Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.1% → 100.0%

Time: 20.2s

Alternatives: 7

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.1% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

   1. add-log-exp 98.8%

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

   2. add-sqr-sqrt 98.8%

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

   3. hypot-1-def 98.8%

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

   4. sqrt-prod 98.8%

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

   5. unpow2 98.8%

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

   6. sqrt-prod 53.2%

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

   7. add-sqr-sqrt 99.0%

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

   8. clear-num 99.0%

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

   9. un-div-inv 99.0%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

   1. *-un-lft-identity 98.8%

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

   2. add-sqr-sqrt 98.8%

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

   3. hypot-1-def 98.8%

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

   4. sqrt-prod 98.8%

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

   5. unpow2 98.8%

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

   6. sqrt-prod 53.2%

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

   7. add-sqr-sqrt 99.0%

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

   8. clear-num 99.0%

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

   9. un-div-inv 99.0%

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

   10. unpow2 99.0%

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

   11. unpow2 99.0%

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

   12. hypot-define 100.0%

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

6. Applied egg-rr 100.0%

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

   1. *-lft-identity 100.0%

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

   2. *-commutative 100.0%

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

   3. associate-/r/ 100.0%

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

8. Simplified 100.0%

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

9. Final simplification 100.0%

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

Alternative 3: 93.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

   1. add-sqr-sqrt 98.8%

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

   2. pow2 98.8%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in kx around 0 92.6%

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

   1. *-un-lft-identity 92.6%

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

   2. unpow2 92.6%

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

   3. add-sqr-sqrt 92.6%

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

   4. associate-*l/ 92.6%

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

   5. *-un-lft-identity 92.6%

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

   6. times-frac 92.6%

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

   7. metadata-eval 92.6%

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

9. Applied egg-rr 92.6%

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

    1. *-lft-identity 92.6%

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

    2. metadata-eval 92.6%

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

    3. times-frac 92.6%

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

    4. *-lft-identity 92.6%

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

11. Simplified 92.6%

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

    1. pow1/2 92.6%

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

    2. pow-to-exp 92.6%

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

    3. associate-*l/ 92.6%

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

    4. metadata-eval 92.6%

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

    5. *-un-lft-identity 92.6%

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

    6. *-commutative 92.6%

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

    7. times-frac 92.6%

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

    8. clear-num 92.6%

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

    9. div-inv 92.6%

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

    10. metadata-eval 92.6%

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

13. Applied egg-rr 92.6%

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

14. Final simplification 92.6%

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

Alternative 4: 74.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq 2.1 \cdot 10^{-211}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \sin kx \cdot \left(\ell \cdot \frac{2}{Om}\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= Om 2.1e-211)
   (sqrt 0.5)
   (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 (* (sin kx) (* l (/ 2.0 Om)))))))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 2.1e-211) {
		tmp = sqrt(0.5);
	} else {
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (sin(kx) * (l * (2.0 / Om)))))));
	}
	return tmp;
}

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 2.1e-211) {
		tmp = Math.sqrt(0.5);
	} else {
		tmp = Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, (Math.sin(kx) * (l * (2.0 / Om)))))));
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if Om <= 2.1e-211:
		tmp = math.sqrt(0.5)
	else:
		tmp = math.sqrt((0.5 + (0.5 / math.hypot(1.0, (math.sin(kx) * (l * (2.0 / Om)))))))
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Om <= 2.1e-211)
		tmp = sqrt(0.5);
	else
		tmp = sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, Float64(sin(kx) * Float64(l * Float64(2.0 / Om)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (Om <= 2.1e-211)
		tmp = sqrt(0.5);
	else
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (sin(kx) * (l * (2.0 / Om)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[Om, 2.1e-211], N[Sqrt[0.5], $MachinePrecision], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + N[(N[Sin[kx], $MachinePrecision] * N[(l * N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Om < 2.10000000000000008e-211

   1. Initial program 98.0%

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

   3. Simplified 98.0%

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

   5. Taylor expanded in l around inf 48.9%

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

      1. *-commutative 48.9%

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

      2. associate-*l* 48.9%

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

      3. *-commutative 48.9%

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

      4. unpow2 48.9%

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

      5. unpow2 48.9%

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

      6. hypot-undefine 50.9%

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

   7. Simplified 50.9%

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

   8. Taylor expanded in l around inf 59.4%

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

   if 2.10000000000000008e-211 < Om

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. add-sqr-sqrt 100.0%

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

      3. hypot-1-def 100.0%

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

      4. sqrt-prod 100.0%

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

      5. unpow2 100.0%

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

      6. sqrt-prod 56.2%

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

      7. add-sqr-sqrt 100.0%

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

      8. clear-num 100.0%

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

      9. un-div-inv 100.0%

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

      10. unpow2 100.0%

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

      11. unpow2 100.0%

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

      12. hypot-define 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. *-lft-identity 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-/r/ 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in ky around 0 94.4%

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

       1. associate-*l/ 94.4%

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

       2. metadata-eval 94.4%

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

       3. associate-/r/ 94.4%

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

       4. *-commutative 94.4%

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

       5. associate-/r/ 94.4%

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

       6. *-commutative 94.4%

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

   11. Applied egg-rr 94.4%

       \[\leadsto \sqrt{0.5 + \color{blue}{\frac{0.5}{\mathsf{hypot}\left(1, \left(\ell \cdot \frac{2}{Om}\right) \cdot \sin kx\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 2.1 \cdot 10^{-211}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \sin kx \cdot \left(\ell \cdot \frac{2}{Om}\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.6% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

   1. add-sqr-sqrt 98.8%

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

   2. pow2 98.8%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in kx around 0 92.6%

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

   1. *-un-lft-identity 92.6%

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

   2. unpow2 92.6%

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

   3. add-sqr-sqrt 92.6%

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

   4. associate-*l/ 92.6%

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

   5. *-un-lft-identity 92.6%

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

   6. times-frac 92.6%

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

   7. metadata-eval 92.6%

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

9. Applied egg-rr 92.6%

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

    1. *-lft-identity 92.6%

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

    2. metadata-eval 92.6%

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

    3. times-frac 92.6%

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

    4. *-lft-identity 92.6%

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

11. Simplified 92.6%

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

    1. *-un-lft-identity 92.6%

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

    2. associate-*l/ 92.6%

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

    3. metadata-eval 92.6%

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

    4. *-un-lft-identity 92.6%

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

    5. *-commutative 92.6%

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

    6. times-frac 92.6%

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

    7. clear-num 92.6%

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

    8. div-inv 92.6%

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

    9. metadata-eval 92.6%

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

13. Applied egg-rr 92.6%

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

    1. *-lft-identity 92.6%

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

    2. *-commutative 92.6%

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

    3. associate-*r/ 92.6%

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

    4. *-commutative 92.6%

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

15. Simplified 92.6%

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

16. Final simplification 92.6%

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

Alternative 6: 66.9% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq 1.22 \cdot 10^{-77}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om \leq 5 \cdot 10^{-66}:\\ \;\;\;\;1\\ \mathbf{elif}\;Om \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= Om 1.22e-77)
   (sqrt 0.5)
   (if (<= Om 5e-66) 1.0 (if (<= Om 2e-7) (sqrt 0.5) 1.0))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 1.22e-77) {
		tmp = sqrt(0.5);
	} else if (Om <= 5e-66) {
		tmp = 1.0;
	} else if (Om <= 2e-7) {
		tmp = sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: tmp
    if (om <= 1.22d-77) then
        tmp = sqrt(0.5d0)
    else if (om <= 5d-66) then
        tmp = 1.0d0
    else if (om <= 2d-7) then
        tmp = sqrt(0.5d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 1.22e-77) {
		tmp = Math.sqrt(0.5);
	} else if (Om <= 5e-66) {
		tmp = 1.0;
	} else if (Om <= 2e-7) {
		tmp = Math.sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if Om <= 1.22e-77:
		tmp = math.sqrt(0.5)
	elif Om <= 5e-66:
		tmp = 1.0
	elif Om <= 2e-7:
		tmp = math.sqrt(0.5)
	else:
		tmp = 1.0
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Om <= 1.22e-77)
		tmp = sqrt(0.5);
	elseif (Om <= 5e-66)
		tmp = 1.0;
	elseif (Om <= 2e-7)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (Om <= 1.22e-77)
		tmp = sqrt(0.5);
	elseif (Om <= 5e-66)
		tmp = 1.0;
	elseif (Om <= 2e-7)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[Om, 1.22e-77], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[Om, 5e-66], 1.0, If[LessEqual[Om, 2e-7], N[Sqrt[0.5], $MachinePrecision], 1.0]]]

Derivation

1. Split input into 2 regimes

2. if Om < 1.22000000000000001e-77 or 4.99999999999999962e-66 < Om < 1.9999999999999999e-7

   1. Initial program 98.4%

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

   3. Simplified 98.4%

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

   5. Taylor expanded in l around inf 55.2%

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

      1. *-commutative 55.2%

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

      2. associate-*l* 55.2%

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

      3. *-commutative 55.2%

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

      4. unpow2 55.2%

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

      5. unpow2 55.2%

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

      6. hypot-undefine 56.8%

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

   7. Simplified 56.8%

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

   8. Taylor expanded in l around inf 64.2%

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

   if 1.22000000000000001e-77 < Om < 4.99999999999999962e-66 or 1.9999999999999999e-7 < Om

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. add-sqr-sqrt 100.0%

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

      3. hypot-1-def 100.0%

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

      4. sqrt-prod 100.0%

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

      5. unpow2 100.0%

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

      6. sqrt-prod 60.9%

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

      7. add-sqr-sqrt 100.0%

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

      8. clear-num 100.0%

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

      9. un-div-inv 100.0%

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

      10. unpow2 100.0%

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

      11. unpow2 100.0%

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

      12. hypot-define 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. *-lft-identity 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-/r/ 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in ky around 0 96.7%

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

   10. Taylor expanded in kx around 0 87.4%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 1.22 \cdot 10^{-77}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om \leq 5 \cdot 10^{-66}:\\ \;\;\;\;1\\ \mathbf{elif}\;Om \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 56.6% accurate, 7.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in l around inf 45.0%

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

   1. *-commutative 45.0%

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

   2. associate-*l* 45.0%

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

   3. *-commutative 45.0%

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

   4. unpow2 45.0%

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

   5. unpow2 45.0%

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

   6. hypot-undefine 46.2%

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

7. Simplified 46.2%

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

8. Taylor expanded in l around inf 55.4%

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

9. Final simplification 55.4%

   \[\leadsto \sqrt{0.5} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024041 
(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))))))))))