Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.7% → 78.4%

Time: 18.2s

Alternatives: 12

Speedup: 3.8×

• 27.1% 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: 55.7% 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: 78.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.5 \cdot 10^{-26} \lor \neg \left(t \leq 1.22 \cdot 10^{-14}\right):\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{\frac{2}{\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)}}{\tan k}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -3.5e-26) (not (<= t 1.22e-14)))
   (*
    l
    (*
     l
     (/
      2.0
      (* (+ 2.0 (pow (/ k t) 2.0)) (* (tan k) (* (pow t 3.0) (sin k)))))))
   (* l (* l (/ (/ 2.0 (* (sin k) (* k (* t k)))) (tan k))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -3.5e-26) || !(t <= 1.22e-14)) {
		tmp = l * (l * (2.0 / ((2.0 + pow((k / t), 2.0)) * (tan(k) * (pow(t, 3.0) * sin(k))))));
	} else {
		tmp = l * (l * ((2.0 / (sin(k) * (k * (t * k)))) / tan(k)));
	}
	return tmp;
}

Fortran

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 ((t <= (-3.5d-26)) .or. (.not. (t <= 1.22d-14))) then
        tmp = l * (l * (2.0d0 / ((2.0d0 + ((k / t) ** 2.0d0)) * (tan(k) * ((t ** 3.0d0) * sin(k))))))
    else
        tmp = l * (l * ((2.0d0 / (sin(k) * (k * (t * k)))) / tan(k)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -3.5e-26) || !(t <= 1.22e-14)) {
		tmp = l * (l * (2.0 / ((2.0 + Math.pow((k / t), 2.0)) * (Math.tan(k) * (Math.pow(t, 3.0) * Math.sin(k))))));
	} else {
		tmp = l * (l * ((2.0 / (Math.sin(k) * (k * (t * k)))) / Math.tan(k)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -3.5e-26) or not (t <= 1.22e-14):
		tmp = l * (l * (2.0 / ((2.0 + math.pow((k / t), 2.0)) * (math.tan(k) * (math.pow(t, 3.0) * math.sin(k))))))
	else:
		tmp = l * (l * ((2.0 / (math.sin(k) * (k * (t * k)))) / math.tan(k)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -3.5e-26) || !(t <= 1.22e-14))
		tmp = Float64(l * Float64(l * Float64(2.0 / Float64(Float64(2.0 + (Float64(k / t) ^ 2.0)) * Float64(tan(k) * Float64((t ^ 3.0) * sin(k)))))));
	else
		tmp = Float64(l * Float64(l * Float64(Float64(2.0 / Float64(sin(k) * Float64(k * Float64(t * k)))) / tan(k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -3.5e-26) || ~((t <= 1.22e-14)))
		tmp = l * (l * (2.0 / ((2.0 + ((k / t) ^ 2.0)) * (tan(k) * ((t ^ 3.0) * sin(k))))));
	else
		tmp = l * (l * ((2.0 / (sin(k) * (k * (t * k)))) / tan(k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -3.5e-26], N[Not[LessEqual[t, 1.22e-14]], $MachinePrecision]], N[(l * N[(l * N[(2.0 / N[(N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(l * N[(N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.49999999999999985e-26 or 1.21999999999999994e-14 < t

   1. Initial program 70.8%

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

      1. associate-/l/ 70.8%

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

      2. associate-*l/ 71.5%

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

      3. associate-*l/ 70.8%

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

      4. associate-/r/ 70.2%

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

      5. *-commutative 70.2%

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

      6. associate-/l/ 70.2%

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

      7. associate-*r* 70.1%

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

      8. *-commutative 70.1%

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

      9. associate-*r* 70.1%

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

      10. *-commutative 70.1%

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

   3. Simplified 70.1%

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

      1. expm1-log1p-u 64.4%

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

      2. expm1-udef 59.0%

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

      3. associate-*l* 62.6%

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

      4. associate-*r* 62.6%

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

      5. *-commutative 62.6%

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

   5. Applied egg-rr 62.6%

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

      1. expm1-def 68.9%

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

      2. expm1-log1p 75.4%

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

      3. associate-*l* 75.4%

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

   7. Simplified 75.4%

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

   if -3.49999999999999985e-26 < t < 1.21999999999999994e-14

   1. Initial program 43.9%

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

      1. associate-/l/ 43.9%

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

      2. associate-*l/ 43.9%

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

      3. associate-*l/ 43.1%

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

      4. associate-/r/ 43.1%

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

      5. *-commutative 43.1%

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

      6. associate-/l/ 43.1%

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

      7. associate-*r* 43.1%

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

      8. *-commutative 43.1%

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

      9. associate-*r* 43.1%

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

      10. *-commutative 43.1%

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

   3. Simplified 43.1%

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

   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. associate-*l* 85.9%

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

      3. *-commutative 85.9%

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

   6. Simplified 85.9%

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

      1. expm1-log1p-u 62.3%

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

      2. expm1-udef 51.1%

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

      3. associate-*r* 51.1%

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

   8. Applied egg-rr 51.1%

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

      1. expm1-def 62.3%

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

      2. expm1-log1p 85.9%

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

      3. associate-*l* 94.9%

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

      4. *-commutative 94.9%

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

      5. associate-/r* 94.8%

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

      6. associate-*r* 94.8%

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

      7. associate-*r* 85.1%

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

      8. unpow2 85.1%

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

      9. *-commutative 85.1%

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

      10. unpow2 85.1%

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

      11. associate-*r* 94.8%

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

   10. Simplified 94.8%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.5 \cdot 10^{-26} \lor \neg \left(t \leq 1.22 \cdot 10^{-14}\right):\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{\frac{2}{\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)}}{\tan k}\right)\\ \end{array} \]

Alternative 2: 79.4% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0)))
   (if (<=
        (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (+ 1.0 (+ 1.0 t_1)))
        INFINITY)
     (* (/ 2.0 (* (sin k) (+ 2.0 t_1))) (/ (/ l (/ (pow t 3.0) l)) (tan k)))
     (* l (* l (/ (/ 2.0 (* (sin k) (* k (* t k)))) (tan k)))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[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[(1.0 + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(l * N[(N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.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)) < +inf.0

   1. Initial program 85.1%

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

      1. associate-*l* 85.1%

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

      2. associate-/l/ 85.1%

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

      3. *-commutative 85.1%

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

      4. associate-*r/ 85.7%

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

      5. associate-/l* 85.2%

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

      6. associate-/r/ 77.8%

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

   3. Simplified 78.9%

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

      1. expm1-log1p-u 62.3%

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

      2. expm1-udef 57.1%

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

      3. associate-*l/ 58.0%

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

      4. associate-*l/ 58.0%

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

   5. Applied egg-rr 58.0%

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

      1. expm1-def 63.2%

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

      2. expm1-log1p 78.7%

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

      3. associate-*r* 78.7%

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

      4. times-frac 85.2%

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

      5. associate-/l* 86.8%

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

   7. Simplified 86.8%

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

   if +inf.0 < (*.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 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l/ 0.0%

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

      3. associate-*l/ 0.0%

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

      4. associate-/r/ 0.1%

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

      5. *-commutative 0.1%

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

      6. associate-/l/ 0.1%

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

      7. associate-*r* 0.1%

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

      8. *-commutative 0.1%

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

      9. associate-*r* 0.1%

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

      10. *-commutative 0.1%

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

   3. Simplified 0.1%

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

   4. Taylor expanded in k around inf 51.2%

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

      1. unpow2 51.2%

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

      2. associate-*l* 59.7%

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

      3. *-commutative 59.7%

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

   6. Simplified 59.7%

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

      1. expm1-log1p-u 48.6%

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

      2. expm1-udef 39.2%

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

      3. associate-*r* 39.2%

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

   8. Applied egg-rr 39.2%

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

      1. expm1-def 48.6%

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

      2. expm1-log1p 59.7%

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

      3. associate-*l* 77.4%

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

      4. *-commutative 77.4%

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

      5. associate-/r* 77.3%

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

      6. associate-*r* 77.4%

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

      7. associate-*r* 64.3%

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

      8. unpow2 64.3%

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

      9. *-commutative 64.3%

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

      10. unpow2 64.3%

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

      11. associate-*r* 77.4%

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

   10. Simplified 77.4%

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

4. Final simplification 83.9%

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

Alternative 3: 75.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -3.3 \cdot 10^{-7} \lor \neg \left(k \leq 4 \cdot 10^{-20}\right):\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{\frac{2}{\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)}}{\tan k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{t}^{3}}{\ell} \cdot \left(\frac{k}{\ell} \cdot \left(k + k\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= k -3.3e-7) (not (<= k 4e-20)))
   (* l (* l (/ (/ 2.0 (* (sin k) (* k (* t k)))) (tan k))))
   (/ 2.0 (* (/ (pow t 3.0) l) (* (/ k l) (+ k k))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((k <= -3.3e-7) || !(k <= 4e-20)) {
		tmp = l * (l * ((2.0 / (sin(k) * (k * (t * k)))) / tan(k)));
	} else {
		tmp = 2.0 / ((pow(t, 3.0) / l) * ((k / l) * (k + k)));
	}
	return tmp;
}

Fortran

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-7)) .or. (.not. (k <= 4d-20))) then
        tmp = l * (l * ((2.0d0 / (sin(k) * (k * (t * k)))) / tan(k)))
    else
        tmp = 2.0d0 / (((t ** 3.0d0) / l) * ((k / l) * (k + k)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((k <= -3.3e-7) || !(k <= 4e-20)) {
		tmp = l * (l * ((2.0 / (Math.sin(k) * (k * (t * k)))) / Math.tan(k)));
	} else {
		tmp = 2.0 / ((Math.pow(t, 3.0) / l) * ((k / l) * (k + k)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (k <= -3.3e-7) or not (k <= 4e-20):
		tmp = l * (l * ((2.0 / (math.sin(k) * (k * (t * k)))) / math.tan(k)))
	else:
		tmp = 2.0 / ((math.pow(t, 3.0) / l) * ((k / l) * (k + k)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((k <= -3.3e-7) || !(k <= 4e-20))
		tmp = Float64(l * Float64(l * Float64(Float64(2.0 / Float64(sin(k) * Float64(k * Float64(t * k)))) / tan(k))));
	else
		tmp = Float64(2.0 / Float64(Float64((t ^ 3.0) / l) * Float64(Float64(k / l) * Float64(k + k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((k <= -3.3e-7) || ~((k <= 4e-20)))
		tmp = l * (l * ((2.0 / (sin(k) * (k * (t * k)))) / tan(k)));
	else
		tmp = 2.0 / (((t ^ 3.0) / l) * ((k / l) * (k + k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[k, -3.3e-7], N[Not[LessEqual[k, 4e-20]], $MachinePrecision]], N[(l * N[(l * N[(N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(k + k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < -3.3000000000000002e-7 or 3.99999999999999978e-20 < k

   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/ 48.7%

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

      5. *-commutative 48.7%

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

      6. associate-/l/ 48.7%

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

      7. associate-*r* 48.6%

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

      8. *-commutative 48.6%

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

      9. associate-*r* 48.6%

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

      10. *-commutative 48.6%

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

   3. Simplified 48.6%

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

   4. Taylor expanded in k around inf 73.1%

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

      1. unpow2 73.1%

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

      2. associate-*l* 78.4%

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

      3. *-commutative 78.4%

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

   6. Simplified 78.4%

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

      1. expm1-log1p-u 71.0%

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

      2. expm1-udef 60.1%

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

      3. associate-*r* 60.1%

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

   8. Applied egg-rr 60.1%

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

      1. expm1-def 71.0%

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

      2. expm1-log1p 78.4%

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

      3. associate-*l* 86.8%

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

      4. *-commutative 86.8%

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

      5. associate-/r* 86.8%

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

      6. associate-*r* 86.8%

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

      7. associate-*r* 78.9%

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

      8. unpow2 78.9%

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

      9. *-commutative 78.9%

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

      10. unpow2 78.9%

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

      11. associate-*r* 86.8%

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

   10. Simplified 86.8%

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

   if -3.3000000000000002e-7 < k < 3.99999999999999978e-20

   1. Initial program 70.7%

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

      1. associate-*l* 70.7%

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

      2. +-commutative 70.7%

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

   3. Simplified 70.7%

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

   4. Taylor expanded in k around 0 63.1%

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

      1. associate-*r/ 63.1%

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

      2. associate-*r* 63.1%

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

      3. unpow2 63.1%

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

      4. times-frac 67.7%

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

      5. *-commutative 67.7%

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

      6. unpow2 67.7%

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

      7. associate-*l* 67.7%

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

   6. Simplified 67.7%

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

      1. expm1-log1p-u 57.7%

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

      2. expm1-udef 33.4%

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

      3. associate-/l* 36.2%

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

   8. Applied egg-rr 36.2%

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

      1. expm1-def 63.0%

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

      2. expm1-log1p 79.4%

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

      3. associate-/r/ 79.4%

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

      4. *-commutative 79.4%

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

      5. count-2 79.4%

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

   10. Simplified 79.4%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -3.3 \cdot 10^{-7} \lor \neg \left(k \leq 4 \cdot 10^{-20}\right):\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{\frac{2}{\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)}}{\tan k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{t}^{3}}{\ell} \cdot \left(\frac{k}{\ell} \cdot \left(k + k\right)\right)}\\ \end{array} \]

Alternative 4: 68.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.45 \cdot 10^{-25}:\\ \;\;\;\;\frac{2}{\frac{{t}^{3}}{\ell} \cdot \left(\frac{k}{\ell} \cdot \left(k + k\right)\right)}\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{-103}:\\ \;\;\;\;\frac{2}{\frac{k}{\cos k} \cdot \left(\frac{k}{\ell} \cdot \left(\left(k \cdot k\right) \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k}{{t}^{3}} \cdot \left(\ell \cdot \frac{\frac{\ell}{k}}{k}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -2.45e-25)
   (/ 2.0 (* (/ (pow t 3.0) l) (* (/ k l) (+ k k))))
   (if (<= t 3.4e-103)
     (/ 2.0 (* (/ k (cos k)) (* (/ k l) (* (* k k) (/ t l)))))
     (* (/ (cos k) (pow t 3.0)) (* l (/ (/ l k) k))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -2.45e-25) {
		tmp = 2.0 / ((pow(t, 3.0) / l) * ((k / l) * (k + k)));
	} else if (t <= 3.4e-103) {
		tmp = 2.0 / ((k / cos(k)) * ((k / l) * ((k * k) * (t / l))));
	} else {
		tmp = (cos(k) / pow(t, 3.0)) * (l * ((l / k) / k));
	}
	return tmp;
}

Fortran

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 (t <= (-2.45d-25)) then
        tmp = 2.0d0 / (((t ** 3.0d0) / l) * ((k / l) * (k + k)))
    else if (t <= 3.4d-103) then
        tmp = 2.0d0 / ((k / cos(k)) * ((k / l) * ((k * k) * (t / l))))
    else
        tmp = (cos(k) / (t ** 3.0d0)) * (l * ((l / k) / k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -2.45e-25) {
		tmp = 2.0 / ((Math.pow(t, 3.0) / l) * ((k / l) * (k + k)));
	} else if (t <= 3.4e-103) {
		tmp = 2.0 / ((k / Math.cos(k)) * ((k / l) * ((k * k) * (t / l))));
	} else {
		tmp = (Math.cos(k) / Math.pow(t, 3.0)) * (l * ((l / k) / k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -2.45e-25:
		tmp = 2.0 / ((math.pow(t, 3.0) / l) * ((k / l) * (k + k)))
	elif t <= 3.4e-103:
		tmp = 2.0 / ((k / math.cos(k)) * ((k / l) * ((k * k) * (t / l))))
	else:
		tmp = (math.cos(k) / math.pow(t, 3.0)) * (l * ((l / k) / k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -2.45e-25)
		tmp = Float64(2.0 / Float64(Float64((t ^ 3.0) / l) * Float64(Float64(k / l) * Float64(k + k))));
	elseif (t <= 3.4e-103)
		tmp = Float64(2.0 / Float64(Float64(k / cos(k)) * Float64(Float64(k / l) * Float64(Float64(k * k) * Float64(t / l)))));
	else
		tmp = Float64(Float64(cos(k) / (t ^ 3.0)) * Float64(l * Float64(Float64(l / k) / k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -2.45e-25)
		tmp = 2.0 / (((t ^ 3.0) / l) * ((k / l) * (k + k)));
	elseif (t <= 3.4e-103)
		tmp = 2.0 / ((k / cos(k)) * ((k / l) * ((k * k) * (t / l))));
	else
		tmp = (cos(k) / (t ^ 3.0)) * (l * ((l / k) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -2.45e-25], N[(2.0 / N[(N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(k + k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.4e-103], N[(2.0 / N[(N[(k / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[k], $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] * N[(l * N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.44999999999999995e-25

   1. Initial program 67.1%

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

      1. associate-*l* 67.1%

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

      2. +-commutative 67.1%

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

   3. Simplified 67.1%

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

   4. Taylor expanded in k around 0 51.5%

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

      1. associate-*r/ 51.5%

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

      2. associate-*r* 51.5%

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

      3. unpow2 51.5%

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

      4. times-frac 56.3%

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

      5. *-commutative 56.3%

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

      6. unpow2 56.3%

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

      7. associate-*l* 56.3%

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

   6. Simplified 56.3%

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

      1. expm1-log1p-u 8.1%

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

      2. expm1-udef 0.0%

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

      3. associate-/l* 0.3%

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

   8. Applied egg-rr 0.3%

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

      1. expm1-def 10.6%

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

      2. expm1-log1p 68.0%

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

      3. associate-/r/ 68.0%

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

      4. *-commutative 68.0%

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

      5. count-2 68.0%

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

   10. Simplified 68.0%

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

   if -2.44999999999999995e-25 < t < 3.40000000000000003e-103

   1. Initial program 39.8%

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

      1. associate-*l* 39.8%

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

      2. +-commutative 39.8%

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

   3. Simplified 39.8%

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

   4. Taylor expanded in t around 0 78.9%

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

      1. times-frac 77.9%

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

      2. unpow2 77.9%

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

      3. *-commutative 77.9%

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

      4. unpow2 77.9%

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

      5. times-frac 84.6%

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

   6. Simplified 84.6%

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

   7. Taylor expanded in k around inf 78.9%

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

      1. unpow2 78.9%

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

      2. associate-*r* 78.8%

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

      3. times-frac 83.7%

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

      4. unpow2 83.7%

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

      5. *-commutative 83.7%

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

      6. associate-*l/ 84.6%

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

      7. *-commutative 84.6%

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

   9. Simplified 84.6%

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

       1. expm1-log1p-u 59.3%

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

       2. expm1-udef 51.9%

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

       3. times-frac 54.7%

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

   11. Applied egg-rr 54.7%

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

       1. expm1-def 63.2%

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

       2. expm1-log1p 88.7%

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

       3. associate-*l* 93.1%

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

   13. Simplified 93.1%

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

   14. Taylor expanded in k around 0 73.6%

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

       1. unpow2 73.6%

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

   16. Simplified 73.6%

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

   if 3.40000000000000003e-103 < t

   1. Initial program 74.3%

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

      1. associate-/l/ 74.4%

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

      2. associate-*l/ 74.2%

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

      3. associate-*l/ 71.9%

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

      4. associate-/r/ 70.7%

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

      5. *-commutative 70.7%

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

      6. associate-/l/ 70.7%

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

      7. associate-*r* 70.7%

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

      8. *-commutative 70.7%

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

      9. associate-*r* 70.7%

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

      10. *-commutative 70.7%

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

   3. Simplified 70.7%

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

   4. Taylor expanded in t around inf 58.3%

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

      1. *-commutative 58.3%

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

      2. times-frac 59.4%

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

      3. unpow2 59.4%

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

   6. Simplified 59.4%

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

   7. Taylor expanded in k around 0 63.2%

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

      1. unpow2 63.2%

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

      2. associate-*r/ 66.4%

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

      3. unpow2 66.4%

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

      4. associate-/r* 71.8%

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

   9. Simplified 71.8%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.45 \cdot 10^{-25}:\\ \;\;\;\;\frac{2}{\frac{{t}^{3}}{\ell} \cdot \left(\frac{k}{\ell} \cdot \left(k + k\right)\right)}\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{-103}:\\ \;\;\;\;\frac{2}{\frac{k}{\cos k} \cdot \left(\frac{k}{\ell} \cdot \left(\left(k \cdot k\right) \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k}{{t}^{3}} \cdot \left(\ell \cdot \frac{\frac{\ell}{k}}{k}\right)\\ \end{array} \]

Alternative 5: 68.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -9.6 \cdot 10^{-26}:\\ \;\;\;\;\frac{2}{\frac{{t}^{3}}{\ell} \cdot \left(\frac{k}{\ell} \cdot \left(k + k\right)\right)}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-68}:\\ \;\;\;\;\frac{2}{\frac{k}{\cos k} \cdot \left(\frac{k}{\ell} \cdot \left(\left(k \cdot k\right) \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k}{{t}^{3}} \cdot \frac{\ell \cdot \frac{\ell}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -9.6e-26)
   (/ 2.0 (* (/ (pow t 3.0) l) (* (/ k l) (+ k k))))
   (if (<= t 8e-68)
     (/ 2.0 (* (/ k (cos k)) (* (/ k l) (* (* k k) (/ t l)))))
     (* (/ (cos k) (pow t 3.0)) (/ (* l (/ l k)) k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -9.6e-26) {
		tmp = 2.0 / ((pow(t, 3.0) / l) * ((k / l) * (k + k)));
	} else if (t <= 8e-68) {
		tmp = 2.0 / ((k / cos(k)) * ((k / l) * ((k * k) * (t / l))));
	} else {
		tmp = (cos(k) / pow(t, 3.0)) * ((l * (l / k)) / k);
	}
	return tmp;
}

Fortran

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 (t <= (-9.6d-26)) then
        tmp = 2.0d0 / (((t ** 3.0d0) / l) * ((k / l) * (k + k)))
    else if (t <= 8d-68) then
        tmp = 2.0d0 / ((k / cos(k)) * ((k / l) * ((k * k) * (t / l))))
    else
        tmp = (cos(k) / (t ** 3.0d0)) * ((l * (l / k)) / k)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -9.6e-26) {
		tmp = 2.0 / ((Math.pow(t, 3.0) / l) * ((k / l) * (k + k)));
	} else if (t <= 8e-68) {
		tmp = 2.0 / ((k / Math.cos(k)) * ((k / l) * ((k * k) * (t / l))));
	} else {
		tmp = (Math.cos(k) / Math.pow(t, 3.0)) * ((l * (l / k)) / k);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -9.6e-26:
		tmp = 2.0 / ((math.pow(t, 3.0) / l) * ((k / l) * (k + k)))
	elif t <= 8e-68:
		tmp = 2.0 / ((k / math.cos(k)) * ((k / l) * ((k * k) * (t / l))))
	else:
		tmp = (math.cos(k) / math.pow(t, 3.0)) * ((l * (l / k)) / k)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -9.6e-26)
		tmp = Float64(2.0 / Float64(Float64((t ^ 3.0) / l) * Float64(Float64(k / l) * Float64(k + k))));
	elseif (t <= 8e-68)
		tmp = Float64(2.0 / Float64(Float64(k / cos(k)) * Float64(Float64(k / l) * Float64(Float64(k * k) * Float64(t / l)))));
	else
		tmp = Float64(Float64(cos(k) / (t ^ 3.0)) * Float64(Float64(l * Float64(l / k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -9.6e-26)
		tmp = 2.0 / (((t ^ 3.0) / l) * ((k / l) * (k + k)));
	elseif (t <= 8e-68)
		tmp = 2.0 / ((k / cos(k)) * ((k / l) * ((k * k) * (t / l))));
	else
		tmp = (cos(k) / (t ^ 3.0)) * ((l * (l / k)) / k);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -9.6e-26], N[(2.0 / N[(N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(k + k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 8e-68], N[(2.0 / N[(N[(k / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[k], $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] * N[(N[(l * N[(l / k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -9.6000000000000004e-26

   1. Initial program 67.1%

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

      1. associate-*l* 67.1%

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

      2. +-commutative 67.1%

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

   3. Simplified 67.1%

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

   4. Taylor expanded in k around 0 51.5%

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

      1. associate-*r/ 51.5%

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

      2. associate-*r* 51.5%

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

      3. unpow2 51.5%

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

      4. times-frac 56.3%

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

      5. *-commutative 56.3%

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

      6. unpow2 56.3%

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

      7. associate-*l* 56.3%

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

   6. Simplified 56.3%

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

      1. expm1-log1p-u 8.1%

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

      2. expm1-udef 0.0%

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

      3. associate-/l* 0.3%

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

   8. Applied egg-rr 0.3%

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

      1. expm1-def 10.6%

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

      2. expm1-log1p 68.0%

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

      3. associate-/r/ 68.0%

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

      4. *-commutative 68.0%

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

      5. count-2 68.0%

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

   10. Simplified 68.0%

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

   if -9.6000000000000004e-26 < t < 8.00000000000000053e-68

   1. Initial program 40.1%

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

      1. associate-*l* 40.1%

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

      2. +-commutative 40.1%

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

   3. Simplified 40.1%

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

   4. Taylor expanded in t around 0 78.4%

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

      1. times-frac 77.5%

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

      2. unpow2 77.5%

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

      3. *-commutative 77.5%

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

      4. unpow2 77.5%

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

      5. times-frac 83.8%

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

   6. Simplified 83.8%

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

   7. Taylor expanded in k around inf 78.4%

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

      1. unpow2 78.4%

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

      2. associate-*r* 78.3%

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

      3. times-frac 83.8%

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

      4. unpow2 83.8%

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

      5. *-commutative 83.8%

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

      6. associate-*l/ 84.7%

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

      7. *-commutative 84.7%

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

   9. Simplified 84.7%

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

       1. expm1-log1p-u 60.1%

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

       2. expm1-udef 50.6%

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

       3. times-frac 53.2%

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

   11. Applied egg-rr 53.2%

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

       1. expm1-def 63.7%

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

       2. expm1-log1p 88.5%

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

       3. associate-*l* 92.6%

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

   13. Simplified 92.6%

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

   14. Taylor expanded in k around 0 71.8%

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

       1. unpow2 71.8%

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

   16. Simplified 71.8%

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

   if 8.00000000000000053e-68 < t

   1. Initial program 77.2%

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

      1. associate-/l/ 77.2%

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

      2. associate-*l/ 77.2%

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

      3. associate-*l/ 74.6%

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

      4. associate-/r/ 73.3%

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

      5. *-commutative 73.3%

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

      6. associate-/l/ 73.3%

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

      7. associate-*r* 73.3%

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

      8. *-commutative 73.3%

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

      9. associate-*r* 73.3%

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

      10. *-commutative 73.3%

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

   3. Simplified 73.3%

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

   4. Taylor expanded in t around inf 61.0%

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

      1. *-commutative 61.0%

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

      2. times-frac 62.2%

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

      3. unpow2 62.2%

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

   6. Simplified 62.2%

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

   7. Taylor expanded in k around 0 66.3%

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

      1. unpow2 66.3%

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

      2. associate-*r/ 69.7%

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

      3. unpow2 69.7%

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

      4. associate-/r* 74.3%

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

   9. Simplified 74.3%

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

       1. associate-*r/ 74.9%

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

   11. Applied egg-rr 74.9%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9.6 \cdot 10^{-26}:\\ \;\;\;\;\frac{2}{\frac{{t}^{3}}{\ell} \cdot \left(\frac{k}{\ell} \cdot \left(k + k\right)\right)}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-68}:\\ \;\;\;\;\frac{2}{\frac{k}{\cos k} \cdot \left(\frac{k}{\ell} \cdot \left(\left(k \cdot k\right) \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k}{{t}^{3}} \cdot \frac{\ell \cdot \frac{\ell}{k}}{k}\\ \end{array} \]

Alternative 6: 67.8% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -9.8 \cdot 10^{-26} \lor \neg \left(t \leq 1.68 \cdot 10^{-18}\right):\\ \;\;\;\;\frac{2}{\frac{{t}^{3}}{\ell} \cdot \left(\frac{k}{\ell} \cdot \left(k + k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\cos k} \cdot \left(\frac{k}{\ell} \cdot \left(\left(k \cdot k\right) \cdot \frac{t}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -9.8e-26) (not (<= t 1.68e-18)))
   (/ 2.0 (* (/ (pow t 3.0) l) (* (/ k l) (+ k k))))
   (/ 2.0 (* (/ k (cos k)) (* (/ k l) (* (* k k) (/ t l)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -9.8e-26) || !(t <= 1.68e-18)) {
		tmp = 2.0 / ((pow(t, 3.0) / l) * ((k / l) * (k + k)));
	} else {
		tmp = 2.0 / ((k / cos(k)) * ((k / l) * ((k * k) * (t / l))));
	}
	return tmp;
}

Fortran

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 ((t <= (-9.8d-26)) .or. (.not. (t <= 1.68d-18))) then
        tmp = 2.0d0 / (((t ** 3.0d0) / l) * ((k / l) * (k + k)))
    else
        tmp = 2.0d0 / ((k / cos(k)) * ((k / l) * ((k * k) * (t / l))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -9.8e-26) || !(t <= 1.68e-18)) {
		tmp = 2.0 / ((Math.pow(t, 3.0) / l) * ((k / l) * (k + k)));
	} else {
		tmp = 2.0 / ((k / Math.cos(k)) * ((k / l) * ((k * k) * (t / l))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -9.8e-26) or not (t <= 1.68e-18):
		tmp = 2.0 / ((math.pow(t, 3.0) / l) * ((k / l) * (k + k)))
	else:
		tmp = 2.0 / ((k / math.cos(k)) * ((k / l) * ((k * k) * (t / l))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -9.8e-26) || !(t <= 1.68e-18))
		tmp = Float64(2.0 / Float64(Float64((t ^ 3.0) / l) * Float64(Float64(k / l) * Float64(k + k))));
	else
		tmp = Float64(2.0 / Float64(Float64(k / cos(k)) * Float64(Float64(k / l) * Float64(Float64(k * k) * Float64(t / l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -9.8e-26) || ~((t <= 1.68e-18)))
		tmp = 2.0 / (((t ^ 3.0) / l) * ((k / l) * (k + k)));
	else
		tmp = 2.0 / ((k / cos(k)) * ((k / l) * ((k * k) * (t / l))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -9.8e-26], N[Not[LessEqual[t, 1.68e-18]], $MachinePrecision]], N[(2.0 / N[(N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(k + k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -9.7999999999999998e-26 or 1.6799999999999999e-18 < t

   1. Initial program 71.0%

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

      1. associate-*l* 71.0%

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

      2. +-commutative 71.0%

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

   3. Simplified 71.0%

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

   4. Taylor expanded in k around 0 56.2%

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

      1. associate-*r/ 56.2%

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

      2. associate-*r* 56.2%

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

      3. unpow2 56.2%

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

      4. times-frac 61.1%

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

      5. *-commutative 61.1%

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

      6. unpow2 61.1%

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

      7. associate-*l* 61.1%

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

   6. Simplified 61.1%

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

      1. expm1-log1p-u 34.6%

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

      2. expm1-udef 29.5%

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

      3. associate-/l* 31.8%

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

   8. Applied egg-rr 31.8%

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

      1. expm1-def 38.1%

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

      2. expm1-log1p 69.7%

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

      3. associate-/r/ 69.7%

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

      4. *-commutative 69.7%

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

      5. count-2 69.7%

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

   10. Simplified 69.7%

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

   if -9.7999999999999998e-26 < t < 1.6799999999999999e-18

   1. Initial program 43.4%

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

      1. associate-*l* 43.5%

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

      2. +-commutative 43.5%

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

   3. Simplified 43.5%

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

   4. Taylor expanded in t around 0 79.9%

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

      1. times-frac 79.1%

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

      2. unpow2 79.1%

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

      3. *-commutative 79.1%

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

      4. unpow2 79.1%

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

      5. times-frac 84.9%

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

   6. Simplified 84.9%

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

   7. Taylor expanded in k around inf 79.9%

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

      1. unpow2 79.9%

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

      2. associate-*r* 79.8%

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

      3. times-frac 84.9%

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

      4. unpow2 84.9%

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

      5. *-commutative 84.9%

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

      6. associate-*l/ 85.8%

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

      7. *-commutative 85.8%

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

   9. Simplified 85.8%

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

       1. expm1-log1p-u 62.0%

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

       2. expm1-udef 52.4%

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

       3. times-frac 54.8%

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

   11. Applied egg-rr 54.8%

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

       1. expm1-def 65.4%

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

       2. expm1-log1p 89.3%

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

       3. associate-*l* 93.1%

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

   13. Simplified 93.1%

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

   14. Taylor expanded in k around 0 73.0%

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

       1. unpow2 73.0%

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

   16. Simplified 73.0%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9.8 \cdot 10^{-26} \lor \neg \left(t \leq 1.68 \cdot 10^{-18}\right):\\ \;\;\;\;\frac{2}{\frac{{t}^{3}}{\ell} \cdot \left(\frac{k}{\ell} \cdot \left(k + k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\cos k} \cdot \left(\frac{k}{\ell} \cdot \left(\left(k \cdot k\right) \cdot \frac{t}{\ell}\right)\right)}\\ \end{array} \]

Alternative 7: 64.0% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -6.6 \cdot 10^{-25}:\\ \;\;\;\;\frac{2}{\frac{{t}^{3}}{\ell} \cdot \left(\frac{k}{\ell} \cdot \left(k + k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{k \cdot k}{\ell \cdot \frac{\ell}{t}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -6.6e-25)
   (/ 2.0 (* (/ (pow t 3.0) l) (* (/ k l) (+ k k))))
   (/ 2.0 (* (/ (* k k) (cos k)) (/ (* k k) (* l (/ l t)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -6.6e-25) {
		tmp = 2.0 / ((pow(t, 3.0) / l) * ((k / l) * (k + k)));
	} else {
		tmp = 2.0 / (((k * k) / cos(k)) * ((k * k) / (l * (l / t))));
	}
	return tmp;
}

Fortran

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 (t <= (-6.6d-25)) then
        tmp = 2.0d0 / (((t ** 3.0d0) / l) * ((k / l) * (k + k)))
    else
        tmp = 2.0d0 / (((k * k) / cos(k)) * ((k * k) / (l * (l / t))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -6.6e-25) {
		tmp = 2.0 / ((Math.pow(t, 3.0) / l) * ((k / l) * (k + k)));
	} else {
		tmp = 2.0 / (((k * k) / Math.cos(k)) * ((k * k) / (l * (l / t))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -6.6e-25:
		tmp = 2.0 / ((math.pow(t, 3.0) / l) * ((k / l) * (k + k)))
	else:
		tmp = 2.0 / (((k * k) / math.cos(k)) * ((k * k) / (l * (l / t))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -6.6e-25)
		tmp = Float64(2.0 / Float64(Float64((t ^ 3.0) / l) * Float64(Float64(k / l) * Float64(k + k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) / cos(k)) * Float64(Float64(k * k) / Float64(l * Float64(l / t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -6.6e-25)
		tmp = 2.0 / (((t ^ 3.0) / l) * ((k / l) * (k + k)));
	else
		tmp = 2.0 / (((k * k) / cos(k)) * ((k * k) / (l * (l / t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -6.6e-25], N[(2.0 / N[(N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(k + k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -6.5999999999999997e-25

   1. Initial program 67.1%

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

      1. associate-*l* 67.1%

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

      2. +-commutative 67.1%

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

   3. Simplified 67.1%

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

   4. Taylor expanded in k around 0 51.5%

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

      1. associate-*r/ 51.5%

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

      2. associate-*r* 51.5%

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

      3. unpow2 51.5%

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

      4. times-frac 56.3%

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

      5. *-commutative 56.3%

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

      6. unpow2 56.3%

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

      7. associate-*l* 56.3%

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

   6. Simplified 56.3%

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

      1. expm1-log1p-u 8.1%

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

      2. expm1-udef 0.0%

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

      3. associate-/l* 0.3%

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

   8. Applied egg-rr 0.3%

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

      1. expm1-def 10.6%

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

      2. expm1-log1p 68.0%

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

      3. associate-/r/ 68.0%

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

      4. *-commutative 68.0%

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

      5. count-2 68.0%

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

   10. Simplified 68.0%

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

   if -6.5999999999999997e-25 < t

   1. Initial program 54.8%

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

      1. associate-*l* 54.8%

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

      2. +-commutative 54.8%

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

   3. Simplified 54.8%

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

   4. Taylor expanded in t around 0 72.1%

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

      1. times-frac 72.0%

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

      2. unpow2 72.0%

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

      3. *-commutative 72.0%

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

      4. unpow2 72.0%

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

      5. times-frac 77.3%

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

   6. Simplified 77.3%

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

   7. Taylor expanded in k around 0 68.6%

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

      1. unpow2 68.6%

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

      2. associate-/l* 68.6%

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

   9. Simplified 68.6%

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

   10. Taylor expanded in t around 0 64.4%

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

       1. associate-/l* 66.7%

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

       2. unpow2 66.7%

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

       3. unpow2 66.7%

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

       4. associate-*r/ 70.5%

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

   12. Simplified 70.5%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.6 \cdot 10^{-25}:\\ \;\;\;\;\frac{2}{\frac{{t}^{3}}{\ell} \cdot \left(\frac{k}{\ell} \cdot \left(k + k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{k \cdot k}{\ell \cdot \frac{\ell}{t}}}\\ \end{array} \]

Alternative 8: 64.8% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5.8 \cdot 10^{-132} \lor \neg \left(t \leq 1.2 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{2}{\frac{{t}^{3}}{\ell} \cdot \left(\frac{k}{\ell} \cdot \left(k + k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \frac{{k}^{4}}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -5.8e-132) (not (<= t 1.2e-19)))
   (/ 2.0 (* (/ (pow t 3.0) l) (* (/ k l) (+ k k))))
   (/ 2.0 (* (/ t l) (/ (pow k 4.0) l)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -5.8e-132) || !(t <= 1.2e-19)) {
		tmp = 2.0 / ((pow(t, 3.0) / l) * ((k / l) * (k + k)));
	} else {
		tmp = 2.0 / ((t / l) * (pow(k, 4.0) / l));
	}
	return tmp;
}

Fortran

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 ((t <= (-5.8d-132)) .or. (.not. (t <= 1.2d-19))) then
        tmp = 2.0d0 / (((t ** 3.0d0) / l) * ((k / l) * (k + k)))
    else
        tmp = 2.0d0 / ((t / l) * ((k ** 4.0d0) / l))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -5.8e-132) || !(t <= 1.2e-19)) {
		tmp = 2.0 / ((Math.pow(t, 3.0) / l) * ((k / l) * (k + k)));
	} else {
		tmp = 2.0 / ((t / l) * (Math.pow(k, 4.0) / l));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -5.8e-132) or not (t <= 1.2e-19):
		tmp = 2.0 / ((math.pow(t, 3.0) / l) * ((k / l) * (k + k)))
	else:
		tmp = 2.0 / ((t / l) * (math.pow(k, 4.0) / l))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -5.8e-132) || !(t <= 1.2e-19))
		tmp = Float64(2.0 / Float64(Float64((t ^ 3.0) / l) * Float64(Float64(k / l) * Float64(k + k))));
	else
		tmp = Float64(2.0 / Float64(Float64(t / l) * Float64((k ^ 4.0) / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -5.8e-132) || ~((t <= 1.2e-19)))
		tmp = 2.0 / (((t ^ 3.0) / l) * ((k / l) * (k + k)));
	else
		tmp = 2.0 / ((t / l) * ((k ^ 4.0) / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -5.8e-132], N[Not[LessEqual[t, 1.2e-19]], $MachinePrecision]], N[(2.0 / N[(N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(k + k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t / l), $MachinePrecision] * N[(N[Power[k, 4.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -5.79999999999999967e-132 or 1.20000000000000011e-19 < t

   1. Initial program 70.2%

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

      1. associate-*l* 70.2%

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

      2. +-commutative 70.2%

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

   3. Simplified 70.2%

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

   4. Taylor expanded in k around 0 55.7%

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

      1. associate-*r/ 55.7%

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

      2. associate-*r* 55.7%

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

      3. unpow2 55.7%

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

      4. times-frac 60.7%

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

      5. *-commutative 60.7%

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

      6. unpow2 60.7%

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

      7. associate-*l* 60.7%

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

   6. Simplified 60.7%

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

      1. expm1-log1p-u 33.6%

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

      2. expm1-udef 27.1%

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

      3. associate-/l* 29.2%

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

   8. Applied egg-rr 29.2%

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

      1. expm1-def 36.8%

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

      2. expm1-log1p 68.4%

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

      3. associate-/r/ 68.4%

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

      4. *-commutative 68.4%

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

      5. count-2 68.4%

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

   10. Simplified 68.4%

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

   if -5.79999999999999967e-132 < t < 1.20000000000000011e-19

   1. Initial program 40.4%

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

      1. associate-*l* 40.4%

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

      2. +-commutative 40.4%

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

   3. Simplified 40.4%

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

   4. Taylor expanded in t around 0 80.6%

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

      1. times-frac 79.7%

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

      2. unpow2 79.7%

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

      3. *-commutative 79.7%

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

      4. unpow2 79.7%

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

      5. times-frac 85.5%

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

   6. Simplified 85.5%

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

   7. Taylor expanded in k around 0 63.4%

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

      1. unpow2 63.4%

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

      2. times-frac 68.9%

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

   9. Simplified 68.9%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.8 \cdot 10^{-132} \lor \neg \left(t \leq 1.2 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{2}{\frac{{t}^{3}}{\ell} \cdot \left(\frac{k}{\ell} \cdot \left(k + k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \frac{{k}^{4}}{\ell}}\\ \end{array} \]

Alternative 9: 62.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -6.4 \cdot 10^{-34} \lor \neg \left(t \leq 1.2 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -6.4e-34) (not (<= t 1.2e-19)))
   (* (/ l (pow t 3.0)) (/ l (* k k)))
   (* 2.0 (* (/ l t) (/ l (pow k 4.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -6.4e-34) || !(t <= 1.2e-19)) {
		tmp = (l / pow(t, 3.0)) * (l / (k * k));
	} else {
		tmp = 2.0 * ((l / t) * (l / pow(k, 4.0)));
	}
	return tmp;
}

Fortran

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 ((t <= (-6.4d-34)) .or. (.not. (t <= 1.2d-19))) then
        tmp = (l / (t ** 3.0d0)) * (l / (k * k))
    else
        tmp = 2.0d0 * ((l / t) * (l / (k ** 4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -6.4e-34) || !(t <= 1.2e-19)) {
		tmp = (l / Math.pow(t, 3.0)) * (l / (k * k));
	} else {
		tmp = 2.0 * ((l / t) * (l / Math.pow(k, 4.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -6.4e-34) or not (t <= 1.2e-19):
		tmp = (l / math.pow(t, 3.0)) * (l / (k * k))
	else:
		tmp = 2.0 * ((l / t) * (l / math.pow(k, 4.0)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -6.4e-34) || !(t <= 1.2e-19))
		tmp = Float64(Float64(l / (t ^ 3.0)) * Float64(l / Float64(k * k)));
	else
		tmp = Float64(2.0 * Float64(Float64(l / t) * Float64(l / (k ^ 4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -6.4e-34) || ~((t <= 1.2e-19)))
		tmp = (l / (t ^ 3.0)) * (l / (k * k));
	else
		tmp = 2.0 * ((l / t) * (l / (k ^ 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -6.4e-34], N[Not[LessEqual[t, 1.2e-19]], $MachinePrecision]], N[(N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l / t), $MachinePrecision] * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -6.40000000000000005e-34 or 1.20000000000000011e-19 < t

   1. Initial program 71.4%

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

      1. associate-/l/ 71.4%

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

      2. associate-*l/ 72.1%

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

      3. associate-*l/ 70.8%

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

      4. associate-/r/ 70.1%

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

      5. *-commutative 70.1%

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

      6. associate-/l/ 70.1%

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

      7. associate-*r* 70.1%

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

      8. *-commutative 70.1%

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

      9. associate-*r* 70.1%

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

      10. *-commutative 70.1%

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

   3. Simplified 70.1%

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

   4. Taylor expanded in k around 0 56.1%

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

      1. unpow2 56.1%

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

      2. *-commutative 56.1%

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

      3. times-frac 61.0%

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

      4. unpow2 61.0%

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

   6. Simplified 61.0%

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

   if -6.40000000000000005e-34 < t < 1.20000000000000011e-19

   1. Initial program 42.4%

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

      1. associate-*l* 42.5%

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

      2. +-commutative 42.5%

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

   3. Simplified 42.5%

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

   4. Taylor expanded in t around 0 79.5%

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

      1. times-frac 78.7%

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

      2. unpow2 78.7%

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

      3. *-commutative 78.7%

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

      4. unpow2 78.7%

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

      5. times-frac 84.7%

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

   6. Simplified 84.7%

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

   7. Taylor expanded in k around inf 79.5%

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

      1. unpow2 79.5%

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

      2. associate-*r* 79.5%

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

      3. times-frac 84.7%

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

      4. unpow2 84.7%

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

      5. *-commutative 84.7%

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

      6. associate-*l/ 85.5%

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

      7. *-commutative 85.5%

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

   9. Simplified 85.5%

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

   10. Taylor expanded in k around 0 61.8%

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

       1. unpow2 61.8%

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

       2. *-commutative 61.8%

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

       3. times-frac 67.6%

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

       4. *-commutative 67.6%

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

   12. Simplified 67.6%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.4 \cdot 10^{-34} \lor \neg \left(t \leq 1.2 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \end{array} \]

Alternative 10: 64.9% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -6.4 \cdot 10^{-34} \lor \neg \left(t \leq 1.2 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{\frac{\ell}{k}}{k} \cdot \frac{\ell}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -6.4e-34) (not (<= t 1.2e-19)))
   (* (/ (/ l k) k) (/ l (pow t 3.0)))
   (* 2.0 (* (/ l t) (/ l (pow k 4.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -6.4e-34) || !(t <= 1.2e-19)) {
		tmp = ((l / k) / k) * (l / pow(t, 3.0));
	} else {
		tmp = 2.0 * ((l / t) * (l / pow(k, 4.0)));
	}
	return tmp;
}

Fortran

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 ((t <= (-6.4d-34)) .or. (.not. (t <= 1.2d-19))) then
        tmp = ((l / k) / k) * (l / (t ** 3.0d0))
    else
        tmp = 2.0d0 * ((l / t) * (l / (k ** 4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -6.4e-34) || !(t <= 1.2e-19)) {
		tmp = ((l / k) / k) * (l / Math.pow(t, 3.0));
	} else {
		tmp = 2.0 * ((l / t) * (l / Math.pow(k, 4.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -6.4e-34) or not (t <= 1.2e-19):
		tmp = ((l / k) / k) * (l / math.pow(t, 3.0))
	else:
		tmp = 2.0 * ((l / t) * (l / math.pow(k, 4.0)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -6.4e-34) || !(t <= 1.2e-19))
		tmp = Float64(Float64(Float64(l / k) / k) * Float64(l / (t ^ 3.0)));
	else
		tmp = Float64(2.0 * Float64(Float64(l / t) * Float64(l / (k ^ 4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -6.4e-34) || ~((t <= 1.2e-19)))
		tmp = ((l / k) / k) * (l / (t ^ 3.0));
	else
		tmp = 2.0 * ((l / t) * (l / (k ^ 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -6.4e-34], N[Not[LessEqual[t, 1.2e-19]], $MachinePrecision]], N[(N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision] * N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l / t), $MachinePrecision] * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -6.40000000000000005e-34 or 1.20000000000000011e-19 < t

   1. Initial program 71.4%

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

      1. associate-/l/ 71.4%

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

      2. associate-*l/ 72.1%

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

      3. associate-*l/ 70.8%

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

      4. associate-/r/ 70.1%

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

      5. *-commutative 70.1%

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

      6. associate-/l/ 70.1%

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

      7. associate-*r* 70.1%

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

      8. *-commutative 70.1%

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

      9. associate-*r* 70.1%

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

      10. *-commutative 70.1%

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

   3. Simplified 70.1%

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

   4. Taylor expanded in k around 0 56.1%

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

      1. unpow2 56.1%

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

      2. *-commutative 56.1%

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

      3. times-frac 61.0%

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

      4. unpow2 61.0%

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

   6. Simplified 61.0%

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

   7. Taylor expanded in l around 0 56.1%

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

      1. unpow2 56.1%

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

      2. *-commutative 56.1%

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

      3. times-frac 61.0%

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

      4. unpow2 61.0%

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

      5. associate-/r* 68.7%

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

   9. Simplified 68.7%

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

   if -6.40000000000000005e-34 < t < 1.20000000000000011e-19

   1. Initial program 42.4%

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

      1. associate-*l* 42.5%

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

      2. +-commutative 42.5%

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

   3. Simplified 42.5%

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

   4. Taylor expanded in t around 0 79.5%

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

      1. times-frac 78.7%

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

      2. unpow2 78.7%

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

      3. *-commutative 78.7%

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

      4. unpow2 78.7%

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

      5. times-frac 84.7%

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

   6. Simplified 84.7%

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

   7. Taylor expanded in k around inf 79.5%

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

      1. unpow2 79.5%

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

      2. associate-*r* 79.5%

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

      3. times-frac 84.7%

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

      4. unpow2 84.7%

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

      5. *-commutative 84.7%

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

      6. associate-*l/ 85.5%

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

      7. *-commutative 85.5%

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

   9. Simplified 85.5%

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

   10. Taylor expanded in k around 0 61.8%

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

       1. unpow2 61.8%

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

       2. *-commutative 61.8%

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

       3. times-frac 67.6%

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

       4. *-commutative 67.6%

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

   12. Simplified 67.6%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.4 \cdot 10^{-34} \lor \neg \left(t \leq 1.2 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{\frac{\ell}{k}}{k} \cdot \frac{\ell}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \end{array} \]

Alternative 11: 64.8% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -9.6 \cdot 10^{-26} \lor \neg \left(t \leq 1.2 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{\frac{\ell}{k}}{k} \cdot \frac{\ell}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \frac{{k}^{4}}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -9.6e-26) (not (<= t 1.2e-19)))
   (* (/ (/ l k) k) (/ l (pow t 3.0)))
   (/ 2.0 (* (/ t l) (/ (pow k 4.0) l)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -9.6e-26) || !(t <= 1.2e-19)) {
		tmp = ((l / k) / k) * (l / pow(t, 3.0));
	} else {
		tmp = 2.0 / ((t / l) * (pow(k, 4.0) / l));
	}
	return tmp;
}

Fortran

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 ((t <= (-9.6d-26)) .or. (.not. (t <= 1.2d-19))) then
        tmp = ((l / k) / k) * (l / (t ** 3.0d0))
    else
        tmp = 2.0d0 / ((t / l) * ((k ** 4.0d0) / l))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -9.6e-26) || !(t <= 1.2e-19)) {
		tmp = ((l / k) / k) * (l / Math.pow(t, 3.0));
	} else {
		tmp = 2.0 / ((t / l) * (Math.pow(k, 4.0) / l));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -9.6e-26) or not (t <= 1.2e-19):
		tmp = ((l / k) / k) * (l / math.pow(t, 3.0))
	else:
		tmp = 2.0 / ((t / l) * (math.pow(k, 4.0) / l))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -9.6e-26) || !(t <= 1.2e-19))
		tmp = Float64(Float64(Float64(l / k) / k) * Float64(l / (t ^ 3.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(t / l) * Float64((k ^ 4.0) / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -9.6e-26) || ~((t <= 1.2e-19)))
		tmp = ((l / k) / k) * (l / (t ^ 3.0));
	else
		tmp = 2.0 / ((t / l) * ((k ^ 4.0) / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -9.6e-26], N[Not[LessEqual[t, 1.2e-19]], $MachinePrecision]], N[(N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision] * N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t / l), $MachinePrecision] * N[(N[Power[k, 4.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -9.6000000000000004e-26 or 1.20000000000000011e-19 < t

   1. Initial program 71.2%

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

      1. associate-/l/ 71.2%

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

      2. associate-*l/ 71.9%

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

      3. associate-*l/ 70.6%

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

      4. associate-/r/ 69.9%

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

      5. *-commutative 69.9%

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

      6. associate-/l/ 69.9%

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

      7. associate-*r* 69.9%

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

      8. *-commutative 69.9%

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

      9. associate-*r* 69.9%

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

      10. *-commutative 69.9%

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

   3. Simplified 69.9%

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

   4. Taylor expanded in k around 0 55.8%

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

      1. unpow2 55.8%

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

      2. *-commutative 55.8%

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

      3. times-frac 60.7%

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

      4. unpow2 60.7%

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

   6. Simplified 60.7%

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

   7. Taylor expanded in l around 0 55.8%

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

      1. unpow2 55.8%

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

      2. *-commutative 55.8%

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

      3. times-frac 60.7%

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

      4. unpow2 60.7%

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

      5. associate-/r* 68.5%

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

   9. Simplified 68.5%

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

   if -9.6000000000000004e-26 < t < 1.20000000000000011e-19

   1. Initial program 42.9%

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

      1. associate-*l* 43.0%

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

      2. +-commutative 43.0%

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

   3. Simplified 43.0%

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

   4. Taylor expanded in t around 0 79.7%

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

      1. times-frac 78.9%

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

      2. unpow2 78.9%

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

      3. *-commutative 78.9%

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

      4. unpow2 78.9%

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

      5. times-frac 84.8%

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

   6. Simplified 84.8%

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

   7. Taylor expanded in k around 0 62.2%

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

      1. unpow2 62.2%

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

      2. times-frac 67.9%

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

   9. Simplified 67.9%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9.6 \cdot 10^{-26} \lor \neg \left(t \leq 1.2 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{\frac{\ell}{k}}{k} \cdot \frac{\ell}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \frac{{k}^{4}}{\ell}}\\ \end{array} \]

Alternative 12: 56.4% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right) \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* 2.0 (* (/ l t) (/ l (pow k 4.0)))))

C

double code(double t, double l, double k) {
	return 2.0 * ((l / t) * (l / pow(k, 4.0)));
}

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 * ((l / t) * (l / (k ** 4.0d0)))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 * ((l / t) * (l / Math.pow(k, 4.0)));
}

Python

def code(t, l, k):
	return 2.0 * ((l / t) * (l / math.pow(k, 4.0)))

Julia

function code(t, l, k)
	return Float64(2.0 * Float64(Float64(l / t) * Float64(l / (k ^ 4.0))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 * ((l / t) * (l / (k ^ 4.0)));
end

Wolfram

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 58.5%

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

   1. associate-*l* 58.5%

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

   2. +-commutative 58.5%

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

3. Simplified 58.5%

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

4. Taylor expanded in t around 0 65.6%

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

   1. times-frac 66.0%

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

   2. unpow2 66.0%

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

   3. *-commutative 66.0%

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

   4. unpow2 66.0%

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

   5. times-frac 70.6%

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

6. Simplified 70.6%

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

7. Taylor expanded in k around inf 65.6%

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

   1. unpow2 65.6%

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

   2. associate-*r* 65.6%

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

   3. times-frac 68.9%

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

   4. unpow2 68.9%

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

   5. *-commutative 68.9%

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

   6. associate-*l/ 71.0%

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

   7. *-commutative 71.0%

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

9. Simplified 71.0%

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

10. Taylor expanded in k around 0 54.8%

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

    1. unpow2 54.8%

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

    2. *-commutative 54.8%

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

    3. times-frac 58.7%

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

    4. *-commutative 58.7%

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

12. Simplified 58.7%

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

13. Final simplification 58.7%

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

Reproduce

herbie shell --seed 2023195 
(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))))