Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.0% → 82.2%

Time: 17.3s

Alternatives: 13

Speedup: 3.8×

• 27.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (*
   (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
   (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))

C

double code(double t, double l, double k) {
	return 2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) + 1.0));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) + 1.0d0))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t), 2.0)) + 1.0));
}

Python

def code(t, l, k):
	return 2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t), 2.0)) + 1.0))

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) + 1.0)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) + 1.0));
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 55.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (*
   (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
   (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))

C

double code(double t, double l, double k) {
	return 2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) + 1.0));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) + 1.0d0))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t), 2.0)) + 1.0));
}

Python

def code(t, l, k):
	return 2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t), 2.0)) + 1.0))

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) + 1.0)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) + 1.0));
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 82.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 2.35 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{1}{\frac{k}{\ell}}}{\frac{\frac{k}{\ell}}{{t}^{-3}}}\\ \mathbf{elif}\;k \leq 3.6 \cdot 10^{+192}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{\frac{2}{\tan k}}{\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \cos k}{t}}{{\sin k}^{2}}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 2.35e-17)
   (/ (/ 1.0 (/ k l)) (/ (/ k l) (pow t -3.0)))
   (if (<= k 3.6e+192)
     (* l (* l (/ (/ 2.0 (tan k)) (* (sin k) (* k (* t k))))))
     (* 2.0 (/ (/ (* (* (/ l k) (/ l k)) (cos k)) t) (pow (sin k) 2.0))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.35e-17) {
		tmp = (1.0 / (k / l)) / ((k / l) / pow(t, -3.0));
	} else if (k <= 3.6e+192) {
		tmp = l * (l * ((2.0 / tan(k)) / (sin(k) * (k * (t * k)))));
	} else {
		tmp = 2.0 * (((((l / k) * (l / k)) * cos(k)) / t) / pow(sin(k), 2.0));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.35d-17) then
        tmp = (1.0d0 / (k / l)) / ((k / l) / (t ** (-3.0d0)))
    else if (k <= 3.6d+192) then
        tmp = l * (l * ((2.0d0 / tan(k)) / (sin(k) * (k * (t * k)))))
    else
        tmp = 2.0d0 * (((((l / k) * (l / k)) * cos(k)) / t) / (sin(k) ** 2.0d0))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.35e-17) {
		tmp = (1.0 / (k / l)) / ((k / l) / Math.pow(t, -3.0));
	} else if (k <= 3.6e+192) {
		tmp = l * (l * ((2.0 / Math.tan(k)) / (Math.sin(k) * (k * (t * k)))));
	} else {
		tmp = 2.0 * (((((l / k) * (l / k)) * Math.cos(k)) / t) / Math.pow(Math.sin(k), 2.0));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 2.35e-17:
		tmp = (1.0 / (k / l)) / ((k / l) / math.pow(t, -3.0))
	elif k <= 3.6e+192:
		tmp = l * (l * ((2.0 / math.tan(k)) / (math.sin(k) * (k * (t * k)))))
	else:
		tmp = 2.0 * (((((l / k) * (l / k)) * math.cos(k)) / t) / math.pow(math.sin(k), 2.0))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 2.35e-17)
		tmp = Float64(Float64(1.0 / Float64(k / l)) / Float64(Float64(k / l) / (t ^ -3.0)));
	elseif (k <= 3.6e+192)
		tmp = Float64(l * Float64(l * Float64(Float64(2.0 / tan(k)) / Float64(sin(k) * Float64(k * Float64(t * k))))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(Float64(Float64(l / k) * Float64(l / k)) * cos(k)) / t) / (sin(k) ^ 2.0)));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2.35e-17)
		tmp = (1.0 / (k / l)) / ((k / l) / (t ^ -3.0));
	elseif (k <= 3.6e+192)
		tmp = l * (l * ((2.0 / tan(k)) / (sin(k) * (k * (t * k)))));
	else
		tmp = 2.0 * (((((l / k) * (l / k)) * cos(k)) / t) / (sin(k) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 2.35e-17], N[(N[(1.0 / N[(k / l), $MachinePrecision]), $MachinePrecision] / N[(N[(k / l), $MachinePrecision] / N[Power[t, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 3.6e+192], N[(l * N[(l * N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 2.35e-17

   1. Initial program 55.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 55.0%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-*l* 47.8%

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

      3. associate-*r* 47.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      4. +-commutative 47.8%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      5. associate-+r+ 47.8%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      6. metadata-eval 47.8%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   3. Simplified 47.8%

      \[\leadsto \color{blue}{\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

   4. Taylor expanded in k around 0 49.6%

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   5. Step-by-step derivation

      1. associate-/l* 50.6%

         \[\leadsto \frac{2}{2 \cdot \color{blue}{\frac{{k}^{2}}{\frac{{\ell}^{2}}{{t}^{3}}}}} \]

      2. unpow2 50.6%

         \[\leadsto \frac{2}{2 \cdot \frac{\color{blue}{k \cdot k}}{\frac{{\ell}^{2}}{{t}^{3}}}} \]

      3. unpow2 50.6%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\frac{\color{blue}{\ell \cdot \ell}}{{t}^{3}}}} \]

   6. Simplified 50.6%

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{k \cdot k}{\frac{\ell \cdot \ell}{{t}^{3}}}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 40.1%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\ell \cdot \ell}{{t}^{3}}\right)\right)}}} \]

      2. expm1-udef 35.6%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{e^{\mathsf{log1p}\left(\frac{\ell \cdot \ell}{{t}^{3}}\right)} - 1}}} \]

      3. div-inv 35.6%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{e^{\mathsf{log1p}\left(\color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{1}{{t}^{3}}}\right)} - 1}} \]

      4. pow-flip 35.6%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \color{blue}{{t}^{\left(-3\right)}}\right)} - 1}} \]

      5. metadata-eval 35.6%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot {t}^{\color{blue}{-3}}\right)} - 1}} \]

   8. Applied egg-rr 35.6%

      \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot {t}^{-3}\right)} - 1}}} \]
   9. Step-by-step derivation

      1. expm1-def 40.1%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot {t}^{-3}\right)\right)}}} \]

      2. expm1-log1p 50.6%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{\left(\ell \cdot \ell\right) \cdot {t}^{-3}}}} \]

      3. associate-*l* 53.6%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{\ell \cdot \left(\ell \cdot {t}^{-3}\right)}}} \]

   10. Simplified 53.6%

       \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{\ell \cdot \left(\ell \cdot {t}^{-3}\right)}}} \]
   11. Step-by-step derivation

       1. expm1-log1p-u 39.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{2 \cdot \frac{k \cdot k}{\ell \cdot \left(\ell \cdot {t}^{-3}\right)}}\right)\right)} \]

       2. expm1-udef 39.2%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{2 \cdot \frac{k \cdot k}{\ell \cdot \left(\ell \cdot {t}^{-3}\right)}}\right)} - 1} \]

       3. associate-/r* 39.2%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{2}{2}}{\frac{k \cdot k}{\ell \cdot \left(\ell \cdot {t}^{-3}\right)}}}\right)} - 1 \]

       4. metadata-eval 39.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{1}}{\frac{k \cdot k}{\ell \cdot \left(\ell \cdot {t}^{-3}\right)}}\right)} - 1 \]

       5. times-frac 46.8%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{\frac{k}{\ell} \cdot \frac{k}{\ell \cdot {t}^{-3}}}}\right)} - 1 \]

   12. Applied egg-rr 46.8%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\frac{k}{\ell} \cdot \frac{k}{\ell \cdot {t}^{-3}}}\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 48.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{k}{\ell} \cdot \frac{k}{\ell \cdot {t}^{-3}}}\right)\right)} \]

       2. expm1-log1p 63.7%

          \[\leadsto \color{blue}{\frac{1}{\frac{k}{\ell} \cdot \frac{k}{\ell \cdot {t}^{-3}}}} \]

       3. associate-/r* 63.7%

          \[\leadsto \color{blue}{\frac{\frac{1}{\frac{k}{\ell}}}{\frac{k}{\ell \cdot {t}^{-3}}}} \]

       4. associate-/r* 64.6%

          \[\leadsto \frac{\frac{1}{\frac{k}{\ell}}}{\color{blue}{\frac{\frac{k}{\ell}}{{t}^{-3}}}} \]

   14. Simplified 64.6%

       \[\leadsto \color{blue}{\frac{\frac{1}{\frac{k}{\ell}}}{\frac{\frac{k}{\ell}}{{t}^{-3}}}} \]

   if 2.35e-17 < k < 3.6000000000000002e192

   1. Initial program 49.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 49.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 49.3%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 49.3%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 49.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 49.3%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 49.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 49.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 49.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 49.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 49.3%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 49.3%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 79.3%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left({k}^{2} \cdot \left(\sin k \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. unpow2 79.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \left(\sin k \cdot t\right)\right)} \]

      2. *-commutative 79.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \sin k\right)}\right)} \]

   6. Simplified 79.3%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 68.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}\right)\right)} \]

      2. expm1-udef 60.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}\right)} - 1} \]

   8. Applied egg-rr 60.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 68.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}\right)\right)} \]

      2. expm1-log1p 79.3%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}} \]

      3. associate-*l* 84.4%

         \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}\right)} \]

      4. associate-/r* 84.4%

         \[\leadsto \ell \cdot \left(\ell \cdot \color{blue}{\frac{\frac{2}{\tan k}}{\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)}}\right) \]

      5. unpow2 84.4%

         \[\leadsto \ell \cdot \left(\ell \cdot \frac{\frac{2}{\tan k}}{\color{blue}{{k}^{2}} \cdot \left(t \cdot \sin k\right)}\right) \]

      6. *-commutative 84.4%

         \[\leadsto \ell \cdot \left(\ell \cdot \frac{\frac{2}{\tan k}}{\color{blue}{\left(t \cdot \sin k\right) \cdot {k}^{2}}}\right) \]

      7. *-commutative 84.4%

         \[\leadsto \ell \cdot \left(\ell \cdot \frac{\frac{2}{\tan k}}{\color{blue}{\left(\sin k \cdot t\right)} \cdot {k}^{2}}\right) \]

      8. associate-*l* 84.4%

         \[\leadsto \ell \cdot \left(\ell \cdot \frac{\frac{2}{\tan k}}{\color{blue}{\sin k \cdot \left(t \cdot {k}^{2}\right)}}\right) \]

      9. *-commutative 84.4%

         \[\leadsto \ell \cdot \left(\ell \cdot \frac{\frac{2}{\tan k}}{\sin k \cdot \color{blue}{\left({k}^{2} \cdot t\right)}}\right) \]

      10. unpow2 84.4%

          \[\leadsto \ell \cdot \left(\ell \cdot \frac{\frac{2}{\tan k}}{\sin k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)}\right) \]

      11. associate-*l* 89.5%

          \[\leadsto \ell \cdot \left(\ell \cdot \frac{\frac{2}{\tan k}}{\sin k \cdot \color{blue}{\left(k \cdot \left(k \cdot t\right)\right)}}\right) \]

   10. Simplified 89.5%

       \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \frac{\frac{2}{\tan k}}{\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)}\right)} \]

   if 3.6000000000000002e192 < k

   1. Initial program 52.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 52.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 52.0%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 52.0%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 56.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 56.0%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 56.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 56.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 56.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 56.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 56.0%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 56.0%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 68.2%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 68.2%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

      2. times-frac 68.2%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

      3. unpow2 68.2%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      4. unpow2 68.2%

         \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      5. times-frac 99.6%

         \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      6. *-commutative 99.6%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

   6. Simplified 99.6%

      \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} \]

   7. Taylor expanded in l around 0 68.2%

      \[\leadsto 2 \cdot \color{blue}{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   8. Step-by-step derivation

      1. associate-/r* 68.2%

         \[\leadsto 2 \cdot \color{blue}{\frac{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2}}}{{\sin k}^{2} \cdot t}} \]

      2. *-commutative 68.2%

         \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2}}}{\color{blue}{t \cdot {\sin k}^{2}}} \]

      3. associate-*r/ 68.2%

         \[\leadsto 2 \cdot \frac{\color{blue}{\cos k \cdot \frac{{\ell}^{2}}{{k}^{2}}}}{t \cdot {\sin k}^{2}} \]

      4. unpow2 68.2%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{t \cdot {\sin k}^{2}} \]

      5. unpow2 68.2%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{t \cdot {\sin k}^{2}} \]

      6. times-frac 99.6%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}}{t \cdot {\sin k}^{2}} \]

      7. unpow2 99.6%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \color{blue}{{\left(\frac{\ell}{k}\right)}^{2}}}{t \cdot {\sin k}^{2}} \]

      8. *-commutative 99.6%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\left(\frac{\ell}{k}\right)}^{2} \cdot \cos k}}{t \cdot {\sin k}^{2}} \]

      9. times-frac 99.7%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)} \]

   9. Simplified 99.7%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)} \]
   10. Step-by-step derivation

       1. frac-times 99.6%

          \[\leadsto 2 \cdot \color{blue}{\frac{{\left(\frac{\ell}{k}\right)}^{2} \cdot \cos k}{t \cdot {\sin k}^{2}}} \]

       2. associate-/r* 99.6%

          \[\leadsto 2 \cdot \color{blue}{\frac{\frac{{\left(\frac{\ell}{k}\right)}^{2} \cdot \cos k}{t}}{{\sin k}^{2}}} \]

   11. Applied egg-rr 99.6%

       \[\leadsto 2 \cdot \color{blue}{\frac{\frac{{\left(\frac{\ell}{k}\right)}^{2} \cdot \cos k}{t}}{{\sin k}^{2}}} \]
   12. Step-by-step derivation

       1. unpow2 99.6%

          \[\leadsto 2 \cdot \frac{\frac{\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \cos k}{t}}{{\sin k}^{2}} \]

   13. Applied egg-rr 99.6%

       \[\leadsto 2 \cdot \frac{\frac{\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \cos k}{t}}{{\sin k}^{2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.35 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{1}{\frac{k}{\ell}}}{\frac{\frac{k}{\ell}}{{t}^{-3}}}\\ \mathbf{elif}\;k \leq 3.6 \cdot 10^{+192}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{\frac{2}{\tan k}}{\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \cos k}{t}}{{\sin k}^{2}}\\ \end{array} \]

Alternative 2: 80.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} t_1 := 1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\\ \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot t_1} \leq 5 \cdot 10^{+298}:\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\tan k \cdot t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \cos k}{t}}{{\sin k}^{2}}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (+ 1.0 (+ 1.0 (pow (/ k t) 2.0)))))
   (if (<=
        (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) t_1))
        5e+298)
     (/ 2.0 (* (* (/ (pow t 3.0) l) (/ k l)) (* (tan k) t_1)))
     (* 2.0 (/ (/ (* (* (/ l k) (/ l k)) (cos k)) t) (pow (sin k) 2.0))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double t_1 = 1.0 + (1.0 + pow((k / t), 2.0));
	double tmp;
	if ((2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * t_1)) <= 5e+298) {
		tmp = 2.0 / (((pow(t, 3.0) / l) * (k / l)) * (tan(k) * t_1));
	} else {
		tmp = 2.0 * (((((l / k) * (l / k)) * cos(k)) / t) / pow(sin(k), 2.0));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 + (1.0d0 + ((k / t) ** 2.0d0))
    if ((2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * t_1)) <= 5d+298) then
        tmp = 2.0d0 / ((((t ** 3.0d0) / l) * (k / l)) * (tan(k) * t_1))
    else
        tmp = 2.0d0 * (((((l / k) * (l / k)) * cos(k)) / t) / (sin(k) ** 2.0d0))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double t_1 = 1.0 + (1.0 + Math.pow((k / t), 2.0));
	double tmp;
	if ((2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * t_1)) <= 5e+298) {
		tmp = 2.0 / (((Math.pow(t, 3.0) / l) * (k / l)) * (Math.tan(k) * t_1));
	} else {
		tmp = 2.0 * (((((l / k) * (l / k)) * Math.cos(k)) / t) / Math.pow(Math.sin(k), 2.0));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	t_1 = 1.0 + (1.0 + math.pow((k / t), 2.0))
	tmp = 0
	if (2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * t_1)) <= 5e+298:
		tmp = 2.0 / (((math.pow(t, 3.0) / l) * (k / l)) * (math.tan(k) * t_1))
	else:
		tmp = 2.0 * (((((l / k) * (l / k)) * math.cos(k)) / t) / math.pow(math.sin(k), 2.0))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	t_1 = Float64(1.0 + Float64(1.0 + (Float64(k / t) ^ 2.0)))
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * t_1)) <= 5e+298)
		tmp = Float64(2.0 / Float64(Float64(Float64((t ^ 3.0) / l) * Float64(k / l)) * Float64(tan(k) * t_1)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(Float64(Float64(l / k) * Float64(l / k)) * cos(k)) / t) / (sin(k) ^ 2.0)));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	t_1 = 1.0 + (1.0 + ((k / t) ^ 2.0));
	tmp = 0.0;
	if ((2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * t_1)) <= 5e+298)
		tmp = 2.0 / ((((t ^ 3.0) / l) * (k / l)) * (tan(k) * t_1));
	else
		tmp = 2.0 * (((((l / k) * (l / k)) * cos(k)) / t) / (sin(k) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[(1.0 + N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 5e+298], N[(2.0 / N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) < 5.0000000000000003e298

   1. Initial program 79.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 79.1%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. +-commutative 79.1%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Simplified 79.1%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]

   4. Taylor expanded in k around 0 79.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot {t}^{3}}{{\ell}^{2}}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. *-commutative 79.1%

         \[\leadsto \frac{2}{\frac{\color{blue}{{t}^{3} \cdot k}}{{\ell}^{2}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      2. unpow2 79.1%

         \[\leadsto \frac{2}{\frac{{t}^{3} \cdot k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      3. times-frac 83.1%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{k}{\ell}\right)} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   6. Simplified 83.1%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{k}{\ell}\right)} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   if 5.0000000000000003e298 < (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))

   1. Initial program 23.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 23.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 23.4%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 23.4%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 24.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 24.3%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 24.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 24.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 24.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 24.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 24.3%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 24.3%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 58.4%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 58.4%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

      2. times-frac 57.6%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

      3. unpow2 57.6%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      4. unpow2 57.6%

         \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      5. times-frac 80.1%

         \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      6. *-commutative 80.1%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

   6. Simplified 80.1%

      \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} \]

   7. Taylor expanded in l around 0 58.4%

      \[\leadsto 2 \cdot \color{blue}{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   8. Step-by-step derivation

      1. associate-/r* 57.6%

         \[\leadsto 2 \cdot \color{blue}{\frac{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2}}}{{\sin k}^{2} \cdot t}} \]

      2. *-commutative 57.6%

         \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2}}}{\color{blue}{t \cdot {\sin k}^{2}}} \]

      3. associate-*r/ 57.6%

         \[\leadsto 2 \cdot \frac{\color{blue}{\cos k \cdot \frac{{\ell}^{2}}{{k}^{2}}}}{t \cdot {\sin k}^{2}} \]

      4. unpow2 57.6%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{t \cdot {\sin k}^{2}} \]

      5. unpow2 57.6%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{t \cdot {\sin k}^{2}} \]

      6. times-frac 80.1%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}}{t \cdot {\sin k}^{2}} \]

      7. unpow2 80.1%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \color{blue}{{\left(\frac{\ell}{k}\right)}^{2}}}{t \cdot {\sin k}^{2}} \]

      8. *-commutative 80.1%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\left(\frac{\ell}{k}\right)}^{2} \cdot \cos k}}{t \cdot {\sin k}^{2}} \]

      9. times-frac 81.7%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)} \]

   9. Simplified 81.7%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)} \]
   10. Step-by-step derivation

       1. frac-times 80.1%

          \[\leadsto 2 \cdot \color{blue}{\frac{{\left(\frac{\ell}{k}\right)}^{2} \cdot \cos k}{t \cdot {\sin k}^{2}}} \]

       2. associate-/r* 81.7%

          \[\leadsto 2 \cdot \color{blue}{\frac{\frac{{\left(\frac{\ell}{k}\right)}^{2} \cdot \cos k}{t}}{{\sin k}^{2}}} \]

   11. Applied egg-rr 81.7%

       \[\leadsto 2 \cdot \color{blue}{\frac{\frac{{\left(\frac{\ell}{k}\right)}^{2} \cdot \cos k}{t}}{{\sin k}^{2}}} \]
   12. Step-by-step derivation

       1. unpow2 81.7%

          \[\leadsto 2 \cdot \frac{\frac{\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \cos k}{t}}{{\sin k}^{2}} \]

   13. Applied egg-rr 81.7%

       \[\leadsto 2 \cdot \frac{\frac{\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \cos k}{t}}{{\sin k}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \leq 5 \cdot 10^{+298}:\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \cos k}{t}}{{\sin k}^{2}}\\ \end{array} \]

Alternative 3: 82.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 2.55 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{1}{\frac{k}{\ell}}}{\frac{\frac{k}{\ell}}{{t}^{-3}}}\\ \mathbf{elif}\;k \leq 5.8 \cdot 10^{+192}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{\frac{2}{\tan k}}{\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 2.55e-15)
   (/ (/ 1.0 (/ k l)) (/ (/ k l) (pow t -3.0)))
   (if (<= k 5.8e+192)
     (* l (* l (/ (/ 2.0 (tan k)) (* (sin k) (* k (* t k))))))
     (* 2.0 (* (* (/ l k) (/ l k)) (/ (cos k) (* t (pow (sin k) 2.0))))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.55e-15) {
		tmp = (1.0 / (k / l)) / ((k / l) / pow(t, -3.0));
	} else if (k <= 5.8e+192) {
		tmp = l * (l * ((2.0 / tan(k)) / (sin(k) * (k * (t * k)))));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * (cos(k) / (t * pow(sin(k), 2.0))));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.55d-15) then
        tmp = (1.0d0 / (k / l)) / ((k / l) / (t ** (-3.0d0)))
    else if (k <= 5.8d+192) then
        tmp = l * (l * ((2.0d0 / tan(k)) / (sin(k) * (k * (t * k)))))
    else
        tmp = 2.0d0 * (((l / k) * (l / k)) * (cos(k) / (t * (sin(k) ** 2.0d0))))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.55e-15) {
		tmp = (1.0 / (k / l)) / ((k / l) / Math.pow(t, -3.0));
	} else if (k <= 5.8e+192) {
		tmp = l * (l * ((2.0 / Math.tan(k)) / (Math.sin(k) * (k * (t * k)))));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * (Math.cos(k) / (t * Math.pow(Math.sin(k), 2.0))));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 2.55e-15:
		tmp = (1.0 / (k / l)) / ((k / l) / math.pow(t, -3.0))
	elif k <= 5.8e+192:
		tmp = l * (l * ((2.0 / math.tan(k)) / (math.sin(k) * (k * (t * k)))))
	else:
		tmp = 2.0 * (((l / k) * (l / k)) * (math.cos(k) / (t * math.pow(math.sin(k), 2.0))))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 2.55e-15)
		tmp = Float64(Float64(1.0 / Float64(k / l)) / Float64(Float64(k / l) / (t ^ -3.0)));
	elseif (k <= 5.8e+192)
		tmp = Float64(l * Float64(l * Float64(Float64(2.0 / tan(k)) / Float64(sin(k) * Float64(k * Float64(t * k))))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(cos(k) / Float64(t * (sin(k) ^ 2.0)))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2.55e-15)
		tmp = (1.0 / (k / l)) / ((k / l) / (t ^ -3.0));
	elseif (k <= 5.8e+192)
		tmp = l * (l * ((2.0 / tan(k)) / (sin(k) * (k * (t * k)))));
	else
		tmp = 2.0 * (((l / k) * (l / k)) * (cos(k) / (t * (sin(k) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 2.55e-15], N[(N[(1.0 / N[(k / l), $MachinePrecision]), $MachinePrecision] / N[(N[(k / l), $MachinePrecision] / N[Power[t, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 5.8e+192], N[(l * N[(l * N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 2.55e-15

   1. Initial program 55.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 55.0%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-*l* 47.8%

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

      3. associate-*r* 47.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      4. +-commutative 47.8%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      5. associate-+r+ 47.8%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      6. metadata-eval 47.8%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   3. Simplified 47.8%

      \[\leadsto \color{blue}{\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

   4. Taylor expanded in k around 0 49.6%

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   5. Step-by-step derivation

      1. associate-/l* 50.6%

         \[\leadsto \frac{2}{2 \cdot \color{blue}{\frac{{k}^{2}}{\frac{{\ell}^{2}}{{t}^{3}}}}} \]

      2. unpow2 50.6%

         \[\leadsto \frac{2}{2 \cdot \frac{\color{blue}{k \cdot k}}{\frac{{\ell}^{2}}{{t}^{3}}}} \]

      3. unpow2 50.6%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\frac{\color{blue}{\ell \cdot \ell}}{{t}^{3}}}} \]

   6. Simplified 50.6%

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{k \cdot k}{\frac{\ell \cdot \ell}{{t}^{3}}}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 40.1%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\ell \cdot \ell}{{t}^{3}}\right)\right)}}} \]

      2. expm1-udef 35.6%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{e^{\mathsf{log1p}\left(\frac{\ell \cdot \ell}{{t}^{3}}\right)} - 1}}} \]

      3. div-inv 35.6%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{e^{\mathsf{log1p}\left(\color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{1}{{t}^{3}}}\right)} - 1}} \]

      4. pow-flip 35.6%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \color{blue}{{t}^{\left(-3\right)}}\right)} - 1}} \]

      5. metadata-eval 35.6%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot {t}^{\color{blue}{-3}}\right)} - 1}} \]

   8. Applied egg-rr 35.6%

      \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot {t}^{-3}\right)} - 1}}} \]
   9. Step-by-step derivation

      1. expm1-def 40.1%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot {t}^{-3}\right)\right)}}} \]

      2. expm1-log1p 50.6%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{\left(\ell \cdot \ell\right) \cdot {t}^{-3}}}} \]

      3. associate-*l* 53.6%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{\ell \cdot \left(\ell \cdot {t}^{-3}\right)}}} \]

   10. Simplified 53.6%

       \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{\ell \cdot \left(\ell \cdot {t}^{-3}\right)}}} \]
   11. Step-by-step derivation

       1. expm1-log1p-u 39.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{2 \cdot \frac{k \cdot k}{\ell \cdot \left(\ell \cdot {t}^{-3}\right)}}\right)\right)} \]

       2. expm1-udef 39.2%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{2 \cdot \frac{k \cdot k}{\ell \cdot \left(\ell \cdot {t}^{-3}\right)}}\right)} - 1} \]

       3. associate-/r* 39.2%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{2}{2}}{\frac{k \cdot k}{\ell \cdot \left(\ell \cdot {t}^{-3}\right)}}}\right)} - 1 \]

       4. metadata-eval 39.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{1}}{\frac{k \cdot k}{\ell \cdot \left(\ell \cdot {t}^{-3}\right)}}\right)} - 1 \]

       5. times-frac 46.8%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{\frac{k}{\ell} \cdot \frac{k}{\ell \cdot {t}^{-3}}}}\right)} - 1 \]

   12. Applied egg-rr 46.8%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\frac{k}{\ell} \cdot \frac{k}{\ell \cdot {t}^{-3}}}\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 48.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{k}{\ell} \cdot \frac{k}{\ell \cdot {t}^{-3}}}\right)\right)} \]

       2. expm1-log1p 63.7%

          \[\leadsto \color{blue}{\frac{1}{\frac{k}{\ell} \cdot \frac{k}{\ell \cdot {t}^{-3}}}} \]

       3. associate-/r* 63.7%

          \[\leadsto \color{blue}{\frac{\frac{1}{\frac{k}{\ell}}}{\frac{k}{\ell \cdot {t}^{-3}}}} \]

       4. associate-/r* 64.6%

          \[\leadsto \frac{\frac{1}{\frac{k}{\ell}}}{\color{blue}{\frac{\frac{k}{\ell}}{{t}^{-3}}}} \]

   14. Simplified 64.6%

       \[\leadsto \color{blue}{\frac{\frac{1}{\frac{k}{\ell}}}{\frac{\frac{k}{\ell}}{{t}^{-3}}}} \]

   if 2.55e-15 < k < 5.8000000000000003e192

   1. Initial program 49.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 49.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 49.3%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 49.3%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 49.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 49.3%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 49.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 49.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 49.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 49.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 49.3%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 49.3%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 79.3%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left({k}^{2} \cdot \left(\sin k \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. unpow2 79.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \left(\sin k \cdot t\right)\right)} \]

      2. *-commutative 79.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \sin k\right)}\right)} \]

   6. Simplified 79.3%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 68.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}\right)\right)} \]

      2. expm1-udef 60.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}\right)} - 1} \]

   8. Applied egg-rr 60.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 68.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}\right)\right)} \]

      2. expm1-log1p 79.3%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}} \]

      3. associate-*l* 84.4%

         \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}\right)} \]

      4. associate-/r* 84.4%

         \[\leadsto \ell \cdot \left(\ell \cdot \color{blue}{\frac{\frac{2}{\tan k}}{\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)}}\right) \]

      5. unpow2 84.4%

         \[\leadsto \ell \cdot \left(\ell \cdot \frac{\frac{2}{\tan k}}{\color{blue}{{k}^{2}} \cdot \left(t \cdot \sin k\right)}\right) \]

      6. *-commutative 84.4%

         \[\leadsto \ell \cdot \left(\ell \cdot \frac{\frac{2}{\tan k}}{\color{blue}{\left(t \cdot \sin k\right) \cdot {k}^{2}}}\right) \]

      7. *-commutative 84.4%

         \[\leadsto \ell \cdot \left(\ell \cdot \frac{\frac{2}{\tan k}}{\color{blue}{\left(\sin k \cdot t\right)} \cdot {k}^{2}}\right) \]

      8. associate-*l* 84.4%

         \[\leadsto \ell \cdot \left(\ell \cdot \frac{\frac{2}{\tan k}}{\color{blue}{\sin k \cdot \left(t \cdot {k}^{2}\right)}}\right) \]

      9. *-commutative 84.4%

         \[\leadsto \ell \cdot \left(\ell \cdot \frac{\frac{2}{\tan k}}{\sin k \cdot \color{blue}{\left({k}^{2} \cdot t\right)}}\right) \]

      10. unpow2 84.4%

          \[\leadsto \ell \cdot \left(\ell \cdot \frac{\frac{2}{\tan k}}{\sin k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)}\right) \]

      11. associate-*l* 89.5%

          \[\leadsto \ell \cdot \left(\ell \cdot \frac{\frac{2}{\tan k}}{\sin k \cdot \color{blue}{\left(k \cdot \left(k \cdot t\right)\right)}}\right) \]

   10. Simplified 89.5%

       \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \frac{\frac{2}{\tan k}}{\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)}\right)} \]

   if 5.8000000000000003e192 < k

   1. Initial program 52.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 52.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 52.0%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 52.0%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 56.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 56.0%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 56.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 56.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 56.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 56.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 56.0%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 56.0%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 68.2%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 68.2%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

      2. times-frac 68.2%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

      3. unpow2 68.2%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      4. unpow2 68.2%

         \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      5. times-frac 99.6%

         \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      6. *-commutative 99.6%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

   6. Simplified 99.6%

      \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.55 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{1}{\frac{k}{\ell}}}{\frac{\frac{k}{\ell}}{{t}^{-3}}}\\ \mathbf{elif}\;k \leq 5.8 \cdot 10^{+192}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{\frac{2}{\tan k}}{\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]

Alternative 4: 82.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 3.8 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{1}{\frac{k}{\ell}}}{\frac{\frac{k}{\ell}}{{t}^{-3}}}\\ \mathbf{elif}\;k \leq 3.6 \cdot 10^{+192}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{\frac{2}{\tan k}}{\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \cos k}{t \cdot {\sin k}^{2}}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 3.8e-13)
   (/ (/ 1.0 (/ k l)) (/ (/ k l) (pow t -3.0)))
   (if (<= k 3.6e+192)
     (* l (* l (/ (/ 2.0 (tan k)) (* (sin k) (* k (* t k))))))
     (* 2.0 (/ (* (* (/ l k) (/ l k)) (cos k)) (* t (pow (sin k) 2.0)))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.8e-13) {
		tmp = (1.0 / (k / l)) / ((k / l) / pow(t, -3.0));
	} else if (k <= 3.6e+192) {
		tmp = l * (l * ((2.0 / tan(k)) / (sin(k) * (k * (t * k)))));
	} else {
		tmp = 2.0 * ((((l / k) * (l / k)) * cos(k)) / (t * pow(sin(k), 2.0)));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 3.8d-13) then
        tmp = (1.0d0 / (k / l)) / ((k / l) / (t ** (-3.0d0)))
    else if (k <= 3.6d+192) then
        tmp = l * (l * ((2.0d0 / tan(k)) / (sin(k) * (k * (t * k)))))
    else
        tmp = 2.0d0 * ((((l / k) * (l / k)) * cos(k)) / (t * (sin(k) ** 2.0d0)))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.8e-13) {
		tmp = (1.0 / (k / l)) / ((k / l) / Math.pow(t, -3.0));
	} else if (k <= 3.6e+192) {
		tmp = l * (l * ((2.0 / Math.tan(k)) / (Math.sin(k) * (k * (t * k)))));
	} else {
		tmp = 2.0 * ((((l / k) * (l / k)) * Math.cos(k)) / (t * Math.pow(Math.sin(k), 2.0)));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 3.8e-13:
		tmp = (1.0 / (k / l)) / ((k / l) / math.pow(t, -3.0))
	elif k <= 3.6e+192:
		tmp = l * (l * ((2.0 / math.tan(k)) / (math.sin(k) * (k * (t * k)))))
	else:
		tmp = 2.0 * ((((l / k) * (l / k)) * math.cos(k)) / (t * math.pow(math.sin(k), 2.0)))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 3.8e-13)
		tmp = Float64(Float64(1.0 / Float64(k / l)) / Float64(Float64(k / l) / (t ^ -3.0)));
	elseif (k <= 3.6e+192)
		tmp = Float64(l * Float64(l * Float64(Float64(2.0 / tan(k)) / Float64(sin(k) * Float64(k * Float64(t * k))))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(Float64(l / k) * Float64(l / k)) * cos(k)) / Float64(t * (sin(k) ^ 2.0))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 3.8e-13)
		tmp = (1.0 / (k / l)) / ((k / l) / (t ^ -3.0));
	elseif (k <= 3.6e+192)
		tmp = l * (l * ((2.0 / tan(k)) / (sin(k) * (k * (t * k)))));
	else
		tmp = 2.0 * ((((l / k) * (l / k)) * cos(k)) / (t * (sin(k) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 3.8e-13], N[(N[(1.0 / N[(k / l), $MachinePrecision]), $MachinePrecision] / N[(N[(k / l), $MachinePrecision] / N[Power[t, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 3.6e+192], N[(l * N[(l * N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 3.8e-13

   1. Initial program 55.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 55.0%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-*l* 47.8%

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

      3. associate-*r* 47.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      4. +-commutative 47.8%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      5. associate-+r+ 47.8%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      6. metadata-eval 47.8%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   3. Simplified 47.8%

      \[\leadsto \color{blue}{\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

   4. Taylor expanded in k around 0 49.6%

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   5. Step-by-step derivation

      1. associate-/l* 50.6%

         \[\leadsto \frac{2}{2 \cdot \color{blue}{\frac{{k}^{2}}{\frac{{\ell}^{2}}{{t}^{3}}}}} \]

      2. unpow2 50.6%

         \[\leadsto \frac{2}{2 \cdot \frac{\color{blue}{k \cdot k}}{\frac{{\ell}^{2}}{{t}^{3}}}} \]

      3. unpow2 50.6%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\frac{\color{blue}{\ell \cdot \ell}}{{t}^{3}}}} \]

   6. Simplified 50.6%

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{k \cdot k}{\frac{\ell \cdot \ell}{{t}^{3}}}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 40.1%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\ell \cdot \ell}{{t}^{3}}\right)\right)}}} \]

      2. expm1-udef 35.6%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{e^{\mathsf{log1p}\left(\frac{\ell \cdot \ell}{{t}^{3}}\right)} - 1}}} \]

      3. div-inv 35.6%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{e^{\mathsf{log1p}\left(\color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{1}{{t}^{3}}}\right)} - 1}} \]

      4. pow-flip 35.6%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \color{blue}{{t}^{\left(-3\right)}}\right)} - 1}} \]

      5. metadata-eval 35.6%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot {t}^{\color{blue}{-3}}\right)} - 1}} \]

   8. Applied egg-rr 35.6%

      \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot {t}^{-3}\right)} - 1}}} \]
   9. Step-by-step derivation

      1. expm1-def 40.1%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot {t}^{-3}\right)\right)}}} \]

      2. expm1-log1p 50.6%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{\left(\ell \cdot \ell\right) \cdot {t}^{-3}}}} \]

      3. associate-*l* 53.6%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{\ell \cdot \left(\ell \cdot {t}^{-3}\right)}}} \]

   10. Simplified 53.6%

       \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{\ell \cdot \left(\ell \cdot {t}^{-3}\right)}}} \]
   11. Step-by-step derivation

       1. expm1-log1p-u 39.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{2 \cdot \frac{k \cdot k}{\ell \cdot \left(\ell \cdot {t}^{-3}\right)}}\right)\right)} \]

       2. expm1-udef 39.2%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{2 \cdot \frac{k \cdot k}{\ell \cdot \left(\ell \cdot {t}^{-3}\right)}}\right)} - 1} \]

       3. associate-/r* 39.2%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{2}{2}}{\frac{k \cdot k}{\ell \cdot \left(\ell \cdot {t}^{-3}\right)}}}\right)} - 1 \]

       4. metadata-eval 39.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{1}}{\frac{k \cdot k}{\ell \cdot \left(\ell \cdot {t}^{-3}\right)}}\right)} - 1 \]

       5. times-frac 46.8%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{\frac{k}{\ell} \cdot \frac{k}{\ell \cdot {t}^{-3}}}}\right)} - 1 \]

   12. Applied egg-rr 46.8%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\frac{k}{\ell} \cdot \frac{k}{\ell \cdot {t}^{-3}}}\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 48.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{k}{\ell} \cdot \frac{k}{\ell \cdot {t}^{-3}}}\right)\right)} \]

       2. expm1-log1p 63.7%

          \[\leadsto \color{blue}{\frac{1}{\frac{k}{\ell} \cdot \frac{k}{\ell \cdot {t}^{-3}}}} \]

       3. associate-/r* 63.7%

          \[\leadsto \color{blue}{\frac{\frac{1}{\frac{k}{\ell}}}{\frac{k}{\ell \cdot {t}^{-3}}}} \]

       4. associate-/r* 64.6%

          \[\leadsto \frac{\frac{1}{\frac{k}{\ell}}}{\color{blue}{\frac{\frac{k}{\ell}}{{t}^{-3}}}} \]

   14. Simplified 64.6%

       \[\leadsto \color{blue}{\frac{\frac{1}{\frac{k}{\ell}}}{\frac{\frac{k}{\ell}}{{t}^{-3}}}} \]

   if 3.8e-13 < k < 3.6000000000000002e192

   1. Initial program 49.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 49.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 49.3%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 49.3%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 49.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 49.3%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 49.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 49.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 49.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 49.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 49.3%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 49.3%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 79.3%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left({k}^{2} \cdot \left(\sin k \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. unpow2 79.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \left(\sin k \cdot t\right)\right)} \]

      2. *-commutative 79.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \sin k\right)}\right)} \]

   6. Simplified 79.3%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 68.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}\right)\right)} \]

      2. expm1-udef 60.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}\right)} - 1} \]

   8. Applied egg-rr 60.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 68.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}\right)\right)} \]

      2. expm1-log1p 79.3%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}} \]

      3. associate-*l* 84.4%

         \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}\right)} \]

      4. associate-/r* 84.4%

         \[\leadsto \ell \cdot \left(\ell \cdot \color{blue}{\frac{\frac{2}{\tan k}}{\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)}}\right) \]

      5. unpow2 84.4%

         \[\leadsto \ell \cdot \left(\ell \cdot \frac{\frac{2}{\tan k}}{\color{blue}{{k}^{2}} \cdot \left(t \cdot \sin k\right)}\right) \]

      6. *-commutative 84.4%

         \[\leadsto \ell \cdot \left(\ell \cdot \frac{\frac{2}{\tan k}}{\color{blue}{\left(t \cdot \sin k\right) \cdot {k}^{2}}}\right) \]

      7. *-commutative 84.4%

         \[\leadsto \ell \cdot \left(\ell \cdot \frac{\frac{2}{\tan k}}{\color{blue}{\left(\sin k \cdot t\right)} \cdot {k}^{2}}\right) \]

      8. associate-*l* 84.4%

         \[\leadsto \ell \cdot \left(\ell \cdot \frac{\frac{2}{\tan k}}{\color{blue}{\sin k \cdot \left(t \cdot {k}^{2}\right)}}\right) \]

      9. *-commutative 84.4%

         \[\leadsto \ell \cdot \left(\ell \cdot \frac{\frac{2}{\tan k}}{\sin k \cdot \color{blue}{\left({k}^{2} \cdot t\right)}}\right) \]

      10. unpow2 84.4%

          \[\leadsto \ell \cdot \left(\ell \cdot \frac{\frac{2}{\tan k}}{\sin k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)}\right) \]

      11. associate-*l* 89.5%

          \[\leadsto \ell \cdot \left(\ell \cdot \frac{\frac{2}{\tan k}}{\sin k \cdot \color{blue}{\left(k \cdot \left(k \cdot t\right)\right)}}\right) \]

   10. Simplified 89.5%

       \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \frac{\frac{2}{\tan k}}{\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)}\right)} \]

   if 3.6000000000000002e192 < k

   1. Initial program 52.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 52.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 52.0%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 52.0%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 56.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 56.0%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 56.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 56.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 56.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 56.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 56.0%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 56.0%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 68.2%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 68.2%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

      2. times-frac 68.2%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

      3. unpow2 68.2%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      4. unpow2 68.2%

         \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      5. times-frac 99.6%

         \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      6. *-commutative 99.6%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

   6. Simplified 99.6%

      \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} \]
   7. Step-by-step derivation

      1. associate-*r/ 99.6%

         \[\leadsto 2 \cdot \color{blue}{\frac{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \cos k}{t \cdot {\sin k}^{2}}} \]

      2. pow2 99.6%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\left(\frac{\ell}{k}\right)}^{2}} \cdot \cos k}{t \cdot {\sin k}^{2}} \]

   8. Applied egg-rr 99.6%

      \[\leadsto 2 \cdot \color{blue}{\frac{{\left(\frac{\ell}{k}\right)}^{2} \cdot \cos k}{t \cdot {\sin k}^{2}}} \]
   9. Step-by-step derivation

      1. unpow2 99.6%

         \[\leadsto 2 \cdot \frac{\frac{\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \cos k}{t}}{{\sin k}^{2}} \]

   10. Applied egg-rr 99.6%

       \[\leadsto 2 \cdot \frac{\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \cos k}{t \cdot {\sin k}^{2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.8 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{1}{\frac{k}{\ell}}}{\frac{\frac{k}{\ell}}{{t}^{-3}}}\\ \mathbf{elif}\;k \leq 3.6 \cdot 10^{+192}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{\frac{2}{\tan k}}{\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \cos k}{t \cdot {\sin k}^{2}}\\ \end{array} \]

Alternative 5: 82.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 4.7 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{1}{\frac{k}{\ell}}}{\frac{\frac{k}{\ell}}{{t}^{-3}}}\\ \mathbf{elif}\;k \leq 3.8 \cdot 10^{+192}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{\frac{2}{\tan k}}{\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot \left(0.5 - \frac{\cos \left(k + k\right)}{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 4.7e-17)
   (/ (/ 1.0 (/ k l)) (/ (/ k l) (pow t -3.0)))
   (if (<= k 3.8e+192)
     (* l (* l (/ (/ 2.0 (tan k)) (* (sin k) (* k (* t k))))))
     (*
      2.0
      (*
       (* (/ l k) (/ l k))
       (/ (cos k) (* t (- 0.5 (/ (cos (+ k k)) 2.0)))))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 4.7e-17) {
		tmp = (1.0 / (k / l)) / ((k / l) / pow(t, -3.0));
	} else if (k <= 3.8e+192) {
		tmp = l * (l * ((2.0 / tan(k)) / (sin(k) * (k * (t * k)))));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * (cos(k) / (t * (0.5 - (cos((k + k)) / 2.0)))));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 4.7d-17) then
        tmp = (1.0d0 / (k / l)) / ((k / l) / (t ** (-3.0d0)))
    else if (k <= 3.8d+192) then
        tmp = l * (l * ((2.0d0 / tan(k)) / (sin(k) * (k * (t * k)))))
    else
        tmp = 2.0d0 * (((l / k) * (l / k)) * (cos(k) / (t * (0.5d0 - (cos((k + k)) / 2.0d0)))))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 4.7e-17) {
		tmp = (1.0 / (k / l)) / ((k / l) / Math.pow(t, -3.0));
	} else if (k <= 3.8e+192) {
		tmp = l * (l * ((2.0 / Math.tan(k)) / (Math.sin(k) * (k * (t * k)))));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * (Math.cos(k) / (t * (0.5 - (Math.cos((k + k)) / 2.0)))));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 4.7e-17:
		tmp = (1.0 / (k / l)) / ((k / l) / math.pow(t, -3.0))
	elif k <= 3.8e+192:
		tmp = l * (l * ((2.0 / math.tan(k)) / (math.sin(k) * (k * (t * k)))))
	else:
		tmp = 2.0 * (((l / k) * (l / k)) * (math.cos(k) / (t * (0.5 - (math.cos((k + k)) / 2.0)))))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 4.7e-17)
		tmp = Float64(Float64(1.0 / Float64(k / l)) / Float64(Float64(k / l) / (t ^ -3.0)));
	elseif (k <= 3.8e+192)
		tmp = Float64(l * Float64(l * Float64(Float64(2.0 / tan(k)) / Float64(sin(k) * Float64(k * Float64(t * k))))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(cos(k) / Float64(t * Float64(0.5 - Float64(cos(Float64(k + k)) / 2.0))))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 4.7e-17)
		tmp = (1.0 / (k / l)) / ((k / l) / (t ^ -3.0));
	elseif (k <= 3.8e+192)
		tmp = l * (l * ((2.0 / tan(k)) / (sin(k) * (k * (t * k)))));
	else
		tmp = 2.0 * (((l / k) * (l / k)) * (cos(k) / (t * (0.5 - (cos((k + k)) / 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 4.7e-17], N[(N[(1.0 / N[(k / l), $MachinePrecision]), $MachinePrecision] / N[(N[(k / l), $MachinePrecision] / N[Power[t, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 3.8e+192], N[(l * N[(l * N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t * N[(0.5 - N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 4.7e-17

   1. Initial program 55.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 55.0%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-*l* 47.8%

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

      3. associate-*r* 47.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      4. +-commutative 47.8%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      5. associate-+r+ 47.8%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      6. metadata-eval 47.8%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   3. Simplified 47.8%

      \[\leadsto \color{blue}{\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

   4. Taylor expanded in k around 0 49.6%

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   5. Step-by-step derivation

      1. associate-/l* 50.6%

         \[\leadsto \frac{2}{2 \cdot \color{blue}{\frac{{k}^{2}}{\frac{{\ell}^{2}}{{t}^{3}}}}} \]

      2. unpow2 50.6%

         \[\leadsto \frac{2}{2 \cdot \frac{\color{blue}{k \cdot k}}{\frac{{\ell}^{2}}{{t}^{3}}}} \]

      3. unpow2 50.6%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\frac{\color{blue}{\ell \cdot \ell}}{{t}^{3}}}} \]

   6. Simplified 50.6%

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{k \cdot k}{\frac{\ell \cdot \ell}{{t}^{3}}}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 40.1%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\ell \cdot \ell}{{t}^{3}}\right)\right)}}} \]

      2. expm1-udef 35.6%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{e^{\mathsf{log1p}\left(\frac{\ell \cdot \ell}{{t}^{3}}\right)} - 1}}} \]

      3. div-inv 35.6%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{e^{\mathsf{log1p}\left(\color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{1}{{t}^{3}}}\right)} - 1}} \]

      4. pow-flip 35.6%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \color{blue}{{t}^{\left(-3\right)}}\right)} - 1}} \]

      5. metadata-eval 35.6%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot {t}^{\color{blue}{-3}}\right)} - 1}} \]

   8. Applied egg-rr 35.6%

      \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot {t}^{-3}\right)} - 1}}} \]
   9. Step-by-step derivation

      1. expm1-def 40.1%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot {t}^{-3}\right)\right)}}} \]

      2. expm1-log1p 50.6%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{\left(\ell \cdot \ell\right) \cdot {t}^{-3}}}} \]

      3. associate-*l* 53.6%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{\ell \cdot \left(\ell \cdot {t}^{-3}\right)}}} \]

   10. Simplified 53.6%

       \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{\ell \cdot \left(\ell \cdot {t}^{-3}\right)}}} \]
   11. Step-by-step derivation

       1. expm1-log1p-u 39.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{2 \cdot \frac{k \cdot k}{\ell \cdot \left(\ell \cdot {t}^{-3}\right)}}\right)\right)} \]

       2. expm1-udef 39.2%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{2 \cdot \frac{k \cdot k}{\ell \cdot \left(\ell \cdot {t}^{-3}\right)}}\right)} - 1} \]

       3. associate-/r* 39.2%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{2}{2}}{\frac{k \cdot k}{\ell \cdot \left(\ell \cdot {t}^{-3}\right)}}}\right)} - 1 \]

       4. metadata-eval 39.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{1}}{\frac{k \cdot k}{\ell \cdot \left(\ell \cdot {t}^{-3}\right)}}\right)} - 1 \]

       5. times-frac 46.8%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{\frac{k}{\ell} \cdot \frac{k}{\ell \cdot {t}^{-3}}}}\right)} - 1 \]

   12. Applied egg-rr 46.8%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\frac{k}{\ell} \cdot \frac{k}{\ell \cdot {t}^{-3}}}\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 48.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{k}{\ell} \cdot \frac{k}{\ell \cdot {t}^{-3}}}\right)\right)} \]

       2. expm1-log1p 63.7%

          \[\leadsto \color{blue}{\frac{1}{\frac{k}{\ell} \cdot \frac{k}{\ell \cdot {t}^{-3}}}} \]

       3. associate-/r* 63.7%

          \[\leadsto \color{blue}{\frac{\frac{1}{\frac{k}{\ell}}}{\frac{k}{\ell \cdot {t}^{-3}}}} \]

       4. associate-/r* 64.6%

          \[\leadsto \frac{\frac{1}{\frac{k}{\ell}}}{\color{blue}{\frac{\frac{k}{\ell}}{{t}^{-3}}}} \]

   14. Simplified 64.6%

       \[\leadsto \color{blue}{\frac{\frac{1}{\frac{k}{\ell}}}{\frac{\frac{k}{\ell}}{{t}^{-3}}}} \]

   if 4.7e-17 < k < 3.7999999999999999e192

   1. Initial program 49.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 49.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 49.3%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 49.3%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 49.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 49.3%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 49.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 49.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 49.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 49.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 49.3%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 49.3%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 79.3%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left({k}^{2} \cdot \left(\sin k \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. unpow2 79.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \left(\sin k \cdot t\right)\right)} \]

      2. *-commutative 79.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \sin k\right)}\right)} \]

   6. Simplified 79.3%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 68.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}\right)\right)} \]

      2. expm1-udef 60.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}\right)} - 1} \]

   8. Applied egg-rr 60.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 68.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}\right)\right)} \]

      2. expm1-log1p 79.3%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}} \]

      3. associate-*l* 84.4%

         \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}\right)} \]

      4. associate-/r* 84.4%

         \[\leadsto \ell \cdot \left(\ell \cdot \color{blue}{\frac{\frac{2}{\tan k}}{\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)}}\right) \]

      5. unpow2 84.4%

         \[\leadsto \ell \cdot \left(\ell \cdot \frac{\frac{2}{\tan k}}{\color{blue}{{k}^{2}} \cdot \left(t \cdot \sin k\right)}\right) \]

      6. *-commutative 84.4%

         \[\leadsto \ell \cdot \left(\ell \cdot \frac{\frac{2}{\tan k}}{\color{blue}{\left(t \cdot \sin k\right) \cdot {k}^{2}}}\right) \]

      7. *-commutative 84.4%

         \[\leadsto \ell \cdot \left(\ell \cdot \frac{\frac{2}{\tan k}}{\color{blue}{\left(\sin k \cdot t\right)} \cdot {k}^{2}}\right) \]

      8. associate-*l* 84.4%

         \[\leadsto \ell \cdot \left(\ell \cdot \frac{\frac{2}{\tan k}}{\color{blue}{\sin k \cdot \left(t \cdot {k}^{2}\right)}}\right) \]

      9. *-commutative 84.4%

         \[\leadsto \ell \cdot \left(\ell \cdot \frac{\frac{2}{\tan k}}{\sin k \cdot \color{blue}{\left({k}^{2} \cdot t\right)}}\right) \]

      10. unpow2 84.4%

          \[\leadsto \ell \cdot \left(\ell \cdot \frac{\frac{2}{\tan k}}{\sin k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)}\right) \]

      11. associate-*l* 89.5%

          \[\leadsto \ell \cdot \left(\ell \cdot \frac{\frac{2}{\tan k}}{\sin k \cdot \color{blue}{\left(k \cdot \left(k \cdot t\right)\right)}}\right) \]

   10. Simplified 89.5%

       \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \frac{\frac{2}{\tan k}}{\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)}\right)} \]

   if 3.7999999999999999e192 < k

   1. Initial program 52.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 52.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 52.0%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 52.0%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 56.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 56.0%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 56.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 56.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 56.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 56.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 56.0%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 56.0%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 68.2%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 68.2%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

      2. times-frac 68.2%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

      3. unpow2 68.2%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      4. unpow2 68.2%

         \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      5. times-frac 99.6%

         \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      6. *-commutative 99.6%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

   6. Simplified 99.6%

      \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} \]
   7. Step-by-step derivation

      1. unpow2 99.6%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot \color{blue}{\left(\sin k \cdot \sin k\right)}}\right) \]

      2. sin-mult 99.4%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot \color{blue}{\frac{\cos \left(k - k\right) - \cos \left(k + k\right)}{2}}}\right) \]

   8. Applied egg-rr 99.4%

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot \color{blue}{\frac{\cos \left(k - k\right) - \cos \left(k + k\right)}{2}}}\right) \]
   9. Step-by-step derivation

      1. div-sub 99.4%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot \color{blue}{\left(\frac{\cos \left(k - k\right)}{2} - \frac{\cos \left(k + k\right)}{2}\right)}}\right) \]

      2. +-inverses 99.4%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot \left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(k + k\right)}{2}\right)}\right) \]

      3. cos-0 99.4%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot \left(\frac{\color{blue}{1}}{2} - \frac{\cos \left(k + k\right)}{2}\right)}\right) \]

      4. metadata-eval 99.4%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot \left(\color{blue}{0.5} - \frac{\cos \left(k + k\right)}{2}\right)}\right) \]

   10. Simplified 99.4%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.7 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{1}{\frac{k}{\ell}}}{\frac{\frac{k}{\ell}}{{t}^{-3}}}\\ \mathbf{elif}\;k \leq 3.8 \cdot 10^{+192}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{\frac{2}{\tan k}}{\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot \left(0.5 - \frac{\cos \left(k + k\right)}{2}\right)}\right)\\ \end{array} \]

Alternative 6: 78.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 3.3 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{1}{\frac{k}{\ell}}}{\frac{\frac{k}{\ell}}{{t}^{-3}}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{\frac{2}{\tan k}}{\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 3.3e-15)
   (/ (/ 1.0 (/ k l)) (/ (/ k l) (pow t -3.0)))
   (* l (* l (/ (/ 2.0 (tan k)) (* (sin k) (* k (* t k))))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.3e-15) {
		tmp = (1.0 / (k / l)) / ((k / l) / pow(t, -3.0));
	} else {
		tmp = l * (l * ((2.0 / tan(k)) / (sin(k) * (k * (t * k)))));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 3.3d-15) then
        tmp = (1.0d0 / (k / l)) / ((k / l) / (t ** (-3.0d0)))
    else
        tmp = l * (l * ((2.0d0 / tan(k)) / (sin(k) * (k * (t * k)))))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.3e-15) {
		tmp = (1.0 / (k / l)) / ((k / l) / Math.pow(t, -3.0));
	} else {
		tmp = l * (l * ((2.0 / Math.tan(k)) / (Math.sin(k) * (k * (t * k)))));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 3.3e-15:
		tmp = (1.0 / (k / l)) / ((k / l) / math.pow(t, -3.0))
	else:
		tmp = l * (l * ((2.0 / math.tan(k)) / (math.sin(k) * (k * (t * k)))))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 3.3e-15)
		tmp = Float64(Float64(1.0 / Float64(k / l)) / Float64(Float64(k / l) / (t ^ -3.0)));
	else
		tmp = Float64(l * Float64(l * Float64(Float64(2.0 / tan(k)) / Float64(sin(k) * Float64(k * Float64(t * k))))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 3.3e-15)
		tmp = (1.0 / (k / l)) / ((k / l) / (t ^ -3.0));
	else
		tmp = l * (l * ((2.0 / tan(k)) / (sin(k) * (k * (t * k)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 3.3e-15], N[(N[(1.0 / N[(k / l), $MachinePrecision]), $MachinePrecision] / N[(N[(k / l), $MachinePrecision] / N[Power[t, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(l * N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.3e-15

   1. Initial program 55.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 55.0%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-*l* 47.8%

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

      3. associate-*r* 47.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      4. +-commutative 47.8%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      5. associate-+r+ 47.8%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      6. metadata-eval 47.8%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   3. Simplified 47.8%

      \[\leadsto \color{blue}{\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

   4. Taylor expanded in k around 0 49.6%

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   5. Step-by-step derivation

      1. associate-/l* 50.6%

         \[\leadsto \frac{2}{2 \cdot \color{blue}{\frac{{k}^{2}}{\frac{{\ell}^{2}}{{t}^{3}}}}} \]

      2. unpow2 50.6%

         \[\leadsto \frac{2}{2 \cdot \frac{\color{blue}{k \cdot k}}{\frac{{\ell}^{2}}{{t}^{3}}}} \]

      3. unpow2 50.6%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\frac{\color{blue}{\ell \cdot \ell}}{{t}^{3}}}} \]

   6. Simplified 50.6%

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{k \cdot k}{\frac{\ell \cdot \ell}{{t}^{3}}}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 40.1%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\ell \cdot \ell}{{t}^{3}}\right)\right)}}} \]

      2. expm1-udef 35.6%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{e^{\mathsf{log1p}\left(\frac{\ell \cdot \ell}{{t}^{3}}\right)} - 1}}} \]

      3. div-inv 35.6%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{e^{\mathsf{log1p}\left(\color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{1}{{t}^{3}}}\right)} - 1}} \]

      4. pow-flip 35.6%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \color{blue}{{t}^{\left(-3\right)}}\right)} - 1}} \]

      5. metadata-eval 35.6%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot {t}^{\color{blue}{-3}}\right)} - 1}} \]

   8. Applied egg-rr 35.6%

      \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot {t}^{-3}\right)} - 1}}} \]
   9. Step-by-step derivation

      1. expm1-def 40.1%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot {t}^{-3}\right)\right)}}} \]

      2. expm1-log1p 50.6%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{\left(\ell \cdot \ell\right) \cdot {t}^{-3}}}} \]

      3. associate-*l* 53.6%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{\ell \cdot \left(\ell \cdot {t}^{-3}\right)}}} \]

   10. Simplified 53.6%

       \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{\ell \cdot \left(\ell \cdot {t}^{-3}\right)}}} \]
   11. Step-by-step derivation

       1. expm1-log1p-u 39.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{2 \cdot \frac{k \cdot k}{\ell \cdot \left(\ell \cdot {t}^{-3}\right)}}\right)\right)} \]

       2. expm1-udef 39.2%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{2 \cdot \frac{k \cdot k}{\ell \cdot \left(\ell \cdot {t}^{-3}\right)}}\right)} - 1} \]

       3. associate-/r* 39.2%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{2}{2}}{\frac{k \cdot k}{\ell \cdot \left(\ell \cdot {t}^{-3}\right)}}}\right)} - 1 \]

       4. metadata-eval 39.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{1}}{\frac{k \cdot k}{\ell \cdot \left(\ell \cdot {t}^{-3}\right)}}\right)} - 1 \]

       5. times-frac 46.8%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{\frac{k}{\ell} \cdot \frac{k}{\ell \cdot {t}^{-3}}}}\right)} - 1 \]

   12. Applied egg-rr 46.8%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\frac{k}{\ell} \cdot \frac{k}{\ell \cdot {t}^{-3}}}\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 48.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{k}{\ell} \cdot \frac{k}{\ell \cdot {t}^{-3}}}\right)\right)} \]

       2. expm1-log1p 63.7%

          \[\leadsto \color{blue}{\frac{1}{\frac{k}{\ell} \cdot \frac{k}{\ell \cdot {t}^{-3}}}} \]

       3. associate-/r* 63.7%

          \[\leadsto \color{blue}{\frac{\frac{1}{\frac{k}{\ell}}}{\frac{k}{\ell \cdot {t}^{-3}}}} \]

       4. associate-/r* 64.6%

          \[\leadsto \frac{\frac{1}{\frac{k}{\ell}}}{\color{blue}{\frac{\frac{k}{\ell}}{{t}^{-3}}}} \]

   14. Simplified 64.6%

       \[\leadsto \color{blue}{\frac{\frac{1}{\frac{k}{\ell}}}{\frac{\frac{k}{\ell}}{{t}^{-3}}}} \]

   if 3.3e-15 < k

   1. Initial program 50.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 50.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 50.4%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 50.4%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 52.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 52.0%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 52.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 52.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 52.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 52.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 52.0%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 52.0%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 74.8%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left({k}^{2} \cdot \left(\sin k \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. unpow2 74.8%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \left(\sin k \cdot t\right)\right)} \]

      2. *-commutative 74.8%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \sin k\right)}\right)} \]

   6. Simplified 74.8%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 68.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}\right)\right)} \]

      2. expm1-udef 63.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}\right)} - 1} \]

   8. Applied egg-rr 63.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 68.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}\right)\right)} \]

      2. expm1-log1p 74.8%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}} \]

      3. associate-*l* 79.8%

         \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}\right)} \]

      4. associate-/r* 79.8%

         \[\leadsto \ell \cdot \left(\ell \cdot \color{blue}{\frac{\frac{2}{\tan k}}{\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)}}\right) \]

      5. unpow2 79.8%

         \[\leadsto \ell \cdot \left(\ell \cdot \frac{\frac{2}{\tan k}}{\color{blue}{{k}^{2}} \cdot \left(t \cdot \sin k\right)}\right) \]

      6. *-commutative 79.8%

         \[\leadsto \ell \cdot \left(\ell \cdot \frac{\frac{2}{\tan k}}{\color{blue}{\left(t \cdot \sin k\right) \cdot {k}^{2}}}\right) \]

      7. *-commutative 79.8%

         \[\leadsto \ell \cdot \left(\ell \cdot \frac{\frac{2}{\tan k}}{\color{blue}{\left(\sin k \cdot t\right)} \cdot {k}^{2}}\right) \]

      8. associate-*l* 79.8%

         \[\leadsto \ell \cdot \left(\ell \cdot \frac{\frac{2}{\tan k}}{\color{blue}{\sin k \cdot \left(t \cdot {k}^{2}\right)}}\right) \]

      9. *-commutative 79.8%

         \[\leadsto \ell \cdot \left(\ell \cdot \frac{\frac{2}{\tan k}}{\sin k \cdot \color{blue}{\left({k}^{2} \cdot t\right)}}\right) \]

      10. unpow2 79.8%

          \[\leadsto \ell \cdot \left(\ell \cdot \frac{\frac{2}{\tan k}}{\sin k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)}\right) \]

      11. associate-*l* 84.4%

          \[\leadsto \ell \cdot \left(\ell \cdot \frac{\frac{2}{\tan k}}{\sin k \cdot \color{blue}{\left(k \cdot \left(k \cdot t\right)\right)}}\right) \]

   10. Simplified 84.4%

       \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \frac{\frac{2}{\tan k}}{\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.3 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{1}{\frac{k}{\ell}}}{\frac{\frac{k}{\ell}}{{t}^{-3}}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{\frac{2}{\tan k}}{\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)}\right)\\ \end{array} \]

Alternative 7: 68.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 4.8 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{1}{\frac{k}{\ell}}}{\frac{\frac{k}{\ell}}{{t}^{-3}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{\cos k}{k \cdot k}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 4.8e-21)
   (/ (/ 1.0 (/ k l)) (/ (/ k l) (pow t -3.0)))
   (* 2.0 (* (/ (pow (/ l k) 2.0) t) (/ (cos k) (* k k))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 4.8e-21) {
		tmp = (1.0 / (k / l)) / ((k / l) / pow(t, -3.0));
	} else {
		tmp = 2.0 * ((pow((l / k), 2.0) / t) * (cos(k) / (k * k)));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 4.8d-21) then
        tmp = (1.0d0 / (k / l)) / ((k / l) / (t ** (-3.0d0)))
    else
        tmp = 2.0d0 * ((((l / k) ** 2.0d0) / t) * (cos(k) / (k * k)))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 4.8e-21) {
		tmp = (1.0 / (k / l)) / ((k / l) / Math.pow(t, -3.0));
	} else {
		tmp = 2.0 * ((Math.pow((l / k), 2.0) / t) * (Math.cos(k) / (k * k)));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 4.8e-21:
		tmp = (1.0 / (k / l)) / ((k / l) / math.pow(t, -3.0))
	else:
		tmp = 2.0 * ((math.pow((l / k), 2.0) / t) * (math.cos(k) / (k * k)))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 4.8e-21)
		tmp = Float64(Float64(1.0 / Float64(k / l)) / Float64(Float64(k / l) / (t ^ -3.0)));
	else
		tmp = Float64(2.0 * Float64(Float64((Float64(l / k) ^ 2.0) / t) * Float64(cos(k) / Float64(k * k))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 4.8e-21)
		tmp = (1.0 / (k / l)) / ((k / l) / (t ^ -3.0));
	else
		tmp = 2.0 * ((((l / k) ^ 2.0) / t) * (cos(k) / (k * k)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 4.8e-21], N[(N[(1.0 / N[(k / l), $MachinePrecision]), $MachinePrecision] / N[(N[(k / l), $MachinePrecision] / N[Power[t, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.7999999999999999e-21

   1. Initial program 55.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 55.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-*l* 48.0%

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

      3. associate-*r* 48.0%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      4. +-commutative 48.0%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      5. associate-+r+ 48.0%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      6. metadata-eval 48.0%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   3. Simplified 48.0%

      \[\leadsto \color{blue}{\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

   4. Taylor expanded in k around 0 49.8%

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   5. Step-by-step derivation

      1. associate-/l* 50.8%

         \[\leadsto \frac{2}{2 \cdot \color{blue}{\frac{{k}^{2}}{\frac{{\ell}^{2}}{{t}^{3}}}}} \]

      2. unpow2 50.8%

         \[\leadsto \frac{2}{2 \cdot \frac{\color{blue}{k \cdot k}}{\frac{{\ell}^{2}}{{t}^{3}}}} \]

      3. unpow2 50.8%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\frac{\color{blue}{\ell \cdot \ell}}{{t}^{3}}}} \]

   6. Simplified 50.8%

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{k \cdot k}{\frac{\ell \cdot \ell}{{t}^{3}}}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 40.1%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\ell \cdot \ell}{{t}^{3}}\right)\right)}}} \]

      2. expm1-udef 35.6%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{e^{\mathsf{log1p}\left(\frac{\ell \cdot \ell}{{t}^{3}}\right)} - 1}}} \]

      3. div-inv 35.6%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{e^{\mathsf{log1p}\left(\color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{1}{{t}^{3}}}\right)} - 1}} \]

      4. pow-flip 35.6%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \color{blue}{{t}^{\left(-3\right)}}\right)} - 1}} \]

      5. metadata-eval 35.6%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot {t}^{\color{blue}{-3}}\right)} - 1}} \]

   8. Applied egg-rr 35.6%

      \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot {t}^{-3}\right)} - 1}}} \]
   9. Step-by-step derivation

      1. expm1-def 40.1%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot {t}^{-3}\right)\right)}}} \]

      2. expm1-log1p 50.8%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{\left(\ell \cdot \ell\right) \cdot {t}^{-3}}}} \]

      3. associate-*l* 53.7%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{\ell \cdot \left(\ell \cdot {t}^{-3}\right)}}} \]

   10. Simplified 53.7%

       \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{\ell \cdot \left(\ell \cdot {t}^{-3}\right)}}} \]
   11. Step-by-step derivation

       1. expm1-log1p-u 39.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{2 \cdot \frac{k \cdot k}{\ell \cdot \left(\ell \cdot {t}^{-3}\right)}}\right)\right)} \]

       2. expm1-udef 39.3%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{2 \cdot \frac{k \cdot k}{\ell \cdot \left(\ell \cdot {t}^{-3}\right)}}\right)} - 1} \]

       3. associate-/r* 39.3%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{2}{2}}{\frac{k \cdot k}{\ell \cdot \left(\ell \cdot {t}^{-3}\right)}}}\right)} - 1 \]

       4. metadata-eval 39.3%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{1}}{\frac{k \cdot k}{\ell \cdot \left(\ell \cdot {t}^{-3}\right)}}\right)} - 1 \]

       5. times-frac 46.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{\frac{k}{\ell} \cdot \frac{k}{\ell \cdot {t}^{-3}}}}\right)} - 1 \]

   12. Applied egg-rr 46.9%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\frac{k}{\ell} \cdot \frac{k}{\ell \cdot {t}^{-3}}}\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 48.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{k}{\ell} \cdot \frac{k}{\ell \cdot {t}^{-3}}}\right)\right)} \]

       2. expm1-log1p 64.0%

          \[\leadsto \color{blue}{\frac{1}{\frac{k}{\ell} \cdot \frac{k}{\ell \cdot {t}^{-3}}}} \]

       3. associate-/r* 64.0%

          \[\leadsto \color{blue}{\frac{\frac{1}{\frac{k}{\ell}}}{\frac{k}{\ell \cdot {t}^{-3}}}} \]

       4. associate-/r* 64.9%

          \[\leadsto \frac{\frac{1}{\frac{k}{\ell}}}{\color{blue}{\frac{\frac{k}{\ell}}{{t}^{-3}}}} \]

   14. Simplified 64.9%

       \[\leadsto \color{blue}{\frac{\frac{1}{\frac{k}{\ell}}}{\frac{\frac{k}{\ell}}{{t}^{-3}}}} \]

   if 4.7999999999999999e-21 < k

   1. Initial program 49.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 49.7%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 49.7%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 49.7%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 51.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 51.2%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 51.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 51.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 51.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 51.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 51.3%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 51.3%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 73.0%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 73.0%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

      2. times-frac 70.1%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

      3. unpow2 70.1%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      4. unpow2 70.1%

         \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      5. times-frac 87.2%

         \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      6. *-commutative 87.2%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

   6. Simplified 87.2%

      \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} \]

   7. Taylor expanded in l around 0 73.0%

      \[\leadsto 2 \cdot \color{blue}{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   8. Step-by-step derivation

      1. associate-/r* 70.0%

         \[\leadsto 2 \cdot \color{blue}{\frac{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2}}}{{\sin k}^{2} \cdot t}} \]

      2. *-commutative 70.0%

         \[\leadsto 2 \cdot \frac{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2}}}{\color{blue}{t \cdot {\sin k}^{2}}} \]

      3. associate-*r/ 70.0%

         \[\leadsto 2 \cdot \frac{\color{blue}{\cos k \cdot \frac{{\ell}^{2}}{{k}^{2}}}}{t \cdot {\sin k}^{2}} \]

      4. unpow2 70.0%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{t \cdot {\sin k}^{2}} \]

      5. unpow2 70.0%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{t \cdot {\sin k}^{2}} \]

      6. times-frac 87.1%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}}{t \cdot {\sin k}^{2}} \]

      7. unpow2 87.1%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \color{blue}{{\left(\frac{\ell}{k}\right)}^{2}}}{t \cdot {\sin k}^{2}} \]

      8. *-commutative 87.1%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\left(\frac{\ell}{k}\right)}^{2} \cdot \cos k}}{t \cdot {\sin k}^{2}} \]

      9. times-frac 88.6%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)} \]

   9. Simplified 88.6%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)} \]

   10. Taylor expanded in k around 0 69.3%

       \[\leadsto 2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{\cos k}{\color{blue}{{k}^{2}}}\right) \]
   11. Step-by-step derivation

       1. unpow2 69.3%

          \[\leadsto 2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{\cos k}{\color{blue}{k \cdot k}}\right) \]

   12. Simplified 69.3%

       \[\leadsto 2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{\cos k}{\color{blue}{k \cdot k}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.8 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{1}{\frac{k}{\ell}}}{\frac{\frac{k}{\ell}}{{t}^{-3}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{\cos k}{k \cdot k}\right)\\ \end{array} \]

Alternative 8: 68.0% accurate, 3.5× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 8.5 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{1}{\frac{k}{\ell}}}{\frac{\frac{k}{\ell}}{{t}^{-3}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{k \cdot \left(t \cdot k\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 8.5e-13)
   (/ (/ 1.0 (/ k l)) (/ (/ k l) (pow t -3.0)))
   (* 2.0 (* (* (/ l k) (/ l k)) (/ (cos k) (* k (* t k)))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 8.5e-13) {
		tmp = (1.0 / (k / l)) / ((k / l) / pow(t, -3.0));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * (cos(k) / (k * (t * k))));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 8.5d-13) then
        tmp = (1.0d0 / (k / l)) / ((k / l) / (t ** (-3.0d0)))
    else
        tmp = 2.0d0 * (((l / k) * (l / k)) * (cos(k) / (k * (t * k))))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 8.5e-13) {
		tmp = (1.0 / (k / l)) / ((k / l) / Math.pow(t, -3.0));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * (Math.cos(k) / (k * (t * k))));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 8.5e-13:
		tmp = (1.0 / (k / l)) / ((k / l) / math.pow(t, -3.0))
	else:
		tmp = 2.0 * (((l / k) * (l / k)) * (math.cos(k) / (k * (t * k))))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 8.5e-13)
		tmp = Float64(Float64(1.0 / Float64(k / l)) / Float64(Float64(k / l) / (t ^ -3.0)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(cos(k) / Float64(k * Float64(t * k)))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 8.5e-13)
		tmp = (1.0 / (k / l)) / ((k / l) / (t ^ -3.0));
	else
		tmp = 2.0 * (((l / k) * (l / k)) * (cos(k) / (k * (t * k))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 8.5e-13], N[(N[(1.0 / N[(k / l), $MachinePrecision]), $MachinePrecision] / N[(N[(k / l), $MachinePrecision] / N[Power[t, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 8.5000000000000001e-13

   1. Initial program 55.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 55.0%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-*l* 47.8%

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

      3. associate-*r* 47.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      4. +-commutative 47.8%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      5. associate-+r+ 47.8%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      6. metadata-eval 47.8%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   3. Simplified 47.8%

      \[\leadsto \color{blue}{\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

   4. Taylor expanded in k around 0 49.6%

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   5. Step-by-step derivation

      1. associate-/l* 50.6%

         \[\leadsto \frac{2}{2 \cdot \color{blue}{\frac{{k}^{2}}{\frac{{\ell}^{2}}{{t}^{3}}}}} \]

      2. unpow2 50.6%

         \[\leadsto \frac{2}{2 \cdot \frac{\color{blue}{k \cdot k}}{\frac{{\ell}^{2}}{{t}^{3}}}} \]

      3. unpow2 50.6%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\frac{\color{blue}{\ell \cdot \ell}}{{t}^{3}}}} \]

   6. Simplified 50.6%

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{k \cdot k}{\frac{\ell \cdot \ell}{{t}^{3}}}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 40.1%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\ell \cdot \ell}{{t}^{3}}\right)\right)}}} \]

      2. expm1-udef 35.6%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{e^{\mathsf{log1p}\left(\frac{\ell \cdot \ell}{{t}^{3}}\right)} - 1}}} \]

      3. div-inv 35.6%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{e^{\mathsf{log1p}\left(\color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{1}{{t}^{3}}}\right)} - 1}} \]

      4. pow-flip 35.6%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \color{blue}{{t}^{\left(-3\right)}}\right)} - 1}} \]

      5. metadata-eval 35.6%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot {t}^{\color{blue}{-3}}\right)} - 1}} \]

   8. Applied egg-rr 35.6%

      \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot {t}^{-3}\right)} - 1}}} \]
   9. Step-by-step derivation

      1. expm1-def 40.1%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot {t}^{-3}\right)\right)}}} \]

      2. expm1-log1p 50.6%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{\left(\ell \cdot \ell\right) \cdot {t}^{-3}}}} \]

      3. associate-*l* 53.6%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{\ell \cdot \left(\ell \cdot {t}^{-3}\right)}}} \]

   10. Simplified 53.6%

       \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{\ell \cdot \left(\ell \cdot {t}^{-3}\right)}}} \]
   11. Step-by-step derivation

       1. expm1-log1p-u 39.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{2 \cdot \frac{k \cdot k}{\ell \cdot \left(\ell \cdot {t}^{-3}\right)}}\right)\right)} \]

       2. expm1-udef 39.2%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{2 \cdot \frac{k \cdot k}{\ell \cdot \left(\ell \cdot {t}^{-3}\right)}}\right)} - 1} \]

       3. associate-/r* 39.2%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{2}{2}}{\frac{k \cdot k}{\ell \cdot \left(\ell \cdot {t}^{-3}\right)}}}\right)} - 1 \]

       4. metadata-eval 39.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{1}}{\frac{k \cdot k}{\ell \cdot \left(\ell \cdot {t}^{-3}\right)}}\right)} - 1 \]

       5. times-frac 46.8%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{\frac{k}{\ell} \cdot \frac{k}{\ell \cdot {t}^{-3}}}}\right)} - 1 \]

   12. Applied egg-rr 46.8%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\frac{k}{\ell} \cdot \frac{k}{\ell \cdot {t}^{-3}}}\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 48.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{k}{\ell} \cdot \frac{k}{\ell \cdot {t}^{-3}}}\right)\right)} \]

       2. expm1-log1p 63.7%

          \[\leadsto \color{blue}{\frac{1}{\frac{k}{\ell} \cdot \frac{k}{\ell \cdot {t}^{-3}}}} \]

       3. associate-/r* 63.7%

          \[\leadsto \color{blue}{\frac{\frac{1}{\frac{k}{\ell}}}{\frac{k}{\ell \cdot {t}^{-3}}}} \]

       4. associate-/r* 64.6%

          \[\leadsto \frac{\frac{1}{\frac{k}{\ell}}}{\color{blue}{\frac{\frac{k}{\ell}}{{t}^{-3}}}} \]

   14. Simplified 64.6%

       \[\leadsto \color{blue}{\frac{\frac{1}{\frac{k}{\ell}}}{\frac{\frac{k}{\ell}}{{t}^{-3}}}} \]

   if 8.5000000000000001e-13 < k

   1. Initial program 50.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 50.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 50.4%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 50.4%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 52.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 52.0%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 52.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 52.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 52.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 52.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 52.0%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 52.0%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 74.8%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 74.8%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

      2. times-frac 71.7%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

      3. unpow2 71.7%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      4. unpow2 71.7%

         \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      5. times-frac 89.5%

         \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      6. *-commutative 89.5%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

   6. Simplified 89.5%

      \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} \]

   7. Taylor expanded in k around 0 69.2%

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{\color{blue}{{k}^{2} \cdot t}}\right) \]
   8. Step-by-step derivation

      1. unpow2 69.2%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{\color{blue}{\left(k \cdot k\right)} \cdot t}\right) \]

      2. associate-*l* 69.2%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{\color{blue}{k \cdot \left(k \cdot t\right)}}\right) \]

   9. Simplified 69.2%

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{\color{blue}{k \cdot \left(k \cdot t\right)}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.5 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{1}{\frac{k}{\ell}}}{\frac{\frac{k}{\ell}}{{t}^{-3}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{k \cdot \left(t \cdot k\right)}\right)\\ \end{array} \]

Alternative 9: 66.2% accurate, 3.7× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 1.06 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{1}{\frac{k}{\ell}}}{\frac{\frac{k}{\ell}}{{t}^{-3}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{t}}{{k}^{4}}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 1.06e-20)
   (/ (/ 1.0 (/ k l)) (/ (/ k l) (pow t -3.0)))
   (* 2.0 (/ (* l (/ l t)) (pow k 4.0)))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.06e-20) {
		tmp = (1.0 / (k / l)) / ((k / l) / pow(t, -3.0));
	} else {
		tmp = 2.0 * ((l * (l / t)) / pow(k, 4.0));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.06d-20) then
        tmp = (1.0d0 / (k / l)) / ((k / l) / (t ** (-3.0d0)))
    else
        tmp = 2.0d0 * ((l * (l / t)) / (k ** 4.0d0))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.06e-20) {
		tmp = (1.0 / (k / l)) / ((k / l) / Math.pow(t, -3.0));
	} else {
		tmp = 2.0 * ((l * (l / t)) / Math.pow(k, 4.0));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 1.06e-20:
		tmp = (1.0 / (k / l)) / ((k / l) / math.pow(t, -3.0))
	else:
		tmp = 2.0 * ((l * (l / t)) / math.pow(k, 4.0))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 1.06e-20)
		tmp = Float64(Float64(1.0 / Float64(k / l)) / Float64(Float64(k / l) / (t ^ -3.0)));
	else
		tmp = Float64(2.0 * Float64(Float64(l * Float64(l / t)) / (k ^ 4.0)));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.06e-20)
		tmp = (1.0 / (k / l)) / ((k / l) / (t ^ -3.0));
	else
		tmp = 2.0 * ((l * (l / t)) / (k ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 1.06e-20], N[(N[(1.0 / N[(k / l), $MachinePrecision]), $MachinePrecision] / N[(N[(k / l), $MachinePrecision] / N[Power[t, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.06e-20

   1. Initial program 55.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 55.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-*l* 48.0%

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

      3. associate-*r* 48.0%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      4. +-commutative 48.0%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      5. associate-+r+ 48.0%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      6. metadata-eval 48.0%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   3. Simplified 48.0%

      \[\leadsto \color{blue}{\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

   4. Taylor expanded in k around 0 49.8%

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   5. Step-by-step derivation

      1. associate-/l* 50.8%

         \[\leadsto \frac{2}{2 \cdot \color{blue}{\frac{{k}^{2}}{\frac{{\ell}^{2}}{{t}^{3}}}}} \]

      2. unpow2 50.8%

         \[\leadsto \frac{2}{2 \cdot \frac{\color{blue}{k \cdot k}}{\frac{{\ell}^{2}}{{t}^{3}}}} \]

      3. unpow2 50.8%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\frac{\color{blue}{\ell \cdot \ell}}{{t}^{3}}}} \]

   6. Simplified 50.8%

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{k \cdot k}{\frac{\ell \cdot \ell}{{t}^{3}}}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 40.1%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\ell \cdot \ell}{{t}^{3}}\right)\right)}}} \]

      2. expm1-udef 35.6%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{e^{\mathsf{log1p}\left(\frac{\ell \cdot \ell}{{t}^{3}}\right)} - 1}}} \]

      3. div-inv 35.6%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{e^{\mathsf{log1p}\left(\color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{1}{{t}^{3}}}\right)} - 1}} \]

      4. pow-flip 35.6%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \color{blue}{{t}^{\left(-3\right)}}\right)} - 1}} \]

      5. metadata-eval 35.6%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot {t}^{\color{blue}{-3}}\right)} - 1}} \]

   8. Applied egg-rr 35.6%

      \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot {t}^{-3}\right)} - 1}}} \]
   9. Step-by-step derivation

      1. expm1-def 40.1%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot {t}^{-3}\right)\right)}}} \]

      2. expm1-log1p 50.8%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{\left(\ell \cdot \ell\right) \cdot {t}^{-3}}}} \]

      3. associate-*l* 53.7%

         \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{\ell \cdot \left(\ell \cdot {t}^{-3}\right)}}} \]

   10. Simplified 53.7%

       \[\leadsto \frac{2}{2 \cdot \frac{k \cdot k}{\color{blue}{\ell \cdot \left(\ell \cdot {t}^{-3}\right)}}} \]
   11. Step-by-step derivation

       1. expm1-log1p-u 39.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{2 \cdot \frac{k \cdot k}{\ell \cdot \left(\ell \cdot {t}^{-3}\right)}}\right)\right)} \]

       2. expm1-udef 39.3%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{2 \cdot \frac{k \cdot k}{\ell \cdot \left(\ell \cdot {t}^{-3}\right)}}\right)} - 1} \]

       3. associate-/r* 39.3%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{2}{2}}{\frac{k \cdot k}{\ell \cdot \left(\ell \cdot {t}^{-3}\right)}}}\right)} - 1 \]

       4. metadata-eval 39.3%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{1}}{\frac{k \cdot k}{\ell \cdot \left(\ell \cdot {t}^{-3}\right)}}\right)} - 1 \]

       5. times-frac 46.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{\frac{k}{\ell} \cdot \frac{k}{\ell \cdot {t}^{-3}}}}\right)} - 1 \]

   12. Applied egg-rr 46.9%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\frac{k}{\ell} \cdot \frac{k}{\ell \cdot {t}^{-3}}}\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 48.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{k}{\ell} \cdot \frac{k}{\ell \cdot {t}^{-3}}}\right)\right)} \]

       2. expm1-log1p 64.0%

          \[\leadsto \color{blue}{\frac{1}{\frac{k}{\ell} \cdot \frac{k}{\ell \cdot {t}^{-3}}}} \]

       3. associate-/r* 64.0%

          \[\leadsto \color{blue}{\frac{\frac{1}{\frac{k}{\ell}}}{\frac{k}{\ell \cdot {t}^{-3}}}} \]

       4. associate-/r* 64.9%

          \[\leadsto \frac{\frac{1}{\frac{k}{\ell}}}{\color{blue}{\frac{\frac{k}{\ell}}{{t}^{-3}}}} \]

   14. Simplified 64.9%

       \[\leadsto \color{blue}{\frac{\frac{1}{\frac{k}{\ell}}}{\frac{\frac{k}{\ell}}{{t}^{-3}}}} \]

   if 1.06e-20 < k

   1. Initial program 49.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 49.7%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 49.7%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 49.7%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 51.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 51.2%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 51.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 51.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 51.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 51.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 51.3%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 51.3%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 73.0%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 73.0%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

      2. times-frac 70.1%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

      3. unpow2 70.1%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      4. unpow2 70.1%

         \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      5. times-frac 87.2%

         \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      6. *-commutative 87.2%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

   6. Simplified 87.2%

      \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} \]

   7. Taylor expanded in k around 0 61.0%

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   8. Step-by-step derivation

      1. unpow2 61.0%

         \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

      2. *-commutative 61.0%

         \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

      3. times-frac 62.7%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

   9. Simplified 62.7%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]
   10. Step-by-step derivation

       1. associate-*r/ 63.8%

          \[\leadsto 2 \cdot \color{blue}{\frac{\frac{\ell}{t} \cdot \ell}{{k}^{4}}} \]

   11. Applied egg-rr 63.8%

       \[\leadsto 2 \cdot \color{blue}{\frac{\frac{\ell}{t} \cdot \ell}{{k}^{4}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.06 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{1}{\frac{k}{\ell}}}{\frac{\frac{k}{\ell}}{{t}^{-3}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{t}}{{k}^{4}}\\ \end{array} \]

Alternative 10: 65.2% accurate, 3.8× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 6 \cdot 10^{-22}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{{t}^{3} \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{t}}{{k}^{4}}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 6e-22)
   (* (/ l k) (/ l (* (pow t 3.0) k)))
   (* 2.0 (/ (* l (/ l t)) (pow k 4.0)))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 6e-22) {
		tmp = (l / k) * (l / (pow(t, 3.0) * k));
	} else {
		tmp = 2.0 * ((l * (l / t)) / pow(k, 4.0));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 6d-22) then
        tmp = (l / k) * (l / ((t ** 3.0d0) * k))
    else
        tmp = 2.0d0 * ((l * (l / t)) / (k ** 4.0d0))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 6e-22) {
		tmp = (l / k) * (l / (Math.pow(t, 3.0) * k));
	} else {
		tmp = 2.0 * ((l * (l / t)) / Math.pow(k, 4.0));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 6e-22:
		tmp = (l / k) * (l / (math.pow(t, 3.0) * k))
	else:
		tmp = 2.0 * ((l * (l / t)) / math.pow(k, 4.0))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 6e-22)
		tmp = Float64(Float64(l / k) * Float64(l / Float64((t ^ 3.0) * k)));
	else
		tmp = Float64(2.0 * Float64(Float64(l * Float64(l / t)) / (k ^ 4.0)));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 6e-22)
		tmp = (l / k) * (l / ((t ^ 3.0) * k));
	else
		tmp = 2.0 * ((l * (l / t)) / (k ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 6e-22], N[(N[(l / k), $MachinePrecision] * N[(l / N[(N[Power[t, 3.0], $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.9999999999999998e-22

   1. Initial program 55.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 55.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 55.8%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 54.2%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 54.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 54.2%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 54.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 54.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 54.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 54.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 54.2%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 54.2%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around 0 49.8%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 49.8%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. unpow2 49.8%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      3. associate-*l* 57.6%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

   6. Simplified 57.6%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot {t}^{3}\right)}} \]
   7. Step-by-step derivation

      1. times-frac 64.5%

         \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}} \]

   8. Applied egg-rr 64.5%

      \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}} \]

   if 5.9999999999999998e-22 < k

   1. Initial program 49.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 49.7%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 49.7%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 49.7%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 51.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 51.2%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 51.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 51.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 51.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 51.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 51.3%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 51.3%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 73.0%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 73.0%

         \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

      2. times-frac 70.1%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

      3. unpow2 70.1%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      4. unpow2 70.1%

         \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      5. times-frac 87.2%

         \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

      6. *-commutative 87.2%

         \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

   6. Simplified 87.2%

      \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} \]

   7. Taylor expanded in k around 0 61.0%

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   8. Step-by-step derivation

      1. unpow2 61.0%

         \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

      2. *-commutative 61.0%

         \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

      3. times-frac 62.7%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

   9. Simplified 62.7%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]
   10. Step-by-step derivation

       1. associate-*r/ 63.8%

          \[\leadsto 2 \cdot \color{blue}{\frac{\frac{\ell}{t} \cdot \ell}{{k}^{4}}} \]

   11. Applied egg-rr 63.8%

       \[\leadsto 2 \cdot \color{blue}{\frac{\frac{\ell}{t} \cdot \ell}{{k}^{4}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6 \cdot 10^{-22}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{{t}^{3} \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{t}}{{k}^{4}}\\ \end{array} \]

Alternative 11: 55.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ 2 \cdot \left(\frac{\ell}{t} \cdot \left(\ell \cdot {k}^{-4}\right)\right) \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k) :precision binary64 (* 2.0 (* (/ l t) (* l (pow k -4.0)))))

C

k = abs(k);
double code(double t, double l, double k) {
	return 2.0 * ((l / t) * (l * pow(k, -4.0)));
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 * ((l / t) * (l * (k ** (-4.0d0))))
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	return 2.0 * ((l / t) * (l * Math.pow(k, -4.0)));
}

Python

k = abs(k)
def code(t, l, k):
	return 2.0 * ((l / t) * (l * math.pow(k, -4.0)))

Julia

k = abs(k)
function code(t, l, k)
	return Float64(2.0 * Float64(Float64(l / t) * Float64(l * (k ^ -4.0))))
end

MATLAB

k = abs(k)
function tmp = code(t, l, k)
	tmp = 2.0 * ((l / t) * (l * (k ^ -4.0)));
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(2.0 * N[(N[(l / t), $MachinePrecision] * N[(l * N[Power[k, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.9%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/l/ 53.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   2. associate-*l/ 54.2%

      \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

   3. associate-*l/ 53.1%

      \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

   4. associate-/r/ 53.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

   5. *-commutative 53.5%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

   6. associate-/l/ 53.5%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

   7. associate-*r* 53.5%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   8. *-commutative 53.5%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

   9. associate-*r* 53.5%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

   10. *-commutative 53.5%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

3. Simplified 53.5%

   \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

4. Taylor expanded in k around inf 57.6%

   \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
5. Step-by-step derivation

   1. *-commutative 57.6%

      \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

   2. times-frac 57.7%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

   3. unpow2 57.7%

      \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

   4. unpow2 57.7%

      \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

   5. times-frac 68.0%

      \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

   6. *-commutative 68.0%

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

6. Simplified 68.0%

   \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} \]

7. Taylor expanded in k around 0 50.5%

   \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
8. Step-by-step derivation

   1. unpow2 50.5%

      \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

   2. *-commutative 50.5%

      \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

   3. times-frac 53.5%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

9. Simplified 53.5%

   \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]
10. Step-by-step derivation

    1. expm1-log1p-u 41.8%

       \[\leadsto 2 \cdot \left(\frac{\ell}{t} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\ell}{{k}^{4}}\right)\right)}\right) \]

    2. expm1-udef 41.5%

       \[\leadsto 2 \cdot \left(\frac{\ell}{t} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\ell}{{k}^{4}}\right)} - 1\right)}\right) \]

    3. div-inv 41.5%

       \[\leadsto 2 \cdot \left(\frac{\ell}{t} \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\ell \cdot \frac{1}{{k}^{4}}}\right)} - 1\right)\right) \]

    4. pow-flip 41.5%

       \[\leadsto 2 \cdot \left(\frac{\ell}{t} \cdot \left(e^{\mathsf{log1p}\left(\ell \cdot \color{blue}{{k}^{\left(-4\right)}}\right)} - 1\right)\right) \]

    5. metadata-eval 41.5%

       \[\leadsto 2 \cdot \left(\frac{\ell}{t} \cdot \left(e^{\mathsf{log1p}\left(\ell \cdot {k}^{\color{blue}{-4}}\right)} - 1\right)\right) \]

11. Applied egg-rr 41.5%

    \[\leadsto 2 \cdot \left(\frac{\ell}{t} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\ell \cdot {k}^{-4}\right)} - 1\right)}\right) \]
12. Step-by-step derivation

    1. expm1-def 41.8%

       \[\leadsto 2 \cdot \left(\frac{\ell}{t} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\ell \cdot {k}^{-4}\right)\right)}\right) \]

    2. expm1-log1p 53.5%

       \[\leadsto 2 \cdot \left(\frac{\ell}{t} \cdot \color{blue}{\left(\ell \cdot {k}^{-4}\right)}\right) \]

13. Simplified 53.5%

    \[\leadsto 2 \cdot \left(\frac{\ell}{t} \cdot \color{blue}{\left(\ell \cdot {k}^{-4}\right)}\right) \]

14. Final simplification 53.5%

    \[\leadsto 2 \cdot \left(\frac{\ell}{t} \cdot \left(\ell \cdot {k}^{-4}\right)\right) \]

Alternative 12: 55.8% accurate, 3.8× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ 2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right) \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k) :precision binary64 (* 2.0 (* (/ l t) (/ l (pow k 4.0)))))

C

k = abs(k);
double code(double t, double l, double k) {
	return 2.0 * ((l / t) * (l / pow(k, 4.0)));
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 * ((l / t) * (l / (k ** 4.0d0)))
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	return 2.0 * ((l / t) * (l / Math.pow(k, 4.0)));
}

Python

k = abs(k)
def code(t, l, k):
	return 2.0 * ((l / t) * (l / math.pow(k, 4.0)))

Julia

k = abs(k)
function code(t, l, k)
	return Float64(2.0 * Float64(Float64(l / t) * Float64(l / (k ^ 4.0))))
end

MATLAB

k = abs(k)
function tmp = code(t, l, k)
	tmp = 2.0 * ((l / t) * (l / (k ^ 4.0)));
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(2.0 * N[(N[(l / t), $MachinePrecision] * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.9%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/l/ 53.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   2. associate-*l/ 54.2%

      \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

   3. associate-*l/ 53.1%

      \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

   4. associate-/r/ 53.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

   5. *-commutative 53.5%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

   6. associate-/l/ 53.5%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

   7. associate-*r* 53.5%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   8. *-commutative 53.5%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

   9. associate-*r* 53.5%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

   10. *-commutative 53.5%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

3. Simplified 53.5%

   \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

4. Taylor expanded in k around inf 57.6%

   \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
5. Step-by-step derivation

   1. *-commutative 57.6%

      \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

   2. times-frac 57.7%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

   3. unpow2 57.7%

      \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

   4. unpow2 57.7%

      \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

   5. times-frac 68.0%

      \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

   6. *-commutative 68.0%

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

6. Simplified 68.0%

   \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} \]

7. Taylor expanded in k around 0 50.5%

   \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
8. Step-by-step derivation

   1. unpow2 50.5%

      \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

   2. *-commutative 50.5%

      \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

   3. times-frac 53.5%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

9. Simplified 53.5%

   \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

10. Final simplification 53.5%

    \[\leadsto 2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right) \]

Alternative 13: 55.2% accurate, 3.8× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ 2 \cdot \frac{\ell \cdot \left(\ell \cdot {k}^{-4}\right)}{t} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k) :precision binary64 (* 2.0 (/ (* l (* l (pow k -4.0))) t)))

C

k = abs(k);
double code(double t, double l, double k) {
	return 2.0 * ((l * (l * pow(k, -4.0))) / t);
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 * ((l * (l * (k ** (-4.0d0)))) / t)
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	return 2.0 * ((l * (l * Math.pow(k, -4.0))) / t);
}

Python

k = abs(k)
def code(t, l, k):
	return 2.0 * ((l * (l * math.pow(k, -4.0))) / t)

Julia

k = abs(k)
function code(t, l, k)
	return Float64(2.0 * Float64(Float64(l * Float64(l * (k ^ -4.0))) / t))
end

MATLAB

k = abs(k)
function tmp = code(t, l, k)
	tmp = 2.0 * ((l * (l * (k ^ -4.0))) / t);
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(2.0 * N[(N[(l * N[(l * N[Power[k, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.9%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/l/ 53.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   2. associate-*l/ 54.2%

      \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

   3. associate-*l/ 53.1%

      \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

   4. associate-/r/ 53.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

   5. *-commutative 53.5%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

   6. associate-/l/ 53.5%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

   7. associate-*r* 53.5%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   8. *-commutative 53.5%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

   9. associate-*r* 53.5%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

   10. *-commutative 53.5%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

3. Simplified 53.5%

   \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

4. Taylor expanded in k around inf 57.6%

   \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
5. Step-by-step derivation

   1. *-commutative 57.6%

      \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

   2. times-frac 57.7%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]

   3. unpow2 57.7%

      \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

   4. unpow2 57.7%

      \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

   5. times-frac 68.0%

      \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

   6. *-commutative 68.0%

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

6. Simplified 68.0%

   \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} \]

7. Taylor expanded in k around 0 50.5%

   \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
8. Step-by-step derivation

   1. unpow2 50.5%

      \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

   2. *-commutative 50.5%

      \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

   3. times-frac 53.5%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

9. Simplified 53.5%

   \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]
10. Step-by-step derivation

    1. associate-*l/ 53.7%

       \[\leadsto 2 \cdot \color{blue}{\frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}} \]

    2. div-inv 53.7%

       \[\leadsto 2 \cdot \frac{\ell \cdot \color{blue}{\left(\ell \cdot \frac{1}{{k}^{4}}\right)}}{t} \]

    3. pow-flip 53.7%

       \[\leadsto 2 \cdot \frac{\ell \cdot \left(\ell \cdot \color{blue}{{k}^{\left(-4\right)}}\right)}{t} \]

    4. metadata-eval 53.7%

       \[\leadsto 2 \cdot \frac{\ell \cdot \left(\ell \cdot {k}^{\color{blue}{-4}}\right)}{t} \]

11. Applied egg-rr 53.7%

    \[\leadsto 2 \cdot \color{blue}{\frac{\ell \cdot \left(\ell \cdot {k}^{-4}\right)}{t}} \]

12. Final simplification 53.7%

    \[\leadsto 2 \cdot \frac{\ell \cdot \left(\ell \cdot {k}^{-4}\right)}{t} \]

Reproduce

herbie shell --seed 2023243 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  :precision binary64
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))