Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.2% → 100.0%

Time: 18.0s

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: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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%

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

   1. expm1-log1p-u 98.0%

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

   2. expm1-udef 98.0%

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

5. Applied egg-rr 100.0%

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

   1. expm1-def 100.0%

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

   2. expm1-log1p 100.0%

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

   3. *-commutative 100.0%

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

   4. associate-*r/ 100.0%

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

   5. associate-/l* 100.0%

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

7. Simplified 100.0%

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

8. Final simplification 100.0%

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

Alternative 2: 93.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

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

Python

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

Julia

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

MATLAB

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

Derivation

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%

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

   1. expm1-log1p-u 98.0%

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

   2. expm1-udef 98.0%

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

5. Applied egg-rr 100.0%

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

   1. expm1-def 100.0%

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

   2. expm1-log1p 100.0%

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

   3. *-commutative 100.0%

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

   4. associate-*r/ 100.0%

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

   5. associate-/l* 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in kx around 0 92.4%

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

   1. add-log-exp 92.4%

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

   2. un-div-inv 92.4%

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

   3. *-commutative 92.4%

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

   4. associate-/r/ 92.4%

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

   5. associate-*l* 92.4%

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

10. Applied egg-rr 92.4%

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

11. Final simplification 92.4%

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

Alternative 3: 90.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 9 \cdot 10^{-115}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{2}{Om} \cdot \left(ky \cdot \ell\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \sin kx \cdot \frac{2 \cdot \ell}{Om}\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= kx 9e-115)
   (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 (* (/ 2.0 Om) (* ky l))))))
   (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 (* (sin kx) (/ (* 2.0 l) Om))))))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (kx <= 9e-115) {
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, ((2.0 / Om) * (ky * l))))));
	} else {
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (sin(kx) * ((2.0 * l) / Om))))));
	}
	return tmp;
}

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (kx <= 9e-115) {
		tmp = Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, ((2.0 / Om) * (ky * l))))));
	} else {
		tmp = Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, (Math.sin(kx) * ((2.0 * l) / Om))))));
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if kx <= 9e-115:
		tmp = math.sqrt((0.5 + (0.5 / math.hypot(1.0, ((2.0 / Om) * (ky * l))))))
	else:
		tmp = math.sqrt((0.5 + (0.5 / math.hypot(1.0, (math.sin(kx) * ((2.0 * l) / Om))))))
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (kx <= 9e-115)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, Float64(Float64(2.0 / Om) * Float64(ky * l))))));
	else
		tmp = sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, Float64(sin(kx) * Float64(Float64(2.0 * l) / Om))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (kx <= 9e-115)
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, ((2.0 / Om) * (ky * l))))));
	else
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (sin(kx) * ((2.0 * l) / Om))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[kx, 9e-115], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + N[(N[(2.0 / Om), $MachinePrecision] * N[(ky * l), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + N[(N[Sin[kx], $MachinePrecision] * N[(N[(2.0 * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 9.00000000000000046e-115

   1. Initial program 97.0%

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

   3. Simplified 97.0%

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

      1. expm1-log1p-u 97.0%

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

      2. expm1-udef 97.0%

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

   5. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*r/ 100.0%

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

      5. associate-/l* 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in kx around 0 93.4%

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

      1. un-div-inv 93.4%

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

      2. *-commutative 93.4%

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

      3. associate-/r/ 93.4%

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

      4. associate-*l* 93.4%

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

   10. Applied egg-rr 93.4%

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

   11. Taylor expanded in ky around 0 89.4%

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

       1. *-commutative 89.4%

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

   13. Simplified 89.4%

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

   if 9.00000000000000046e-115 < kx

   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 93.7%

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

      1. associate-/l* 92.5%

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

      2. associate-/r/ 93.7%

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

      3. associate-*l* 93.7%

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

      4. metadata-eval 93.7%

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

      5. unpow2 93.7%

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

      6. unpow2 93.7%

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

      7. times-frac 98.6%

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

      8. swap-sqr 98.6%

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

      9. unpow2 98.6%

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

      10. swap-sqr 98.6%

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

      11. hypot-1-def 98.6%

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

      12. *-commutative 98.6%

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

      13. associate-*r/ 98.6%

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

      14. associate-*r/ 98.6%

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

   6. Simplified 98.6%

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

      1. expm1-log1p-u 98.6%

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

      2. expm1-udef 98.6%

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

      3. un-div-inv 98.6%

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

      4. associate-/l* 98.6%

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

      5. div-inv 98.6%

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

      6. associate-/l/ 98.6%

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

      7. clear-num 98.6%

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

      8. associate-/r/ 98.6%

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

      9. *-commutative 98.6%

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

   8. Applied egg-rr 98.6%

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

      1. expm1-def 98.6%

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

      2. expm1-log1p 98.6%

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

      3. associate-*r/ 98.6%

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

   10. Simplified 98.6%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 9 \cdot 10^{-115}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{2}{Om} \cdot \left(ky \cdot \ell\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \sin kx \cdot \frac{2 \cdot \ell}{Om}\right)}}\\ \end{array} \]

Alternative 4: 93.5% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

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%

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

   1. expm1-log1p-u 98.0%

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

   2. expm1-udef 98.0%

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

5. Applied egg-rr 100.0%

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

   1. expm1-def 100.0%

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

   2. expm1-log1p 100.0%

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

   3. *-commutative 100.0%

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

   4. associate-*r/ 100.0%

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

   5. associate-/l* 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in kx around 0 92.4%

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

   1. un-div-inv 92.4%

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

   2. *-commutative 92.4%

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

   3. associate-/r/ 92.4%

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

   4. associate-*l* 92.4%

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

10. Applied egg-rr 92.4%

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

11. Final simplification 92.4%

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

Alternative 5: 85.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq 5.6 \cdot 10^{+73}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{2}{Om} \cdot \left(ky \cdot \ell\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= Om 5.6e+73)
   (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 (* (/ 2.0 Om) (* ky l))))))
   1.0))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 5.6e+73) {
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, ((2.0 / Om) * (ky * l))))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 5.6e+73) {
		tmp = Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, ((2.0 / Om) * (ky * l))))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if Om <= 5.6e+73:
		tmp = math.sqrt((0.5 + (0.5 / math.hypot(1.0, ((2.0 / Om) * (ky * l))))))
	else:
		tmp = 1.0
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Om <= 5.6e+73)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, Float64(Float64(2.0 / Om) * Float64(ky * l))))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (Om <= 5.6e+73)
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, ((2.0 / Om) * (ky * l))))));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[Om, 5.6e+73], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + N[(N[(2.0 / Om), $MachinePrecision] * N[(ky * l), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if Om < 5.60000000000000016e73

   1. Initial program 97.7%

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

   3. Simplified 97.7%

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

      1. expm1-log1p-u 97.7%

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

      2. expm1-udef 97.7%

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

   5. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*r/ 100.0%

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

      5. associate-/l* 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in kx around 0 91.2%

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

      1. un-div-inv 91.2%

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

      2. *-commutative 91.2%

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

      3. associate-/r/ 91.2%

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

      4. associate-*l* 91.2%

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

   10. Applied egg-rr 91.2%

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

   11. Taylor expanded in ky around 0 86.7%

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

       1. *-commutative 86.7%

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

   13. Simplified 86.7%

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

   if 5.60000000000000016e73 < 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. Step-by-step derivation

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*r/ 100.0%

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

      5. associate-/l* 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in kx around 0 98.4%

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

      1. add-log-exp 98.4%

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

      2. un-div-inv 98.4%

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

      3. *-commutative 98.4%

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

      4. associate-/r/ 98.4%

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

      5. associate-*l* 98.4%

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

   10. Applied egg-rr 98.4%

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

   11. Taylor expanded in Om around inf 92.2%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 5.6 \cdot 10^{+73}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{2}{Om} \cdot \left(ky \cdot \ell\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6: 71.1% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 10^{-17}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 4.4 \cdot 10^{+78}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;\ell \leq 9.5 \cdot 10^{+94}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= l 1e-17)
   1.0
   (if (<= l 4.4e+78) (sqrt 0.5) (if (<= l 9.5e+94) 1.0 (sqrt 0.5)))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 1e-17) {
		tmp = 1.0;
	} else if (l <= 4.4e+78) {
		tmp = sqrt(0.5);
	} else if (l <= 9.5e+94) {
		tmp = 1.0;
	} else {
		tmp = sqrt(0.5);
	}
	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-17) then
        tmp = 1.0d0
    else if (l <= 4.4d+78) then
        tmp = sqrt(0.5d0)
    else if (l <= 9.5d+94) then
        tmp = 1.0d0
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 1e-17) {
		tmp = 1.0;
	} else if (l <= 4.4e+78) {
		tmp = Math.sqrt(0.5);
	} else if (l <= 9.5e+94) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if l <= 1e-17:
		tmp = 1.0
	elif l <= 4.4e+78:
		tmp = math.sqrt(0.5)
	elif l <= 9.5e+94:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (l <= 1e-17)
		tmp = 1.0;
	elseif (l <= 4.4e+78)
		tmp = sqrt(0.5);
	elseif (l <= 9.5e+94)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (l <= 1e-17)
		tmp = 1.0;
	elseif (l <= 4.4e+78)
		tmp = sqrt(0.5);
	elseif (l <= 9.5e+94)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[l, 1e-17], 1.0, If[LessEqual[l, 4.4e+78], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[l, 9.5e+94], 1.0, N[Sqrt[0.5], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if l < 1.00000000000000007e-17 or 4.40000000000000028e78 < l < 9.4999999999999998e94

   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%

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

      1. expm1-log1p-u 98.0%

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

      2. expm1-udef 98.0%

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

   5. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*r/ 100.0%

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

      5. associate-/l* 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in kx around 0 92.9%

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

      1. add-log-exp 92.9%

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

      2. un-div-inv 92.9%

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

      3. *-commutative 92.9%

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

      4. associate-/r/ 92.9%

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

      5. associate-*l* 92.9%

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

   10. Applied egg-rr 92.9%

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

   11. Taylor expanded in Om around inf 70.6%

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

   if 1.00000000000000007e-17 < l < 4.40000000000000028e78 or 9.4999999999999998e94 < l

   1. Initial program 98.3%

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

   3. Simplified 98.3%

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

   4. Taylor expanded in Om around 0 84.4%

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

      1. associate-*r* 84.4%

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

      2. *-commutative 84.4%

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

      3. unpow2 84.4%

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

      4. unpow2 84.4%

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

      5. hypot-def 86.1%

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

      6. associate-*r/ 86.1%

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

      7. associate-/l* 86.1%

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

   6. Simplified 86.1%

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

   7. Taylor expanded in Om around 0 87.9%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 10^{-17}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 4.4 \cdot 10^{+78}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;\ell \leq 9.5 \cdot 10^{+94}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 7: 62.7% accurate, 722.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (l Om kx ky) :precision binary64 1.0)

C

double code(double l, double Om, double kx, double ky) {
	return 1.0;
}

Fortran

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    code = 1.0d0
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	return 1.0;
}

Python

def code(l, Om, kx, ky):
	return 1.0

Julia

function code(l, Om, kx, ky)
	return 1.0
end

MATLAB

function tmp = code(l, Om, kx, ky)
	tmp = 1.0;
end

Wolfram

code[l_, Om_, kx_, ky_] := 1.0

Derivation

1. Initial program 98.0%

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

3. Simplified 98.0%

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

   1. expm1-log1p-u 98.0%

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

   2. expm1-udef 98.0%

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

5. Applied egg-rr 100.0%

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

   1. expm1-def 100.0%

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

   2. expm1-log1p 100.0%

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

   3. *-commutative 100.0%

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

   4. associate-*r/ 100.0%

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

   5. associate-/l* 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in kx around 0 92.4%

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

   1. add-log-exp 92.4%

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

   2. un-div-inv 92.4%

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

   3. *-commutative 92.4%

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

   4. associate-/r/ 92.4%

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

   5. associate-*l* 92.4%

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

10. Applied egg-rr 92.4%

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

11. Taylor expanded in Om around inf 61.5%

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

12. Final simplification 61.5%

    \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023313 
(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))))))))))