Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.9% → 92.1%

Time: 31.1s

Alternatives: 23

Speedup: 2.0×

• 40.6% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. t = 4.1278944987786847e+214
  2. l = 1.8969096063425044e-168
  3. k = 1.1876625591001954e+192

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: 35.9% 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: 92.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

code[t_, l_, k_] := N[(2.0 / N[Power[N[(N[(N[Power[N[(N[Power[k, 1/3], $MachinePrecision] * N[Power[N[(1.0 / t), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.7%

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

3. Simplified 43.4%

   \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. unpow3 43.4%

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

   2. times-frac 57.3%

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

   3. pow2 57.3%

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

6. Applied egg-rr 57.3%

   \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)} \]
7. Step-by-step derivation

   1. add-cube-cbrt 57.2%

      \[\leadsto \frac{2}{\color{blue}{\left(\sqrt[3]{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)} \cdot \sqrt[3]{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}\right) \cdot \sqrt[3]{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}}} \]

   2. pow3 57.3%

      \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt[3]{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}\right)}^{3}}} \]

8. Applied egg-rr 76.8%

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

   1. +-rgt-identity 76.8%

      \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k \cdot \left(\tan k \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 0\right)}\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

   2. unpow2 76.8%

      \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 0\right)\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

   3. frac-2neg 76.8%

      \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(\color{blue}{\frac{-k}{-t}} \cdot \frac{k}{t} + 0\right)\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

   4. times-frac 51.7%

      \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(\color{blue}{\frac{\left(-k\right) \cdot k}{\left(-t\right) \cdot t}} + 0\right)\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

   5. associate-*l* 51.7%

      \[\leadsto \frac{2}{{\left(\sqrt[3]{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{\left(-k\right) \cdot k}{\left(-t\right) \cdot t} + 0\right)}} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

   6. *-commutative 51.7%

      \[\leadsto \frac{2}{{\left(\sqrt[3]{\color{blue}{\left(\frac{\left(-k\right) \cdot k}{\left(-t\right) \cdot t} + 0\right) \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

   7. times-frac 76.8%

      \[\leadsto \frac{2}{{\left(\sqrt[3]{\left(\color{blue}{\frac{-k}{-t} \cdot \frac{k}{t}} + 0\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

   8. frac-2neg 76.8%

      \[\leadsto \frac{2}{{\left(\sqrt[3]{\left(\color{blue}{\frac{k}{t}} \cdot \frac{k}{t} + 0\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

   9. unpow2 76.8%

      \[\leadsto \frac{2}{{\left(\sqrt[3]{\left(\color{blue}{{\left(\frac{k}{t}\right)}^{2}} + 0\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

   10. +-rgt-identity 76.8%

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

   11. cbrt-prod 77.4%

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

10. Applied egg-rr 88.6%

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

    1. pow1/3 50.6%

       \[\leadsto \frac{2}{{\left(\left({\color{blue}{\left({\left(\frac{k}{t}\right)}^{0.3333333333333333}\right)}}^{2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right) \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

    2. div-inv 50.6%

       \[\leadsto \frac{2}{{\left(\left({\left({\color{blue}{\left(k \cdot \frac{1}{t}\right)}}^{0.3333333333333333}\right)}^{2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right) \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

    3. unpow-prod-down 22.7%

       \[\leadsto \frac{2}{{\left(\left({\color{blue}{\left({k}^{0.3333333333333333} \cdot {\left(\frac{1}{t}\right)}^{0.3333333333333333}\right)}}^{2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right) \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

    4. pow1/3 43.5%

       \[\leadsto \frac{2}{{\left(\left({\left(\color{blue}{\sqrt[3]{k}} \cdot {\left(\frac{1}{t}\right)}^{0.3333333333333333}\right)}^{2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right) \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

12. Applied egg-rr 43.5%

    \[\leadsto \frac{2}{{\left(\left({\color{blue}{\left(\sqrt[3]{k} \cdot {\left(\frac{1}{t}\right)}^{0.3333333333333333}\right)}}^{2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right) \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]
13. Step-by-step derivation

    1. unpow1/3 93.2%

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

14. Simplified 93.2%

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

15. Final simplification 93.2%

    \[\leadsto \frac{2}{{\left(\left({\left(\sqrt[3]{k} \cdot \sqrt[3]{\frac{1}{t}}\right)}^{2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right) \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]
16. Add Preprocessing

Alternative 2: 68.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\\ \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-316}:\\ \;\;\;\;\frac{2}{{\left(t\_1 \cdot \left({\left(\sqrt[3]{\frac{k}{t}}\right)}^{2} \cdot \left(\sqrt[3]{\tan k} \cdot \sqrt[3]{\sin k}\right)\right)\right)}^{3}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 5 \cdot 10^{+242}:\\ \;\;\;\;\frac{2}{\frac{{\left(k \cdot \sqrt{t}\right)}^{2} \cdot {\sin k}^{2}}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t\_1 \cdot \left(\sqrt[3]{\sin k \cdot \tan k} \cdot {\left(\frac{1}{\sqrt[3]{\frac{t}{k}}}\right)}^{2}\right)\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ t (pow (cbrt l) 2.0))))
   (if (<= (* l l) 2e-316)
     (/
      2.0
      (pow
       (* t_1 (* (pow (cbrt (/ k t)) 2.0) (* (cbrt (tan k)) (cbrt (sin k)))))
       3.0))
     (if (<= (* l l) 5e+242)
       (/
        2.0
        (/
         (* (pow (* k (sqrt t)) 2.0) (pow (sin k) 2.0))
         (* (cos k) (pow l 2.0))))
       (/
        2.0
        (pow
         (*
          t_1
          (* (cbrt (* (sin k) (tan k))) (pow (/ 1.0 (cbrt (/ t k))) 2.0)))
         3.0))))))

C

double code(double t, double l, double k) {
	double t_1 = t / pow(cbrt(l), 2.0);
	double tmp;
	if ((l * l) <= 2e-316) {
		tmp = 2.0 / pow((t_1 * (pow(cbrt((k / t)), 2.0) * (cbrt(tan(k)) * cbrt(sin(k))))), 3.0);
	} else if ((l * l) <= 5e+242) {
		tmp = 2.0 / ((pow((k * sqrt(t)), 2.0) * pow(sin(k), 2.0)) / (cos(k) * pow(l, 2.0)));
	} else {
		tmp = 2.0 / pow((t_1 * (cbrt((sin(k) * tan(k))) * pow((1.0 / cbrt((t / k))), 2.0))), 3.0);
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = t / Math.pow(Math.cbrt(l), 2.0);
	double tmp;
	if ((l * l) <= 2e-316) {
		tmp = 2.0 / Math.pow((t_1 * (Math.pow(Math.cbrt((k / t)), 2.0) * (Math.cbrt(Math.tan(k)) * Math.cbrt(Math.sin(k))))), 3.0);
	} else if ((l * l) <= 5e+242) {
		tmp = 2.0 / ((Math.pow((k * Math.sqrt(t)), 2.0) * Math.pow(Math.sin(k), 2.0)) / (Math.cos(k) * Math.pow(l, 2.0)));
	} else {
		tmp = 2.0 / Math.pow((t_1 * (Math.cbrt((Math.sin(k) * Math.tan(k))) * Math.pow((1.0 / Math.cbrt((t / k))), 2.0))), 3.0);
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(t / (cbrt(l) ^ 2.0))
	tmp = 0.0
	if (Float64(l * l) <= 2e-316)
		tmp = Float64(2.0 / (Float64(t_1 * Float64((cbrt(Float64(k / t)) ^ 2.0) * Float64(cbrt(tan(k)) * cbrt(sin(k))))) ^ 3.0));
	elseif (Float64(l * l) <= 5e+242)
		tmp = Float64(2.0 / Float64(Float64((Float64(k * sqrt(t)) ^ 2.0) * (sin(k) ^ 2.0)) / Float64(cos(k) * (l ^ 2.0))));
	else
		tmp = Float64(2.0 / (Float64(t_1 * Float64(cbrt(Float64(sin(k) * tan(k))) * (Float64(1.0 / cbrt(Float64(t / k))) ^ 2.0))) ^ 3.0));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(l * l), $MachinePrecision], 2e-316], N[(2.0 / N[Power[N[(t$95$1 * N[(N[Power[N[Power[N[(k / t), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[N[Tan[k], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 5e+242], N[(2.0 / N[(N[(N[Power[N[(k * N[Sqrt[t], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(t$95$1 * N[(N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(1.0 / N[Power[N[(t / k), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 2.000000017e-316

   1. Initial program 15.9%

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

   3. Simplified 26.5%

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow3 26.5%

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

      2. times-frac 52.7%

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

      3. pow2 52.7%

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

   6. Applied egg-rr 52.7%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)} \]
   7. Step-by-step derivation

      1. add-cube-cbrt 52.7%

         \[\leadsto \frac{2}{\color{blue}{\left(\sqrt[3]{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)} \cdot \sqrt[3]{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}\right) \cdot \sqrt[3]{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}}} \]

      2. pow3 52.7%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt[3]{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}\right)}^{3}}} \]

   8. Applied egg-rr 77.4%

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

      1. +-rgt-identity 77.4%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k \cdot \left(\tan k \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 0\right)}\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

      2. unpow2 77.4%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 0\right)\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

      3. frac-2neg 77.4%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(\color{blue}{\frac{-k}{-t}} \cdot \frac{k}{t} + 0\right)\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

      4. times-frac 52.7%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(\color{blue}{\frac{\left(-k\right) \cdot k}{\left(-t\right) \cdot t}} + 0\right)\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

      5. associate-*l* 52.7%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{\left(-k\right) \cdot k}{\left(-t\right) \cdot t} + 0\right)}} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

      6. *-commutative 52.7%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\color{blue}{\left(\frac{\left(-k\right) \cdot k}{\left(-t\right) \cdot t} + 0\right) \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

      7. times-frac 77.4%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\left(\color{blue}{\frac{-k}{-t} \cdot \frac{k}{t}} + 0\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

      8. frac-2neg 77.4%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\left(\color{blue}{\frac{k}{t}} \cdot \frac{k}{t} + 0\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

      9. unpow2 77.4%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\left(\color{blue}{{\left(\frac{k}{t}\right)}^{2}} + 0\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

      10. +-rgt-identity 77.4%

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

      11. cbrt-prod 80.2%

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

   10. Applied egg-rr 88.4%

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

       1. *-commutative 88.4%

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

       2. cbrt-prod 91.2%

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

   12. Applied egg-rr 91.2%

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

   if 2.000000017e-316 < (*.f64 l l) < 5.0000000000000004e242

   1. Initial program 44.6%

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

   3. Simplified 53.3%

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow3 53.3%

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

      2. times-frac 59.1%

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

      3. pow2 59.1%

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

   6. Applied egg-rr 59.1%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)} \]
   7. Step-by-step derivation

      1. unpow2 59.1%

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

      2. frac-2neg 59.1%

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

      3. frac-times 35.5%

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

   8. Applied egg-rr 35.5%

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

   9. Taylor expanded in t around 0 92.3%

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

       1. associate-*r* 92.4%

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

       2. *-commutative 92.4%

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

       3. *-commutative 92.4%

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

   11. Simplified 92.4%

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

       1. add-sqr-sqrt 46.6%

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

       2. pow2 46.6%

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

       3. *-commutative 46.6%

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

       4. sqrt-prod 46.5%

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

       5. unpow2 46.5%

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

       6. sqrt-prod 26.6%

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

       7. add-sqr-sqrt 48.0%

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

   13. Applied egg-rr 48.0%

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

   if 5.0000000000000004e242 < (*.f64 l l)

   1. Initial program 39.2%

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

   3. Simplified 39.2%

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow3 39.2%

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

      2. times-frac 57.8%

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

      3. pow2 57.8%

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

   6. Applied egg-rr 57.8%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)} \]
   7. Step-by-step derivation

      1. add-cube-cbrt 57.7%

         \[\leadsto \frac{2}{\color{blue}{\left(\sqrt[3]{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)} \cdot \sqrt[3]{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}\right) \cdot \sqrt[3]{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}}} \]

      2. pow3 57.7%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt[3]{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}\right)}^{3}}} \]

   8. Applied egg-rr 83.0%

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

      1. +-rgt-identity 83.0%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k \cdot \left(\tan k \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 0\right)}\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

      2. unpow2 83.0%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 0\right)\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

      3. frac-2neg 83.0%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(\color{blue}{\frac{-k}{-t}} \cdot \frac{k}{t} + 0\right)\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

      4. times-frac 54.8%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(\color{blue}{\frac{\left(-k\right) \cdot k}{\left(-t\right) \cdot t}} + 0\right)\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

      5. associate-*l* 54.8%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{\left(-k\right) \cdot k}{\left(-t\right) \cdot t} + 0\right)}} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

      6. *-commutative 54.8%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\color{blue}{\left(\frac{\left(-k\right) \cdot k}{\left(-t\right) \cdot t} + 0\right) \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

      7. times-frac 83.0%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\left(\color{blue}{\frac{-k}{-t} \cdot \frac{k}{t}} + 0\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

      8. frac-2neg 83.0%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\left(\color{blue}{\frac{k}{t}} \cdot \frac{k}{t} + 0\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

      9. unpow2 83.0%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\left(\color{blue}{{\left(\frac{k}{t}\right)}^{2}} + 0\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

      10. +-rgt-identity 83.0%

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

      11. cbrt-prod 82.9%

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

   10. Applied egg-rr 90.8%

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

       1. clear-num 90.9%

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

       2. metadata-eval 90.9%

          \[\leadsto \frac{2}{{\left(\left({\left(\sqrt[3]{\frac{\color{blue}{3 \cdot 0.3333333333333333}}{\frac{t}{k}}}\right)}^{2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right) \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

       3. cbrt-div 91.0%

          \[\leadsto \frac{2}{{\left(\left({\color{blue}{\left(\frac{\sqrt[3]{3 \cdot 0.3333333333333333}}{\sqrt[3]{\frac{t}{k}}}\right)}}^{2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right) \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

       4. metadata-eval 91.0%

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

       5. metadata-eval 91.0%

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

   12. Applied egg-rr 91.0%

       \[\leadsto \frac{2}{{\left(\left({\color{blue}{\left(\frac{1}{\sqrt[3]{\frac{t}{k}}}\right)}}^{2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right) \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-316}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left({\left(\sqrt[3]{\frac{k}{t}}\right)}^{2} \cdot \left(\sqrt[3]{\tan k} \cdot \sqrt[3]{\sin k}\right)\right)\right)}^{3}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 5 \cdot 10^{+242}:\\ \;\;\;\;\frac{2}{\frac{{\left(k \cdot \sqrt{t}\right)}^{2} \cdot {\sin k}^{2}}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\sqrt[3]{\sin k \cdot \tan k} \cdot {\left(\frac{1}{\sqrt[3]{\frac{t}{k}}}\right)}^{2}\right)\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

code[t_, l_, k_] := N[(2.0 / N[Power[N[(N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(N[Power[k, 1/3], $MachinePrecision] / N[Power[t, 1/3], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.7%

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

3. Simplified 43.4%

   \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. unpow3 43.4%

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

   2. times-frac 57.3%

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

   3. pow2 57.3%

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

6. Applied egg-rr 57.3%

   \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)} \]
7. Step-by-step derivation

   1. add-cube-cbrt 57.2%

      \[\leadsto \frac{2}{\color{blue}{\left(\sqrt[3]{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)} \cdot \sqrt[3]{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}\right) \cdot \sqrt[3]{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}}} \]

   2. pow3 57.3%

      \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt[3]{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}\right)}^{3}}} \]

8. Applied egg-rr 76.8%

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

   1. +-rgt-identity 76.8%

      \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k \cdot \left(\tan k \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 0\right)}\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

   2. unpow2 76.8%

      \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 0\right)\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

   3. frac-2neg 76.8%

      \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(\color{blue}{\frac{-k}{-t}} \cdot \frac{k}{t} + 0\right)\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

   4. times-frac 51.7%

      \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(\color{blue}{\frac{\left(-k\right) \cdot k}{\left(-t\right) \cdot t}} + 0\right)\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

   5. associate-*l* 51.7%

      \[\leadsto \frac{2}{{\left(\sqrt[3]{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{\left(-k\right) \cdot k}{\left(-t\right) \cdot t} + 0\right)}} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

   6. *-commutative 51.7%

      \[\leadsto \frac{2}{{\left(\sqrt[3]{\color{blue}{\left(\frac{\left(-k\right) \cdot k}{\left(-t\right) \cdot t} + 0\right) \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

   7. times-frac 76.8%

      \[\leadsto \frac{2}{{\left(\sqrt[3]{\left(\color{blue}{\frac{-k}{-t} \cdot \frac{k}{t}} + 0\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

   8. frac-2neg 76.8%

      \[\leadsto \frac{2}{{\left(\sqrt[3]{\left(\color{blue}{\frac{k}{t}} \cdot \frac{k}{t} + 0\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

   9. unpow2 76.8%

      \[\leadsto \frac{2}{{\left(\sqrt[3]{\left(\color{blue}{{\left(\frac{k}{t}\right)}^{2}} + 0\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

   10. +-rgt-identity 76.8%

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

   11. cbrt-prod 77.4%

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

10. Applied egg-rr 88.6%

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

    1. cbrt-div 93.0%

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

12. Applied egg-rr 93.0%

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

13. Final simplification 93.0%

    \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\sqrt[3]{\sin k \cdot \tan k} \cdot {\left(\frac{\sqrt[3]{k}}{\sqrt[3]{t}}\right)}^{2}\right)\right)}^{3}} \]
14. Add Preprocessing

Alternative 4: 68.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-316} \lor \neg \left(\ell \cdot \ell \leq 5 \cdot 10^{+242}\right):\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\sqrt[3]{\sin k \cdot \tan k} \cdot {\left(\frac{1}{\sqrt[3]{\frac{t}{k}}}\right)}^{2}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(k \cdot \sqrt{t}\right)}^{2} \cdot {\sin k}^{2}}{\cos k \cdot {\ell}^{2}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= (* l l) 2e-316) (not (<= (* l l) 5e+242)))
   (/
    2.0
    (pow
     (*
      (/ t (pow (cbrt l) 2.0))
      (* (cbrt (* (sin k) (tan k))) (pow (/ 1.0 (cbrt (/ t k))) 2.0)))
     3.0))
   (/
    2.0
    (/
     (* (pow (* k (sqrt t)) 2.0) (pow (sin k) 2.0))
     (* (cos k) (pow l 2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (((l * l) <= 2e-316) || !((l * l) <= 5e+242)) {
		tmp = 2.0 / pow(((t / pow(cbrt(l), 2.0)) * (cbrt((sin(k) * tan(k))) * pow((1.0 / cbrt((t / k))), 2.0))), 3.0);
	} else {
		tmp = 2.0 / ((pow((k * sqrt(t)), 2.0) * pow(sin(k), 2.0)) / (cos(k) * pow(l, 2.0)));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (((l * l) <= 2e-316) || !((l * l) <= 5e+242)) {
		tmp = 2.0 / Math.pow(((t / Math.pow(Math.cbrt(l), 2.0)) * (Math.cbrt((Math.sin(k) * Math.tan(k))) * Math.pow((1.0 / Math.cbrt((t / k))), 2.0))), 3.0);
	} else {
		tmp = 2.0 / ((Math.pow((k * Math.sqrt(t)), 2.0) * Math.pow(Math.sin(k), 2.0)) / (Math.cos(k) * Math.pow(l, 2.0)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if ((Float64(l * l) <= 2e-316) || !(Float64(l * l) <= 5e+242))
		tmp = Float64(2.0 / (Float64(Float64(t / (cbrt(l) ^ 2.0)) * Float64(cbrt(Float64(sin(k) * tan(k))) * (Float64(1.0 / cbrt(Float64(t / k))) ^ 2.0))) ^ 3.0));
	else
		tmp = Float64(2.0 / Float64(Float64((Float64(k * sqrt(t)) ^ 2.0) * (sin(k) ^ 2.0)) / Float64(cos(k) * (l ^ 2.0))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[N[(l * l), $MachinePrecision], 2e-316], N[Not[LessEqual[N[(l * l), $MachinePrecision], 5e+242]], $MachinePrecision]], N[(2.0 / N[Power[N[(N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(1.0 / N[Power[N[(t / k), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[N[(k * N[Sqrt[t], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 2.000000017e-316 or 5.0000000000000004e242 < (*.f64 l l)

   1. Initial program 28.9%

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

   3. Simplified 33.6%

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow3 33.6%

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

      2. times-frac 55.6%

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

      3. pow2 55.6%

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

   6. Applied egg-rr 55.6%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)} \]
   7. Step-by-step derivation

      1. add-cube-cbrt 55.5%

         \[\leadsto \frac{2}{\color{blue}{\left(\sqrt[3]{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)} \cdot \sqrt[3]{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}\right) \cdot \sqrt[3]{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}}} \]

      2. pow3 55.5%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt[3]{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}\right)}^{3}}} \]

   8. Applied egg-rr 80.5%

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

      1. +-rgt-identity 80.5%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k \cdot \left(\tan k \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 0\right)}\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

      2. unpow2 80.5%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 0\right)\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

      3. frac-2neg 80.5%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(\color{blue}{\frac{-k}{-t}} \cdot \frac{k}{t} + 0\right)\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

      4. times-frac 53.9%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(\color{blue}{\frac{\left(-k\right) \cdot k}{\left(-t\right) \cdot t}} + 0\right)\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

      5. associate-*l* 53.9%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{\left(-k\right) \cdot k}{\left(-t\right) \cdot t} + 0\right)}} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

      6. *-commutative 53.9%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\color{blue}{\left(\frac{\left(-k\right) \cdot k}{\left(-t\right) \cdot t} + 0\right) \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

      7. times-frac 80.5%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\left(\color{blue}{\frac{-k}{-t} \cdot \frac{k}{t}} + 0\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

      8. frac-2neg 80.5%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\left(\color{blue}{\frac{k}{t}} \cdot \frac{k}{t} + 0\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

      9. unpow2 80.5%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\left(\color{blue}{{\left(\frac{k}{t}\right)}^{2}} + 0\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

      10. +-rgt-identity 80.5%

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

      11. cbrt-prod 81.7%

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

   10. Applied egg-rr 89.7%

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

       1. clear-num 89.8%

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

       2. metadata-eval 89.8%

          \[\leadsto \frac{2}{{\left(\left({\left(\sqrt[3]{\frac{\color{blue}{3 \cdot 0.3333333333333333}}{\frac{t}{k}}}\right)}^{2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right) \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

       3. cbrt-div 89.9%

          \[\leadsto \frac{2}{{\left(\left({\color{blue}{\left(\frac{\sqrt[3]{3 \cdot 0.3333333333333333}}{\sqrt[3]{\frac{t}{k}}}\right)}}^{2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right) \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

       4. metadata-eval 89.9%

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

       5. metadata-eval 89.9%

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

   12. Applied egg-rr 89.9%

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

   if 2.000000017e-316 < (*.f64 l l) < 5.0000000000000004e242

   1. Initial program 44.6%

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

   3. Simplified 53.3%

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow3 53.3%

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

      2. times-frac 59.1%

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

      3. pow2 59.1%

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

   6. Applied egg-rr 59.1%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)} \]
   7. Step-by-step derivation

      1. unpow2 59.1%

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

      2. frac-2neg 59.1%

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

      3. frac-times 35.5%

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

   8. Applied egg-rr 35.5%

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

   9. Taylor expanded in t around 0 92.3%

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

       1. associate-*r* 92.4%

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

       2. *-commutative 92.4%

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

       3. *-commutative 92.4%

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

   11. Simplified 92.4%

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

       1. add-sqr-sqrt 46.6%

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

       2. pow2 46.6%

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

       3. *-commutative 46.6%

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

       4. sqrt-prod 46.5%

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

       5. unpow2 46.5%

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

       6. sqrt-prod 26.6%

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

       7. add-sqr-sqrt 48.0%

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

   13. Applied egg-rr 48.0%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-316} \lor \neg \left(\ell \cdot \ell \leq 5 \cdot 10^{+242}\right):\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\sqrt[3]{\sin k \cdot \tan k} \cdot {\left(\frac{1}{\sqrt[3]{\frac{t}{k}}}\right)}^{2}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(k \cdot \sqrt{t}\right)}^{2} \cdot {\sin k}^{2}}{\cos k \cdot {\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

code[t_, l_, k_] := N[(2.0 / N[Power[N[(N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[Power[N[(k / t), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.7%

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

3. Simplified 43.4%

   \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. unpow3 43.4%

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

   2. times-frac 57.3%

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

   3. pow2 57.3%

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

6. Applied egg-rr 57.3%

   \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)} \]
7. Step-by-step derivation

   1. add-cube-cbrt 57.2%

      \[\leadsto \frac{2}{\color{blue}{\left(\sqrt[3]{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)} \cdot \sqrt[3]{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}\right) \cdot \sqrt[3]{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}}} \]

   2. pow3 57.3%

      \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt[3]{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}\right)}^{3}}} \]

8. Applied egg-rr 76.8%

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

   1. +-rgt-identity 76.8%

      \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k \cdot \left(\tan k \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 0\right)}\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

   2. unpow2 76.8%

      \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 0\right)\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

   3. frac-2neg 76.8%

      \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(\color{blue}{\frac{-k}{-t}} \cdot \frac{k}{t} + 0\right)\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

   4. times-frac 51.7%

      \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(\color{blue}{\frac{\left(-k\right) \cdot k}{\left(-t\right) \cdot t}} + 0\right)\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

   5. associate-*l* 51.7%

      \[\leadsto \frac{2}{{\left(\sqrt[3]{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{\left(-k\right) \cdot k}{\left(-t\right) \cdot t} + 0\right)}} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

   6. *-commutative 51.7%

      \[\leadsto \frac{2}{{\left(\sqrt[3]{\color{blue}{\left(\frac{\left(-k\right) \cdot k}{\left(-t\right) \cdot t} + 0\right) \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

   7. times-frac 76.8%

      \[\leadsto \frac{2}{{\left(\sqrt[3]{\left(\color{blue}{\frac{-k}{-t} \cdot \frac{k}{t}} + 0\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

   8. frac-2neg 76.8%

      \[\leadsto \frac{2}{{\left(\sqrt[3]{\left(\color{blue}{\frac{k}{t}} \cdot \frac{k}{t} + 0\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

   9. unpow2 76.8%

      \[\leadsto \frac{2}{{\left(\sqrt[3]{\left(\color{blue}{{\left(\frac{k}{t}\right)}^{2}} + 0\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

   10. +-rgt-identity 76.8%

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

   11. cbrt-prod 77.4%

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

10. Applied egg-rr 88.6%

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

11. Final simplification 88.6%

    \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\sqrt[3]{\sin k \cdot \tan k} \cdot {\left(\sqrt[3]{\frac{k}{t}}\right)}^{2}\right)\right)}^{3}} \]
12. Add Preprocessing

Alternative 6: 63.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4.7 \cdot 10^{-188} \lor \neg \left(\ell \leq 2.8 \cdot 10^{+123}\right):\\ \;\;\;\;\frac{2}{{\left(t \cdot \left(\sqrt[3]{\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)} \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(k \cdot \sqrt{t}\right)}^{2} \cdot {\sin k}^{2}}{\cos k \cdot {\ell}^{2}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= l 4.7e-188) (not (<= l 2.8e+123)))
   (/
    2.0
    (pow
     (*
      t
      (* (cbrt (* (sin k) (* (tan k) (pow (/ k t) 2.0)))) (pow (cbrt l) -2.0)))
     3.0))
   (/
    2.0
    (/
     (* (pow (* k (sqrt t)) 2.0) (pow (sin k) 2.0))
     (* (cos k) (pow l 2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l <= 4.7e-188) || !(l <= 2.8e+123)) {
		tmp = 2.0 / pow((t * (cbrt((sin(k) * (tan(k) * pow((k / t), 2.0)))) * pow(cbrt(l), -2.0))), 3.0);
	} else {
		tmp = 2.0 / ((pow((k * sqrt(t)), 2.0) * pow(sin(k), 2.0)) / (cos(k) * pow(l, 2.0)));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((l <= 4.7e-188) || !(l <= 2.8e+123)) {
		tmp = 2.0 / Math.pow((t * (Math.cbrt((Math.sin(k) * (Math.tan(k) * Math.pow((k / t), 2.0)))) * Math.pow(Math.cbrt(l), -2.0))), 3.0);
	} else {
		tmp = 2.0 / ((Math.pow((k * Math.sqrt(t)), 2.0) * Math.pow(Math.sin(k), 2.0)) / (Math.cos(k) * Math.pow(l, 2.0)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if ((l <= 4.7e-188) || !(l <= 2.8e+123))
		tmp = Float64(2.0 / (Float64(t * Float64(cbrt(Float64(sin(k) * Float64(tan(k) * (Float64(k / t) ^ 2.0)))) * (cbrt(l) ^ -2.0))) ^ 3.0));
	else
		tmp = Float64(2.0 / Float64(Float64((Float64(k * sqrt(t)) ^ 2.0) * (sin(k) ^ 2.0)) / Float64(cos(k) * (l ^ 2.0))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[l, 4.7e-188], N[Not[LessEqual[l, 2.8e+123]], $MachinePrecision]], N[(2.0 / N[Power[N[(t * N[(N[Power[N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[N[(k * N[Sqrt[t], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.69999999999999998e-188 or 2.80000000000000011e123 < l

   1. Initial program 31.9%

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

   3. Simplified 39.3%

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow3 39.3%

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

      2. times-frac 55.6%

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

      3. pow2 55.6%

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

   6. Applied egg-rr 55.6%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)} \]
   7. Step-by-step derivation

      1. add-cube-cbrt 55.5%

         \[\leadsto \frac{2}{\color{blue}{\left(\sqrt[3]{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)} \cdot \sqrt[3]{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}\right) \cdot \sqrt[3]{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}}} \]

      2. pow3 55.5%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt[3]{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}\right)}^{3}}} \]

   8. Applied egg-rr 77.8%

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

      1. +-rgt-identity 77.8%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k \cdot \left(\tan k \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 0\right)}\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

      2. unpow2 77.8%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 0\right)\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

      3. frac-2neg 77.8%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(\color{blue}{\frac{-k}{-t}} \cdot \frac{k}{t} + 0\right)\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

      4. times-frac 52.1%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(\color{blue}{\frac{\left(-k\right) \cdot k}{\left(-t\right) \cdot t}} + 0\right)\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

      5. associate-*l* 52.1%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{\left(-k\right) \cdot k}{\left(-t\right) \cdot t} + 0\right)}} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

      6. *-commutative 52.1%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\color{blue}{\left(\frac{\left(-k\right) \cdot k}{\left(-t\right) \cdot t} + 0\right) \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

      7. times-frac 77.8%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\left(\color{blue}{\frac{-k}{-t} \cdot \frac{k}{t}} + 0\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

      8. frac-2neg 77.8%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\left(\color{blue}{\frac{k}{t}} \cdot \frac{k}{t} + 0\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

      9. unpow2 77.8%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\left(\color{blue}{{\left(\frac{k}{t}\right)}^{2}} + 0\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

      10. +-rgt-identity 77.8%

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

      11. cbrt-prod 78.6%

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

   10. Applied egg-rr 88.0%

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

       1. pow1 88.0%

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

   12. Applied egg-rr 77.8%

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

       1. unpow1 77.8%

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

       2. associate-*r* 78.9%

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

       3. *-commutative 78.9%

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

       4. associate-*l* 78.9%

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

       5. associate-*l* 78.9%

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

   14. Simplified 78.9%

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

   if 4.69999999999999998e-188 < l < 2.80000000000000011e123

   1. Initial program 50.9%

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

   3. Simplified 55.7%

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow3 55.7%

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

      2. times-frac 62.6%

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

      3. pow2 62.6%

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

   6. Applied egg-rr 62.6%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)} \]
   7. Step-by-step derivation

      1. unpow2 62.6%

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

      2. frac-2neg 62.6%

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

      3. frac-times 39.1%

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

   8. Applied egg-rr 39.1%

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

   9. Taylor expanded in t around 0 96.7%

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

       1. associate-*r* 96.7%

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

       2. *-commutative 96.7%

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

       3. *-commutative 96.7%

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

   11. Simplified 96.7%

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

       1. add-sqr-sqrt 49.9%

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

       2. pow2 49.9%

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

       3. *-commutative 49.9%

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

       4. sqrt-prod 49.9%

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

       5. unpow2 49.9%

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

       6. sqrt-prod 28.0%

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

       7. add-sqr-sqrt 51.4%

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

   13. Applied egg-rr 51.4%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.7 \cdot 10^{-188} \lor \neg \left(\ell \leq 2.8 \cdot 10^{+123}\right):\\ \;\;\;\;\frac{2}{{\left(t \cdot \left(\sqrt[3]{\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)} \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(k \cdot \sqrt{t}\right)}^{2} \cdot {\sin k}^{2}}{\cos k \cdot {\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt[3]{\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\\ \mathbf{if}\;\ell \leq 4.8 \cdot 10^{-188}:\\ \;\;\;\;\frac{2}{{\left(t \cdot \left(t\_1 \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)}^{3}}\\ \mathbf{elif}\;\ell \leq 1.02 \cdot 10^{+125}:\\ \;\;\;\;\frac{2}{\frac{{\left(k \cdot \sqrt{t}\right)}^{2} \cdot {\sin k}^{2}}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot t\_1\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (cbrt (* (sin k) (* (tan k) (pow (/ k t) 2.0))))))
   (if (<= l 4.8e-188)
     (/ 2.0 (pow (* t (* t_1 (pow (cbrt l) -2.0))) 3.0))
     (if (<= l 1.02e+125)
       (/
        2.0
        (/
         (* (pow (* k (sqrt t)) 2.0) (pow (sin k) 2.0))
         (* (cos k) (pow l 2.0))))
       (/ 2.0 (pow (* (/ t (pow (cbrt l) 2.0)) t_1) 3.0))))))

C

double code(double t, double l, double k) {
	double t_1 = cbrt((sin(k) * (tan(k) * pow((k / t), 2.0))));
	double tmp;
	if (l <= 4.8e-188) {
		tmp = 2.0 / pow((t * (t_1 * pow(cbrt(l), -2.0))), 3.0);
	} else if (l <= 1.02e+125) {
		tmp = 2.0 / ((pow((k * sqrt(t)), 2.0) * pow(sin(k), 2.0)) / (cos(k) * pow(l, 2.0)));
	} else {
		tmp = 2.0 / pow(((t / pow(cbrt(l), 2.0)) * t_1), 3.0);
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.cbrt((Math.sin(k) * (Math.tan(k) * Math.pow((k / t), 2.0))));
	double tmp;
	if (l <= 4.8e-188) {
		tmp = 2.0 / Math.pow((t * (t_1 * Math.pow(Math.cbrt(l), -2.0))), 3.0);
	} else if (l <= 1.02e+125) {
		tmp = 2.0 / ((Math.pow((k * Math.sqrt(t)), 2.0) * Math.pow(Math.sin(k), 2.0)) / (Math.cos(k) * Math.pow(l, 2.0)));
	} else {
		tmp = 2.0 / Math.pow(((t / Math.pow(Math.cbrt(l), 2.0)) * t_1), 3.0);
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = cbrt(Float64(sin(k) * Float64(tan(k) * (Float64(k / t) ^ 2.0))))
	tmp = 0.0
	if (l <= 4.8e-188)
		tmp = Float64(2.0 / (Float64(t * Float64(t_1 * (cbrt(l) ^ -2.0))) ^ 3.0));
	elseif (l <= 1.02e+125)
		tmp = Float64(2.0 / Float64(Float64((Float64(k * sqrt(t)) ^ 2.0) * (sin(k) ^ 2.0)) / Float64(cos(k) * (l ^ 2.0))));
	else
		tmp = Float64(2.0 / (Float64(Float64(t / (cbrt(l) ^ 2.0)) * t_1) ^ 3.0));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[l, 4.8e-188], N[(2.0 / N[Power[N[(t * N[(t$95$1 * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.02e+125], N[(2.0 / N[(N[(N[Power[N[(k * N[Sqrt[t], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < 4.8e-188

   1. Initial program 29.3%

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

   3. Simplified 38.0%

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow3 38.0%

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

      2. times-frac 53.7%

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

      3. pow2 53.7%

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

   6. Applied egg-rr 53.7%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)} \]
   7. Step-by-step derivation

      1. add-cube-cbrt 53.6%

         \[\leadsto \frac{2}{\color{blue}{\left(\sqrt[3]{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)} \cdot \sqrt[3]{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}\right) \cdot \sqrt[3]{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}}} \]

      2. pow3 53.6%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt[3]{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}\right)}^{3}}} \]

   8. Applied egg-rr 74.9%

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

      1. +-rgt-identity 74.9%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k \cdot \left(\tan k \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 0\right)}\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

      2. unpow2 74.9%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 0\right)\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

      3. frac-2neg 74.9%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(\color{blue}{\frac{-k}{-t}} \cdot \frac{k}{t} + 0\right)\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

      4. times-frac 49.8%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(\color{blue}{\frac{\left(-k\right) \cdot k}{\left(-t\right) \cdot t}} + 0\right)\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

      5. associate-*l* 49.7%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{\left(-k\right) \cdot k}{\left(-t\right) \cdot t} + 0\right)}} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

      6. *-commutative 49.7%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\color{blue}{\left(\frac{\left(-k\right) \cdot k}{\left(-t\right) \cdot t} + 0\right) \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

      7. times-frac 74.9%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\left(\color{blue}{\frac{-k}{-t} \cdot \frac{k}{t}} + 0\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

      8. frac-2neg 74.9%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\left(\color{blue}{\frac{k}{t}} \cdot \frac{k}{t} + 0\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

      9. unpow2 74.9%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\left(\color{blue}{{\left(\frac{k}{t}\right)}^{2}} + 0\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \]

      10. +-rgt-identity 74.9%

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

      11. cbrt-prod 75.9%

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

   10. Applied egg-rr 86.5%

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

       1. pow1 86.5%

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

   12. Applied egg-rr 74.9%

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

       1. unpow1 74.9%

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

       2. associate-*r* 76.3%

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

       3. *-commutative 76.3%

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

       4. associate-*l* 76.2%

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

       5. associate-*l* 76.2%

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

   14. Simplified 76.2%

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

   if 4.8e-188 < l < 1.02e125

   1. Initial program 50.9%

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

   3. Simplified 55.7%

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow3 55.7%

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

      2. times-frac 62.6%

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

      3. pow2 62.6%

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

   6. Applied egg-rr 62.6%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)} \]
   7. Step-by-step derivation

      1. unpow2 62.6%

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

      2. frac-2neg 62.6%

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

      3. frac-times 39.1%

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

   8. Applied egg-rr 39.1%

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

   9. Taylor expanded in t around 0 96.7%

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

       1. associate-*r* 96.7%

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

       2. *-commutative 96.7%

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

       3. *-commutative 96.7%

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

   11. Simplified 96.7%

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

       1. add-sqr-sqrt 49.9%

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

       2. pow2 49.9%

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

       3. *-commutative 49.9%

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

       4. sqrt-prod 49.9%

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

       5. unpow2 49.9%

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

       6. sqrt-prod 28.0%

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

       7. add-sqr-sqrt 51.4%

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

   13. Applied egg-rr 51.4%

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

   if 1.02e125 < l

   1. Initial program 45.8%

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

   3. Simplified 45.8%

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow3 45.8%

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

      2. times-frac 65.4%

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

      3. pow2 65.4%

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

   6. Applied egg-rr 65.4%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)} \]
   7. Step-by-step derivation

      1. add-cube-cbrt 65.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\sqrt[3]{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)} \cdot \sqrt[3]{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}\right) \cdot \sqrt[3]{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}}} \]

      2. pow3 65.3%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt[3]{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}\right)}^{3}}} \]

   8. Applied egg-rr 92.8%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.8 \cdot 10^{-188}:\\ \;\;\;\;\frac{2}{{\left(t \cdot \left(\sqrt[3]{\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)} \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)}^{3}}\\ \mathbf{elif}\;\ell \leq 1.02 \cdot 10^{+125}:\\ \;\;\;\;\frac{2}{\frac{{\left(k \cdot \sqrt{t}\right)}^{2} \cdot {\sin k}^{2}}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 59.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 9.5 \cdot 10^{-167}:\\ \;\;\;\;\frac{2}{t \cdot \log \left({\left(e^{{k}^{4}}\right)}^{\left({\ell}^{-2}\right)}\right)}\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+145}:\\ \;\;\;\;\frac{2}{\frac{{\left(k \cdot \sqrt{t}\right)}^{2} \cdot {\sin k}^{2}}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= l 9.5e-167)
   (/ 2.0 (* t (log (pow (exp (pow k 4.0)) (pow l -2.0)))))
   (if (<= l 5e+145)
     (/
      2.0
      (/
       (* (pow (* k (sqrt t)) 2.0) (pow (sin k) 2.0))
       (* (cos k) (pow l 2.0))))
     (/
      2.0
      (*
       (* (* (sin k) (tan k)) (pow (/ k t) 2.0))
       (pow (/ t (pow (cbrt l) 2.0)) 3.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (l <= 9.5e-167) {
		tmp = 2.0 / (t * log(pow(exp(pow(k, 4.0)), pow(l, -2.0))));
	} else if (l <= 5e+145) {
		tmp = 2.0 / ((pow((k * sqrt(t)), 2.0) * pow(sin(k), 2.0)) / (cos(k) * pow(l, 2.0)));
	} else {
		tmp = 2.0 / (((sin(k) * tan(k)) * pow((k / t), 2.0)) * pow((t / pow(cbrt(l), 2.0)), 3.0));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (l <= 9.5e-167) {
		tmp = 2.0 / (t * Math.log(Math.pow(Math.exp(Math.pow(k, 4.0)), Math.pow(l, -2.0))));
	} else if (l <= 5e+145) {
		tmp = 2.0 / ((Math.pow((k * Math.sqrt(t)), 2.0) * Math.pow(Math.sin(k), 2.0)) / (Math.cos(k) * Math.pow(l, 2.0)));
	} else {
		tmp = 2.0 / (((Math.sin(k) * Math.tan(k)) * Math.pow((k / t), 2.0)) * Math.pow((t / Math.pow(Math.cbrt(l), 2.0)), 3.0));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (l <= 9.5e-167)
		tmp = Float64(2.0 / Float64(t * log((exp((k ^ 4.0)) ^ (l ^ -2.0)))));
	elseif (l <= 5e+145)
		tmp = Float64(2.0 / Float64(Float64((Float64(k * sqrt(t)) ^ 2.0) * (sin(k) ^ 2.0)) / Float64(cos(k) * (l ^ 2.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(sin(k) * tan(k)) * (Float64(k / t) ^ 2.0)) * (Float64(t / (cbrt(l) ^ 2.0)) ^ 3.0)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[l, 9.5e-167], N[(2.0 / N[(t * N[Log[N[Power[N[Exp[N[Power[k, 4.0], $MachinePrecision]], $MachinePrecision], N[Power[l, -2.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5e+145], N[(2.0 / N[(N[(N[Power[N[(k * N[Sqrt[t], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 9.49999999999999955e-167

   1. Initial program 29.5%

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

   3. Simplified 38.2%

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 59.3%

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

      1. *-commutative 59.3%

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

      2. associate-/l* 59.6%

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

   7. Simplified 59.6%

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

      1. add-log-exp 56.8%

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

      2. div-inv 56.8%

         \[\leadsto \frac{2}{t \cdot \log \left(e^{\color{blue}{{k}^{4} \cdot \frac{1}{{\ell}^{2}}}}\right)} \]

      3. exp-prod 64.7%

         \[\leadsto \frac{2}{t \cdot \log \color{blue}{\left({\left(e^{{k}^{4}}\right)}^{\left(\frac{1}{{\ell}^{2}}\right)}\right)}} \]

      4. pow-flip 65.3%

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

      5. metadata-eval 65.3%

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

   9. Applied egg-rr 65.3%

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

   if 9.49999999999999955e-167 < l < 4.99999999999999967e145

   1. Initial program 52.4%

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

   3. Simplified 56.8%

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow3 56.8%

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

      2. times-frac 61.8%

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

      3. pow2 61.8%

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

   6. Applied egg-rr 61.8%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)} \]
   7. Step-by-step derivation

      1. unpow2 61.8%

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

      2. frac-2neg 61.8%

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

      3. frac-times 35.3%

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

   8. Applied egg-rr 35.3%

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

   9. Taylor expanded in t around 0 96.9%

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

       1. associate-*r* 96.9%

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

       2. *-commutative 96.9%

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

       3. *-commutative 96.9%

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

   11. Simplified 96.9%

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

       1. add-sqr-sqrt 49.9%

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

       2. pow2 49.9%

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

       3. *-commutative 49.9%

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

       4. sqrt-prod 49.9%

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

       5. unpow2 49.9%

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

       6. sqrt-prod 27.8%

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

       7. add-sqr-sqrt 51.3%

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

   13. Applied egg-rr 51.3%

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

   if 4.99999999999999967e145 < l

   1. Initial program 40.7%

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

   3. Simplified 40.7%

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow3 40.7%

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

      2. times-frac 65.1%

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

      3. pow2 65.1%

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

   6. Applied egg-rr 65.1%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)} \]
   7. Step-by-step derivation

      1. add-cube-cbrt 65.0%

         \[\leadsto \frac{2}{\color{blue}{\left(\sqrt[3]{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)} \cdot \sqrt[3]{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}\right) \cdot \sqrt[3]{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}}} \]

      2. pow3 65.0%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt[3]{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}\right)}^{3}}} \]

   8. Applied egg-rr 91.1%

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

      1. cube-prod 80.4%

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

      2. rem-cube-cbrt 80.5%

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

      3. associate-*r* 80.6%

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

   10. Simplified 80.6%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 9.5 \cdot 10^{-167}:\\ \;\;\;\;\frac{2}{t \cdot \log \left({\left(e^{{k}^{4}}\right)}^{\left({\ell}^{-2}\right)}\right)}\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+145}:\\ \;\;\;\;\frac{2}{\frac{{\left(k \cdot \sqrt{t}\right)}^{2} \cdot {\sin k}^{2}}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 58.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.3 \cdot 10^{-151}:\\ \;\;\;\;\frac{2}{t \cdot \log \left({\left(e^{{k}^{4}}\right)}^{\left({\ell}^{-2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(k \cdot \sqrt{t}\right)}^{2} \cdot {\sin k}^{2}}{\cos k \cdot {\ell}^{2}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 2.3e-151)
   (/ 2.0 (* t (log (pow (exp (pow k 4.0)) (pow l -2.0)))))
   (/
    2.0
    (/
     (* (pow (* k (sqrt t)) 2.0) (pow (sin k) 2.0))
     (* (cos k) (pow l 2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.3e-151) {
		tmp = 2.0 / (t * log(pow(exp(pow(k, 4.0)), pow(l, -2.0))));
	} else {
		tmp = 2.0 / ((pow((k * sqrt(t)), 2.0) * pow(sin(k), 2.0)) / (cos(k) * pow(l, 2.0)));
	}
	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 (k <= 2.3d-151) then
        tmp = 2.0d0 / (t * log((exp((k ** 4.0d0)) ** (l ** (-2.0d0)))))
    else
        tmp = 2.0d0 / ((((k * sqrt(t)) ** 2.0d0) * (sin(k) ** 2.0d0)) / (cos(k) * (l ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.3e-151) {
		tmp = 2.0 / (t * Math.log(Math.pow(Math.exp(Math.pow(k, 4.0)), Math.pow(l, -2.0))));
	} else {
		tmp = 2.0 / ((Math.pow((k * Math.sqrt(t)), 2.0) * Math.pow(Math.sin(k), 2.0)) / (Math.cos(k) * Math.pow(l, 2.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 2.3e-151:
		tmp = 2.0 / (t * math.log(math.pow(math.exp(math.pow(k, 4.0)), math.pow(l, -2.0))))
	else:
		tmp = 2.0 / ((math.pow((k * math.sqrt(t)), 2.0) * math.pow(math.sin(k), 2.0)) / (math.cos(k) * math.pow(l, 2.0)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 2.3e-151)
		tmp = Float64(2.0 / Float64(t * log((exp((k ^ 4.0)) ^ (l ^ -2.0)))));
	else
		tmp = Float64(2.0 / Float64(Float64((Float64(k * sqrt(t)) ^ 2.0) * (sin(k) ^ 2.0)) / Float64(cos(k) * (l ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2.3e-151)
		tmp = 2.0 / (t * log((exp((k ^ 4.0)) ^ (l ^ -2.0))));
	else
		tmp = 2.0 / ((((k * sqrt(t)) ^ 2.0) * (sin(k) ^ 2.0)) / (cos(k) * (l ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 2.3e-151], N[(2.0 / N[(t * N[Log[N[Power[N[Exp[N[Power[k, 4.0], $MachinePrecision]], $MachinePrecision], N[Power[l, -2.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[N[(k * N[Sqrt[t], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.29999999999999996e-151

   1. Initial program 39.4%

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

   3. Simplified 43.0%

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 64.2%

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

      1. *-commutative 64.2%

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

      2. associate-/l* 62.3%

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

   7. Simplified 62.3%

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

      1. add-log-exp 61.1%

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

      2. div-inv 61.1%

         \[\leadsto \frac{2}{t \cdot \log \left(e^{\color{blue}{{k}^{4} \cdot \frac{1}{{\ell}^{2}}}}\right)} \]

      3. exp-prod 69.2%

         \[\leadsto \frac{2}{t \cdot \log \color{blue}{\left({\left(e^{{k}^{4}}\right)}^{\left(\frac{1}{{\ell}^{2}}\right)}\right)}} \]

      4. pow-flip 69.7%

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

      5. metadata-eval 69.7%

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

   9. Applied egg-rr 69.7%

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

   if 2.29999999999999996e-151 < k

   1. Initial program 31.6%

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

   3. Simplified 44.1%

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow3 44.1%

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

      2. times-frac 62.7%

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

      3. pow2 62.7%

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

   6. Applied egg-rr 62.7%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)} \]
   7. Step-by-step derivation

      1. unpow2 62.7%

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

      2. frac-2neg 62.7%

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

      3. frac-times 42.2%

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

   8. Applied egg-rr 42.2%

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

   9. Taylor expanded in t around 0 81.5%

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

       1. associate-*r* 81.5%

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

       2. *-commutative 81.5%

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

       3. *-commutative 81.5%

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

   11. Simplified 81.5%

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

       1. add-sqr-sqrt 43.8%

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

       2. pow2 43.8%

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

       3. *-commutative 43.8%

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

       4. sqrt-prod 43.8%

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

       5. unpow2 43.8%

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

       6. sqrt-prod 44.7%

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

       7. add-sqr-sqrt 44.9%

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

   13. Applied egg-rr 44.9%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.3 \cdot 10^{-151}:\\ \;\;\;\;\frac{2}{t \cdot \log \left({\left(e^{{k}^{4}}\right)}^{\left({\ell}^{-2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(k \cdot \sqrt{t}\right)}^{2} \cdot {\sin k}^{2}}{\cos k \cdot {\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.3 \cdot 10^{-151}:\\ \;\;\;\;\frac{2}{t \cdot \log \left({\left(e^{{k}^{4}}\right)}^{\left({\ell}^{-2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{1}{\left(t \cdot {k}^{2}\right) \cdot \frac{{\sin k}^{2}}{\cos k}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 2.3e-151)
   (/ 2.0 (* t (log (pow (exp (pow k 4.0)) (pow l -2.0)))))
   (*
    2.0
    (*
     (pow l 2.0)
     (/ 1.0 (* (* t (pow k 2.0)) (/ (pow (sin k) 2.0) (cos k))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.3e-151) {
		tmp = 2.0 / (t * log(pow(exp(pow(k, 4.0)), pow(l, -2.0))));
	} else {
		tmp = 2.0 * (pow(l, 2.0) * (1.0 / ((t * pow(k, 2.0)) * (pow(sin(k), 2.0) / cos(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 <= 2.3d-151) then
        tmp = 2.0d0 / (t * log((exp((k ** 4.0d0)) ** (l ** (-2.0d0)))))
    else
        tmp = 2.0d0 * ((l ** 2.0d0) * (1.0d0 / ((t * (k ** 2.0d0)) * ((sin(k) ** 2.0d0) / cos(k)))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.3e-151) {
		tmp = 2.0 / (t * Math.log(Math.pow(Math.exp(Math.pow(k, 4.0)), Math.pow(l, -2.0))));
	} else {
		tmp = 2.0 * (Math.pow(l, 2.0) * (1.0 / ((t * Math.pow(k, 2.0)) * (Math.pow(Math.sin(k), 2.0) / Math.cos(k)))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 2.3e-151:
		tmp = 2.0 / (t * math.log(math.pow(math.exp(math.pow(k, 4.0)), math.pow(l, -2.0))))
	else:
		tmp = 2.0 * (math.pow(l, 2.0) * (1.0 / ((t * math.pow(k, 2.0)) * (math.pow(math.sin(k), 2.0) / math.cos(k)))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 2.3e-151)
		tmp = Float64(2.0 / Float64(t * log((exp((k ^ 4.0)) ^ (l ^ -2.0)))));
	else
		tmp = Float64(2.0 * Float64((l ^ 2.0) * Float64(1.0 / Float64(Float64(t * (k ^ 2.0)) * Float64((sin(k) ^ 2.0) / cos(k))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2.3e-151)
		tmp = 2.0 / (t * log((exp((k ^ 4.0)) ^ (l ^ -2.0))));
	else
		tmp = 2.0 * ((l ^ 2.0) * (1.0 / ((t * (k ^ 2.0)) * ((sin(k) ^ 2.0) / cos(k)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 2.3e-151], N[(2.0 / N[(t * N[Log[N[Power[N[Exp[N[Power[k, 4.0], $MachinePrecision]], $MachinePrecision], N[Power[l, -2.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] * N[(1.0 / N[(N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.29999999999999996e-151

   1. Initial program 39.4%

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

   3. Simplified 43.0%

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 64.2%

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

      1. *-commutative 64.2%

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

      2. associate-/l* 62.3%

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

   7. Simplified 62.3%

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

      1. add-log-exp 61.1%

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

      2. div-inv 61.1%

         \[\leadsto \frac{2}{t \cdot \log \left(e^{\color{blue}{{k}^{4} \cdot \frac{1}{{\ell}^{2}}}}\right)} \]

      3. exp-prod 69.2%

         \[\leadsto \frac{2}{t \cdot \log \color{blue}{\left({\left(e^{{k}^{4}}\right)}^{\left(\frac{1}{{\ell}^{2}}\right)}\right)}} \]

      4. pow-flip 69.7%

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

      5. metadata-eval 69.7%

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

   9. Applied egg-rr 69.7%

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

   if 2.29999999999999996e-151 < k

   1. Initial program 31.6%

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

      1. associate-/r* 31.6%

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

      2. associate-/l/ 31.6%

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

      3. associate-*l/ 31.5%

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

      4. associate-/r/ 30.9%

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

      5. +-commutative 30.9%

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

      6. unpow2 30.9%

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

      7. sqr-neg 30.9%

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

      8. distribute-frac-neg2 30.9%

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

      9. distribute-frac-neg2 30.9%

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

      10. unpow2 30.9%

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

      11. +-rgt-identity 30.9%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

      12. metadata-eval 30.9%

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

      13. associate--l+ 30.9%

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

      14. +-commutative 30.9%

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

      15. associate--l+ 30.9%

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

   3. Simplified 43.4%

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

   5. Taylor expanded in k around inf 81.5%

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

      1. associate-/l* 81.5%

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

   7. Simplified 81.5%

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

      1. clear-num 81.5%

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

      2. inv-pow 81.5%

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

      3. associate-*r* 81.6%

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

   9. Applied egg-rr 81.6%

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

       1. unpow-1 81.6%

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

       2. associate-/l* 81.6%

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

   11. Simplified 81.6%

       \[\leadsto 2 \cdot \left({\ell}^{2} \cdot \color{blue}{\frac{1}{\left({k}^{2} \cdot t\right) \cdot \frac{{\sin k}^{2}}{\cos k}}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.3 \cdot 10^{-151}:\\ \;\;\;\;\frac{2}{t \cdot \log \left({\left(e^{{k}^{4}}\right)}^{\left({\ell}^{-2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{1}{\left(t \cdot {k}^{2}\right) \cdot \frac{{\sin k}^{2}}{\cos k}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 71.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.3 \cdot 10^{-151}:\\ \;\;\;\;\frac{2}{t \cdot \log \left({\left(e^{{k}^{4}}\right)}^{\left({\ell}^{-2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 2.3e-151)
   (/ 2.0 (* t (log (pow (exp (pow k 4.0)) (pow l -2.0)))))
   (*
    2.0
    (* (pow l 2.0) (/ (cos k) (* (pow k 2.0) (* t (pow (sin k) 2.0))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.3e-151) {
		tmp = 2.0 / (t * log(pow(exp(pow(k, 4.0)), pow(l, -2.0))));
	} else {
		tmp = 2.0 * (pow(l, 2.0) * (cos(k) / (pow(k, 2.0) * (t * pow(sin(k), 2.0)))));
	}
	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 (k <= 2.3d-151) then
        tmp = 2.0d0 / (t * log((exp((k ** 4.0d0)) ** (l ** (-2.0d0)))))
    else
        tmp = 2.0d0 * ((l ** 2.0d0) * (cos(k) / ((k ** 2.0d0) * (t * (sin(k) ** 2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.3e-151) {
		tmp = 2.0 / (t * Math.log(Math.pow(Math.exp(Math.pow(k, 4.0)), Math.pow(l, -2.0))));
	} else {
		tmp = 2.0 * (Math.pow(l, 2.0) * (Math.cos(k) / (Math.pow(k, 2.0) * (t * Math.pow(Math.sin(k), 2.0)))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 2.3e-151:
		tmp = 2.0 / (t * math.log(math.pow(math.exp(math.pow(k, 4.0)), math.pow(l, -2.0))))
	else:
		tmp = 2.0 * (math.pow(l, 2.0) * (math.cos(k) / (math.pow(k, 2.0) * (t * math.pow(math.sin(k), 2.0)))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 2.3e-151)
		tmp = Float64(2.0 / Float64(t * log((exp((k ^ 4.0)) ^ (l ^ -2.0)))));
	else
		tmp = Float64(2.0 * Float64((l ^ 2.0) * Float64(cos(k) / Float64((k ^ 2.0) * Float64(t * (sin(k) ^ 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2.3e-151)
		tmp = 2.0 / (t * log((exp((k ^ 4.0)) ^ (l ^ -2.0))));
	else
		tmp = 2.0 * ((l ^ 2.0) * (cos(k) / ((k ^ 2.0) * (t * (sin(k) ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 2.3e-151], N[(2.0 / N[(t * N[Log[N[Power[N[Exp[N[Power[k, 4.0], $MachinePrecision]], $MachinePrecision], N[Power[l, -2.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.29999999999999996e-151

   1. Initial program 39.4%

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

   3. Simplified 43.0%

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 64.2%

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

      1. *-commutative 64.2%

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

      2. associate-/l* 62.3%

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

   7. Simplified 62.3%

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

      1. add-log-exp 61.1%

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

      2. div-inv 61.1%

         \[\leadsto \frac{2}{t \cdot \log \left(e^{\color{blue}{{k}^{4} \cdot \frac{1}{{\ell}^{2}}}}\right)} \]

      3. exp-prod 69.2%

         \[\leadsto \frac{2}{t \cdot \log \color{blue}{\left({\left(e^{{k}^{4}}\right)}^{\left(\frac{1}{{\ell}^{2}}\right)}\right)}} \]

      4. pow-flip 69.7%

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

      5. metadata-eval 69.7%

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

   9. Applied egg-rr 69.7%

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

   if 2.29999999999999996e-151 < k

   1. Initial program 31.6%

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

      1. associate-/r* 31.6%

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

      2. associate-/l/ 31.6%

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

      3. associate-*l/ 31.5%

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

      4. associate-/r/ 30.9%

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

      5. +-commutative 30.9%

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

      6. unpow2 30.9%

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

      7. sqr-neg 30.9%

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

      8. distribute-frac-neg2 30.9%

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

      9. distribute-frac-neg2 30.9%

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

      10. unpow2 30.9%

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

      11. +-rgt-identity 30.9%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

      12. metadata-eval 30.9%

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

      13. associate--l+ 30.9%

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

      14. +-commutative 30.9%

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

      15. associate--l+ 30.9%

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

   3. Simplified 43.4%

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

   5. Taylor expanded in k around inf 81.5%

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

      1. associate-/l* 81.5%

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

   7. Simplified 81.5%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.3 \cdot 10^{-151}:\\ \;\;\;\;\frac{2}{t \cdot \log \left({\left(e^{{k}^{4}}\right)}^{\left({\ell}^{-2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 71.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.12 \cdot 10^{-151}:\\ \;\;\;\;\frac{2}{t \cdot \log \left({\left(e^{{k}^{4}}\right)}^{\left({\ell}^{-2}\right)}\right)}\\ \mathbf{elif}\;k \leq 5.2 \cdot 10^{-14}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {k}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{\frac{\cos k}{t}}{{\left(k \cdot \sin k\right)}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.12e-151)
   (/ 2.0 (* t (log (pow (exp (pow k 4.0)) (pow l -2.0)))))
   (if (<= k 5.2e-14)
     (* 2.0 (* (pow l 2.0) (/ (cos k) (* (pow k 2.0) (* t (pow k 2.0))))))
     (* 2.0 (* (pow l 2.0) (/ (/ (cos k) t) (pow (* k (sin k)) 2.0)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.12e-151) {
		tmp = 2.0 / (t * log(pow(exp(pow(k, 4.0)), pow(l, -2.0))));
	} else if (k <= 5.2e-14) {
		tmp = 2.0 * (pow(l, 2.0) * (cos(k) / (pow(k, 2.0) * (t * pow(k, 2.0)))));
	} else {
		tmp = 2.0 * (pow(l, 2.0) * ((cos(k) / t) / pow((k * sin(k)), 2.0)));
	}
	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 (k <= 1.12d-151) then
        tmp = 2.0d0 / (t * log((exp((k ** 4.0d0)) ** (l ** (-2.0d0)))))
    else if (k <= 5.2d-14) then
        tmp = 2.0d0 * ((l ** 2.0d0) * (cos(k) / ((k ** 2.0d0) * (t * (k ** 2.0d0)))))
    else
        tmp = 2.0d0 * ((l ** 2.0d0) * ((cos(k) / t) / ((k * sin(k)) ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.12e-151) {
		tmp = 2.0 / (t * Math.log(Math.pow(Math.exp(Math.pow(k, 4.0)), Math.pow(l, -2.0))));
	} else if (k <= 5.2e-14) {
		tmp = 2.0 * (Math.pow(l, 2.0) * (Math.cos(k) / (Math.pow(k, 2.0) * (t * Math.pow(k, 2.0)))));
	} else {
		tmp = 2.0 * (Math.pow(l, 2.0) * ((Math.cos(k) / t) / Math.pow((k * Math.sin(k)), 2.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.12e-151:
		tmp = 2.0 / (t * math.log(math.pow(math.exp(math.pow(k, 4.0)), math.pow(l, -2.0))))
	elif k <= 5.2e-14:
		tmp = 2.0 * (math.pow(l, 2.0) * (math.cos(k) / (math.pow(k, 2.0) * (t * math.pow(k, 2.0)))))
	else:
		tmp = 2.0 * (math.pow(l, 2.0) * ((math.cos(k) / t) / math.pow((k * math.sin(k)), 2.0)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.12e-151)
		tmp = Float64(2.0 / Float64(t * log((exp((k ^ 4.0)) ^ (l ^ -2.0)))));
	elseif (k <= 5.2e-14)
		tmp = Float64(2.0 * Float64((l ^ 2.0) * Float64(cos(k) / Float64((k ^ 2.0) * Float64(t * (k ^ 2.0))))));
	else
		tmp = Float64(2.0 * Float64((l ^ 2.0) * Float64(Float64(cos(k) / t) / (Float64(k * sin(k)) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.12e-151)
		tmp = 2.0 / (t * log((exp((k ^ 4.0)) ^ (l ^ -2.0))));
	elseif (k <= 5.2e-14)
		tmp = 2.0 * ((l ^ 2.0) * (cos(k) / ((k ^ 2.0) * (t * (k ^ 2.0)))));
	else
		tmp = 2.0 * ((l ^ 2.0) * ((cos(k) / t) / ((k * sin(k)) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.12e-151], N[(2.0 / N[(t * N[Log[N[Power[N[Exp[N[Power[k, 4.0], $MachinePrecision]], $MachinePrecision], N[Power[l, -2.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 5.2e-14], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision] / N[Power[N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.11999999999999994e-151

   1. Initial program 39.4%

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

   3. Simplified 43.0%

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 64.2%

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

      1. *-commutative 64.2%

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

      2. associate-/l* 62.3%

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

   7. Simplified 62.3%

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

      1. add-log-exp 61.1%

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

      2. div-inv 61.1%

         \[\leadsto \frac{2}{t \cdot \log \left(e^{\color{blue}{{k}^{4} \cdot \frac{1}{{\ell}^{2}}}}\right)} \]

      3. exp-prod 69.2%

         \[\leadsto \frac{2}{t \cdot \log \color{blue}{\left({\left(e^{{k}^{4}}\right)}^{\left(\frac{1}{{\ell}^{2}}\right)}\right)}} \]

      4. pow-flip 69.7%

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

      5. metadata-eval 69.7%

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

   9. Applied egg-rr 69.7%

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

   if 1.11999999999999994e-151 < k < 5.19999999999999993e-14

   1. Initial program 44.6%

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

      1. associate-/r* 44.6%

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

      2. associate-/l/ 44.6%

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

      3. associate-*l/ 44.6%

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

      4. associate-/r/ 44.6%

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

      5. +-commutative 44.6%

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

      6. unpow2 44.6%

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

      7. sqr-neg 44.6%

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

      8. distribute-frac-neg2 44.6%

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

      9. distribute-frac-neg2 44.6%

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

      10. unpow2 44.6%

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

      11. +-rgt-identity 44.6%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

      12. metadata-eval 44.6%

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

      13. associate--l+ 44.6%

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

      14. +-commutative 44.6%

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

      15. associate--l+ 44.6%

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

   3. Simplified 48.3%

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

   5. Taylor expanded in k around inf 85.4%

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

      1. associate-/l* 85.4%

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

   7. Simplified 85.4%

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

   8. Taylor expanded in k around 0 85.4%

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

   if 5.19999999999999993e-14 < k

   1. Initial program 25.8%

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

      1. associate-/r* 25.8%

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

      2. associate-/l/ 25.8%

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

      3. associate-*l/ 25.7%

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

      4. associate-/r/ 24.9%

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

      5. +-commutative 24.9%

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

      6. unpow2 24.9%

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

      7. sqr-neg 24.9%

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

      8. distribute-frac-neg2 24.9%

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

      9. distribute-frac-neg2 24.9%

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

      10. unpow2 24.9%

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

      11. +-rgt-identity 24.9%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

      12. metadata-eval 24.9%

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

      13. associate--l+ 24.9%

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

      14. +-commutative 24.9%

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

      15. associate--l+ 24.9%

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

   3. Simplified 41.2%

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

      1. associate-*r* 42.0%

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

      2. unpow2 42.0%

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

      3. times-frac 50.2%

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

      4. associate-/l/ 50.2%

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

      5. associate-*l/ 51.0%

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

      6. associate-*l* 51.0%

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

   6. Applied egg-rr 51.0%

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

   7. Taylor expanded in l around 0 79.7%

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

      1. associate-/l* 79.8%

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

      2. *-commutative 79.8%

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

      3. associate-*r* 79.8%

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

      4. unpow2 79.8%

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

      5. unpow2 79.8%

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

      6. swap-sqr 79.8%

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

      7. unpow2 79.8%

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

      8. *-commutative 79.8%

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

      9. associate-/r* 79.9%

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

   9. Simplified 79.9%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.12 \cdot 10^{-151}:\\ \;\;\;\;\frac{2}{t \cdot \log \left({\left(e^{{k}^{4}}\right)}^{\left({\ell}^{-2}\right)}\right)}\\ \mathbf{elif}\;k \leq 5.2 \cdot 10^{-14}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {k}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{\frac{\cos k}{t}}{{\left(k \cdot \sin k\right)}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 70.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2}{\frac{{\sin k}^{2} \cdot \left(t \cdot {k}^{2}\right)}{{\ell}^{2}}}\\ t_2 := \sin k \cdot \tan k\\ \mathbf{if}\;t \leq 6.5 \cdot 10^{-116}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{+102}:\\ \;\;\;\;\frac{\ell}{\frac{k}{t}} \cdot \left(\frac{2}{{t}^{3}} \cdot \left(t \cdot \frac{\ell}{k \cdot t\_2}\right)\right)\\ \mathbf{elif}\;t \leq 1.04 \cdot 10^{+139}:\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(t\_2 \cdot \frac{1}{\frac{t}{k} \cdot \frac{t}{k}}\right)}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{+197}:\\ \;\;\;\;\frac{2}{{\ell}^{-2} \cdot \left(t \cdot {k}^{4}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ 2.0 (/ (* (pow (sin k) 2.0) (* t (pow k 2.0))) (pow l 2.0))))
        (t_2 (* (sin k) (tan k))))
   (if (<= t 6.5e-116)
     t_1
     (if (<= t 5.8e+102)
       (* (/ l (/ k t)) (* (/ 2.0 (pow t 3.0)) (* t (/ l (* k t_2)))))
       (if (<= t 1.04e+139)
         (/
          2.0
          (*
           (* (/ (pow t 2.0) l) (/ t l))
           (* t_2 (/ 1.0 (* (/ t k) (/ t k))))))
         (if (<= t 2e+197) (/ 2.0 (* (pow l -2.0) (* t (pow k 4.0)))) t_1))))))

C

double code(double t, double l, double k) {
	double t_1 = 2.0 / ((pow(sin(k), 2.0) * (t * pow(k, 2.0))) / pow(l, 2.0));
	double t_2 = sin(k) * tan(k);
	double tmp;
	if (t <= 6.5e-116) {
		tmp = t_1;
	} else if (t <= 5.8e+102) {
		tmp = (l / (k / t)) * ((2.0 / pow(t, 3.0)) * (t * (l / (k * t_2))));
	} else if (t <= 1.04e+139) {
		tmp = 2.0 / (((pow(t, 2.0) / l) * (t / l)) * (t_2 * (1.0 / ((t / k) * (t / k)))));
	} else if (t <= 2e+197) {
		tmp = 2.0 / (pow(l, -2.0) * (t * pow(k, 4.0)));
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = 2.0d0 / (((sin(k) ** 2.0d0) * (t * (k ** 2.0d0))) / (l ** 2.0d0))
    t_2 = sin(k) * tan(k)
    if (t <= 6.5d-116) then
        tmp = t_1
    else if (t <= 5.8d+102) then
        tmp = (l / (k / t)) * ((2.0d0 / (t ** 3.0d0)) * (t * (l / (k * t_2))))
    else if (t <= 1.04d+139) then
        tmp = 2.0d0 / ((((t ** 2.0d0) / l) * (t / l)) * (t_2 * (1.0d0 / ((t / k) * (t / k)))))
    else if (t <= 2d+197) then
        tmp = 2.0d0 / ((l ** (-2.0d0)) * (t * (k ** 4.0d0)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = 2.0 / ((Math.pow(Math.sin(k), 2.0) * (t * Math.pow(k, 2.0))) / Math.pow(l, 2.0));
	double t_2 = Math.sin(k) * Math.tan(k);
	double tmp;
	if (t <= 6.5e-116) {
		tmp = t_1;
	} else if (t <= 5.8e+102) {
		tmp = (l / (k / t)) * ((2.0 / Math.pow(t, 3.0)) * (t * (l / (k * t_2))));
	} else if (t <= 1.04e+139) {
		tmp = 2.0 / (((Math.pow(t, 2.0) / l) * (t / l)) * (t_2 * (1.0 / ((t / k) * (t / k)))));
	} else if (t <= 2e+197) {
		tmp = 2.0 / (Math.pow(l, -2.0) * (t * Math.pow(k, 4.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = 2.0 / ((math.pow(math.sin(k), 2.0) * (t * math.pow(k, 2.0))) / math.pow(l, 2.0))
	t_2 = math.sin(k) * math.tan(k)
	tmp = 0
	if t <= 6.5e-116:
		tmp = t_1
	elif t <= 5.8e+102:
		tmp = (l / (k / t)) * ((2.0 / math.pow(t, 3.0)) * (t * (l / (k * t_2))))
	elif t <= 1.04e+139:
		tmp = 2.0 / (((math.pow(t, 2.0) / l) * (t / l)) * (t_2 * (1.0 / ((t / k) * (t / k)))))
	elif t <= 2e+197:
		tmp = 2.0 / (math.pow(l, -2.0) * (t * math.pow(k, 4.0)))
	else:
		tmp = t_1
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) * Float64(t * (k ^ 2.0))) / (l ^ 2.0)))
	t_2 = Float64(sin(k) * tan(k))
	tmp = 0.0
	if (t <= 6.5e-116)
		tmp = t_1;
	elseif (t <= 5.8e+102)
		tmp = Float64(Float64(l / Float64(k / t)) * Float64(Float64(2.0 / (t ^ 3.0)) * Float64(t * Float64(l / Float64(k * t_2)))));
	elseif (t <= 1.04e+139)
		tmp = Float64(2.0 / Float64(Float64(Float64((t ^ 2.0) / l) * Float64(t / l)) * Float64(t_2 * Float64(1.0 / Float64(Float64(t / k) * Float64(t / k))))));
	elseif (t <= 2e+197)
		tmp = Float64(2.0 / Float64((l ^ -2.0) * Float64(t * (k ^ 4.0))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = 2.0 / (((sin(k) ^ 2.0) * (t * (k ^ 2.0))) / (l ^ 2.0));
	t_2 = sin(k) * tan(k);
	tmp = 0.0;
	if (t <= 6.5e-116)
		tmp = t_1;
	elseif (t <= 5.8e+102)
		tmp = (l / (k / t)) * ((2.0 / (t ^ 3.0)) * (t * (l / (k * t_2))));
	elseif (t <= 1.04e+139)
		tmp = 2.0 / ((((t ^ 2.0) / l) * (t / l)) * (t_2 * (1.0 / ((t / k) * (t / k)))));
	elseif (t <= 2e+197)
		tmp = 2.0 / ((l ^ -2.0) * (t * (k ^ 4.0)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 6.5e-116], t$95$1, If[LessEqual[t, 5.8e+102], N[(N[(l / N[(k / t), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] * N[(t * N[(l / N[(k * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.04e+139], N[(2.0 / N[(N[(N[(N[Power[t, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * N[(1.0 / N[(N[(t / k), $MachinePrecision] * N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2e+197], N[(2.0 / N[(N[Power[l, -2.0], $MachinePrecision] * N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < 6.5000000000000001e-116 or 1.9999999999999999e197 < t

   1. Initial program 31.8%

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

   3. Simplified 39.2%

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow3 39.2%

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

      2. times-frac 52.3%

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

      3. pow2 52.3%

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

   6. Applied egg-rr 52.3%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)} \]
   7. Step-by-step derivation

      1. unpow2 52.3%

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

      2. frac-2neg 52.3%

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

      3. frac-times 27.9%

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

   8. Applied egg-rr 27.9%

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

   9. Taylor expanded in t around 0 77.4%

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

       1. associate-*r* 77.4%

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

       2. *-commutative 77.4%

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

       3. *-commutative 77.4%

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

   11. Simplified 77.4%

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

   12. Taylor expanded in k around 0 71.0%

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

   if 6.5000000000000001e-116 < t < 5.8000000000000005e102

   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-/r* 70.2%

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

      2. associate-/l/ 70.2%

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

      3. associate-*l/ 72.7%

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

      4. associate-/r/ 71.4%

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

      5. +-commutative 71.4%

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

      6. unpow2 71.4%

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

      7. sqr-neg 71.4%

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

      8. distribute-frac-neg2 71.4%

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

      9. distribute-frac-neg2 71.4%

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

      10. unpow2 71.4%

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

      11. +-rgt-identity 71.4%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

      12. metadata-eval 71.4%

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

      13. associate--l+ 71.4%

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

      14. +-commutative 71.4%

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

      15. associate--l+ 71.4%

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

   3. Simplified 71.4%

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

      1. associate-*r* 79.5%

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

      2. unpow2 79.5%

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

      3. times-frac 82.3%

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

      4. associate-/l/ 82.2%

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

      5. associate-*l/ 83.5%

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

      6. associate-*l* 83.5%

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

   6. Applied egg-rr 83.5%

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

      1. div-inv 83.5%

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

      2. times-frac 84.7%

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

   8. Applied egg-rr 84.7%

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

      1. associate-*r/ 84.7%

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

      2. *-rgt-identity 84.7%

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

      3. associate-/r/ 84.5%

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

      4. associate-/l* 87.1%

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

      5. associate-*l* 87.3%

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

      6. associate-/l/ 87.3%

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

   10. Simplified 87.3%

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

   if 5.8000000000000005e102 < t < 1.04e139

   1. Initial program 12.5%

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

   3. Simplified 25.3%

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow3 25.3%

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

      2. times-frac 86.7%

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

      3. pow2 86.7%

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

   6. Applied egg-rr 86.7%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)} \]
   7. Step-by-step derivation

      1. unpow2 86.7%

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

      2. clear-num 86.7%

         \[\leadsto \frac{2}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\color{blue}{\frac{1}{\frac{t}{k}}} \cdot \frac{k}{t} + 0\right)\right)} \]

      3. clear-num 86.9%

         \[\leadsto \frac{2}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\frac{1}{\frac{t}{k}} \cdot \color{blue}{\frac{1}{\frac{t}{k}}} + 0\right)\right)} \]

      4. frac-times 87.1%

         \[\leadsto \frac{2}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\color{blue}{\frac{1 \cdot 1}{\frac{t}{k} \cdot \frac{t}{k}}} + 0\right)\right)} \]

      5. metadata-eval 87.1%

         \[\leadsto \frac{2}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\frac{\color{blue}{1}}{\frac{t}{k} \cdot \frac{t}{k}} + 0\right)\right)} \]

   8. Applied egg-rr 87.1%

      \[\leadsto \frac{2}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\color{blue}{\frac{1}{\frac{t}{k} \cdot \frac{t}{k}}} + 0\right)\right)} \]

   if 1.04e139 < t < 1.9999999999999999e197

   1. Initial program 25.0%

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

   3. Simplified 37.5%

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 75.0%

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

      1. *-commutative 75.0%

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

      2. associate-/l* 74.8%

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

   7. Simplified 74.8%

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

      1. pow1 74.8%

         \[\leadsto \frac{2}{\color{blue}{{\left(t \cdot \frac{{k}^{4}}{{\ell}^{2}}\right)}^{1}}} \]

      2. div-inv 74.8%

         \[\leadsto \frac{2}{{\left(t \cdot \color{blue}{\left({k}^{4} \cdot \frac{1}{{\ell}^{2}}\right)}\right)}^{1}} \]

      3. pow-flip 74.8%

         \[\leadsto \frac{2}{{\left(t \cdot \left({k}^{4} \cdot \color{blue}{{\ell}^{\left(-2\right)}}\right)\right)}^{1}} \]

      4. metadata-eval 74.8%

         \[\leadsto \frac{2}{{\left(t \cdot \left({k}^{4} \cdot {\ell}^{\color{blue}{-2}}\right)\right)}^{1}} \]

   9. Applied egg-rr 74.8%

      \[\leadsto \frac{2}{\color{blue}{{\left(t \cdot \left({k}^{4} \cdot {\ell}^{-2}\right)\right)}^{1}}} \]
   10. Step-by-step derivation

       1. unpow1 74.8%

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

       2. associate-*r* 75.0%

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

   11. Simplified 75.0%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.5 \cdot 10^{-116}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2} \cdot \left(t \cdot {k}^{2}\right)}{{\ell}^{2}}}\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{+102}:\\ \;\;\;\;\frac{\ell}{\frac{k}{t}} \cdot \left(\frac{2}{{t}^{3}} \cdot \left(t \cdot \frac{\ell}{k \cdot \left(\sin k \cdot \tan k\right)}\right)\right)\\ \mathbf{elif}\;t \leq 1.04 \cdot 10^{+139}:\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \frac{1}{\frac{t}{k} \cdot \frac{t}{k}}\right)}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{+197}:\\ \;\;\;\;\frac{2}{{\ell}^{-2} \cdot \left(t \cdot {k}^{4}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2} \cdot \left(t \cdot {k}^{2}\right)}{{\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 71.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 8.5 \cdot 10^{-155}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{{\left(t \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{3}}}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}}\\ \mathbf{elif}\;k \leq 2 \cdot 10^{-14}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {k}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{\frac{\cos k}{t}}{{\left(k \cdot \sin k\right)}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 8.5e-155)
   (*
    (/ (/ (* 2.0 l) (pow (* t (cbrt (* (sin k) (tan k)))) 3.0)) (/ k t))
    (/ l (/ k t)))
   (if (<= k 2e-14)
     (* 2.0 (* (pow l 2.0) (/ (cos k) (* (pow k 2.0) (* t (pow k 2.0))))))
     (* 2.0 (* (pow l 2.0) (/ (/ (cos k) t) (pow (* k (sin k)) 2.0)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 8.5e-155) {
		tmp = (((2.0 * l) / pow((t * cbrt((sin(k) * tan(k)))), 3.0)) / (k / t)) * (l / (k / t));
	} else if (k <= 2e-14) {
		tmp = 2.0 * (pow(l, 2.0) * (cos(k) / (pow(k, 2.0) * (t * pow(k, 2.0)))));
	} else {
		tmp = 2.0 * (pow(l, 2.0) * ((cos(k) / t) / pow((k * sin(k)), 2.0)));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 8.5e-155) {
		tmp = (((2.0 * l) / Math.pow((t * Math.cbrt((Math.sin(k) * Math.tan(k)))), 3.0)) / (k / t)) * (l / (k / t));
	} else if (k <= 2e-14) {
		tmp = 2.0 * (Math.pow(l, 2.0) * (Math.cos(k) / (Math.pow(k, 2.0) * (t * Math.pow(k, 2.0)))));
	} else {
		tmp = 2.0 * (Math.pow(l, 2.0) * ((Math.cos(k) / t) / Math.pow((k * Math.sin(k)), 2.0)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 8.5e-155)
		tmp = Float64(Float64(Float64(Float64(2.0 * l) / (Float64(t * cbrt(Float64(sin(k) * tan(k)))) ^ 3.0)) / Float64(k / t)) * Float64(l / Float64(k / t)));
	elseif (k <= 2e-14)
		tmp = Float64(2.0 * Float64((l ^ 2.0) * Float64(cos(k) / Float64((k ^ 2.0) * Float64(t * (k ^ 2.0))))));
	else
		tmp = Float64(2.0 * Float64((l ^ 2.0) * Float64(Float64(cos(k) / t) / (Float64(k * sin(k)) ^ 2.0))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 8.5e-155], N[(N[(N[(N[(2.0 * l), $MachinePrecision] / N[Power[N[(t * N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / N[(k / t), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2e-14], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision] / N[Power[N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 8.4999999999999996e-155

   1. Initial program 39.4%

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

      1. associate-/r* 39.4%

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

      2. associate-/l/ 39.4%

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

      3. associate-*l/ 40.6%

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

      4. associate-/r/ 40.6%

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

      5. +-commutative 40.6%

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

      6. unpow2 40.6%

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

      7. sqr-neg 40.6%

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

      8. distribute-frac-neg2 40.6%

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

      9. distribute-frac-neg2 40.6%

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

      10. unpow2 40.6%

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

      11. +-rgt-identity 40.6%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

      12. metadata-eval 40.6%

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

      13. associate--l+ 40.6%

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

      14. +-commutative 40.6%

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

      15. associate--l+ 40.6%

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

   3. Simplified 44.8%

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

      1. associate-*r* 52.8%

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

      2. unpow2 52.8%

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

      3. times-frac 56.4%

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

      4. associate-/l/ 56.4%

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

      5. associate-*l/ 56.4%

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

      6. associate-*l* 56.4%

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

   6. Applied egg-rr 56.4%

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

      1. add-cube-cbrt 56.4%

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

      2. pow3 56.4%

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

      3. cbrt-prod 56.4%

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

      4. unpow3 56.4%

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

      5. add-cbrt-cube 69.0%

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

   8. Applied egg-rr 69.0%

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

   if 8.4999999999999996e-155 < k < 2e-14

   1. Initial program 44.6%

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

      1. associate-/r* 44.6%

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

      2. associate-/l/ 44.6%

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

      3. associate-*l/ 44.6%

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

      4. associate-/r/ 44.6%

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

      5. +-commutative 44.6%

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

      6. unpow2 44.6%

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

      7. sqr-neg 44.6%

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

      8. distribute-frac-neg2 44.6%

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

      9. distribute-frac-neg2 44.6%

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

      10. unpow2 44.6%

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

      11. +-rgt-identity 44.6%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

      12. metadata-eval 44.6%

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

      13. associate--l+ 44.6%

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

      14. +-commutative 44.6%

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

      15. associate--l+ 44.6%

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

   3. Simplified 48.3%

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

   5. Taylor expanded in k around inf 85.4%

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

      1. associate-/l* 85.4%

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

   7. Simplified 85.4%

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

   8. Taylor expanded in k around 0 85.4%

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

   if 2e-14 < k

   1. Initial program 25.8%

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

      1. associate-/r* 25.8%

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

      2. associate-/l/ 25.8%

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

      3. associate-*l/ 25.7%

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

      4. associate-/r/ 24.9%

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

      5. +-commutative 24.9%

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

      6. unpow2 24.9%

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

      7. sqr-neg 24.9%

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

      8. distribute-frac-neg2 24.9%

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

      9. distribute-frac-neg2 24.9%

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

      10. unpow2 24.9%

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

      11. +-rgt-identity 24.9%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

      12. metadata-eval 24.9%

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

      13. associate--l+ 24.9%

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

      14. +-commutative 24.9%

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

      15. associate--l+ 24.9%

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

   3. Simplified 41.2%

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

      1. associate-*r* 42.0%

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

      2. unpow2 42.0%

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

      3. times-frac 50.2%

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

      4. associate-/l/ 50.2%

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

      5. associate-*l/ 51.0%

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

      6. associate-*l* 51.0%

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

   6. Applied egg-rr 51.0%

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

   7. Taylor expanded in l around 0 79.7%

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

      1. associate-/l* 79.8%

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

      2. *-commutative 79.8%

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

      3. associate-*r* 79.8%

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

      4. unpow2 79.8%

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

      5. unpow2 79.8%

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

      6. swap-sqr 79.8%

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

      7. unpow2 79.8%

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

      8. *-commutative 79.8%

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

      9. associate-/r* 79.9%

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

   9. Simplified 79.9%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.5 \cdot 10^{-155}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{{\left(t \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{3}}}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}}\\ \mathbf{elif}\;k \leq 2 \cdot 10^{-14}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {k}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{\frac{\cos k}{t}}{{\left(k \cdot \sin k\right)}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 76.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2.9 \cdot 10^{-105}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{\frac{\cos k}{t}}{{\left(k \cdot \sin k\right)}^{2}}\right)\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{+102}:\\ \;\;\;\;\frac{\ell}{\frac{k}{t}} \cdot \left(\frac{2}{{t}^{3}} \cdot \left(t \cdot \frac{\ell}{k \cdot \left(\sin k \cdot \tan k\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2} \cdot \left(t \cdot {k}^{2}\right)}{{\ell}^{2}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 2.9e-105)
   (* 2.0 (* (pow l 2.0) (/ (/ (cos k) t) (pow (* k (sin k)) 2.0))))
   (if (<= t 5.8e+102)
     (*
      (/ l (/ k t))
      (* (/ 2.0 (pow t 3.0)) (* t (/ l (* k (* (sin k) (tan k)))))))
     (/ 2.0 (/ (* (pow (sin k) 2.0) (* t (pow k 2.0))) (pow l 2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 2.9e-105) {
		tmp = 2.0 * (pow(l, 2.0) * ((cos(k) / t) / pow((k * sin(k)), 2.0)));
	} else if (t <= 5.8e+102) {
		tmp = (l / (k / t)) * ((2.0 / pow(t, 3.0)) * (t * (l / (k * (sin(k) * tan(k))))));
	} else {
		tmp = 2.0 / ((pow(sin(k), 2.0) * (t * pow(k, 2.0))) / pow(l, 2.0));
	}
	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 <= 2.9d-105) then
        tmp = 2.0d0 * ((l ** 2.0d0) * ((cos(k) / t) / ((k * sin(k)) ** 2.0d0)))
    else if (t <= 5.8d+102) then
        tmp = (l / (k / t)) * ((2.0d0 / (t ** 3.0d0)) * (t * (l / (k * (sin(k) * tan(k))))))
    else
        tmp = 2.0d0 / (((sin(k) ** 2.0d0) * (t * (k ** 2.0d0))) / (l ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 2.9e-105) {
		tmp = 2.0 * (Math.pow(l, 2.0) * ((Math.cos(k) / t) / Math.pow((k * Math.sin(k)), 2.0)));
	} else if (t <= 5.8e+102) {
		tmp = (l / (k / t)) * ((2.0 / Math.pow(t, 3.0)) * (t * (l / (k * (Math.sin(k) * Math.tan(k))))));
	} else {
		tmp = 2.0 / ((Math.pow(Math.sin(k), 2.0) * (t * Math.pow(k, 2.0))) / Math.pow(l, 2.0));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 2.9e-105:
		tmp = 2.0 * (math.pow(l, 2.0) * ((math.cos(k) / t) / math.pow((k * math.sin(k)), 2.0)))
	elif t <= 5.8e+102:
		tmp = (l / (k / t)) * ((2.0 / math.pow(t, 3.0)) * (t * (l / (k * (math.sin(k) * math.tan(k))))))
	else:
		tmp = 2.0 / ((math.pow(math.sin(k), 2.0) * (t * math.pow(k, 2.0))) / math.pow(l, 2.0))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 2.9e-105)
		tmp = Float64(2.0 * Float64((l ^ 2.0) * Float64(Float64(cos(k) / t) / (Float64(k * sin(k)) ^ 2.0))));
	elseif (t <= 5.8e+102)
		tmp = Float64(Float64(l / Float64(k / t)) * Float64(Float64(2.0 / (t ^ 3.0)) * Float64(t * Float64(l / Float64(k * Float64(sin(k) * tan(k)))))));
	else
		tmp = Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) * Float64(t * (k ^ 2.0))) / (l ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 2.9e-105)
		tmp = 2.0 * ((l ^ 2.0) * ((cos(k) / t) / ((k * sin(k)) ^ 2.0)));
	elseif (t <= 5.8e+102)
		tmp = (l / (k / t)) * ((2.0 / (t ^ 3.0)) * (t * (l / (k * (sin(k) * tan(k))))));
	else
		tmp = 2.0 / (((sin(k) ^ 2.0) * (t * (k ^ 2.0))) / (l ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 2.9e-105], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision] / N[Power[N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.8e+102], N[(N[(l / N[(k / t), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] * N[(t * N[(l / N[(k * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 2.90000000000000003e-105

   1. Initial program 35.6%

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

      1. associate-/r* 35.6%

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

      2. associate-/l/ 35.6%

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

      3. associate-*l/ 36.2%

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

      4. associate-/r/ 35.9%

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

      5. +-commutative 35.9%

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

      6. unpow2 35.9%

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

      7. sqr-neg 35.9%

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

      8. distribute-frac-neg2 35.9%

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

      9. distribute-frac-neg2 35.9%

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

      10. unpow2 35.9%

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

      11. +-rgt-identity 35.9%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

      12. metadata-eval 35.9%

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

      13. associate--l+ 35.9%

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

      14. +-commutative 35.9%

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

      15. associate--l+ 35.9%

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

   3. Simplified 43.6%

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

      1. associate-*r* 49.9%

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

      2. unpow2 49.9%

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

      3. times-frac 54.3%

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

      4. associate-/l/ 54.3%

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

      5. associate-*l/ 54.6%

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

      6. associate-*l* 54.6%

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

   6. Applied egg-rr 54.6%

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

   7. Taylor expanded in l around 0 79.2%

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

      1. associate-/l* 79.2%

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

      2. *-commutative 79.2%

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

      3. associate-*r* 76.9%

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

      4. unpow2 76.9%

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

      5. unpow2 76.9%

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

      6. swap-sqr 76.9%

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

      7. unpow2 76.9%

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

      8. *-commutative 76.9%

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

      9. associate-/r* 76.9%

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

   9. Simplified 76.9%

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

   if 2.90000000000000003e-105 < t < 5.8000000000000005e102

   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-/r* 70.8%

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

      2. associate-/l/ 70.9%

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

      3. associate-*l/ 73.8%

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

      4. associate-/r/ 73.8%

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

      5. +-commutative 73.8%

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

      6. unpow2 73.8%

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

      7. sqr-neg 73.8%

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

      8. distribute-frac-neg2 73.8%

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

      9. distribute-frac-neg2 73.8%

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

      10. unpow2 73.8%

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

      11. +-rgt-identity 73.8%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

      12. metadata-eval 73.8%

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

      13. associate--l+ 73.8%

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

      14. +-commutative 73.8%

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

      15. associate--l+ 73.8%

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

   3. Simplified 73.8%

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

      1. associate-*r* 82.9%

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

      2. unpow2 82.9%

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

      3. times-frac 85.9%

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

      4. associate-/l/ 85.8%

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

      5. associate-*l/ 85.9%

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

      6. associate-*l* 85.8%

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

   6. Applied egg-rr 85.8%

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

      1. div-inv 85.8%

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

      2. times-frac 88.5%

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

   8. Applied egg-rr 88.5%

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

      1. associate-*r/ 88.6%

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

      2. *-rgt-identity 88.6%

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

      3. associate-/r/ 88.4%

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

      4. associate-/l* 91.2%

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

      5. associate-*l* 91.5%

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

      6. associate-/l/ 91.5%

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

   10. Simplified 91.5%

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

   if 5.8000000000000005e102 < t

   1. Initial program 12.5%

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

   3. Simplified 22.6%

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow3 22.6%

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

      2. times-frac 40.5%

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

      3. pow2 40.5%

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

   6. Applied egg-rr 40.5%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)} \]
   7. Step-by-step derivation

      1. unpow2 40.5%

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

      2. frac-2neg 40.5%

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

      3. frac-times 20.2%

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

   8. Applied egg-rr 20.2%

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

   9. Taylor expanded in t around 0 65.6%

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

       1. associate-*r* 65.6%

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

       2. *-commutative 65.6%

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

       3. *-commutative 65.6%

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

   11. Simplified 65.6%

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

   12. Taylor expanded in k around 0 63.2%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.9 \cdot 10^{-105}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{\frac{\cos k}{t}}{{\left(k \cdot \sin k\right)}^{2}}\right)\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{+102}:\\ \;\;\;\;\frac{\ell}{\frac{k}{t}} \cdot \left(\frac{2}{{t}^{3}} \cdot \left(t \cdot \frac{\ell}{k \cdot \left(\sin k \cdot \tan k\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2} \cdot \left(t \cdot {k}^{2}\right)}{{\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 68.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin k \cdot \tan k\\ \mathbf{if}\;t \leq 2.25 \cdot 10^{-116}:\\ \;\;\;\;\frac{\frac{2 \cdot {\ell}^{2}}{{k}^{4}}}{t}\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{+102}:\\ \;\;\;\;\frac{\ell}{\frac{k}{t}} \cdot \left(\frac{2}{{t}^{3}} \cdot \left(t \cdot \frac{\ell}{k \cdot t\_1}\right)\right)\\ \mathbf{elif}\;t \leq 9.8 \cdot 10^{+138}:\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(t\_1 \cdot \frac{1}{\frac{t}{k} \cdot \frac{t}{k}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\ell}^{-2} \cdot \left(t \cdot {k}^{4}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (sin k) (tan k))))
   (if (<= t 2.25e-116)
     (/ (/ (* 2.0 (pow l 2.0)) (pow k 4.0)) t)
     (if (<= t 5.8e+102)
       (* (/ l (/ k t)) (* (/ 2.0 (pow t 3.0)) (* t (/ l (* k t_1)))))
       (if (<= t 9.8e+138)
         (/
          2.0
          (*
           (* (/ (pow t 2.0) l) (/ t l))
           (* t_1 (/ 1.0 (* (/ t k) (/ t k))))))
         (/ 2.0 (* (pow l -2.0) (* t (pow k 4.0)))))))))

C

double code(double t, double l, double k) {
	double t_1 = sin(k) * tan(k);
	double tmp;
	if (t <= 2.25e-116) {
		tmp = ((2.0 * pow(l, 2.0)) / pow(k, 4.0)) / t;
	} else if (t <= 5.8e+102) {
		tmp = (l / (k / t)) * ((2.0 / pow(t, 3.0)) * (t * (l / (k * t_1))));
	} else if (t <= 9.8e+138) {
		tmp = 2.0 / (((pow(t, 2.0) / l) * (t / l)) * (t_1 * (1.0 / ((t / k) * (t / k)))));
	} else {
		tmp = 2.0 / (pow(l, -2.0) * (t * 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) :: t_1
    real(8) :: tmp
    t_1 = sin(k) * tan(k)
    if (t <= 2.25d-116) then
        tmp = ((2.0d0 * (l ** 2.0d0)) / (k ** 4.0d0)) / t
    else if (t <= 5.8d+102) then
        tmp = (l / (k / t)) * ((2.0d0 / (t ** 3.0d0)) * (t * (l / (k * t_1))))
    else if (t <= 9.8d+138) then
        tmp = 2.0d0 / ((((t ** 2.0d0) / l) * (t / l)) * (t_1 * (1.0d0 / ((t / k) * (t / k)))))
    else
        tmp = 2.0d0 / ((l ** (-2.0d0)) * (t * (k ** 4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.sin(k) * Math.tan(k);
	double tmp;
	if (t <= 2.25e-116) {
		tmp = ((2.0 * Math.pow(l, 2.0)) / Math.pow(k, 4.0)) / t;
	} else if (t <= 5.8e+102) {
		tmp = (l / (k / t)) * ((2.0 / Math.pow(t, 3.0)) * (t * (l / (k * t_1))));
	} else if (t <= 9.8e+138) {
		tmp = 2.0 / (((Math.pow(t, 2.0) / l) * (t / l)) * (t_1 * (1.0 / ((t / k) * (t / k)))));
	} else {
		tmp = 2.0 / (Math.pow(l, -2.0) * (t * Math.pow(k, 4.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.sin(k) * math.tan(k)
	tmp = 0
	if t <= 2.25e-116:
		tmp = ((2.0 * math.pow(l, 2.0)) / math.pow(k, 4.0)) / t
	elif t <= 5.8e+102:
		tmp = (l / (k / t)) * ((2.0 / math.pow(t, 3.0)) * (t * (l / (k * t_1))))
	elif t <= 9.8e+138:
		tmp = 2.0 / (((math.pow(t, 2.0) / l) * (t / l)) * (t_1 * (1.0 / ((t / k) * (t / k)))))
	else:
		tmp = 2.0 / (math.pow(l, -2.0) * (t * math.pow(k, 4.0)))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(sin(k) * tan(k))
	tmp = 0.0
	if (t <= 2.25e-116)
		tmp = Float64(Float64(Float64(2.0 * (l ^ 2.0)) / (k ^ 4.0)) / t);
	elseif (t <= 5.8e+102)
		tmp = Float64(Float64(l / Float64(k / t)) * Float64(Float64(2.0 / (t ^ 3.0)) * Float64(t * Float64(l / Float64(k * t_1)))));
	elseif (t <= 9.8e+138)
		tmp = Float64(2.0 / Float64(Float64(Float64((t ^ 2.0) / l) * Float64(t / l)) * Float64(t_1 * Float64(1.0 / Float64(Float64(t / k) * Float64(t / k))))));
	else
		tmp = Float64(2.0 / Float64((l ^ -2.0) * Float64(t * (k ^ 4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = sin(k) * tan(k);
	tmp = 0.0;
	if (t <= 2.25e-116)
		tmp = ((2.0 * (l ^ 2.0)) / (k ^ 4.0)) / t;
	elseif (t <= 5.8e+102)
		tmp = (l / (k / t)) * ((2.0 / (t ^ 3.0)) * (t * (l / (k * t_1))));
	elseif (t <= 9.8e+138)
		tmp = 2.0 / ((((t ^ 2.0) / l) * (t / l)) * (t_1 * (1.0 / ((t / k) * (t / k)))));
	else
		tmp = 2.0 / ((l ^ -2.0) * (t * (k ^ 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 2.25e-116], N[(N[(N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[t, 5.8e+102], N[(N[(l / N[(k / t), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] * N[(t * N[(l / N[(k * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 9.8e+138], N[(2.0 / N[(N[(N[(N[Power[t, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(1.0 / N[(N[(t / k), $MachinePrecision] * N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[l, -2.0], $MachinePrecision] * N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if t < 2.25000000000000006e-116

   1. Initial program 35.0%

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

      1. associate-/r* 35.0%

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

      2. associate-/l/ 35.0%

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

      3. associate-*l/ 35.5%

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

      4. associate-/r/ 35.5%

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

      5. +-commutative 35.5%

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

      6. unpow2 35.5%

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

      7. sqr-neg 35.5%

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

      8. distribute-frac-neg2 35.5%

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

      9. distribute-frac-neg2 35.5%

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

      10. unpow2 35.5%

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

      11. +-rgt-identity 35.5%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

      12. metadata-eval 35.5%

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

      13. associate--l+ 35.5%

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

      14. +-commutative 35.5%

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

      15. associate--l+ 35.5%

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

   3. Simplified 43.4%

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

   5. Taylor expanded in k around 0 67.1%

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

      1. associate-*r/ 67.1%

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

      2. associate-/r* 67.5%

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

   7. Simplified 67.5%

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

   if 2.25000000000000006e-116 < t < 5.8000000000000005e102

   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-/r* 70.2%

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

      2. associate-/l/ 70.2%

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

      3. associate-*l/ 72.7%

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

      4. associate-/r/ 71.4%

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

      5. +-commutative 71.4%

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

      6. unpow2 71.4%

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

      7. sqr-neg 71.4%

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

      8. distribute-frac-neg2 71.4%

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

      9. distribute-frac-neg2 71.4%

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

      10. unpow2 71.4%

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

      11. +-rgt-identity 71.4%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

      12. metadata-eval 71.4%

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

      13. associate--l+ 71.4%

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

      14. +-commutative 71.4%

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

      15. associate--l+ 71.4%

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

   3. Simplified 71.4%

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

      1. associate-*r* 79.5%

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

      2. unpow2 79.5%

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

      3. times-frac 82.3%

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

      4. associate-/l/ 82.2%

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

      5. associate-*l/ 83.5%

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

      6. associate-*l* 83.5%

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

   6. Applied egg-rr 83.5%

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

      1. div-inv 83.5%

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

      2. times-frac 84.7%

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

   8. Applied egg-rr 84.7%

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

      1. associate-*r/ 84.7%

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

      2. *-rgt-identity 84.7%

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

      3. associate-/r/ 84.5%

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

      4. associate-/l* 87.1%

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

      5. associate-*l* 87.3%

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

      6. associate-/l/ 87.3%

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

   10. Simplified 87.3%

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

   if 5.8000000000000005e102 < t < 9.79999999999999966e138

   1. Initial program 12.5%

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

   3. Simplified 25.3%

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow3 25.3%

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

      2. times-frac 86.7%

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

      3. pow2 86.7%

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

   6. Applied egg-rr 86.7%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)} \]
   7. Step-by-step derivation

      1. unpow2 86.7%

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

      2. clear-num 86.7%

         \[\leadsto \frac{2}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\color{blue}{\frac{1}{\frac{t}{k}}} \cdot \frac{k}{t} + 0\right)\right)} \]

      3. clear-num 86.9%

         \[\leadsto \frac{2}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\frac{1}{\frac{t}{k}} \cdot \color{blue}{\frac{1}{\frac{t}{k}}} + 0\right)\right)} \]

      4. frac-times 87.1%

         \[\leadsto \frac{2}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\color{blue}{\frac{1 \cdot 1}{\frac{t}{k} \cdot \frac{t}{k}}} + 0\right)\right)} \]

      5. metadata-eval 87.1%

         \[\leadsto \frac{2}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\frac{\color{blue}{1}}{\frac{t}{k} \cdot \frac{t}{k}} + 0\right)\right)} \]

   8. Applied egg-rr 87.1%

      \[\leadsto \frac{2}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\color{blue}{\frac{1}{\frac{t}{k} \cdot \frac{t}{k}}} + 0\right)\right)} \]

   if 9.79999999999999966e138 < t

   1. Initial program 12.5%

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

   3. Simplified 21.9%

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 60.5%

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

      1. *-commutative 60.5%

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

      2. associate-/l* 54.7%

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

   7. Simplified 54.7%

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

      1. pow1 54.7%

         \[\leadsto \frac{2}{\color{blue}{{\left(t \cdot \frac{{k}^{4}}{{\ell}^{2}}\right)}^{1}}} \]

      2. div-inv 54.7%

         \[\leadsto \frac{2}{{\left(t \cdot \color{blue}{\left({k}^{4} \cdot \frac{1}{{\ell}^{2}}\right)}\right)}^{1}} \]

      3. pow-flip 54.9%

         \[\leadsto \frac{2}{{\left(t \cdot \left({k}^{4} \cdot \color{blue}{{\ell}^{\left(-2\right)}}\right)\right)}^{1}} \]

      4. metadata-eval 54.9%

         \[\leadsto \frac{2}{{\left(t \cdot \left({k}^{4} \cdot {\ell}^{\color{blue}{-2}}\right)\right)}^{1}} \]

   9. Applied egg-rr 54.9%

      \[\leadsto \frac{2}{\color{blue}{{\left(t \cdot \left({k}^{4} \cdot {\ell}^{-2}\right)\right)}^{1}}} \]
   10. Step-by-step derivation

       1. unpow1 54.9%

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

       2. associate-*r* 60.8%

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

   11. Simplified 60.8%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.25 \cdot 10^{-116}:\\ \;\;\;\;\frac{\frac{2 \cdot {\ell}^{2}}{{k}^{4}}}{t}\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{+102}:\\ \;\;\;\;\frac{\ell}{\frac{k}{t}} \cdot \left(\frac{2}{{t}^{3}} \cdot \left(t \cdot \frac{\ell}{k \cdot \left(\sin k \cdot \tan k\right)}\right)\right)\\ \mathbf{elif}\;t \leq 9.8 \cdot 10^{+138}:\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \frac{1}{\frac{t}{k} \cdot \frac{t}{k}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\ell}^{-2} \cdot \left(t \cdot {k}^{4}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 68.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin k \cdot \tan k\\ \mathbf{if}\;t \leq 5.8 \cdot 10^{-116}:\\ \;\;\;\;\frac{\frac{2 \cdot {\ell}^{2}}{{k}^{4}}}{t}\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{+102}:\\ \;\;\;\;\frac{\ell}{\frac{k}{t}} \cdot \left(\frac{2}{{t}^{3}} \cdot \left(t \cdot \frac{\ell}{k \cdot t\_1}\right)\right)\\ \mathbf{elif}\;t \leq 3.65 \cdot 10^{+140}:\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(t\_1 \cdot \frac{\frac{k}{t}}{\frac{t}{k}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\ell}^{-2} \cdot \left(t \cdot {k}^{4}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (sin k) (tan k))))
   (if (<= t 5.8e-116)
     (/ (/ (* 2.0 (pow l 2.0)) (pow k 4.0)) t)
     (if (<= t 5.8e+102)
       (* (/ l (/ k t)) (* (/ 2.0 (pow t 3.0)) (* t (/ l (* k t_1)))))
       (if (<= t 3.65e+140)
         (/ 2.0 (* (* (/ (pow t 2.0) l) (/ t l)) (* t_1 (/ (/ k t) (/ t k)))))
         (/ 2.0 (* (pow l -2.0) (* t (pow k 4.0)))))))))

C

double code(double t, double l, double k) {
	double t_1 = sin(k) * tan(k);
	double tmp;
	if (t <= 5.8e-116) {
		tmp = ((2.0 * pow(l, 2.0)) / pow(k, 4.0)) / t;
	} else if (t <= 5.8e+102) {
		tmp = (l / (k / t)) * ((2.0 / pow(t, 3.0)) * (t * (l / (k * t_1))));
	} else if (t <= 3.65e+140) {
		tmp = 2.0 / (((pow(t, 2.0) / l) * (t / l)) * (t_1 * ((k / t) / (t / k))));
	} else {
		tmp = 2.0 / (pow(l, -2.0) * (t * 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) :: t_1
    real(8) :: tmp
    t_1 = sin(k) * tan(k)
    if (t <= 5.8d-116) then
        tmp = ((2.0d0 * (l ** 2.0d0)) / (k ** 4.0d0)) / t
    else if (t <= 5.8d+102) then
        tmp = (l / (k / t)) * ((2.0d0 / (t ** 3.0d0)) * (t * (l / (k * t_1))))
    else if (t <= 3.65d+140) then
        tmp = 2.0d0 / ((((t ** 2.0d0) / l) * (t / l)) * (t_1 * ((k / t) / (t / k))))
    else
        tmp = 2.0d0 / ((l ** (-2.0d0)) * (t * (k ** 4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.sin(k) * Math.tan(k);
	double tmp;
	if (t <= 5.8e-116) {
		tmp = ((2.0 * Math.pow(l, 2.0)) / Math.pow(k, 4.0)) / t;
	} else if (t <= 5.8e+102) {
		tmp = (l / (k / t)) * ((2.0 / Math.pow(t, 3.0)) * (t * (l / (k * t_1))));
	} else if (t <= 3.65e+140) {
		tmp = 2.0 / (((Math.pow(t, 2.0) / l) * (t / l)) * (t_1 * ((k / t) / (t / k))));
	} else {
		tmp = 2.0 / (Math.pow(l, -2.0) * (t * Math.pow(k, 4.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.sin(k) * math.tan(k)
	tmp = 0
	if t <= 5.8e-116:
		tmp = ((2.0 * math.pow(l, 2.0)) / math.pow(k, 4.0)) / t
	elif t <= 5.8e+102:
		tmp = (l / (k / t)) * ((2.0 / math.pow(t, 3.0)) * (t * (l / (k * t_1))))
	elif t <= 3.65e+140:
		tmp = 2.0 / (((math.pow(t, 2.0) / l) * (t / l)) * (t_1 * ((k / t) / (t / k))))
	else:
		tmp = 2.0 / (math.pow(l, -2.0) * (t * math.pow(k, 4.0)))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(sin(k) * tan(k))
	tmp = 0.0
	if (t <= 5.8e-116)
		tmp = Float64(Float64(Float64(2.0 * (l ^ 2.0)) / (k ^ 4.0)) / t);
	elseif (t <= 5.8e+102)
		tmp = Float64(Float64(l / Float64(k / t)) * Float64(Float64(2.0 / (t ^ 3.0)) * Float64(t * Float64(l / Float64(k * t_1)))));
	elseif (t <= 3.65e+140)
		tmp = Float64(2.0 / Float64(Float64(Float64((t ^ 2.0) / l) * Float64(t / l)) * Float64(t_1 * Float64(Float64(k / t) / Float64(t / k)))));
	else
		tmp = Float64(2.0 / Float64((l ^ -2.0) * Float64(t * (k ^ 4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = sin(k) * tan(k);
	tmp = 0.0;
	if (t <= 5.8e-116)
		tmp = ((2.0 * (l ^ 2.0)) / (k ^ 4.0)) / t;
	elseif (t <= 5.8e+102)
		tmp = (l / (k / t)) * ((2.0 / (t ^ 3.0)) * (t * (l / (k * t_1))));
	elseif (t <= 3.65e+140)
		tmp = 2.0 / ((((t ^ 2.0) / l) * (t / l)) * (t_1 * ((k / t) / (t / k))));
	else
		tmp = 2.0 / ((l ^ -2.0) * (t * (k ^ 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 5.8e-116], N[(N[(N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[t, 5.8e+102], N[(N[(l / N[(k / t), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] * N[(t * N[(l / N[(k * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.65e+140], N[(2.0 / N[(N[(N[(N[Power[t, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(N[(k / t), $MachinePrecision] / N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[l, -2.0], $MachinePrecision] * N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if t < 5.7999999999999996e-116

   1. Initial program 35.0%

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

      1. associate-/r* 35.0%

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

      2. associate-/l/ 35.0%

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

      3. associate-*l/ 35.5%

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

      4. associate-/r/ 35.5%

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

      5. +-commutative 35.5%

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

      6. unpow2 35.5%

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

      7. sqr-neg 35.5%

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

      8. distribute-frac-neg2 35.5%

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

      9. distribute-frac-neg2 35.5%

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

      10. unpow2 35.5%

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

      11. +-rgt-identity 35.5%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

      12. metadata-eval 35.5%

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

      13. associate--l+ 35.5%

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

      14. +-commutative 35.5%

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

      15. associate--l+ 35.5%

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

   3. Simplified 43.4%

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

   5. Taylor expanded in k around 0 67.1%

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

      1. associate-*r/ 67.1%

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

      2. associate-/r* 67.5%

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

   7. Simplified 67.5%

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

   if 5.7999999999999996e-116 < t < 5.8000000000000005e102

   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-/r* 70.2%

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

      2. associate-/l/ 70.2%

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

      3. associate-*l/ 72.7%

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

      4. associate-/r/ 71.4%

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

      5. +-commutative 71.4%

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

      6. unpow2 71.4%

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

      7. sqr-neg 71.4%

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

      8. distribute-frac-neg2 71.4%

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

      9. distribute-frac-neg2 71.4%

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

      10. unpow2 71.4%

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

      11. +-rgt-identity 71.4%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

      12. metadata-eval 71.4%

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

      13. associate--l+ 71.4%

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

      14. +-commutative 71.4%

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

      15. associate--l+ 71.4%

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

   3. Simplified 71.4%

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

      1. associate-*r* 79.5%

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

      2. unpow2 79.5%

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

      3. times-frac 82.3%

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

      4. associate-/l/ 82.2%

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

      5. associate-*l/ 83.5%

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

      6. associate-*l* 83.5%

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

   6. Applied egg-rr 83.5%

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

      1. div-inv 83.5%

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

      2. times-frac 84.7%

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

   8. Applied egg-rr 84.7%

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

      1. associate-*r/ 84.7%

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

      2. *-rgt-identity 84.7%

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

      3. associate-/r/ 84.5%

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

      4. associate-/l* 87.1%

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

      5. associate-*l* 87.3%

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

      6. associate-/l/ 87.3%

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

   10. Simplified 87.3%

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

   if 5.8000000000000005e102 < t < 3.6500000000000002e140

   1. Initial program 12.5%

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

   3. Simplified 25.3%

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. unpow3 25.3%

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

      2. times-frac 86.7%

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

      3. pow2 86.7%

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

   6. Applied egg-rr 86.7%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)} \]
   7. Step-by-step derivation

      1. unpow2 86.7%

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

      2. clear-num 86.7%

         \[\leadsto \frac{2}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\frac{k}{t} \cdot \color{blue}{\frac{1}{\frac{t}{k}}} + 0\right)\right)} \]

      3. un-div-inv 86.9%

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

   8. Applied egg-rr 86.9%

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

   if 3.6500000000000002e140 < t

   1. Initial program 12.5%

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

   3. Simplified 21.9%

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 60.5%

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

      1. *-commutative 60.5%

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

      2. associate-/l* 54.7%

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

   7. Simplified 54.7%

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

      1. pow1 54.7%

         \[\leadsto \frac{2}{\color{blue}{{\left(t \cdot \frac{{k}^{4}}{{\ell}^{2}}\right)}^{1}}} \]

      2. div-inv 54.7%

         \[\leadsto \frac{2}{{\left(t \cdot \color{blue}{\left({k}^{4} \cdot \frac{1}{{\ell}^{2}}\right)}\right)}^{1}} \]

      3. pow-flip 54.9%

         \[\leadsto \frac{2}{{\left(t \cdot \left({k}^{4} \cdot \color{blue}{{\ell}^{\left(-2\right)}}\right)\right)}^{1}} \]

      4. metadata-eval 54.9%

         \[\leadsto \frac{2}{{\left(t \cdot \left({k}^{4} \cdot {\ell}^{\color{blue}{-2}}\right)\right)}^{1}} \]

   9. Applied egg-rr 54.9%

      \[\leadsto \frac{2}{\color{blue}{{\left(t \cdot \left({k}^{4} \cdot {\ell}^{-2}\right)\right)}^{1}}} \]
   10. Step-by-step derivation

       1. unpow1 54.9%

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

       2. associate-*r* 60.8%

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

   11. Simplified 60.8%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.8 \cdot 10^{-116}:\\ \;\;\;\;\frac{\frac{2 \cdot {\ell}^{2}}{{k}^{4}}}{t}\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{+102}:\\ \;\;\;\;\frac{\ell}{\frac{k}{t}} \cdot \left(\frac{2}{{t}^{3}} \cdot \left(t \cdot \frac{\ell}{k \cdot \left(\sin k \cdot \tan k\right)}\right)\right)\\ \mathbf{elif}\;t \leq 3.65 \cdot 10^{+140}:\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \frac{\frac{k}{t}}{\frac{t}{k}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\ell}^{-2} \cdot \left(t \cdot {k}^{4}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 68.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.55 \cdot 10^{-116}:\\ \;\;\;\;\frac{\frac{2 \cdot {\ell}^{2}}{{k}^{4}}}{t}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{+116}:\\ \;\;\;\;\frac{\ell}{\frac{k}{t}} \cdot \left(\frac{2}{{t}^{3}} \cdot \left(t \cdot \frac{\ell}{k \cdot \left(\sin k \cdot \tan k\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\ell}^{-2} \cdot \left(t \cdot {k}^{4}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 1.55e-116)
   (/ (/ (* 2.0 (pow l 2.0)) (pow k 4.0)) t)
   (if (<= t 1.1e+116)
     (*
      (/ l (/ k t))
      (* (/ 2.0 (pow t 3.0)) (* t (/ l (* k (* (sin k) (tan k)))))))
     (/ 2.0 (* (pow l -2.0) (* t (pow k 4.0)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 1.55e-116) {
		tmp = ((2.0 * pow(l, 2.0)) / pow(k, 4.0)) / t;
	} else if (t <= 1.1e+116) {
		tmp = (l / (k / t)) * ((2.0 / pow(t, 3.0)) * (t * (l / (k * (sin(k) * tan(k))))));
	} else {
		tmp = 2.0 / (pow(l, -2.0) * (t * 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 <= 1.55d-116) then
        tmp = ((2.0d0 * (l ** 2.0d0)) / (k ** 4.0d0)) / t
    else if (t <= 1.1d+116) then
        tmp = (l / (k / t)) * ((2.0d0 / (t ** 3.0d0)) * (t * (l / (k * (sin(k) * tan(k))))))
    else
        tmp = 2.0d0 / ((l ** (-2.0d0)) * (t * (k ** 4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 1.55e-116) {
		tmp = ((2.0 * Math.pow(l, 2.0)) / Math.pow(k, 4.0)) / t;
	} else if (t <= 1.1e+116) {
		tmp = (l / (k / t)) * ((2.0 / Math.pow(t, 3.0)) * (t * (l / (k * (Math.sin(k) * Math.tan(k))))));
	} else {
		tmp = 2.0 / (Math.pow(l, -2.0) * (t * Math.pow(k, 4.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 1.55e-116:
		tmp = ((2.0 * math.pow(l, 2.0)) / math.pow(k, 4.0)) / t
	elif t <= 1.1e+116:
		tmp = (l / (k / t)) * ((2.0 / math.pow(t, 3.0)) * (t * (l / (k * (math.sin(k) * math.tan(k))))))
	else:
		tmp = 2.0 / (math.pow(l, -2.0) * (t * math.pow(k, 4.0)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 1.55e-116)
		tmp = Float64(Float64(Float64(2.0 * (l ^ 2.0)) / (k ^ 4.0)) / t);
	elseif (t <= 1.1e+116)
		tmp = Float64(Float64(l / Float64(k / t)) * Float64(Float64(2.0 / (t ^ 3.0)) * Float64(t * Float64(l / Float64(k * Float64(sin(k) * tan(k)))))));
	else
		tmp = Float64(2.0 / Float64((l ^ -2.0) * Float64(t * (k ^ 4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 1.55e-116)
		tmp = ((2.0 * (l ^ 2.0)) / (k ^ 4.0)) / t;
	elseif (t <= 1.1e+116)
		tmp = (l / (k / t)) * ((2.0 / (t ^ 3.0)) * (t * (l / (k * (sin(k) * tan(k))))));
	else
		tmp = 2.0 / ((l ^ -2.0) * (t * (k ^ 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 1.55e-116], N[(N[(N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[t, 1.1e+116], N[(N[(l / N[(k / t), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] * N[(t * N[(l / N[(k * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[l, -2.0], $MachinePrecision] * N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.55000000000000009e-116

   1. Initial program 35.0%

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

      1. associate-/r* 35.0%

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

      2. associate-/l/ 35.0%

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

      3. associate-*l/ 35.5%

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

      4. associate-/r/ 35.5%

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

      5. +-commutative 35.5%

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

      6. unpow2 35.5%

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

      7. sqr-neg 35.5%

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

      8. distribute-frac-neg2 35.5%

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

      9. distribute-frac-neg2 35.5%

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

      10. unpow2 35.5%

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

      11. +-rgt-identity 35.5%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

      12. metadata-eval 35.5%

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

      13. associate--l+ 35.5%

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

      14. +-commutative 35.5%

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

      15. associate--l+ 35.5%

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

   3. Simplified 43.4%

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

   5. Taylor expanded in k around 0 67.1%

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

      1. associate-*r/ 67.1%

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

      2. associate-/r* 67.5%

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

   7. Simplified 67.5%

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

   if 1.55000000000000009e-116 < t < 1.1e116

   1. Initial program 65.1%

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

      1. associate-/r* 65.1%

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

      2. associate-/l/ 65.1%

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

      3. associate-*l/ 67.4%

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

      4. associate-/r/ 66.1%

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

      5. +-commutative 66.1%

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

      6. unpow2 66.1%

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

      7. sqr-neg 66.1%

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

      8. distribute-frac-neg2 66.1%

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

      9. distribute-frac-neg2 66.1%

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

      10. unpow2 66.1%

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

      11. +-rgt-identity 66.1%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

      12. metadata-eval 66.1%

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

      13. associate--l+ 66.1%

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

      14. +-commutative 66.1%

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

      15. associate--l+ 66.1%

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

   3. Simplified 68.6%

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

      1. associate-*r* 76.3%

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

      2. unpow2 76.3%

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

      3. times-frac 79.0%

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

      4. associate-/l/ 78.9%

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

      5. associate-*l/ 80.1%

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

      6. associate-*l* 80.1%

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

   6. Applied egg-rr 80.1%

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

      1. div-inv 80.0%

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

      2. times-frac 81.1%

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

   8. Applied egg-rr 81.1%

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

      1. associate-*r/ 81.2%

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

      2. *-rgt-identity 81.2%

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

      3. associate-/r/ 81.0%

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

      4. associate-/l* 83.4%

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

      5. associate-*l* 83.6%

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

      6. associate-/l/ 83.6%

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

   10. Simplified 83.6%

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

   if 1.1e116 < t

   1. Initial program 13.5%

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

   3. Simplified 21.7%

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 58.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
   6. Step-by-step derivation

      1. *-commutative 58.0%

         \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {k}^{4}}}{{\ell}^{2}}} \]

      2. associate-/l* 52.9%

         \[\leadsto \frac{2}{\color{blue}{t \cdot \frac{{k}^{4}}{{\ell}^{2}}}} \]

   7. Simplified 52.9%

      \[\leadsto \frac{2}{\color{blue}{t \cdot \frac{{k}^{4}}{{\ell}^{2}}}} \]
   8. Step-by-step derivation

      1. pow1 52.9%

         \[\leadsto \frac{2}{\color{blue}{{\left(t \cdot \frac{{k}^{4}}{{\ell}^{2}}\right)}^{1}}} \]

      2. div-inv 52.9%

         \[\leadsto \frac{2}{{\left(t \cdot \color{blue}{\left({k}^{4} \cdot \frac{1}{{\ell}^{2}}\right)}\right)}^{1}} \]

      3. pow-flip 53.1%

         \[\leadsto \frac{2}{{\left(t \cdot \left({k}^{4} \cdot \color{blue}{{\ell}^{\left(-2\right)}}\right)\right)}^{1}} \]

      4. metadata-eval 53.1%

         \[\leadsto \frac{2}{{\left(t \cdot \left({k}^{4} \cdot {\ell}^{\color{blue}{-2}}\right)\right)}^{1}} \]

   9. Applied egg-rr 53.1%

      \[\leadsto \frac{2}{\color{blue}{{\left(t \cdot \left({k}^{4} \cdot {\ell}^{-2}\right)\right)}^{1}}} \]
   10. Step-by-step derivation

       1. unpow1 53.1%

          \[\leadsto \frac{2}{\color{blue}{t \cdot \left({k}^{4} \cdot {\ell}^{-2}\right)}} \]

       2. associate-*r* 58.2%

          \[\leadsto \frac{2}{\color{blue}{\left(t \cdot {k}^{4}\right) \cdot {\ell}^{-2}}} \]

   11. Simplified 58.2%

       \[\leadsto \frac{2}{\color{blue}{\left(t \cdot {k}^{4}\right) \cdot {\ell}^{-2}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.55 \cdot 10^{-116}:\\ \;\;\;\;\frac{\frac{2 \cdot {\ell}^{2}}{{k}^{4}}}{t}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{+116}:\\ \;\;\;\;\frac{\ell}{\frac{k}{t}} \cdot \left(\frac{2}{{t}^{3}} \cdot \left(t \cdot \frac{\ell}{k \cdot \left(\sin k \cdot \tan k\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\ell}^{-2} \cdot \left(t \cdot {k}^{4}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 61.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.4 \cdot 10^{-194}:\\ \;\;\;\;\frac{\ell}{\frac{k}{t}} \cdot \left(2 \cdot \frac{\frac{\ell}{{k}^{3}}}{{t}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{\cos k}{t \cdot {k}^{4}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= l 2.4e-194)
   (* (/ l (/ k t)) (* 2.0 (/ (/ l (pow k 3.0)) (pow t 2.0))))
   (* 2.0 (* (pow l 2.0) (/ (cos k) (* t (pow k 4.0)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (l <= 2.4e-194) {
		tmp = (l / (k / t)) * (2.0 * ((l / pow(k, 3.0)) / pow(t, 2.0)));
	} else {
		tmp = 2.0 * (pow(l, 2.0) * (cos(k) / (t * 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 (l <= 2.4d-194) then
        tmp = (l / (k / t)) * (2.0d0 * ((l / (k ** 3.0d0)) / (t ** 2.0d0)))
    else
        tmp = 2.0d0 * ((l ** 2.0d0) * (cos(k) / (t * (k ** 4.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (l <= 2.4e-194) {
		tmp = (l / (k / t)) * (2.0 * ((l / Math.pow(k, 3.0)) / Math.pow(t, 2.0)));
	} else {
		tmp = 2.0 * (Math.pow(l, 2.0) * (Math.cos(k) / (t * Math.pow(k, 4.0))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if l <= 2.4e-194:
		tmp = (l / (k / t)) * (2.0 * ((l / math.pow(k, 3.0)) / math.pow(t, 2.0)))
	else:
		tmp = 2.0 * (math.pow(l, 2.0) * (math.cos(k) / (t * math.pow(k, 4.0))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (l <= 2.4e-194)
		tmp = Float64(Float64(l / Float64(k / t)) * Float64(2.0 * Float64(Float64(l / (k ^ 3.0)) / (t ^ 2.0))));
	else
		tmp = Float64(2.0 * Float64((l ^ 2.0) * Float64(cos(k) / Float64(t * (k ^ 4.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (l <= 2.4e-194)
		tmp = (l / (k / t)) * (2.0 * ((l / (k ^ 3.0)) / (t ^ 2.0)));
	else
		tmp = 2.0 * ((l ^ 2.0) * (cos(k) / (t * (k ^ 4.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[l, 2.4e-194], N[(N[(l / N[(k / t), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[(l / N[Power[k, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[t, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.4e-194

   1. Initial program 29.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 29.5%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

      2. associate-/l/ 29.5%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      3. associate-*l/ 30.7%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      4. associate-/r/ 30.7%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      5. +-commutative 30.7%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \]

      6. unpow2 30.7%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) - 1} \]

      7. sqr-neg 30.7%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)} + 1\right) - 1} \]

      8. distribute-frac-neg2 30.7%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\left(-\frac{k}{t}\right) \cdot \color{blue}{\frac{k}{-t}} + 1\right) - 1} \]

      9. distribute-frac-neg2 30.7%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\frac{k}{-t}} \cdot \frac{k}{-t} + 1\right) - 1} \]

      10. unpow2 30.7%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{{\left(\frac{k}{-t}\right)}^{2}} + 1\right) - 1} \]

      11. +-rgt-identity 30.7%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

      12. metadata-eval 30.7%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\left({\left(\frac{k}{-t}\right)}^{2} + \color{blue}{\left(1 - 1\right)}\right) + 1\right) - 1} \]

      13. associate--l+ 30.7%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left(\left({\left(\frac{k}{-t}\right)}^{2} + 1\right) - 1\right)} + 1\right) - 1} \]

      14. +-commutative 30.7%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\left(\color{blue}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)} - 1\right) + 1\right) - 1} \]

      15. associate--l+ 30.7%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) - 1\right) + \left(1 - 1\right)}} \]

   3. Simplified 40.1%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{{\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* 48.7%

         \[\leadsto \frac{\color{blue}{\left(\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell\right) \cdot \ell}}{{\left(\frac{k}{t}\right)}^{2}} \]

      2. unpow2 48.7%

         \[\leadsto \frac{\left(\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell\right) \cdot \ell}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]

      3. times-frac 52.6%

         \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}}} \]

      4. associate-/l/ 52.6%

         \[\leadsto \frac{\color{blue}{\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \cdot \ell}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}} \]

      5. associate-*l/ 52.6%

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot \ell}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}} \]

      6. associate-*l* 52.6%

         \[\leadsto \frac{\frac{2 \cdot \ell}{\color{blue}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}}}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}} \]

   6. Applied egg-rr 52.6%

      \[\leadsto \color{blue}{\frac{\frac{2 \cdot \ell}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}}} \]

   7. Taylor expanded in k around 0 57.7%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{\ell}{{k}^{3} \cdot {t}^{2}}\right)} \cdot \frac{\ell}{\frac{k}{t}} \]
   8. Step-by-step derivation

      1. associate-/r* 58.4%

         \[\leadsto \left(2 \cdot \color{blue}{\frac{\frac{\ell}{{k}^{3}}}{{t}^{2}}}\right) \cdot \frac{\ell}{\frac{k}{t}} \]

   9. Simplified 58.4%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{\frac{\ell}{{k}^{3}}}{{t}^{2}}\right)} \cdot \frac{\ell}{\frac{k}{t}} \]

   if 2.4e-194 < l

   1. Initial program 48.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 48.7%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

      2. associate-/l/ 48.7%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      3. associate-*l/ 48.7%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      4. associate-/r/ 48.1%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      5. +-commutative 48.1%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \]

      6. unpow2 48.1%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) - 1} \]

      7. sqr-neg 48.1%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)} + 1\right) - 1} \]

      8. distribute-frac-neg2 48.1%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\left(-\frac{k}{t}\right) \cdot \color{blue}{\frac{k}{-t}} + 1\right) - 1} \]

      9. distribute-frac-neg2 48.1%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\frac{k}{-t}} \cdot \frac{k}{-t} + 1\right) - 1} \]

      10. unpow2 48.1%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{{\left(\frac{k}{-t}\right)}^{2}} + 1\right) - 1} \]

      11. +-rgt-identity 48.1%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

      12. metadata-eval 48.1%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\left({\left(\frac{k}{-t}\right)}^{2} + \color{blue}{\left(1 - 1\right)}\right) + 1\right) - 1} \]

      13. associate--l+ 48.1%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left(\left({\left(\frac{k}{-t}\right)}^{2} + 1\right) - 1\right)} + 1\right) - 1} \]

      14. +-commutative 48.1%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\left(\color{blue}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)} - 1\right) + 1\right) - 1} \]

      15. associate--l+ 48.1%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) - 1\right) + \left(1 - 1\right)}} \]

   3. Simplified 51.3%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{{\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around inf 89.8%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* 89.8%

         \[\leadsto 2 \cdot \color{blue}{\left({\ell}^{2} \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\right)} \]

   7. Simplified 89.8%

      \[\leadsto \color{blue}{2 \cdot \left({\ell}^{2} \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\right)} \]

   8. Taylor expanded in k around 0 75.8%

      \[\leadsto 2 \cdot \left({\ell}^{2} \cdot \frac{\cos k}{\color{blue}{{k}^{4} \cdot t}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.4 \cdot 10^{-194}:\\ \;\;\;\;\frac{\ell}{\frac{k}{t}} \cdot \left(2 \cdot \frac{\frac{\ell}{{k}^{3}}}{{t}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{\cos k}{t \cdot {k}^{4}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 65.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 3.3 \cdot 10^{-161}:\\ \;\;\;\;\frac{2}{t \cdot \frac{{k}^{4}}{{\ell}^{2}}}\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{+171}:\\ \;\;\;\;\frac{\ell}{\frac{k}{t}} \cdot \left(2 \cdot \frac{\frac{\ell}{{k}^{3}}}{{t}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\ell}^{-2} \cdot \left(t \cdot {k}^{4}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 3.3e-161)
   (/ 2.0 (* t (/ (pow k 4.0) (pow l 2.0))))
   (if (<= t 2.8e+171)
     (* (/ l (/ k t)) (* 2.0 (/ (/ l (pow k 3.0)) (pow t 2.0))))
     (/ 2.0 (* (pow l -2.0) (* t (pow k 4.0)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 3.3e-161) {
		tmp = 2.0 / (t * (pow(k, 4.0) / pow(l, 2.0)));
	} else if (t <= 2.8e+171) {
		tmp = (l / (k / t)) * (2.0 * ((l / pow(k, 3.0)) / pow(t, 2.0)));
	} else {
		tmp = 2.0 / (pow(l, -2.0) * (t * 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 <= 3.3d-161) then
        tmp = 2.0d0 / (t * ((k ** 4.0d0) / (l ** 2.0d0)))
    else if (t <= 2.8d+171) then
        tmp = (l / (k / t)) * (2.0d0 * ((l / (k ** 3.0d0)) / (t ** 2.0d0)))
    else
        tmp = 2.0d0 / ((l ** (-2.0d0)) * (t * (k ** 4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 3.3e-161) {
		tmp = 2.0 / (t * (Math.pow(k, 4.0) / Math.pow(l, 2.0)));
	} else if (t <= 2.8e+171) {
		tmp = (l / (k / t)) * (2.0 * ((l / Math.pow(k, 3.0)) / Math.pow(t, 2.0)));
	} else {
		tmp = 2.0 / (Math.pow(l, -2.0) * (t * Math.pow(k, 4.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 3.3e-161:
		tmp = 2.0 / (t * (math.pow(k, 4.0) / math.pow(l, 2.0)))
	elif t <= 2.8e+171:
		tmp = (l / (k / t)) * (2.0 * ((l / math.pow(k, 3.0)) / math.pow(t, 2.0)))
	else:
		tmp = 2.0 / (math.pow(l, -2.0) * (t * math.pow(k, 4.0)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 3.3e-161)
		tmp = Float64(2.0 / Float64(t * Float64((k ^ 4.0) / (l ^ 2.0))));
	elseif (t <= 2.8e+171)
		tmp = Float64(Float64(l / Float64(k / t)) * Float64(2.0 * Float64(Float64(l / (k ^ 3.0)) / (t ^ 2.0))));
	else
		tmp = Float64(2.0 / Float64((l ^ -2.0) * Float64(t * (k ^ 4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 3.3e-161)
		tmp = 2.0 / (t * ((k ^ 4.0) / (l ^ 2.0)));
	elseif (t <= 2.8e+171)
		tmp = (l / (k / t)) * (2.0 * ((l / (k ^ 3.0)) / (t ^ 2.0)));
	else
		tmp = 2.0 / ((l ^ -2.0) * (t * (k ^ 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 3.3e-161], N[(2.0 / N[(t * N[(N[Power[k, 4.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.8e+171], N[(N[(l / N[(k / t), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[(l / N[Power[k, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[t, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[l, -2.0], $MachinePrecision] * N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 3.2999999999999998e-161

   1. Initial program 36.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 44.3%

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 67.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
   6. Step-by-step derivation

      1. *-commutative 67.0%

         \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {k}^{4}}}{{\ell}^{2}}} \]

      2. associate-/l* 67.4%

         \[\leadsto \frac{2}{\color{blue}{t \cdot \frac{{k}^{4}}{{\ell}^{2}}}} \]

   7. Simplified 67.4%

      \[\leadsto \frac{2}{\color{blue}{t \cdot \frac{{k}^{4}}{{\ell}^{2}}}} \]

   if 3.2999999999999998e-161 < t < 2.80000000000000004e171

   1. Initial program 47.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 47.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

      2. associate-/l/ 47.4%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      3. associate-*l/ 48.8%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      4. associate-/r/ 48.1%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      5. +-commutative 48.1%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \]

      6. unpow2 48.1%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) - 1} \]

      7. sqr-neg 48.1%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)} + 1\right) - 1} \]

      8. distribute-frac-neg2 48.1%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\left(-\frac{k}{t}\right) \cdot \color{blue}{\frac{k}{-t}} + 1\right) - 1} \]

      9. distribute-frac-neg2 48.1%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\frac{k}{-t}} \cdot \frac{k}{-t} + 1\right) - 1} \]

      10. unpow2 48.1%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{{\left(\frac{k}{-t}\right)}^{2}} + 1\right) - 1} \]

      11. +-rgt-identity 48.1%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

      12. metadata-eval 48.1%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\left({\left(\frac{k}{-t}\right)}^{2} + \color{blue}{\left(1 - 1\right)}\right) + 1\right) - 1} \]

      13. associate--l+ 48.1%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left(\left({\left(\frac{k}{-t}\right)}^{2} + 1\right) - 1\right)} + 1\right) - 1} \]

      14. +-commutative 48.1%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\left(\color{blue}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)} - 1\right) + 1\right) - 1} \]

      15. associate--l+ 48.1%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) - 1\right) + \left(1 - 1\right)}} \]

   3. Simplified 49.6%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{{\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* 56.0%

         \[\leadsto \frac{\color{blue}{\left(\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell\right) \cdot \ell}}{{\left(\frac{k}{t}\right)}^{2}} \]

      2. unpow2 56.0%

         \[\leadsto \frac{\left(\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell\right) \cdot \ell}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]

      3. times-frac 59.2%

         \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}}} \]

      4. associate-/l/ 59.2%

         \[\leadsto \frac{\color{blue}{\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \cdot \ell}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}} \]

      5. associate-*l/ 59.9%

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot \ell}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}} \]

      6. associate-*l* 59.9%

         \[\leadsto \frac{\frac{2 \cdot \ell}{\color{blue}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}}}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}} \]

   6. Applied egg-rr 59.9%

      \[\leadsto \color{blue}{\frac{\frac{2 \cdot \ell}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}}} \]

   7. Taylor expanded in k around 0 67.2%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{\ell}{{k}^{3} \cdot {t}^{2}}\right)} \cdot \frac{\ell}{\frac{k}{t}} \]
   8. Step-by-step derivation

      1. associate-/r* 70.1%

         \[\leadsto \left(2 \cdot \color{blue}{\frac{\frac{\ell}{{k}^{3}}}{{t}^{2}}}\right) \cdot \frac{\ell}{\frac{k}{t}} \]

   9. Simplified 70.1%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{\frac{\ell}{{k}^{3}}}{{t}^{2}}\right)} \cdot \frac{\ell}{\frac{k}{t}} \]

   if 2.80000000000000004e171 < t

   1. Initial program 11.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   3. Simplified 23.1%

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 59.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
   6. Step-by-step derivation

      1. *-commutative 59.1%

         \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {k}^{4}}}{{\ell}^{2}}} \]

      2. associate-/l* 52.0%

         \[\leadsto \frac{2}{\color{blue}{t \cdot \frac{{k}^{4}}{{\ell}^{2}}}} \]

   7. Simplified 52.0%

      \[\leadsto \frac{2}{\color{blue}{t \cdot \frac{{k}^{4}}{{\ell}^{2}}}} \]
   8. Step-by-step derivation

      1. pow1 52.0%

         \[\leadsto \frac{2}{\color{blue}{{\left(t \cdot \frac{{k}^{4}}{{\ell}^{2}}\right)}^{1}}} \]

      2. div-inv 52.0%

         \[\leadsto \frac{2}{{\left(t \cdot \color{blue}{\left({k}^{4} \cdot \frac{1}{{\ell}^{2}}\right)}\right)}^{1}} \]

      3. pow-flip 52.3%

         \[\leadsto \frac{2}{{\left(t \cdot \left({k}^{4} \cdot \color{blue}{{\ell}^{\left(-2\right)}}\right)\right)}^{1}} \]

      4. metadata-eval 52.3%

         \[\leadsto \frac{2}{{\left(t \cdot \left({k}^{4} \cdot {\ell}^{\color{blue}{-2}}\right)\right)}^{1}} \]

   9. Applied egg-rr 52.3%

      \[\leadsto \frac{2}{\color{blue}{{\left(t \cdot \left({k}^{4} \cdot {\ell}^{-2}\right)\right)}^{1}}} \]
   10. Step-by-step derivation

       1. unpow1 52.3%

          \[\leadsto \frac{2}{\color{blue}{t \cdot \left({k}^{4} \cdot {\ell}^{-2}\right)}} \]

       2. associate-*r* 59.4%

          \[\leadsto \frac{2}{\color{blue}{\left(t \cdot {k}^{4}\right) \cdot {\ell}^{-2}}} \]

   11. Simplified 59.4%

       \[\leadsto \frac{2}{\color{blue}{\left(t \cdot {k}^{4}\right) \cdot {\ell}^{-2}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.3 \cdot 10^{-161}:\\ \;\;\;\;\frac{2}{t \cdot \frac{{k}^{4}}{{\ell}^{2}}}\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{+171}:\\ \;\;\;\;\frac{\ell}{\frac{k}{t}} \cdot \left(2 \cdot \frac{\frac{\ell}{{k}^{3}}}{{t}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\ell}^{-2} \cdot \left(t \cdot {k}^{4}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 61.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{2}{{k}^{4}} \cdot \frac{{\ell}^{2}}{t} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* (/ 2.0 (pow k 4.0)) (/ (pow l 2.0) t)))

C

double code(double t, double l, double k) {
	return (2.0 / pow(k, 4.0)) * (pow(l, 2.0) / t);
}

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 / (k ** 4.0d0)) * ((l ** 2.0d0) / t)
end function

Java

public static double code(double t, double l, double k) {
	return (2.0 / Math.pow(k, 4.0)) * (Math.pow(l, 2.0) / t);
}

Python

def code(t, l, k):
	return (2.0 / math.pow(k, 4.0)) * (math.pow(l, 2.0) / t)

Julia

function code(t, l, k)
	return Float64(Float64(2.0 / (k ^ 4.0)) * Float64((l ^ 2.0) / t))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (2.0 / (k ^ 4.0)) * ((l ^ 2.0) / t);
end

Wolfram

code[t_, l_, k_] := N[(N[(2.0 / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.7%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Step-by-step derivation

   1. associate-/r* 36.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

   2. associate-/l/ 36.7%

      \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

   3. associate-*l/ 37.5%

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

   4. associate-/r/ 37.3%

      \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

   5. +-commutative 37.3%

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \]

   6. unpow2 37.3%

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) - 1} \]

   7. sqr-neg 37.3%

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)} + 1\right) - 1} \]

   8. distribute-frac-neg2 37.3%

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\left(-\frac{k}{t}\right) \cdot \color{blue}{\frac{k}{-t}} + 1\right) - 1} \]

   9. distribute-frac-neg2 37.3%

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\frac{k}{-t}} \cdot \frac{k}{-t} + 1\right) - 1} \]

   10. unpow2 37.3%

       \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{{\left(\frac{k}{-t}\right)}^{2}} + 1\right) - 1} \]

   11. +-rgt-identity 37.3%

       \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

   12. metadata-eval 37.3%

       \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\left({\left(\frac{k}{-t}\right)}^{2} + \color{blue}{\left(1 - 1\right)}\right) + 1\right) - 1} \]

   13. associate--l+ 37.3%

       \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left(\left({\left(\frac{k}{-t}\right)}^{2} + 1\right) - 1\right)} + 1\right) - 1} \]

   14. +-commutative 37.3%

       \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\left(\color{blue}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)} - 1\right) + 1\right) - 1} \]

   15. associate--l+ 37.3%

       \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) - 1\right) + \left(1 - 1\right)}} \]

3. Simplified 44.3%

   \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{{\left(\frac{k}{t}\right)}^{2}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*r* 50.2%

      \[\leadsto \frac{\color{blue}{\left(\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell\right) \cdot \ell}}{{\left(\frac{k}{t}\right)}^{2}} \]

   2. unpow2 50.2%

      \[\leadsto \frac{\left(\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell\right) \cdot \ell}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]

   3. times-frac 54.9%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}}} \]

   4. associate-/l/ 54.9%

      \[\leadsto \frac{\color{blue}{\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \cdot \ell}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}} \]

   5. associate-*l/ 55.1%

      \[\leadsto \frac{\color{blue}{\frac{2 \cdot \ell}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}} \]

   6. associate-*l* 55.1%

      \[\leadsto \frac{\frac{2 \cdot \ell}{\color{blue}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}}}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}} \]

6. Applied egg-rr 55.1%

   \[\leadsto \color{blue}{\frac{\frac{2 \cdot \ell}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}}} \]

7. Taylor expanded in k around 0 64.9%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
8. Step-by-step derivation

   1. associate-*r/ 64.9%

      \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

   2. times-frac 64.3%

      \[\leadsto \color{blue}{\frac{2}{{k}^{4}} \cdot \frac{{\ell}^{2}}{t}} \]

9. Simplified 64.3%

   \[\leadsto \color{blue}{\frac{2}{{k}^{4}} \cdot \frac{{\ell}^{2}}{t}} \]

10. Final simplification 64.3%

    \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{{\ell}^{2}}{t} \]
11. Add Preprocessing

Alternative 22: 63.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{2}{t \cdot \frac{{k}^{4}}{{\ell}^{2}}} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (/ 2.0 (* t (/ (pow k 4.0) (pow l 2.0)))))

C

double code(double t, double l, double k) {
	return 2.0 / (t * (pow(k, 4.0) / pow(l, 2.0)));
}

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 * ((k ** 4.0d0) / (l ** 2.0d0)))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / (t * (Math.pow(k, 4.0) / Math.pow(l, 2.0)));
}

Python

def code(t, l, k):
	return 2.0 / (t * (math.pow(k, 4.0) / math.pow(l, 2.0)))

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(t * Float64((k ^ 4.0) / (l ^ 2.0))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (t * ((k ^ 4.0) / (l ^ 2.0)));
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(t * N[(N[Power[k, 4.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.7%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

3. Simplified 43.4%

   \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in k around 0 64.9%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
6. Step-by-step derivation

   1. *-commutative 64.9%

      \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {k}^{4}}}{{\ell}^{2}}} \]

   2. associate-/l* 64.5%

      \[\leadsto \frac{2}{\color{blue}{t \cdot \frac{{k}^{4}}{{\ell}^{2}}}} \]

7. Simplified 64.5%

   \[\leadsto \frac{2}{\color{blue}{t \cdot \frac{{k}^{4}}{{\ell}^{2}}}} \]

8. Final simplification 64.5%

   \[\leadsto \frac{2}{t \cdot \frac{{k}^{4}}{{\ell}^{2}}} \]
9. Add Preprocessing

Alternative 23: 63.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{2}{{\ell}^{-2} \cdot \left(t \cdot {k}^{4}\right)} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (/ 2.0 (* (pow l -2.0) (* t (pow k 4.0)))))

C

double code(double t, double l, double k) {
	return 2.0 / (pow(l, -2.0) * (t * 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 ** (-2.0d0)) * (t * (k ** 4.0d0)))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / (Math.pow(l, -2.0) * (t * Math.pow(k, 4.0)));
}

Python

def code(t, l, k):
	return 2.0 / (math.pow(l, -2.0) * (t * math.pow(k, 4.0)))

Julia

function code(t, l, k)
	return Float64(2.0 / Float64((l ^ -2.0) * Float64(t * (k ^ 4.0))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / ((l ^ -2.0) * (t * (k ^ 4.0)));
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[Power[l, -2.0], $MachinePrecision] * N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.7%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

3. Simplified 43.4%

   \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in k around 0 64.9%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
6. Step-by-step derivation

   1. *-commutative 64.9%

      \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {k}^{4}}}{{\ell}^{2}}} \]

   2. associate-/l* 64.5%

      \[\leadsto \frac{2}{\color{blue}{t \cdot \frac{{k}^{4}}{{\ell}^{2}}}} \]

7. Simplified 64.5%

   \[\leadsto \frac{2}{\color{blue}{t \cdot \frac{{k}^{4}}{{\ell}^{2}}}} \]
8. Step-by-step derivation

   1. pow1 64.5%

      \[\leadsto \frac{2}{\color{blue}{{\left(t \cdot \frac{{k}^{4}}{{\ell}^{2}}\right)}^{1}}} \]

   2. div-inv 64.5%

      \[\leadsto \frac{2}{{\left(t \cdot \color{blue}{\left({k}^{4} \cdot \frac{1}{{\ell}^{2}}\right)}\right)}^{1}} \]

   3. pow-flip 64.9%

      \[\leadsto \frac{2}{{\left(t \cdot \left({k}^{4} \cdot \color{blue}{{\ell}^{\left(-2\right)}}\right)\right)}^{1}} \]

   4. metadata-eval 64.9%

      \[\leadsto \frac{2}{{\left(t \cdot \left({k}^{4} \cdot {\ell}^{\color{blue}{-2}}\right)\right)}^{1}} \]

9. Applied egg-rr 64.9%

   \[\leadsto \frac{2}{\color{blue}{{\left(t \cdot \left({k}^{4} \cdot {\ell}^{-2}\right)\right)}^{1}}} \]
10. Step-by-step derivation

    1. unpow1 64.9%

       \[\leadsto \frac{2}{\color{blue}{t \cdot \left({k}^{4} \cdot {\ell}^{-2}\right)}} \]

    2. associate-*r* 65.3%

       \[\leadsto \frac{2}{\color{blue}{\left(t \cdot {k}^{4}\right) \cdot {\ell}^{-2}}} \]

11. Simplified 65.3%

    \[\leadsto \frac{2}{\color{blue}{\left(t \cdot {k}^{4}\right) \cdot {\ell}^{-2}}} \]

12. Final simplification 65.3%

    \[\leadsto \frac{2}{{\ell}^{-2} \cdot \left(t \cdot {k}^{4}\right)} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024052 
(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))))