Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.1% → 100.0%

Time: 25.3s

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

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

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (sqrt
  (+
   0.5
   (*
    0.5
    (/
     1.0
     (pow
      (sqrt (hypot 1.0 (* (* 2.0 (/ l Om)) (hypot (sin ky) (sin kx)))))
      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, ((2.0 * (l / Om)) * hypot(sin(ky), sin(kx))))), 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, ((2.0 * (l / Om)) * Math.hypot(Math.sin(ky), Math.sin(kx))))), 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, ((2.0 * (l / Om)) * math.hypot(math.sin(ky), math.sin(kx))))), 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(2.0 * Float64(l / Om)) * hypot(sin(ky), sin(kx))))) ^ 2.0)))))
end

MATLAB

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

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

6. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

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, ((2.0 * (l / Om)) * Math.hypot(Math.sin(ky), Math.sin(kx))))))));
}

Python

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

Julia

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

MATLAB

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

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

   2. hypot-1-def 100.0%

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

   3. sqrt-prod 100.0%

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

   4. unpow2 100.0%

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

   5. sqrt-prod 57.0%

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

   6. add-sqr-sqrt 100.0%

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

   7. div-inv 100.0%

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

   8. clear-num 100.0%

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

   9. +-commutative 100.0%

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

   10. unpow2 100.0%

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

   11. unpow2 100.0%

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

   12. hypot-def 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 3: 94.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

4. Taylor expanded in kx around 0 78.0%

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

   1. associate-/l* 78.1%

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

   2. unpow2 78.1%

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

   3. unpow2 78.1%

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

6. Simplified 78.1%

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

   1. add-sqr-sqrt 78.1%

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

   2. pow2 78.1%

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

8. Applied egg-rr 92.4%

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

9. Final simplification 92.4%

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

Alternative 4: 94.4% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

4. Taylor expanded in kx around 0 78.0%

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

   1. associate-/l* 78.1%

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

   2. unpow2 78.1%

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

   3. unpow2 78.1%

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

6. Simplified 78.1%

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

   1. expm1-log1p-u 78.1%

      \[\leadsto \sqrt{0.5 + \frac{0.5}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 + 4 \cdot \frac{\ell \cdot \ell}{\frac{Om \cdot Om}{{\sin ky}^{2}}}}\right)\right)}}} \]

   2. expm1-udef 78.1%

      \[\leadsto \sqrt{0.5 + \frac{0.5}{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{1 + 4 \cdot \frac{\ell \cdot \ell}{\frac{Om \cdot Om}{{\sin ky}^{2}}}}\right)} - 1}}} \]

8. Applied egg-rr 92.4%

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

   1. expm1-def 92.4%

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

   2. expm1-log1p 92.4%

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

   3. associate-*r* 92.4%

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

   4. associate-*r/ 92.4%

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

10. Simplified 92.4%

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

11. Final simplification 92.4%

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

Alternative 5: 78.9% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2 \cdot \ell}{Om}\\ \mathbf{if}\;t_0 \leq 0.0002:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, ky \cdot t_0\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (let* ((t_0 (/ (* 2.0 l) Om)))
   (if (<= t_0 0.0002) 1.0 (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 (* ky t_0))))))))

C

double code(double l, double Om, double kx, double ky) {
	double t_0 = (2.0 * l) / Om;
	double tmp;
	if (t_0 <= 0.0002) {
		tmp = 1.0;
	} else {
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (ky * t_0)))));
	}
	return tmp;
}

Java

public static double code(double l, double Om, double kx, double ky) {
	double t_0 = (2.0 * l) / Om;
	double tmp;
	if (t_0 <= 0.0002) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, (ky * t_0)))));
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	t_0 = (2.0 * l) / Om
	tmp = 0
	if t_0 <= 0.0002:
		tmp = 1.0
	else:
		tmp = math.sqrt((0.5 + (0.5 / math.hypot(1.0, (ky * t_0)))))
	return tmp

Julia

function code(l, Om, kx, ky)
	t_0 = Float64(Float64(2.0 * l) / Om)
	tmp = 0.0
	if (t_0 <= 0.0002)
		tmp = 1.0;
	else
		tmp = sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, Float64(ky * t_0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	t_0 = (2.0 * l) / Om;
	tmp = 0.0;
	if (t_0 <= 0.0002)
		tmp = 1.0;
	else
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (ky * t_0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := Block[{t$95$0 = N[(N[(2.0 * l), $MachinePrecision] / Om), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0002], 1.0, N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + N[(ky * t$95$0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 2 l) Om) < 2.0000000000000001e-4

   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%

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

   4. Taylor expanded in kx around 0 80.8%

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

      1. associate-/l* 80.9%

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

      2. unpow2 80.9%

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

      3. unpow2 80.9%

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

   6. Simplified 80.9%

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

      1. expm1-log1p-u 80.9%

         \[\leadsto \sqrt{0.5 + \frac{0.5}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 + 4 \cdot \frac{\ell \cdot \ell}{\frac{Om \cdot Om}{{\sin ky}^{2}}}}\right)\right)}}} \]

      2. expm1-udef 80.9%

         \[\leadsto \sqrt{0.5 + \frac{0.5}{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{1 + 4 \cdot \frac{\ell \cdot \ell}{\frac{Om \cdot Om}{{\sin ky}^{2}}}}\right)} - 1}}} \]

   8. Applied egg-rr 94.3%

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

      1. expm1-def 94.3%

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

      2. expm1-log1p 94.3%

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

      3. associate-*r* 94.3%

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

      4. associate-*r/ 94.3%

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

   10. Simplified 94.3%

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

   11. Taylor expanded in l around 0 77.2%

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

   if 2.0000000000000001e-4 < (/.f64 (*.f64 2 l) 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 100.0%

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

   4. Taylor expanded in kx around 0 69.2%

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

      1. associate-/l* 69.2%

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

      2. unpow2 69.2%

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

      3. unpow2 69.2%

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

   6. Simplified 69.2%

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

      1. expm1-log1p-u 69.2%

         \[\leadsto \sqrt{0.5 + \frac{0.5}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 + 4 \cdot \frac{\ell \cdot \ell}{\frac{Om \cdot Om}{{\sin ky}^{2}}}}\right)\right)}}} \]

      2. expm1-udef 69.2%

         \[\leadsto \sqrt{0.5 + \frac{0.5}{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{1 + 4 \cdot \frac{\ell \cdot \ell}{\frac{Om \cdot Om}{{\sin ky}^{2}}}}\right)} - 1}}} \]

   8. Applied egg-rr 86.3%

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

      1. expm1-def 86.3%

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

      2. expm1-log1p 86.3%

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

      3. associate-*r* 86.3%

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

      4. associate-*r/ 86.3%

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

   10. Simplified 86.3%

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

   11. Taylor expanded in ky around 0 85.0%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 \cdot \ell}{Om} \leq 0.0002:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, ky \cdot \frac{2 \cdot \ell}{Om}\right)}}\\ \end{array} \]

Alternative 6: 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 Om around 0 52.4%

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

      1. unpow2 52.4%

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

      2. unpow2 52.4%

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

      3. hypot-def 52.4%

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

   6. Simplified 52.4%

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

   7. Taylor expanded in l around inf 60.3%

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

   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%

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

   4. Taylor expanded in kx around 0 84.5%

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

      1. associate-/l* 84.5%

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

      2. unpow2 84.5%

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

      3. unpow2 84.5%

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

   6. Simplified 84.5%

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

      1. expm1-log1p-u 84.5%

         \[\leadsto \sqrt{0.5 + \frac{0.5}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 + 4 \cdot \frac{\ell \cdot \ell}{\frac{Om \cdot Om}{{\sin ky}^{2}}}}\right)\right)}}} \]

      2. expm1-udef 84.5%

         \[\leadsto \sqrt{0.5 + \frac{0.5}{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{1 + 4 \cdot \frac{\ell \cdot \ell}{\frac{Om \cdot Om}{{\sin ky}^{2}}}}\right)} - 1}}} \]

   8. Applied egg-rr 93.5%

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

      1. expm1-def 93.5%

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

      2. expm1-log1p 93.5%

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

      3. associate-*r* 93.5%

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

      4. associate-*r/ 93.5%

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

   10. Simplified 93.5%

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

   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 7: 54.7% 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 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in Om around 0 44.2%

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

   1. unpow2 44.2%

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

   2. unpow2 44.2%

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

   3. hypot-def 44.2%

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

6. Simplified 44.2%

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

7. Taylor expanded in l around inf 53.6%

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

8. Final simplification 53.6%

   \[\leadsto \sqrt{0.5} \]

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))))))))))