = 1.0
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Om <= 2e-19)
		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-19)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[Om, 2e-19], N[Sqrt[0.5], $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if Om < 2e-19

   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 74.3%

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

      1. associate-*r/ 74.3%

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

      2. associate-*l/ 73.7%

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

      3. unpow2 73.7%

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

      4. *-commutative 73.7%

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

      5. *-commutative 73.7%

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

      6. unpow2 73.7%

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

      7. unpow2 73.7%

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

      8. unswap-sqr 78.6%

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

      9. associate-/r* 78.6%

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

      10. metadata-eval 78.6%

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

      11. associate-*l/ 78.6%

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

      12. associate-*r/ 78.6%

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

      13. swap-sqr 92.8%

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

      14. associate-*r* 92.8%

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

      15. associate-*r* 92.8%

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

   6. Simplified 92.8%

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

   7. Taylor expanded in l around inf 60.3%

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

   if 2e-19 < 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 84.5%

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

      1. associate-*r/ 84.5%

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

      2. associate-*l/ 84.5%

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

      3. unpow2 84.5%

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

      4. *-commutative 84.5%

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

      5. *-commutative 84.5%

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

      6. unpow2 84.5%

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

      7. unpow2 84.5%

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

      8. unswap-sqr 90.4%

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

      9. associate-/r* 90.4%

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

      10. metadata-eval 90.4%

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

      11. associate-*l/ 90.4%

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

      12. associate-*r/ 90.4%

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

      13. swap-sqr 93.5%

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

      14. associate-*r* 93.5%

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

      15. associate-*r* 93.5%

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

   6. Simplified 93.5%

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

      1. expm1-log1p-u 93.4%

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

      2. expm1-udef 93.4%

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

      3. un-div-inv 93.4%

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

      4. div-inv 93.4%

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

      5. clear-num 93.4%

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

   8. Applied egg-rr 93.4%

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

      1. expm1-def 93.4%

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

      2. expm1-log1p 93.5%

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

      3. associate-*r* 93.5%

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

      4. *-commutative 93.5%

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

   10. Simplified 93.5%

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

   11. Taylor expanded in l around 0 84.4%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6: 45.9% accurate, 42.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 10^{+61}:\\ \;\;\;\;1 + -0.5 \cdot \frac{ky}{\frac{Om}{\ell \cdot \ell} \cdot \frac{Om}{ky}}\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(\frac{\ell}{\frac{Om}{\ell}} \cdot \left(ky \cdot \frac{ky}{Om}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= l 1e+61)
   (+ 1.0 (* -0.5 (/ ky (* (/ Om (* l l)) (/ Om ky)))))
   (+ 1.0 (* -0.5 (* (/ l (/ Om l)) (* ky (/ ky Om)))))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 1e+61) {
		tmp = 1.0 + (-0.5 * (ky / ((Om / (l * l)) * (Om / ky))));
	} else {
		tmp = 1.0 + (-0.5 * ((l / (Om / l)) * (ky * (ky / Om))));
	}
	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 (l <= 1d+61) then
        tmp = 1.0d0 + ((-0.5d0) * (ky / ((om / (l * l)) * (om / ky))))
    else
        tmp = 1.0d0 + ((-0.5d0) * ((l / (om / l)) * (ky * (ky / om))))
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 1e+61) {
		tmp = 1.0 + (-0.5 * (ky / ((Om / (l * l)) * (Om / ky))));
	} else {
		tmp = 1.0 + (-0.5 * ((l / (Om / l)) * (ky * (ky / Om))));
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if l <= 1e+61:
		tmp = 1.0 + (-0.5 * (ky / ((Om / (l * l)) * (Om / ky))))
	else:
		tmp = 1.0 + (-0.5 * ((l / (Om / l)) * (ky * (ky / Om))))
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (l <= 1e+61)
		tmp = Float64(1.0 + Float64(-0.5 * Float64(ky / Float64(Float64(Om / Float64(l * l)) * Float64(Om / ky)))));
	else
		tmp = Float64(1.0 + Float64(-0.5 * Float64(Float64(l / Float64(Om / l)) * Float64(ky * Float64(ky / Om)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (l <= 1e+61)
		tmp = 1.0 + (-0.5 * (ky / ((Om / (l * l)) * (Om / ky))));
	else
		tmp = 1.0 + (-0.5 * ((l / (Om / l)) * (ky * (ky / Om))));
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[l, 1e+61], N[(1.0 + N[(-0.5 * N[(ky / N[(N[(Om / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(Om / ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.5 * N[(N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision] * N[(ky * N[(ky / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 9.99999999999999949e60

   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 79.9%

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

      1. associate-*r/ 79.9%

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

      2. associate-*l/ 79.3%

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

      3. unpow2 79.3%

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

      4. *-commutative 79.3%

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

      5. *-commutative 79.3%

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

      6. unpow2 79.3%

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

      7. unpow2 79.3%

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

      8. unswap-sqr 82.4%

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

      9. associate-/r* 82.4%

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

      10. metadata-eval 82.4%

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

      11. associate-*l/ 82.4%

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

      12. associate-*r/ 82.4%

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

      13. swap-sqr 92.5%

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

      14. associate-*r* 92.5%

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

      15. associate-*r* 92.5%

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

   6. Simplified 92.5%

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

   7. Taylor expanded in ky around 0 44.5%

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

      1. unpow2 44.5%

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

      2. times-frac 46.2%

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

      3. unpow2 46.2%

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

      4. unpow2 46.2%

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

   9. Simplified 46.2%

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

       1. *-commutative 46.2%

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

       2. clear-num 46.2%

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

       3. associate-/l* 49.1%

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

       4. frac-times 55.9%

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

       5. *-un-lft-identity 55.9%

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

   11. Applied egg-rr 55.9%

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

   if 9.99999999999999949e60 < l

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in kx around 0 70.2%

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

      1. associate-*r/ 70.2%

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

      2. associate-*l/ 70.2%

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

      3. unpow2 70.2%

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

      4. *-commutative 70.2%

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

      5. *-commutative 70.2%

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

      6. unpow2 70.2%

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

      7. unpow2 70.2%

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

      8. unswap-sqr 85.7%

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

      9. associate-/r* 85.7%

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

      10. metadata-eval 85.7%

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

      11. associate-*l/ 85.7%

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

      12. associate-*r/ 85.7%

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

      13. swap-sqr 91.8%

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

      14. associate-*r* 91.8%

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

      15. associate-*r* 91.8%

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

   6. Simplified 91.8%

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

   7. Taylor expanded in ky around 0 9.1%

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

      1. unpow2 9.1%

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

      2. times-frac 10.4%

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

      3. unpow2 10.4%

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

      4. unpow2 10.4%

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

   9. Simplified 10.4%

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

       1. *-commutative 10.4%

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

       2. associate-/l* 21.4%

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

       3. associate-/l* 21.5%

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

       4. frac-times 22.3%

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

   11. Applied egg-rr 22.3%

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

       1. times-frac 21.5%

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

       2. associate-/r/ 21.5%

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

   13. Applied egg-rr 21.5%

       \[\leadsto 1 + -0.5 \cdot \color{blue}{\left(\frac{\ell}{\frac{Om}{\ell}} \cdot \left(\frac{ky}{Om} \cdot ky\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 10^{+61}:\\ \;\;\;\;1 + -0.5 \cdot \frac{ky}{\frac{Om}{\ell \cdot \ell} \cdot \frac{Om}{ky}}\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(\frac{\ell}{\frac{Om}{\ell}} \cdot \left(ky \cdot \frac{ky}{Om}\right)\right)\\ \end{array} \]

Alternative 7: 46.4% accurate, 42.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.12 \cdot 10^{+156}:\\ \;\;\;\;1 + -0.5 \cdot \frac{ky}{\frac{Om}{\ell \cdot \ell} \cdot \frac{Om}{ky}}\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \frac{\ell}{\frac{Om}{\ell} \cdot \frac{Om}{ky \cdot ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= l 1.12e+156)
   (+ 1.0 (* -0.5 (/ ky (* (/ Om (* l l)) (/ Om ky)))))
   (+ 1.0 (* -0.5 (/ l (* (/ Om l) (/ Om (* ky ky))))))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 1.12e+156) {
		tmp = 1.0 + (-0.5 * (ky / ((Om / (l * l)) * (Om / ky))));
	} else {
		tmp = 1.0 + (-0.5 * (l / ((Om / l) * (Om / (ky * ky)))));
	}
	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 (l <= 1.12d+156) then
        tmp = 1.0d0 + ((-0.5d0) * (ky / ((om / (l * l)) * (om / ky))))
    else
        tmp = 1.0d0 + ((-0.5d0) * (l / ((om / l) * (om / (ky * ky)))))
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 1.12e+156) {
		tmp = 1.0 + (-0.5 * (ky / ((Om / (l * l)) * (Om / ky))));
	} else {
		tmp = 1.0 + (-0.5 * (l / ((Om / l) * (Om / (ky * ky)))));
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if l <= 1.12e+156:
		tmp = 1.0 + (-0.5 * (ky / ((Om / (l * l)) * (Om / ky))))
	else:
		tmp = 1.0 + (-0.5 * (l / ((Om / l) * (Om / (ky * ky)))))
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (l <= 1.12e+156)
		tmp = Float64(1.0 + Float64(-0.5 * Float64(ky / Float64(Float64(Om / Float64(l * l)) * Float64(Om / ky)))));
	else
		tmp = Float64(1.0 + Float64(-0.5 * Float64(l / Float64(Float64(Om / l) * Float64(Om / Float64(ky * ky))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (l <= 1.12e+156)
		tmp = 1.0 + (-0.5 * (ky / ((Om / (l * l)) * (Om / ky))));
	else
		tmp = 1.0 + (-0.5 * (l / ((Om / l) * (Om / (ky * ky)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[l, 1.12e+156], N[(1.0 + N[(-0.5 * N[(ky / N[(N[(Om / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(Om / ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.5 * N[(l / N[(N[(Om / l), $MachinePrecision] * N[(Om / N[(ky * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.12000000000000007e156

   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 80.7%

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

      1. associate-*r/ 80.7%

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

      2. associate-*l/ 80.2%

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

      3. unpow2 80.2%

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

      4. *-commutative 80.2%

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

      5. *-commutative 80.2%

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

      6. unpow2 80.2%

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

      7. unpow2 80.2%

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

      8. unswap-sqr 83.8%

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

      9. associate-/r* 83.8%

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

      10. metadata-eval 83.8%

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

      11. associate-*l/ 83.8%

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

      12. associate-*r/ 83.8%

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

      13. swap-sqr 92.9%

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

      14. associate-*r* 92.9%

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

      15. associate-*r* 92.9%

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

   6. Simplified 92.9%

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

   7. Taylor expanded in ky around 0 41.9%

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

      1. unpow2 41.9%

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

      2. times-frac 43.8%

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

      3. unpow2 43.8%

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

      4. unpow2 43.8%

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

   9. Simplified 43.8%

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

       1. *-commutative 43.8%

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

       2. clear-num 43.8%

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

       3. associate-/l* 46.4%

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

       4. frac-times 52.5%

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

       5. *-un-lft-identity 52.5%

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

   11. Applied egg-rr 52.5%

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

   if 1.12000000000000007e156 < l

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in kx around 0 53.8%

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

      1. associate-*r/ 53.8%

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

      2. associate-*l/ 53.8%

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

      3. unpow2 53.8%

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

      4. *-commutative 53.8%

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

      5. *-commutative 53.8%

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

      6. unpow2 53.8%

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

      7. unpow2 53.8%

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

      8. unswap-sqr 76.1%

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

      9. associate-/r* 76.1%

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

      10. metadata-eval 76.1%

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

      11. associate-*l/ 76.1%

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

      12. associate-*r/ 76.1%

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

      13. swap-sqr 87.6%

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

      14. associate-*r* 87.6%

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

      15. associate-*r* 87.6%

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

   6. Simplified 87.6%

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

   7. Taylor expanded in ky around 0 0.4%

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

      1. unpow2 0.4%

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

      2. times-frac 0.4%

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

      3. unpow2 0.4%

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

      4. unpow2 0.4%

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

   9. Simplified 0.4%

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

       1. clear-num 0.4%

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

       2. associate-/l* 17.3%

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

       3. frac-times 19.7%

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

       4. *-un-lft-identity 19.7%

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

   11. Applied egg-rr 19.7%

       \[\leadsto 1 + -0.5 \cdot \color{blue}{\frac{\ell}{\frac{Om}{ky \cdot ky} \cdot \frac{Om}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.12 \cdot 10^{+156}:\\ \;\;\;\;1 + -0.5 \cdot \frac{ky}{\frac{Om}{\ell \cdot \ell} \cdot \frac{Om}{ky}}\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \frac{\ell}{\frac{Om}{\ell} \cdot \frac{Om}{ky \cdot ky}}\\ \end{array} \]

Alternative 8: 47.0% accurate, 42.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 5 \cdot 10^{-66}:\\ \;\;\;\;1 + -0.5 \cdot \frac{ky}{\frac{Om}{\ell \cdot \ell} \cdot \frac{Om}{ky}}\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \frac{\ell \cdot ky}{\frac{Om \cdot Om}{\ell \cdot ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= l 5e-66)
   (+ 1.0 (* -0.5 (/ ky (* (/ Om (* l l)) (/ Om ky)))))
   (+ 1.0 (* -0.5 (/ (* l ky) (/ (* Om Om) (* l ky)))))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 5e-66) {
		tmp = 1.0 + (-0.5 * (ky / ((Om / (l * l)) * (Om / ky))));
	} else {
		tmp = 1.0 + (-0.5 * ((l * ky) / ((Om * Om) / (l * ky))));
	}
	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 (l <= 5d-66) then
        tmp = 1.0d0 + ((-0.5d0) * (ky / ((om / (l * l)) * (om / ky))))
    else
        tmp = 1.0d0 + ((-0.5d0) * ((l * ky) / ((om * om) / (l * ky))))
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 5e-66) {
		tmp = 1.0 + (-0.5 * (ky / ((Om / (l * l)) * (Om / ky))));
	} else {
		tmp = 1.0 + (-0.5 * ((l * ky) / ((Om * Om) / (l * ky))));
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if l <= 5e-66:
		tmp = 1.0 + (-0.5 * (ky / ((Om / (l * l)) * (Om / ky))))
	else:
		tmp = 1.0 + (-0.5 * ((l * ky) / ((Om * Om) / (l * ky))))
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (l <= 5e-66)
		tmp = Float64(1.0 + Float64(-0.5 * Float64(ky / Float64(Float64(Om / Float64(l * l)) * Float64(Om / ky)))));
	else
		tmp = Float64(1.0 + Float64(-0.5 * Float64(Float64(l * ky) / Float64(Float64(Om * Om) / Float64(l * ky)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (l <= 5e-66)
		tmp = 1.0 + (-0.5 * (ky / ((Om / (l * l)) * (Om / ky))));
	else
		tmp = 1.0 + (-0.5 * ((l * ky) / ((Om * Om) / (l * ky))));
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[l, 5e-66], N[(1.0 + N[(-0.5 * N[(ky / N[(N[(Om / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(Om / ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.5 * N[(N[(l * ky), $MachinePrecision] / N[(N[(Om * Om), $MachinePrecision] / N[(l * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.99999999999999962e-66

   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 76.9%

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

      1. associate-*r/ 76.9%

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

      2. associate-*l/ 76.2%

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

      3. unpow2 76.2%

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

      4. *-commutative 76.2%

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

      5. *-commutative 76.2%

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

      6. unpow2 76.2%

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

      7. unpow2 76.2%

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

      8. unswap-sqr 79.8%

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

      9. associate-/r* 79.8%

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

      10. metadata-eval 79.8%

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

      11. associate-*l/ 79.8%

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

      12. associate-*r/ 79.8%

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

      13. swap-sqr 91.7%

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

      14. associate-*r* 91.7%

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

      15. associate-*r* 91.7%

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

   6. Simplified 91.7%

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

   7. Taylor expanded in ky around 0 42.8%

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

      1. unpow2 42.8%

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

      2. times-frac 44.8%

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

      3. unpow2 44.8%

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

      4. unpow2 44.8%

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

   9. Simplified 44.8%

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

       1. *-commutative 44.8%

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

       2. clear-num 44.8%

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

       3. associate-/l* 48.2%

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

       4. frac-times 56.2%

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

       5. *-un-lft-identity 56.2%

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

   11. Applied egg-rr 56.2%

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

   if 4.99999999999999962e-66 < 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 kx around 0 80.5%

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

      1. associate-*r/ 80.5%

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

      2. associate-*l/ 80.5%

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

      3. unpow2 80.5%

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

      4. *-commutative 80.5%

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

      5. *-commutative 80.5%

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

      6. unpow2 80.5%

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

      7. unpow2 80.5%

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

      8. unswap-sqr 90.1%

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

      9. associate-/r* 90.1%

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

      10. metadata-eval 90.1%

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

      11. associate-*l/ 90.1%

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

      12. associate-*r/ 90.1%

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

      13. swap-sqr 93.9%

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

      14. associate-*r* 93.9%

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

      15. associate-*r* 93.9%

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

   6. Simplified 93.9%

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

   7. Taylor expanded in ky around 0 26.3%

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

      1. unpow2 26.3%

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

      2. times-frac 27.1%

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

      3. unpow2 27.1%

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

      4. unpow2 27.1%

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

   9. Simplified 27.1%

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

       1. *-commutative 27.1%

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

       2. associate-/l* 33.9%

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

       3. associate-/l* 33.9%

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

       4. frac-times 34.5%

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

   11. Applied egg-rr 34.5%

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

   12. Taylor expanded in Om around 0 35.9%

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

       1. *-commutative 35.9%

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

       2. unpow2 35.9%

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

   14. Simplified 35.9%

       \[\leadsto 1 + -0.5 \cdot \frac{\ell \cdot ky}{\color{blue}{\frac{Om \cdot Om}{\ell \cdot ky}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5 \cdot 10^{-66}:\\ \;\;\;\;1 + -0.5 \cdot \frac{ky}{\frac{Om}{\ell \cdot \ell} \cdot \frac{Om}{ky}}\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \frac{\ell \cdot ky}{\frac{Om \cdot Om}{\ell \cdot ky}}\\ \end{array} \]

Alternative 9: 40.7% accurate, 48.1× speedup

Math

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

FPCore

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

C

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

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

Java

public static double code(double l, double Om, double kx, double ky) {
	return 1.0 + (-0.5 * ((ky / (Om / ky)) * ((l * l) / Om)));
}

Python

def code(l, Om, kx, ky):
	return 1.0 + (-0.5 * ((ky / (Om / ky)) * ((l * l) / Om)))

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 78.0%

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

   1. associate-*r/ 78.0%

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

   2. associate-*l/ 77.5%

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

   3. unpow2 77.5%

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

   4. *-commutative 77.5%

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

   5. *-commutative 77.5%

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

   6. unpow2 77.5%

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

   7. unpow2 77.5%

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

   8. unswap-sqr 83.0%

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

   9. associate-/r* 83.0%

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

   10. metadata-eval 83.0%

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

   11. associate-*l/ 83.0%

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

   12. associate-*r/ 83.0%

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

   13. swap-sqr 92.4%

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

   14. associate-*r* 92.4%

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

   15. associate-*r* 92.4%

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

6. Simplified 92.4%

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

7. Taylor expanded in ky around 0 37.7%

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

   1. unpow2 37.7%

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

   2. times-frac 39.4%

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

   3. unpow2 39.4%

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

   4. unpow2 39.4%

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

9. Simplified 39.4%

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

10. Taylor expanded in ky around 0 39.4%

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

    1. unpow2 39.4%

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

    2. associate-/l* 41.7%

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

12. Simplified 41.7%

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

13. Final simplification 41.7%

    \[\leadsto 1 + -0.5 \cdot \left(\frac{ky}{\frac{Om}{ky}} \cdot \frac{\ell \cdot \ell}{Om}\right) \]

Alternative 10: 43.1% accurate, 48.1× speedup

Math

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

FPCore

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

C

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

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

Java

public static double code(double l, double Om, double kx, double ky) {
	return 1.0 + (-0.5 * ((l / (Om / l)) * (ky * (ky / Om))));
}

Python

def code(l, Om, kx, ky):
	return 1.0 + (-0.5 * ((l / (Om / l)) * (ky * (ky / Om))))

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 78.0%

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

   1. associate-*r/ 78.0%

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

   2. associate-*l/ 77.5%

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

   3. unpow2 77.5%

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

   4. *-commutative 77.5%

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

   5. *-commutative 77.5%

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

   6. unpow2 77.5%

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

   7. unpow2 77.5%

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

   8. unswap-sqr 83.0%

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

   9. associate-/r* 83.0%

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

   10. metadata-eval 83.0%

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

   11. associate-*l/ 83.0%

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

   12. associate-*r/ 83.0%

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

   13. swap-sqr 92.4%

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

   14. associate-*r* 92.4%

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

   15. associate-*r* 92.4%

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

6. Simplified 92.4%

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

7. Taylor expanded in ky around 0 37.7%

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

   1. unpow2 37.7%

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

   2. times-frac 39.4%

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

   3. unpow2 39.4%

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

   4. unpow2 39.4%

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

9. Simplified 39.4%

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

    1. *-commutative 39.4%

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

    2. associate-/l* 41.4%

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

    3. associate-/l* 43.8%

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

    4. frac-times 48.0%

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

11. Applied egg-rr 48.0%

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

    1. times-frac 43.8%

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

    2. associate-/r/ 43.8%

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

13. Applied egg-rr 43.8%

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

14. Final simplification 43.8%

    \[\leadsto 1 + -0.5 \cdot \left(\frac{\ell}{\frac{Om}{\ell}} \cdot \left(ky \cdot \frac{ky}{Om}\right)\right) \]

Reproduce

herbie shell --seed 2023285 
(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))))))))))