Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.5% → 86.5%

Time: 29.2s

Alternatives: 25

Speedup: 2.0×

• 27.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.5% 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: 86.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2}{{\left(\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)}^{3}}\\ \mathbf{if}\;t \leq -1.95 \cdot 10^{-140}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3.85 \cdot 10^{-267}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{t \cdot {k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-178}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1
         (/
          2.0
          (pow
           (*
            (* (/ t (pow (cbrt l) 2.0)) (cbrt (sin k)))
            (cbrt (* (tan k) (+ 2.0 (pow (/ k t) 2.0)))))
           3.0))))
   (if (<= t -1.95e-140)
     t_1
     (if (<= t 3.85e-267)
       (*
        2.0
        (* (/ (pow l 2.0) (* t (pow k 2.0))) (/ (cos k) (pow (sin k) 2.0))))
       (if (<= t 7.5e-178)
         (/ 2.0 (pow (* (/ (* k (sin k)) l) (sqrt (/ t (cos k)))) 2.0))
         t_1)))))

C

double code(double t, double l, double k) {
	double t_1 = 2.0 / pow((((t / pow(cbrt(l), 2.0)) * cbrt(sin(k))) * cbrt((tan(k) * (2.0 + pow((k / t), 2.0))))), 3.0);
	double tmp;
	if (t <= -1.95e-140) {
		tmp = t_1;
	} else if (t <= 3.85e-267) {
		tmp = 2.0 * ((pow(l, 2.0) / (t * pow(k, 2.0))) * (cos(k) / pow(sin(k), 2.0)));
	} else if (t <= 7.5e-178) {
		tmp = 2.0 / pow((((k * sin(k)) / l) * sqrt((t / cos(k)))), 2.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = 2.0 / Math.pow((((t / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt(Math.sin(k))) * Math.cbrt((Math.tan(k) * (2.0 + Math.pow((k / t), 2.0))))), 3.0);
	double tmp;
	if (t <= -1.95e-140) {
		tmp = t_1;
	} else if (t <= 3.85e-267) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / (t * Math.pow(k, 2.0))) * (Math.cos(k) / Math.pow(Math.sin(k), 2.0)));
	} else if (t <= 7.5e-178) {
		tmp = 2.0 / Math.pow((((k * Math.sin(k)) / l) * Math.sqrt((t / Math.cos(k)))), 2.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(2.0 / (Float64(Float64(Float64(t / (cbrt(l) ^ 2.0)) * cbrt(sin(k))) * cbrt(Float64(tan(k) * Float64(2.0 + (Float64(k / t) ^ 2.0))))) ^ 3.0))
	tmp = 0.0
	if (t <= -1.95e-140)
		tmp = t_1;
	elseif (t <= 3.85e-267)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / Float64(t * (k ^ 2.0))) * Float64(cos(k) / (sin(k) ^ 2.0))));
	elseif (t <= 7.5e-178)
		tmp = Float64(2.0 / (Float64(Float64(Float64(k * sin(k)) / l) * sqrt(Float64(t / cos(k)))) ^ 2.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(2.0 / N[Power[N[(N[(N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.95e-140], t$95$1, If[LessEqual[t, 3.85e-267], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 7.5e-178], N[(2.0 / N[Power[N[(N[(N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.9500000000000001e-140 or 7.50000000000000019e-178 < t

   1. Initial program 65.5%

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

      1. associate-/r* 68.9%

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

      2. add-cube-cbrt 68.8%

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

      3. *-un-lft-identity 68.8%

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

      4. times-frac 68.8%

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

      5. pow2 68.8%

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

      6. cbrt-div 68.8%

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

      7. rem-cbrt-cube 68.8%

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

      8. cbrt-div 68.8%

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

      9. rem-cbrt-cube 79.9%

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

   3. Applied egg-rr 79.9%

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

      1. add-cube-cbrt 79.9%

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

      2. pow3 79.9%

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

      3. cbrt-prod 79.9%

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

      4. frac-times 76.6%

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

      5. unpow2 76.6%

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

      6. *-un-lft-identity 76.6%

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

      7. cbrt-div 78.4%

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

      8. add-cbrt-cube 82.6%

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

   5. Applied egg-rr 82.6%

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

      1. add-cube-cbrt 82.5%

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

      2. pow3 82.5%

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

   7. Applied egg-rr 88.4%

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

      1. associate-+r+ 88.4%

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

      2. metadata-eval 88.4%

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

   9. Simplified 88.4%

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

   if -1.9500000000000001e-140 < t < 3.85e-267

   1. Initial program 30.0%

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

   3. Simplified 30.0%

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

   4. Taylor expanded in t around 0 81.3%

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

      1. associate-*r* 81.2%

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

      2. times-frac 81.4%

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

   6. Simplified 81.4%

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

   if 3.85e-267 < t < 7.50000000000000019e-178

   1. Initial program 23.5%

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

      1. add-sqr-sqrt 23.5%

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

      2. pow2 23.5%

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

   3. Applied egg-rr 62.8%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}} \]

   4. Taylor expanded in k around inf 93.8%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.95 \cdot 10^{-140}:\\ \;\;\;\;\frac{2}{{\left(\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)}^{3}}\\ \mathbf{elif}\;t \leq 3.85 \cdot 10^{-267}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{t \cdot {k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-178}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)}^{3}}\\ \end{array} \]

Alternative 2: 72.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)) < +inf.0

   1. Initial program 86.1%

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

   3. Simplified 79.9%

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

      1. associate-*r* 81.7%

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

      2. associate-*r* 81.7%

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

      3. times-frac 90.7%

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

   5. Applied egg-rr 90.7%

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

   if +inf.0 < (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))

   1. Initial program 0.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. pow2 0.0%

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

   3. Applied egg-rr 23.0%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}} \]

   4. Taylor expanded in k around inf 44.1%

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

4. Final simplification 75.1%

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

Alternative 3: 82.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2}{\left(\tan k \cdot {\left(\sqrt[3]{\sin k} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{3}\right) \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)}\\ \mathbf{if}\;t \leq -4.3 \cdot 10^{-66}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-267}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{-86}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1
         (/
          2.0
          (*
           (* (tan k) (pow (* (cbrt (sin k)) (/ (/ t (cbrt l)) (cbrt l))) 3.0))
           (+ 1.0 (+ (pow (/ k t) 2.0) 1.0))))))
   (if (<= t -4.3e-66)
     t_1
     (if (<= t 4e-267)
       (*
        2.0
        (/ (* (pow l 2.0) (cos k)) (* (pow k 2.0) (* t (pow (sin k) 2.0)))))
       (if (<= t 2.05e-86)
         (/ 2.0 (pow (* (/ (* k (sin k)) l) (sqrt (/ t (cos k)))) 2.0))
         t_1)))))

C

double code(double t, double l, double k) {
	double t_1 = 2.0 / ((tan(k) * pow((cbrt(sin(k)) * ((t / cbrt(l)) / cbrt(l))), 3.0)) * (1.0 + (pow((k / t), 2.0) + 1.0)));
	double tmp;
	if (t <= -4.3e-66) {
		tmp = t_1;
	} else if (t <= 4e-267) {
		tmp = 2.0 * ((pow(l, 2.0) * cos(k)) / (pow(k, 2.0) * (t * pow(sin(k), 2.0))));
	} else if (t <= 2.05e-86) {
		tmp = 2.0 / pow((((k * sin(k)) / l) * sqrt((t / cos(k)))), 2.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = 2.0 / ((Math.tan(k) * Math.pow((Math.cbrt(Math.sin(k)) * ((t / Math.cbrt(l)) / Math.cbrt(l))), 3.0)) * (1.0 + (Math.pow((k / t), 2.0) + 1.0)));
	double tmp;
	if (t <= -4.3e-66) {
		tmp = t_1;
	} else if (t <= 4e-267) {
		tmp = 2.0 * ((Math.pow(l, 2.0) * Math.cos(k)) / (Math.pow(k, 2.0) * (t * Math.pow(Math.sin(k), 2.0))));
	} else if (t <= 2.05e-86) {
		tmp = 2.0 / Math.pow((((k * Math.sin(k)) / l) * Math.sqrt((t / Math.cos(k)))), 2.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(2.0 / Float64(Float64(tan(k) * (Float64(cbrt(sin(k)) * Float64(Float64(t / cbrt(l)) / cbrt(l))) ^ 3.0)) * Float64(1.0 + Float64((Float64(k / t) ^ 2.0) + 1.0))))
	tmp = 0.0
	if (t <= -4.3e-66)
		tmp = t_1;
	elseif (t <= 4e-267)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k)) / Float64((k ^ 2.0) * Float64(t * (sin(k) ^ 2.0)))));
	elseif (t <= 2.05e-86)
		tmp = Float64(2.0 / (Float64(Float64(Float64(k * sin(k)) / l) * sqrt(Float64(t / cos(k)))) ^ 2.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[Power[N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(N[(t / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -4.3e-66], t$95$1, If[LessEqual[t, 4e-267], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.05e-86], N[(2.0 / N[Power[N[(N[(N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.30000000000000013e-66 or 2.0499999999999999e-86 < t

   1. Initial program 68.8%

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

      1. associate-/r* 71.7%

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

      2. add-cube-cbrt 71.7%

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

      3. *-un-lft-identity 71.7%

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

      4. times-frac 71.7%

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

      5. pow2 71.7%

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

      6. cbrt-div 71.7%

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

      7. rem-cbrt-cube 71.7%

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

      8. cbrt-div 71.6%

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

      9. rem-cbrt-cube 82.8%

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

   3. Applied egg-rr 82.8%

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

      1. add-cube-cbrt 82.7%

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

      2. pow3 82.7%

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

      3. cbrt-prod 82.7%

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

      4. frac-times 78.6%

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

      5. unpow2 78.6%

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

      6. *-un-lft-identity 78.6%

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

      7. cbrt-div 80.8%

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

      8. add-cbrt-cube 86.0%

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

   5. Applied egg-rr 86.0%

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

   if -4.30000000000000013e-66 < t < 3.9999999999999999e-267

   1. Initial program 37.0%

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

   3. Simplified 35.9%

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

   4. Taylor expanded in t around 0 81.5%

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

   if 3.9999999999999999e-267 < t < 2.0499999999999999e-86

   1. Initial program 38.1%

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

      1. add-sqr-sqrt 33.4%

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

      2. pow2 33.4%

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

   3. Applied egg-rr 56.0%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}} \]

   4. Taylor expanded in k around inf 76.6%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.3 \cdot 10^{-66}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot {\left(\sqrt[3]{\sin k} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{3}\right) \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-267}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{-86}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot {\left(\sqrt[3]{\sin k} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{3}\right) \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)}\\ \end{array} \]

Alternative 4: 82.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3}}\\ \mathbf{if}\;t \leq -2.2 \cdot 10^{-61}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{-267}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-88}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1
         (/
          2.0
          (*
           (* (tan k) (+ 2.0 (pow (/ k t) 2.0)))
           (pow (* (/ t (pow (cbrt l) 2.0)) (cbrt (sin k))) 3.0)))))
   (if (<= t -2.2e-61)
     t_1
     (if (<= t 1.45e-267)
       (*
        2.0
        (/ (* (pow l 2.0) (cos k)) (* (pow k 2.0) (* t (pow (sin k) 2.0)))))
       (if (<= t 8e-88)
         (/ 2.0 (pow (* (/ (* k (sin k)) l) (sqrt (/ t (cos k)))) 2.0))
         t_1)))))

C

double code(double t, double l, double k) {
	double t_1 = 2.0 / ((tan(k) * (2.0 + pow((k / t), 2.0))) * pow(((t / pow(cbrt(l), 2.0)) * cbrt(sin(k))), 3.0));
	double tmp;
	if (t <= -2.2e-61) {
		tmp = t_1;
	} else if (t <= 1.45e-267) {
		tmp = 2.0 * ((pow(l, 2.0) * cos(k)) / (pow(k, 2.0) * (t * pow(sin(k), 2.0))));
	} else if (t <= 8e-88) {
		tmp = 2.0 / pow((((k * sin(k)) / l) * sqrt((t / cos(k)))), 2.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = 2.0 / ((Math.tan(k) * (2.0 + Math.pow((k / t), 2.0))) * Math.pow(((t / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt(Math.sin(k))), 3.0));
	double tmp;
	if (t <= -2.2e-61) {
		tmp = t_1;
	} else if (t <= 1.45e-267) {
		tmp = 2.0 * ((Math.pow(l, 2.0) * Math.cos(k)) / (Math.pow(k, 2.0) * (t * Math.pow(Math.sin(k), 2.0))));
	} else if (t <= 8e-88) {
		tmp = 2.0 / Math.pow((((k * Math.sin(k)) / l) * Math.sqrt((t / Math.cos(k)))), 2.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(2.0 / Float64(Float64(tan(k) * Float64(2.0 + (Float64(k / t) ^ 2.0))) * (Float64(Float64(t / (cbrt(l) ^ 2.0)) * cbrt(sin(k))) ^ 3.0)))
	tmp = 0.0
	if (t <= -2.2e-61)
		tmp = t_1;
	elseif (t <= 1.45e-267)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k)) / Float64((k ^ 2.0) * Float64(t * (sin(k) ^ 2.0)))));
	elseif (t <= 8e-88)
		tmp = Float64(2.0 / (Float64(Float64(Float64(k * sin(k)) / l) * sqrt(Float64(t / cos(k)))) ^ 2.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.2e-61], t$95$1, If[LessEqual[t, 1.45e-267], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 8e-88], N[(2.0 / N[Power[N[(N[(N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.20000000000000009e-61 or 7.99999999999999947e-88 < t

   1. Initial program 68.8%

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

      1. associate-/r* 71.7%

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

      2. add-cube-cbrt 71.7%

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

      3. *-un-lft-identity 71.7%

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

      4. times-frac 71.7%

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

      5. pow2 71.7%

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

      6. cbrt-div 71.7%

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

      7. rem-cbrt-cube 71.7%

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

      8. cbrt-div 71.6%

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

      9. rem-cbrt-cube 82.8%

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

   3. Applied egg-rr 82.8%

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

      1. add-cube-cbrt 82.7%

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

      2. pow3 82.7%

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

      3. cbrt-prod 82.7%

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

      4. frac-times 78.6%

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

      5. unpow2 78.6%

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

      6. *-un-lft-identity 78.6%

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

      7. cbrt-div 80.8%

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

      8. add-cbrt-cube 86.0%

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

   5. Applied egg-rr 86.0%

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

      1. add-cube-cbrt 86.0%

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

      2. pow3 86.0%

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

   7. Applied egg-rr 92.3%

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

      1. *-commutative 92.3%

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

      2. cube-prod 86.0%

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

      3. rem-cube-cbrt 86.0%

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

      4. associate-+r+ 86.0%

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

      5. metadata-eval 86.0%

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

   9. Simplified 86.0%

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

   if -2.20000000000000009e-61 < t < 1.45000000000000011e-267

   1. Initial program 37.0%

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

   3. Simplified 35.9%

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

   4. Taylor expanded in t around 0 81.5%

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

   if 1.45000000000000011e-267 < t < 7.99999999999999947e-88

   1. Initial program 38.1%

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

      1. add-sqr-sqrt 33.4%

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

      2. pow2 33.4%

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

   3. Applied egg-rr 56.0%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}} \]

   4. Taylor expanded in k around inf 76.6%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.2 \cdot 10^{-61}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3}}\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{-267}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-88}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3}}\\ \end{array} \]

Alternative 5: 73.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4.2 \cdot 10^{+24}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\sin k \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{t \cdot {k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 4.2e+24)
   (/
    2.0
    (*
     (* (tan k) (+ 2.0 (pow (/ k t) 2.0)))
     (* (sin k) (pow (/ t (pow (cbrt l) 2.0)) 3.0))))
   (*
    2.0
    (* (/ (pow l 2.0) (* t (pow k 2.0))) (/ (cos k) (pow (sin k) 2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 4.2e+24) {
		tmp = 2.0 / ((tan(k) * (2.0 + pow((k / t), 2.0))) * (sin(k) * pow((t / pow(cbrt(l), 2.0)), 3.0)));
	} else {
		tmp = 2.0 * ((pow(l, 2.0) / (t * pow(k, 2.0))) * (cos(k) / pow(sin(k), 2.0)));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 4.2e+24) {
		tmp = 2.0 / ((Math.tan(k) * (2.0 + Math.pow((k / t), 2.0))) * (Math.sin(k) * Math.pow((t / Math.pow(Math.cbrt(l), 2.0)), 3.0)));
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) / (t * Math.pow(k, 2.0))) * (Math.cos(k) / Math.pow(Math.sin(k), 2.0)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 4.2e+24)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(2.0 + (Float64(k / t) ^ 2.0))) * Float64(sin(k) * (Float64(t / (cbrt(l) ^ 2.0)) ^ 3.0))));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / Float64(t * (k ^ 2.0))) * Float64(cos(k) / (sin(k) ^ 2.0))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 4.2e+24], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.2000000000000003e24

   1. Initial program 59.0%

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

      1. associate-/r* 64.4%

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

      2. add-cube-cbrt 64.4%

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

      3. *-un-lft-identity 64.4%

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

      4. times-frac 64.4%

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

      5. pow2 64.4%

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

      6. cbrt-div 64.4%

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

      7. rem-cbrt-cube 64.4%

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

      8. cbrt-div 64.4%

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

      9. rem-cbrt-cube 74.6%

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

   3. Applied egg-rr 74.6%

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

      1. add-cube-cbrt 74.5%

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

      2. pow3 74.5%

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

      3. cbrt-prod 74.5%

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

      4. frac-times 70.2%

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

      5. unpow2 70.2%

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

      6. *-un-lft-identity 70.2%

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

      7. cbrt-div 72.0%

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

      8. add-cbrt-cube 77.2%

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

   5. Applied egg-rr 77.2%

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

      1. distribute-lft-in 77.2%

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

      2. *-commutative 77.2%

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

      3. unpow-prod-down 74.4%

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

      4. pow3 74.4%

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

      5. add-cube-cbrt 74.4%

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

      6. associate-/l/ 74.4%

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

      7. pow2 74.4%

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

   7. Applied egg-rr 74.5%

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

      1. distribute-lft-out 74.5%

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

      2. +-commutative 74.5%

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

      3. associate-*r* 75.4%

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

      4. associate-+r+ 75.4%

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

      5. metadata-eval 75.4%

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

   9. Simplified 75.4%

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

   if 4.2000000000000003e24 < k

   1. Initial program 50.8%

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

   3. Simplified 50.7%

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

   4. Taylor expanded in t around 0 73.2%

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

      1. associate-*r* 73.1%

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

      2. times-frac 73.2%

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

   6. Simplified 73.2%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.2 \cdot 10^{+24}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\sin k \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{t \cdot {k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \end{array} \]

Alternative 6: 78.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{\sqrt[3]{\ell}}\\ t_2 := \frac{2}{\left(\tan k \cdot \left(\sin k \cdot \left({t_1}^{2} \cdot \frac{t_1}{\ell}\right)\right)\right) \cdot \left(1 + \left(1 + \frac{k}{t} \cdot \frac{k}{t}\right)\right)}\\ \mathbf{if}\;t \leq -4 \cdot 10^{-56}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 2.25 \cdot 10^{-267}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{-85}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ t (cbrt l)))
        (t_2
         (/
          2.0
          (*
           (* (tan k) (* (sin k) (* (pow t_1 2.0) (/ t_1 l))))
           (+ 1.0 (+ 1.0 (* (/ k t) (/ k t))))))))
   (if (<= t -4e-56)
     t_2
     (if (<= t 2.25e-267)
       (*
        2.0
        (/ (* (pow l 2.0) (cos k)) (* (pow k 2.0) (* t (pow (sin k) 2.0)))))
       (if (<= t 5.5e-85)
         (/ 2.0 (pow (* (/ (* k (sin k)) l) (sqrt (/ t (cos k)))) 2.0))
         t_2)))))

C

double code(double t, double l, double k) {
	double t_1 = t / cbrt(l);
	double t_2 = 2.0 / ((tan(k) * (sin(k) * (pow(t_1, 2.0) * (t_1 / l)))) * (1.0 + (1.0 + ((k / t) * (k / t)))));
	double tmp;
	if (t <= -4e-56) {
		tmp = t_2;
	} else if (t <= 2.25e-267) {
		tmp = 2.0 * ((pow(l, 2.0) * cos(k)) / (pow(k, 2.0) * (t * pow(sin(k), 2.0))));
	} else if (t <= 5.5e-85) {
		tmp = 2.0 / pow((((k * sin(k)) / l) * sqrt((t / cos(k)))), 2.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = t / Math.cbrt(l);
	double t_2 = 2.0 / ((Math.tan(k) * (Math.sin(k) * (Math.pow(t_1, 2.0) * (t_1 / l)))) * (1.0 + (1.0 + ((k / t) * (k / t)))));
	double tmp;
	if (t <= -4e-56) {
		tmp = t_2;
	} else if (t <= 2.25e-267) {
		tmp = 2.0 * ((Math.pow(l, 2.0) * Math.cos(k)) / (Math.pow(k, 2.0) * (t * Math.pow(Math.sin(k), 2.0))));
	} else if (t <= 5.5e-85) {
		tmp = 2.0 / Math.pow((((k * Math.sin(k)) / l) * Math.sqrt((t / Math.cos(k)))), 2.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(t / cbrt(l))
	t_2 = Float64(2.0 / Float64(Float64(tan(k) * Float64(sin(k) * Float64((t_1 ^ 2.0) * Float64(t_1 / l)))) * Float64(1.0 + Float64(1.0 + Float64(Float64(k / t) * Float64(k / t))))))
	tmp = 0.0
	if (t <= -4e-56)
		tmp = t_2;
	elseif (t <= 2.25e-267)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k)) / Float64((k ^ 2.0) * Float64(t * (sin(k) ^ 2.0)))));
	elseif (t <= 5.5e-85)
		tmp = Float64(2.0 / (Float64(Float64(Float64(k * sin(k)) / l) * sqrt(Float64(t / cos(k)))) ^ 2.0));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(t / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$1, 2.0], $MachinePrecision] * N[(t$95$1 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + N[(N[(k / t), $MachinePrecision] * N[(k / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -4e-56], t$95$2, If[LessEqual[t, 2.25e-267], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.5e-85], N[(2.0 / N[Power[N[(N[(N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.0000000000000002e-56 or 5.4999999999999997e-85 < t

   1. Initial program 68.8%

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

      1. associate-/r* 71.7%

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

      2. add-cube-cbrt 71.7%

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

      3. *-un-lft-identity 71.7%

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

      4. times-frac 71.7%

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

      5. pow2 71.7%

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

      6. cbrt-div 71.7%

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

      7. rem-cbrt-cube 71.7%

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

      8. cbrt-div 71.6%

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

      9. rem-cbrt-cube 82.8%

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

   3. Applied egg-rr 82.8%

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

      1. unpow2 82.8%

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

   5. Applied egg-rr 82.8%

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

   if -4.0000000000000002e-56 < t < 2.25e-267

   1. Initial program 37.0%

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

   3. Simplified 35.9%

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

   4. Taylor expanded in t around 0 81.5%

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

   if 2.25e-267 < t < 5.4999999999999997e-85

   1. Initial program 38.1%

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

      1. add-sqr-sqrt 33.4%

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

      2. pow2 33.4%

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

   3. Applied egg-rr 56.0%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}} \]

   4. Taylor expanded in k around inf 76.6%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4 \cdot 10^{-56}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \left({\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right)\right)\right) \cdot \left(1 + \left(1 + \frac{k}{t} \cdot \frac{k}{t}\right)\right)}\\ \mathbf{elif}\;t \leq 2.25 \cdot 10^{-267}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{-85}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \left({\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right)\right)\right) \cdot \left(1 + \left(1 + \frac{k}{t} \cdot \frac{k}{t}\right)\right)}\\ \end{array} \]

Alternative 7: 76.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.6 \cdot 10^{-60}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\sin k \cdot \frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-268}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{-83}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{\left(k \cdot \sqrt{2}\right) \cdot {t}^{1.5}}{\ell}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -4.6e-60)
   (/
    (/ 2.0 (* (tan k) (* (sin k) (/ (pow (/ t (cbrt l)) 3.0) l))))
    (+ 2.0 (pow (/ k t) 2.0)))
   (if (<= t 3.2e-268)
     (*
      2.0
      (/ (* (pow l 2.0) (cos k)) (* (pow k 2.0) (* t (pow (sin k) 2.0)))))
     (if (<= t 1.7e-83)
       (/ 2.0 (pow (* (/ (* k (sin k)) l) (sqrt (/ t (cos k)))) 2.0))
       (/ 2.0 (pow (/ (* (* k (sqrt 2.0)) (pow t 1.5)) l) 2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -4.6e-60) {
		tmp = (2.0 / (tan(k) * (sin(k) * (pow((t / cbrt(l)), 3.0) / l)))) / (2.0 + pow((k / t), 2.0));
	} else if (t <= 3.2e-268) {
		tmp = 2.0 * ((pow(l, 2.0) * cos(k)) / (pow(k, 2.0) * (t * pow(sin(k), 2.0))));
	} else if (t <= 1.7e-83) {
		tmp = 2.0 / pow((((k * sin(k)) / l) * sqrt((t / cos(k)))), 2.0);
	} else {
		tmp = 2.0 / pow((((k * sqrt(2.0)) * pow(t, 1.5)) / l), 2.0);
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -4.6e-60) {
		tmp = (2.0 / (Math.tan(k) * (Math.sin(k) * (Math.pow((t / Math.cbrt(l)), 3.0) / l)))) / (2.0 + Math.pow((k / t), 2.0));
	} else if (t <= 3.2e-268) {
		tmp = 2.0 * ((Math.pow(l, 2.0) * Math.cos(k)) / (Math.pow(k, 2.0) * (t * Math.pow(Math.sin(k), 2.0))));
	} else if (t <= 1.7e-83) {
		tmp = 2.0 / Math.pow((((k * Math.sin(k)) / l) * Math.sqrt((t / Math.cos(k)))), 2.0);
	} else {
		tmp = 2.0 / Math.pow((((k * Math.sqrt(2.0)) * Math.pow(t, 1.5)) / l), 2.0);
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -4.6e-60)
		tmp = Float64(Float64(2.0 / Float64(tan(k) * Float64(sin(k) * Float64((Float64(t / cbrt(l)) ^ 3.0) / l)))) / Float64(2.0 + (Float64(k / t) ^ 2.0)));
	elseif (t <= 3.2e-268)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k)) / Float64((k ^ 2.0) * Float64(t * (sin(k) ^ 2.0)))));
	elseif (t <= 1.7e-83)
		tmp = Float64(2.0 / (Float64(Float64(Float64(k * sin(k)) / l) * sqrt(Float64(t / cos(k)))) ^ 2.0));
	else
		tmp = Float64(2.0 / (Float64(Float64(Float64(k * sqrt(2.0)) * (t ^ 1.5)) / l) ^ 2.0));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -4.6e-60], N[(N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[N[(t / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.2e-268], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.7e-83], N[(2.0 / N[Power[N[(N[(N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(N[(k * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[t, 1.5], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < -4.6000000000000003e-60

   1. Initial program 74.9%

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

   3. Simplified 77.7%

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

      1. add-cube-cbrt 77.7%

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

      2. pow3 77.7%

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

      3. cbrt-div 77.6%

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

      4. rem-cbrt-cube 82.5%

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

   5. Applied egg-rr 82.5%

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

   if -4.6000000000000003e-60 < t < 3.1999999999999999e-268

   1. Initial program 37.0%

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

   3. Simplified 35.9%

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

   4. Taylor expanded in t around 0 81.5%

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

   if 3.1999999999999999e-268 < t < 1.6999999999999999e-83

   1. Initial program 38.1%

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

      1. add-sqr-sqrt 33.4%

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

      2. pow2 33.4%

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

   3. Applied egg-rr 56.0%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}} \]

   4. Taylor expanded in k around inf 76.6%

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

   if 1.6999999999999999e-83 < t

   1. Initial program 63.0%

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

      1. add-sqr-sqrt 39.7%

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

      2. pow2 39.7%

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

   3. Applied egg-rr 53.3%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}} \]

   4. Taylor expanded in k around 0 68.4%

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

      1. associate-*l/ 68.4%

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

      2. sqrt-pow1 78.1%

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

      3. metadata-eval 78.1%

         \[\leadsto \frac{2}{{\left(\frac{\left(k \cdot \sqrt{2}\right) \cdot {t}^{\color{blue}{1.5}}}{\ell}\right)}^{2}} \]

   6. Applied egg-rr 78.1%

      \[\leadsto \frac{2}{{\color{blue}{\left(\frac{\left(k \cdot \sqrt{2}\right) \cdot {t}^{1.5}}{\ell}\right)}}^{2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.6 \cdot 10^{-60}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\sin k \cdot \frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-268}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{-83}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{\left(k \cdot \sqrt{2}\right) \cdot {t}^{1.5}}{\ell}\right)}^{2}}\\ \end{array} \]

Alternative 8: 75.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.2 \cdot 10^{-63}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{{t}^{3}}}{\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{\ell}{\tan k}\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{-267}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{t \cdot {k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \mathbf{elif}\;t \leq 3.95 \cdot 10^{-84}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{\left(k \cdot \sqrt{2}\right) \cdot {t}^{1.5}}{\ell}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -3.2e-63)
   (*
    (/ (* l (/ 2.0 (pow t 3.0))) (* (sin k) (+ 2.0 (pow (/ k t) 2.0))))
    (/ l (tan k)))
   (if (<= t 1.2e-267)
     (*
      2.0
      (* (/ (pow l 2.0) (* t (pow k 2.0))) (/ (cos k) (pow (sin k) 2.0))))
     (if (<= t 3.95e-84)
       (/ 2.0 (pow (* (/ (* k (sin k)) l) (sqrt (/ t (cos k)))) 2.0))
       (/ 2.0 (pow (/ (* (* k (sqrt 2.0)) (pow t 1.5)) l) 2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -3.2e-63) {
		tmp = ((l * (2.0 / pow(t, 3.0))) / (sin(k) * (2.0 + pow((k / t), 2.0)))) * (l / tan(k));
	} else if (t <= 1.2e-267) {
		tmp = 2.0 * ((pow(l, 2.0) / (t * pow(k, 2.0))) * (cos(k) / pow(sin(k), 2.0)));
	} else if (t <= 3.95e-84) {
		tmp = 2.0 / pow((((k * sin(k)) / l) * sqrt((t / cos(k)))), 2.0);
	} else {
		tmp = 2.0 / pow((((k * sqrt(2.0)) * pow(t, 1.5)) / 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 <= (-3.2d-63)) then
        tmp = ((l * (2.0d0 / (t ** 3.0d0))) / (sin(k) * (2.0d0 + ((k / t) ** 2.0d0)))) * (l / tan(k))
    else if (t <= 1.2d-267) then
        tmp = 2.0d0 * (((l ** 2.0d0) / (t * (k ** 2.0d0))) * (cos(k) / (sin(k) ** 2.0d0)))
    else if (t <= 3.95d-84) then
        tmp = 2.0d0 / ((((k * sin(k)) / l) * sqrt((t / cos(k)))) ** 2.0d0)
    else
        tmp = 2.0d0 / ((((k * sqrt(2.0d0)) * (t ** 1.5d0)) / l) ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -3.2e-63) {
		tmp = ((l * (2.0 / Math.pow(t, 3.0))) / (Math.sin(k) * (2.0 + Math.pow((k / t), 2.0)))) * (l / Math.tan(k));
	} else if (t <= 1.2e-267) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / (t * Math.pow(k, 2.0))) * (Math.cos(k) / Math.pow(Math.sin(k), 2.0)));
	} else if (t <= 3.95e-84) {
		tmp = 2.0 / Math.pow((((k * Math.sin(k)) / l) * Math.sqrt((t / Math.cos(k)))), 2.0);
	} else {
		tmp = 2.0 / Math.pow((((k * Math.sqrt(2.0)) * Math.pow(t, 1.5)) / l), 2.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -3.2e-63:
		tmp = ((l * (2.0 / math.pow(t, 3.0))) / (math.sin(k) * (2.0 + math.pow((k / t), 2.0)))) * (l / math.tan(k))
	elif t <= 1.2e-267:
		tmp = 2.0 * ((math.pow(l, 2.0) / (t * math.pow(k, 2.0))) * (math.cos(k) / math.pow(math.sin(k), 2.0)))
	elif t <= 3.95e-84:
		tmp = 2.0 / math.pow((((k * math.sin(k)) / l) * math.sqrt((t / math.cos(k)))), 2.0)
	else:
		tmp = 2.0 / math.pow((((k * math.sqrt(2.0)) * math.pow(t, 1.5)) / l), 2.0)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -3.2e-63)
		tmp = Float64(Float64(Float64(l * Float64(2.0 / (t ^ 3.0))) / Float64(sin(k) * Float64(2.0 + (Float64(k / t) ^ 2.0)))) * Float64(l / tan(k)));
	elseif (t <= 1.2e-267)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / Float64(t * (k ^ 2.0))) * Float64(cos(k) / (sin(k) ^ 2.0))));
	elseif (t <= 3.95e-84)
		tmp = Float64(2.0 / (Float64(Float64(Float64(k * sin(k)) / l) * sqrt(Float64(t / cos(k)))) ^ 2.0));
	else
		tmp = Float64(2.0 / (Float64(Float64(Float64(k * sqrt(2.0)) * (t ^ 1.5)) / l) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -3.2e-63)
		tmp = ((l * (2.0 / (t ^ 3.0))) / (sin(k) * (2.0 + ((k / t) ^ 2.0)))) * (l / tan(k));
	elseif (t <= 1.2e-267)
		tmp = 2.0 * (((l ^ 2.0) / (t * (k ^ 2.0))) * (cos(k) / (sin(k) ^ 2.0)));
	elseif (t <= 3.95e-84)
		tmp = 2.0 / ((((k * sin(k)) / l) * sqrt((t / cos(k)))) ^ 2.0);
	else
		tmp = 2.0 / ((((k * sqrt(2.0)) * (t ^ 1.5)) / l) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -3.2e-63], N[(N[(N[(l * N[(2.0 / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.2e-267], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.95e-84], N[(2.0 / N[Power[N[(N[(N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(N[(k * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[t, 1.5], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < -3.19999999999999989e-63

   1. Initial program 74.9%

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

   3. Simplified 67.0%

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

      1. associate-*r* 69.9%

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

      2. associate-*r* 69.9%

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

      3. times-frac 81.3%

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

   5. Applied egg-rr 81.3%

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

   if -3.19999999999999989e-63 < t < 1.1999999999999999e-267

   1. Initial program 37.0%

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

   3. Simplified 35.9%

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

   4. Taylor expanded in t around 0 81.5%

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

      1. associate-*r* 81.4%

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

      2. times-frac 81.4%

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

   6. Simplified 81.4%

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

   if 1.1999999999999999e-267 < t < 3.94999999999999995e-84

   1. Initial program 38.1%

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

      1. add-sqr-sqrt 33.4%

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

      2. pow2 33.4%

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

   3. Applied egg-rr 56.0%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}} \]

   4. Taylor expanded in k around inf 76.6%

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

   if 3.94999999999999995e-84 < t

   1. Initial program 63.0%

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

      1. add-sqr-sqrt 39.7%

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

      2. pow2 39.7%

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

   3. Applied egg-rr 53.3%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}} \]

   4. Taylor expanded in k around 0 68.4%

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

      1. associate-*l/ 68.4%

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

      2. sqrt-pow1 78.1%

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

      3. metadata-eval 78.1%

         \[\leadsto \frac{2}{{\left(\frac{\left(k \cdot \sqrt{2}\right) \cdot {t}^{\color{blue}{1.5}}}{\ell}\right)}^{2}} \]

   6. Applied egg-rr 78.1%

      \[\leadsto \frac{2}{{\color{blue}{\left(\frac{\left(k \cdot \sqrt{2}\right) \cdot {t}^{1.5}}{\ell}\right)}}^{2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.2 \cdot 10^{-63}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{{t}^{3}}}{\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{\ell}{\tan k}\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{-267}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{t \cdot {k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \mathbf{elif}\;t \leq 3.95 \cdot 10^{-84}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{\left(k \cdot \sqrt{2}\right) \cdot {t}^{1.5}}{\ell}\right)}^{2}}\\ \end{array} \]

Alternative 9: 75.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.15 \cdot 10^{-63}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{{t}^{3}}}{\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{\ell}{\tan k}\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-267}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{elif}\;t \leq 5.9 \cdot 10^{-83}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{\left(k \cdot \sqrt{2}\right) \cdot {t}^{1.5}}{\ell}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -2.15e-63)
   (*
    (/ (* l (/ 2.0 (pow t 3.0))) (* (sin k) (+ 2.0 (pow (/ k t) 2.0))))
    (/ l (tan k)))
   (if (<= t 3.5e-267)
     (*
      2.0
      (/ (* (pow l 2.0) (cos k)) (* (pow k 2.0) (* t (pow (sin k) 2.0)))))
     (if (<= t 5.9e-83)
       (/ 2.0 (pow (* (/ (* k (sin k)) l) (sqrt (/ t (cos k)))) 2.0))
       (/ 2.0 (pow (/ (* (* k (sqrt 2.0)) (pow t 1.5)) l) 2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -2.15e-63) {
		tmp = ((l * (2.0 / pow(t, 3.0))) / (sin(k) * (2.0 + pow((k / t), 2.0)))) * (l / tan(k));
	} else if (t <= 3.5e-267) {
		tmp = 2.0 * ((pow(l, 2.0) * cos(k)) / (pow(k, 2.0) * (t * pow(sin(k), 2.0))));
	} else if (t <= 5.9e-83) {
		tmp = 2.0 / pow((((k * sin(k)) / l) * sqrt((t / cos(k)))), 2.0);
	} else {
		tmp = 2.0 / pow((((k * sqrt(2.0)) * pow(t, 1.5)) / 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 <= (-2.15d-63)) then
        tmp = ((l * (2.0d0 / (t ** 3.0d0))) / (sin(k) * (2.0d0 + ((k / t) ** 2.0d0)))) * (l / tan(k))
    else if (t <= 3.5d-267) then
        tmp = 2.0d0 * (((l ** 2.0d0) * cos(k)) / ((k ** 2.0d0) * (t * (sin(k) ** 2.0d0))))
    else if (t <= 5.9d-83) then
        tmp = 2.0d0 / ((((k * sin(k)) / l) * sqrt((t / cos(k)))) ** 2.0d0)
    else
        tmp = 2.0d0 / ((((k * sqrt(2.0d0)) * (t ** 1.5d0)) / l) ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -2.15e-63) {
		tmp = ((l * (2.0 / Math.pow(t, 3.0))) / (Math.sin(k) * (2.0 + Math.pow((k / t), 2.0)))) * (l / Math.tan(k));
	} else if (t <= 3.5e-267) {
		tmp = 2.0 * ((Math.pow(l, 2.0) * Math.cos(k)) / (Math.pow(k, 2.0) * (t * Math.pow(Math.sin(k), 2.0))));
	} else if (t <= 5.9e-83) {
		tmp = 2.0 / Math.pow((((k * Math.sin(k)) / l) * Math.sqrt((t / Math.cos(k)))), 2.0);
	} else {
		tmp = 2.0 / Math.pow((((k * Math.sqrt(2.0)) * Math.pow(t, 1.5)) / l), 2.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -2.15e-63:
		tmp = ((l * (2.0 / math.pow(t, 3.0))) / (math.sin(k) * (2.0 + math.pow((k / t), 2.0)))) * (l / math.tan(k))
	elif t <= 3.5e-267:
		tmp = 2.0 * ((math.pow(l, 2.0) * math.cos(k)) / (math.pow(k, 2.0) * (t * math.pow(math.sin(k), 2.0))))
	elif t <= 5.9e-83:
		tmp = 2.0 / math.pow((((k * math.sin(k)) / l) * math.sqrt((t / math.cos(k)))), 2.0)
	else:
		tmp = 2.0 / math.pow((((k * math.sqrt(2.0)) * math.pow(t, 1.5)) / l), 2.0)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -2.15e-63)
		tmp = Float64(Float64(Float64(l * Float64(2.0 / (t ^ 3.0))) / Float64(sin(k) * Float64(2.0 + (Float64(k / t) ^ 2.0)))) * Float64(l / tan(k)));
	elseif (t <= 3.5e-267)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k)) / Float64((k ^ 2.0) * Float64(t * (sin(k) ^ 2.0)))));
	elseif (t <= 5.9e-83)
		tmp = Float64(2.0 / (Float64(Float64(Float64(k * sin(k)) / l) * sqrt(Float64(t / cos(k)))) ^ 2.0));
	else
		tmp = Float64(2.0 / (Float64(Float64(Float64(k * sqrt(2.0)) * (t ^ 1.5)) / l) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -2.15e-63)
		tmp = ((l * (2.0 / (t ^ 3.0))) / (sin(k) * (2.0 + ((k / t) ^ 2.0)))) * (l / tan(k));
	elseif (t <= 3.5e-267)
		tmp = 2.0 * (((l ^ 2.0) * cos(k)) / ((k ^ 2.0) * (t * (sin(k) ^ 2.0))));
	elseif (t <= 5.9e-83)
		tmp = 2.0 / ((((k * sin(k)) / l) * sqrt((t / cos(k)))) ^ 2.0);
	else
		tmp = 2.0 / ((((k * sqrt(2.0)) * (t ^ 1.5)) / l) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -2.15e-63], N[(N[(N[(l * N[(2.0 / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.5e-267], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.9e-83], N[(2.0 / N[Power[N[(N[(N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(N[(k * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[t, 1.5], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < -2.1499999999999999e-63

   1. Initial program 74.9%

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

   3. Simplified 67.0%

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

      1. associate-*r* 69.9%

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

      2. associate-*r* 69.9%

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

      3. times-frac 81.3%

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

   5. Applied egg-rr 81.3%

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

   if -2.1499999999999999e-63 < t < 3.4999999999999999e-267

   1. Initial program 37.0%

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

   3. Simplified 35.9%

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

   4. Taylor expanded in t around 0 81.5%

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

   if 3.4999999999999999e-267 < t < 5.8999999999999997e-83

   1. Initial program 38.1%

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

      1. add-sqr-sqrt 33.4%

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

      2. pow2 33.4%

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

   3. Applied egg-rr 56.0%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}} \]

   4. Taylor expanded in k around inf 76.6%

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

   if 5.8999999999999997e-83 < t

   1. Initial program 63.0%

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

      1. add-sqr-sqrt 39.7%

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

      2. pow2 39.7%

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

   3. Applied egg-rr 53.3%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}} \]

   4. Taylor expanded in k around 0 68.4%

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

      1. associate-*l/ 68.4%

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

      2. sqrt-pow1 78.1%

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

      3. metadata-eval 78.1%

         \[\leadsto \frac{2}{{\left(\frac{\left(k \cdot \sqrt{2}\right) \cdot {t}^{\color{blue}{1.5}}}{\ell}\right)}^{2}} \]

   6. Applied egg-rr 78.1%

      \[\leadsto \frac{2}{{\color{blue}{\left(\frac{\left(k \cdot \sqrt{2}\right) \cdot {t}^{1.5}}{\ell}\right)}}^{2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.15 \cdot 10^{-63}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{{t}^{3}}}{\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{\ell}{\tan k}\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-267}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{elif}\;t \leq 5.9 \cdot 10^{-83}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{\left(k \cdot \sqrt{2}\right) \cdot {t}^{1.5}}{\ell}\right)}^{2}}\\ \end{array} \]

Alternative 10: 69.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\frac{t}{\cos k}}\\ t_2 := k \cdot \sqrt{2}\\ \mathbf{if}\;t \leq -1.02 \cdot 10^{+121}:\\ \;\;\;\;\frac{2}{{\left(\frac{t_2}{\ell} \cdot {\left({t}^{3}\right)}^{0.5}\right)}^{2}}\\ \mathbf{elif}\;t \leq -2.6 \cdot 10^{-180}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{{t}^{3}}}{\sin k \cdot \tan k} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;t \leq -2.4 \cdot 10^{-232}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot t_1}{\frac{\ell}{\sin k}}\right)}^{2}}\\ \mathbf{elif}\;t \leq 3.9 \cdot 10^{-268}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{2}}{\cos k} \cdot \frac{{k}^{2}}{{\ell}^{2}}}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-85}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sin k}{\ell} \cdot t_1\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t_2 \cdot {t}^{1.5}}{\ell}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (sqrt (/ t (cos k)))) (t_2 (* k (sqrt 2.0))))
   (if (<= t -1.02e+121)
     (/ 2.0 (pow (* (/ t_2 l) (pow (pow t 3.0) 0.5)) 2.0))
     (if (<= t -2.6e-180)
       (*
        (/ (* l (/ 2.0 (pow t 3.0))) (* (sin k) (tan k)))
        (/ l (+ 2.0 (pow (/ k t) 2.0))))
       (if (<= t -2.4e-232)
         (/ 2.0 (pow (/ (* k t_1) (/ l (sin k))) 2.0))
         (if (<= t 3.9e-268)
           (/
            2.0
            (* (/ (* t (pow k 2.0)) (cos k)) (/ (pow k 2.0) (pow l 2.0))))
           (if (<= t 8e-85)
             (/ 2.0 (pow (* (/ (* k (sin k)) l) t_1) 2.0))
             (/ 2.0 (pow (/ (* t_2 (pow t 1.5)) l) 2.0)))))))))

C

double code(double t, double l, double k) {
	double t_1 = sqrt((t / cos(k)));
	double t_2 = k * sqrt(2.0);
	double tmp;
	if (t <= -1.02e+121) {
		tmp = 2.0 / pow(((t_2 / l) * pow(pow(t, 3.0), 0.5)), 2.0);
	} else if (t <= -2.6e-180) {
		tmp = ((l * (2.0 / pow(t, 3.0))) / (sin(k) * tan(k))) * (l / (2.0 + pow((k / t), 2.0)));
	} else if (t <= -2.4e-232) {
		tmp = 2.0 / pow(((k * t_1) / (l / sin(k))), 2.0);
	} else if (t <= 3.9e-268) {
		tmp = 2.0 / (((t * pow(k, 2.0)) / cos(k)) * (pow(k, 2.0) / pow(l, 2.0)));
	} else if (t <= 8e-85) {
		tmp = 2.0 / pow((((k * sin(k)) / l) * t_1), 2.0);
	} else {
		tmp = 2.0 / pow(((t_2 * pow(t, 1.5)) / 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((t / cos(k)))
    t_2 = k * sqrt(2.0d0)
    if (t <= (-1.02d+121)) then
        tmp = 2.0d0 / (((t_2 / l) * ((t ** 3.0d0) ** 0.5d0)) ** 2.0d0)
    else if (t <= (-2.6d-180)) then
        tmp = ((l * (2.0d0 / (t ** 3.0d0))) / (sin(k) * tan(k))) * (l / (2.0d0 + ((k / t) ** 2.0d0)))
    else if (t <= (-2.4d-232)) then
        tmp = 2.0d0 / (((k * t_1) / (l / sin(k))) ** 2.0d0)
    else if (t <= 3.9d-268) then
        tmp = 2.0d0 / (((t * (k ** 2.0d0)) / cos(k)) * ((k ** 2.0d0) / (l ** 2.0d0)))
    else if (t <= 8d-85) then
        tmp = 2.0d0 / ((((k * sin(k)) / l) * t_1) ** 2.0d0)
    else
        tmp = 2.0d0 / (((t_2 * (t ** 1.5d0)) / l) ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.sqrt((t / Math.cos(k)));
	double t_2 = k * Math.sqrt(2.0);
	double tmp;
	if (t <= -1.02e+121) {
		tmp = 2.0 / Math.pow(((t_2 / l) * Math.pow(Math.pow(t, 3.0), 0.5)), 2.0);
	} else if (t <= -2.6e-180) {
		tmp = ((l * (2.0 / Math.pow(t, 3.0))) / (Math.sin(k) * Math.tan(k))) * (l / (2.0 + Math.pow((k / t), 2.0)));
	} else if (t <= -2.4e-232) {
		tmp = 2.0 / Math.pow(((k * t_1) / (l / Math.sin(k))), 2.0);
	} else if (t <= 3.9e-268) {
		tmp = 2.0 / (((t * Math.pow(k, 2.0)) / Math.cos(k)) * (Math.pow(k, 2.0) / Math.pow(l, 2.0)));
	} else if (t <= 8e-85) {
		tmp = 2.0 / Math.pow((((k * Math.sin(k)) / l) * t_1), 2.0);
	} else {
		tmp = 2.0 / Math.pow(((t_2 * Math.pow(t, 1.5)) / l), 2.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.sqrt((t / math.cos(k)))
	t_2 = k * math.sqrt(2.0)
	tmp = 0
	if t <= -1.02e+121:
		tmp = 2.0 / math.pow(((t_2 / l) * math.pow(math.pow(t, 3.0), 0.5)), 2.0)
	elif t <= -2.6e-180:
		tmp = ((l * (2.0 / math.pow(t, 3.0))) / (math.sin(k) * math.tan(k))) * (l / (2.0 + math.pow((k / t), 2.0)))
	elif t <= -2.4e-232:
		tmp = 2.0 / math.pow(((k * t_1) / (l / math.sin(k))), 2.0)
	elif t <= 3.9e-268:
		tmp = 2.0 / (((t * math.pow(k, 2.0)) / math.cos(k)) * (math.pow(k, 2.0) / math.pow(l, 2.0)))
	elif t <= 8e-85:
		tmp = 2.0 / math.pow((((k * math.sin(k)) / l) * t_1), 2.0)
	else:
		tmp = 2.0 / math.pow(((t_2 * math.pow(t, 1.5)) / l), 2.0)
	return tmp

Julia

function code(t, l, k)
	t_1 = sqrt(Float64(t / cos(k)))
	t_2 = Float64(k * sqrt(2.0))
	tmp = 0.0
	if (t <= -1.02e+121)
		tmp = Float64(2.0 / (Float64(Float64(t_2 / l) * ((t ^ 3.0) ^ 0.5)) ^ 2.0));
	elseif (t <= -2.6e-180)
		tmp = Float64(Float64(Float64(l * Float64(2.0 / (t ^ 3.0))) / Float64(sin(k) * tan(k))) * Float64(l / Float64(2.0 + (Float64(k / t) ^ 2.0))));
	elseif (t <= -2.4e-232)
		tmp = Float64(2.0 / (Float64(Float64(k * t_1) / Float64(l / sin(k))) ^ 2.0));
	elseif (t <= 3.9e-268)
		tmp = Float64(2.0 / Float64(Float64(Float64(t * (k ^ 2.0)) / cos(k)) * Float64((k ^ 2.0) / (l ^ 2.0))));
	elseif (t <= 8e-85)
		tmp = Float64(2.0 / (Float64(Float64(Float64(k * sin(k)) / l) * t_1) ^ 2.0));
	else
		tmp = Float64(2.0 / (Float64(Float64(t_2 * (t ^ 1.5)) / l) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = sqrt((t / cos(k)));
	t_2 = k * sqrt(2.0);
	tmp = 0.0;
	if (t <= -1.02e+121)
		tmp = 2.0 / (((t_2 / l) * ((t ^ 3.0) ^ 0.5)) ^ 2.0);
	elseif (t <= -2.6e-180)
		tmp = ((l * (2.0 / (t ^ 3.0))) / (sin(k) * tan(k))) * (l / (2.0 + ((k / t) ^ 2.0)));
	elseif (t <= -2.4e-232)
		tmp = 2.0 / (((k * t_1) / (l / sin(k))) ^ 2.0);
	elseif (t <= 3.9e-268)
		tmp = 2.0 / (((t * (k ^ 2.0)) / cos(k)) * ((k ^ 2.0) / (l ^ 2.0)));
	elseif (t <= 8e-85)
		tmp = 2.0 / ((((k * sin(k)) / l) * t_1) ^ 2.0);
	else
		tmp = 2.0 / (((t_2 * (t ^ 1.5)) / l) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Sqrt[N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(k * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.02e+121], N[(2.0 / N[Power[N[(N[(t$95$2 / l), $MachinePrecision] * N[Power[N[Power[t, 3.0], $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -2.6e-180], N[(N[(N[(l * N[(2.0 / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -2.4e-232], N[(2.0 / N[Power[N[(N[(k * t$95$1), $MachinePrecision] / N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.9e-268], N[(2.0 / N[(N[(N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[Power[k, 2.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 8e-85], N[(2.0 / N[Power[N[(N[(N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(t$95$2 * N[Power[t, 1.5], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if t < -1.02000000000000005e121

   1. Initial program 72.7%

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

      1. add-sqr-sqrt 8.1%

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

      2. pow2 8.1%

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

   3. Applied egg-rr 0.0%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}} \]

   4. Taylor expanded in k around 0 0.0%

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

      1. pow1/2 73.3%

         \[\leadsto \frac{2}{{\left(\frac{k \cdot \sqrt{2}}{\ell} \cdot \color{blue}{{\left({t}^{3}\right)}^{0.5}}\right)}^{2}} \]

   6. Applied egg-rr 73.3%

      \[\leadsto \frac{2}{{\left(\frac{k \cdot \sqrt{2}}{\ell} \cdot \color{blue}{{\left({t}^{3}\right)}^{0.5}}\right)}^{2}} \]

   if -1.02000000000000005e121 < t < -2.5999999999999999e-180

   1. Initial program 71.4%

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

   3. Simplified 67.1%

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

      1. associate-*r* 71.9%

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

      2. *-commutative 71.9%

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

      3. times-frac 73.7%

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

   5. Applied egg-rr 73.7%

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

   if -2.5999999999999999e-180 < t < -2.39999999999999999e-232

   1. Initial program 0.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. pow2 0.0%

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

   3. Applied egg-rr 0.0%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}} \]

   4. Taylor expanded in k around inf 54.9%

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

      1. *-commutative 54.9%

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

      2. associate-/l* 54.8%

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

      3. associate-*r/ 55.1%

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

   6. Simplified 55.1%

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

   if -2.39999999999999999e-232 < t < 3.8999999999999998e-268

   1. Initial program 30.8%

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

   2. Taylor expanded in t around 0 82.4%

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

      1. associate-*r* 82.4%

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

      2. *-commutative 82.4%

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

      3. times-frac 81.1%

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

   4. Simplified 81.1%

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

   5. Taylor expanded in k around 0 70.5%

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

   if 3.8999999999999998e-268 < t < 7.9999999999999998e-85

   1. Initial program 38.1%

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

      1. add-sqr-sqrt 33.4%

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

      2. pow2 33.4%

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

   3. Applied egg-rr 56.0%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}} \]

   4. Taylor expanded in k around inf 76.6%

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

   if 7.9999999999999998e-85 < t

   1. Initial program 63.0%

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

      1. add-sqr-sqrt 39.7%

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

      2. pow2 39.7%

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

   3. Applied egg-rr 53.3%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}} \]

   4. Taylor expanded in k around 0 68.4%

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

      1. associate-*l/ 68.4%

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

      2. sqrt-pow1 78.1%

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

      3. metadata-eval 78.1%

         \[\leadsto \frac{2}{{\left(\frac{\left(k \cdot \sqrt{2}\right) \cdot {t}^{\color{blue}{1.5}}}{\ell}\right)}^{2}} \]

   6. Applied egg-rr 78.1%

      \[\leadsto \frac{2}{{\color{blue}{\left(\frac{\left(k \cdot \sqrt{2}\right) \cdot {t}^{1.5}}{\ell}\right)}}^{2}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.02 \cdot 10^{+121}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sqrt{2}}{\ell} \cdot {\left({t}^{3}\right)}^{0.5}\right)}^{2}}\\ \mathbf{elif}\;t \leq -2.6 \cdot 10^{-180}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{{t}^{3}}}{\sin k \cdot \tan k} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;t \leq -2.4 \cdot 10^{-232}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sqrt{\frac{t}{\cos k}}}{\frac{\ell}{\sin k}}\right)}^{2}}\\ \mathbf{elif}\;t \leq 3.9 \cdot 10^{-268}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{2}}{\cos k} \cdot \frac{{k}^{2}}{{\ell}^{2}}}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-85}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{\left(k \cdot \sqrt{2}\right) \cdot {t}^{1.5}}{\ell}\right)}^{2}}\\ \end{array} \]

Alternative 11: 70.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.051:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \left({t}^{3} \cdot \frac{k}{{\ell}^{2}}\right)\right)}\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{-268}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{elif}\;t \leq 1.85 \cdot 10^{-85}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{\left(k \cdot \sqrt{2}\right) \cdot {t}^{1.5}}{\ell}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -0.051)
   (/
    2.0
    (*
     (+ 2.0 (pow (/ k t) 2.0))
     (* (tan k) (* (pow t 3.0) (/ k (pow l 2.0))))))
   (if (<= t 3.1e-268)
     (* 2.0 (/ (pow (* l (pow k -2.0)) 2.0) t))
     (if (<= t 1.85e-85)
       (/ 2.0 (pow (* (/ (* k (sin k)) l) (sqrt (/ t (cos k)))) 2.0))
       (/ 2.0 (pow (/ (* (* k (sqrt 2.0)) (pow t 1.5)) l) 2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -0.051) {
		tmp = 2.0 / ((2.0 + pow((k / t), 2.0)) * (tan(k) * (pow(t, 3.0) * (k / pow(l, 2.0)))));
	} else if (t <= 3.1e-268) {
		tmp = 2.0 * (pow((l * pow(k, -2.0)), 2.0) / t);
	} else if (t <= 1.85e-85) {
		tmp = 2.0 / pow((((k * sin(k)) / l) * sqrt((t / cos(k)))), 2.0);
	} else {
		tmp = 2.0 / pow((((k * sqrt(2.0)) * pow(t, 1.5)) / 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 <= (-0.051d0)) then
        tmp = 2.0d0 / ((2.0d0 + ((k / t) ** 2.0d0)) * (tan(k) * ((t ** 3.0d0) * (k / (l ** 2.0d0)))))
    else if (t <= 3.1d-268) then
        tmp = 2.0d0 * (((l * (k ** (-2.0d0))) ** 2.0d0) / t)
    else if (t <= 1.85d-85) then
        tmp = 2.0d0 / ((((k * sin(k)) / l) * sqrt((t / cos(k)))) ** 2.0d0)
    else
        tmp = 2.0d0 / ((((k * sqrt(2.0d0)) * (t ** 1.5d0)) / l) ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -0.051) {
		tmp = 2.0 / ((2.0 + Math.pow((k / t), 2.0)) * (Math.tan(k) * (Math.pow(t, 3.0) * (k / Math.pow(l, 2.0)))));
	} else if (t <= 3.1e-268) {
		tmp = 2.0 * (Math.pow((l * Math.pow(k, -2.0)), 2.0) / t);
	} else if (t <= 1.85e-85) {
		tmp = 2.0 / Math.pow((((k * Math.sin(k)) / l) * Math.sqrt((t / Math.cos(k)))), 2.0);
	} else {
		tmp = 2.0 / Math.pow((((k * Math.sqrt(2.0)) * Math.pow(t, 1.5)) / l), 2.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -0.051:
		tmp = 2.0 / ((2.0 + math.pow((k / t), 2.0)) * (math.tan(k) * (math.pow(t, 3.0) * (k / math.pow(l, 2.0)))))
	elif t <= 3.1e-268:
		tmp = 2.0 * (math.pow((l * math.pow(k, -2.0)), 2.0) / t)
	elif t <= 1.85e-85:
		tmp = 2.0 / math.pow((((k * math.sin(k)) / l) * math.sqrt((t / math.cos(k)))), 2.0)
	else:
		tmp = 2.0 / math.pow((((k * math.sqrt(2.0)) * math.pow(t, 1.5)) / l), 2.0)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -0.051)
		tmp = Float64(2.0 / Float64(Float64(2.0 + (Float64(k / t) ^ 2.0)) * Float64(tan(k) * Float64((t ^ 3.0) * Float64(k / (l ^ 2.0))))));
	elseif (t <= 3.1e-268)
		tmp = Float64(2.0 * Float64((Float64(l * (k ^ -2.0)) ^ 2.0) / t));
	elseif (t <= 1.85e-85)
		tmp = Float64(2.0 / (Float64(Float64(Float64(k * sin(k)) / l) * sqrt(Float64(t / cos(k)))) ^ 2.0));
	else
		tmp = Float64(2.0 / (Float64(Float64(Float64(k * sqrt(2.0)) * (t ^ 1.5)) / l) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -0.051)
		tmp = 2.0 / ((2.0 + ((k / t) ^ 2.0)) * (tan(k) * ((t ^ 3.0) * (k / (l ^ 2.0)))));
	elseif (t <= 3.1e-268)
		tmp = 2.0 * (((l * (k ^ -2.0)) ^ 2.0) / t);
	elseif (t <= 1.85e-85)
		tmp = 2.0 / ((((k * sin(k)) / l) * sqrt((t / cos(k)))) ^ 2.0);
	else
		tmp = 2.0 / ((((k * sqrt(2.0)) * (t ^ 1.5)) / l) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -0.051], N[(2.0 / N[(N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] * N[(k / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.1e-268], N[(2.0 * N[(N[Power[N[(l * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.85e-85], N[(2.0 / N[Power[N[(N[(N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(N[(k * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[t, 1.5], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < -0.0509999999999999967

   1. Initial program 75.8%

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

   2. Taylor expanded in k around 0 70.5%

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

      1. distribute-lft-in 70.5%

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

      2. *-commutative 70.5%

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

      3. associate-/l* 70.0%

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

      4. *-commutative 70.0%

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

      5. associate-/l* 70.0%

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

   4. Applied egg-rr 70.0%

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

      1. distribute-lft-out 70.0%

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

      2. +-commutative 70.0%

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

      3. *-commutative 70.0%

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

      4. associate-+r+ 70.0%

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

      5. metadata-eval 70.0%

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

      6. associate-/r/ 70.4%

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

   6. Simplified 70.4%

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

   if -0.0509999999999999967 < t < 3.0999999999999998e-268

   1. Initial program 43.2%

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

   2. Taylor expanded in t around 0 79.3%

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

      1. associate-*r* 79.3%

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

      2. *-commutative 79.3%

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

      3. times-frac 77.5%

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

   4. Simplified 77.5%

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

   5. Taylor expanded in k around 0 59.4%

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

      1. associate-/r* 59.5%

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

   7. Simplified 59.5%

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

      1. expm1-log1p-u 59.5%

         \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)\right)}}{t} \]

      2. expm1-udef 57.9%

         \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)} - 1}}{t} \]

      3. div-inv 57.9%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left(\color{blue}{{\ell}^{2} \cdot \frac{1}{{k}^{4}}}\right)} - 1}{t} \]

      4. pow-flip 57.9%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-4\right)}}\right)} - 1}{t} \]

      5. metadata-eval 57.9%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{\color{blue}{-4}}\right)} - 1}{t} \]

   9. Applied egg-rr 57.9%

      \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)} - 1}}{t} \]
   10. Step-by-step derivation

       1. expm1-def 59.5%

          \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)\right)}}{t} \]

       2. expm1-log1p 59.5%

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

   11. Simplified 59.5%

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

       1. add-sqr-sqrt 59.5%

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

       2. pow2 59.5%

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

       3. sqrt-prod 59.5%

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

       4. unpow2 59.5%

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

       5. sqrt-prod 37.2%

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

       6. add-sqr-sqrt 65.4%

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

       7. sqrt-pow1 66.7%

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

       8. metadata-eval 66.7%

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

   13. Applied egg-rr 66.7%

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

   if 3.0999999999999998e-268 < t < 1.84999999999999992e-85

   1. Initial program 38.1%

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

      1. add-sqr-sqrt 33.4%

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

      2. pow2 33.4%

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

   3. Applied egg-rr 56.0%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}} \]

   4. Taylor expanded in k around inf 76.6%

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

   if 1.84999999999999992e-85 < t

   1. Initial program 63.0%

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

      1. add-sqr-sqrt 39.7%

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

      2. pow2 39.7%

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

   3. Applied egg-rr 53.3%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}} \]

   4. Taylor expanded in k around 0 68.4%

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

      1. associate-*l/ 68.4%

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

      2. sqrt-pow1 78.1%

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

      3. metadata-eval 78.1%

         \[\leadsto \frac{2}{{\left(\frac{\left(k \cdot \sqrt{2}\right) \cdot {t}^{\color{blue}{1.5}}}{\ell}\right)}^{2}} \]

   6. Applied egg-rr 78.1%

      \[\leadsto \frac{2}{{\color{blue}{\left(\frac{\left(k \cdot \sqrt{2}\right) \cdot {t}^{1.5}}{\ell}\right)}}^{2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.051:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \left({t}^{3} \cdot \frac{k}{{\ell}^{2}}\right)\right)}\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{-268}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{elif}\;t \leq 1.85 \cdot 10^{-85}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{\left(k \cdot \sqrt{2}\right) \cdot {t}^{1.5}}{\ell}\right)}^{2}}\\ \end{array} \]

Alternative 12: 70.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.0105:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{\tan k}{\frac{{\ell}^{2}}{k \cdot {t}^{3}}}}\\ \mathbf{elif}\;t \leq 7.9 \cdot 10^{-268}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-84}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{\left(k \cdot \sqrt{2}\right) \cdot {t}^{1.5}}{\ell}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -0.0105)
   (/
    2.0
    (*
     (+ 2.0 (pow (/ k t) 2.0))
     (/ (tan k) (/ (pow l 2.0) (* k (pow t 3.0))))))
   (if (<= t 7.9e-268)
     (* 2.0 (/ (pow (* l (pow k -2.0)) 2.0) t))
     (if (<= t 1.15e-84)
       (/ 2.0 (pow (* (/ (* k (sin k)) l) (sqrt (/ t (cos k)))) 2.0))
       (/ 2.0 (pow (/ (* (* k (sqrt 2.0)) (pow t 1.5)) l) 2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -0.0105) {
		tmp = 2.0 / ((2.0 + pow((k / t), 2.0)) * (tan(k) / (pow(l, 2.0) / (k * pow(t, 3.0)))));
	} else if (t <= 7.9e-268) {
		tmp = 2.0 * (pow((l * pow(k, -2.0)), 2.0) / t);
	} else if (t <= 1.15e-84) {
		tmp = 2.0 / pow((((k * sin(k)) / l) * sqrt((t / cos(k)))), 2.0);
	} else {
		tmp = 2.0 / pow((((k * sqrt(2.0)) * pow(t, 1.5)) / 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 <= (-0.0105d0)) then
        tmp = 2.0d0 / ((2.0d0 + ((k / t) ** 2.0d0)) * (tan(k) / ((l ** 2.0d0) / (k * (t ** 3.0d0)))))
    else if (t <= 7.9d-268) then
        tmp = 2.0d0 * (((l * (k ** (-2.0d0))) ** 2.0d0) / t)
    else if (t <= 1.15d-84) then
        tmp = 2.0d0 / ((((k * sin(k)) / l) * sqrt((t / cos(k)))) ** 2.0d0)
    else
        tmp = 2.0d0 / ((((k * sqrt(2.0d0)) * (t ** 1.5d0)) / l) ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -0.0105) {
		tmp = 2.0 / ((2.0 + Math.pow((k / t), 2.0)) * (Math.tan(k) / (Math.pow(l, 2.0) / (k * Math.pow(t, 3.0)))));
	} else if (t <= 7.9e-268) {
		tmp = 2.0 * (Math.pow((l * Math.pow(k, -2.0)), 2.0) / t);
	} else if (t <= 1.15e-84) {
		tmp = 2.0 / Math.pow((((k * Math.sin(k)) / l) * Math.sqrt((t / Math.cos(k)))), 2.0);
	} else {
		tmp = 2.0 / Math.pow((((k * Math.sqrt(2.0)) * Math.pow(t, 1.5)) / l), 2.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -0.0105:
		tmp = 2.0 / ((2.0 + math.pow((k / t), 2.0)) * (math.tan(k) / (math.pow(l, 2.0) / (k * math.pow(t, 3.0)))))
	elif t <= 7.9e-268:
		tmp = 2.0 * (math.pow((l * math.pow(k, -2.0)), 2.0) / t)
	elif t <= 1.15e-84:
		tmp = 2.0 / math.pow((((k * math.sin(k)) / l) * math.sqrt((t / math.cos(k)))), 2.0)
	else:
		tmp = 2.0 / math.pow((((k * math.sqrt(2.0)) * math.pow(t, 1.5)) / l), 2.0)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -0.0105)
		tmp = Float64(2.0 / Float64(Float64(2.0 + (Float64(k / t) ^ 2.0)) * Float64(tan(k) / Float64((l ^ 2.0) / Float64(k * (t ^ 3.0))))));
	elseif (t <= 7.9e-268)
		tmp = Float64(2.0 * Float64((Float64(l * (k ^ -2.0)) ^ 2.0) / t));
	elseif (t <= 1.15e-84)
		tmp = Float64(2.0 / (Float64(Float64(Float64(k * sin(k)) / l) * sqrt(Float64(t / cos(k)))) ^ 2.0));
	else
		tmp = Float64(2.0 / (Float64(Float64(Float64(k * sqrt(2.0)) * (t ^ 1.5)) / l) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -0.0105)
		tmp = 2.0 / ((2.0 + ((k / t) ^ 2.0)) * (tan(k) / ((l ^ 2.0) / (k * (t ^ 3.0)))));
	elseif (t <= 7.9e-268)
		tmp = 2.0 * (((l * (k ^ -2.0)) ^ 2.0) / t);
	elseif (t <= 1.15e-84)
		tmp = 2.0 / ((((k * sin(k)) / l) * sqrt((t / cos(k)))) ^ 2.0);
	else
		tmp = 2.0 / ((((k * sqrt(2.0)) * (t ^ 1.5)) / l) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -0.0105], N[(2.0 / N[(N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] / N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 7.9e-268], N[(2.0 * N[(N[Power[N[(l * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.15e-84], N[(2.0 / N[Power[N[(N[(N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(N[(k * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[t, 1.5], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < -0.0105000000000000007

   1. Initial program 75.8%

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

   2. Taylor expanded in k around 0 70.5%

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

      1. distribute-lft-in 70.5%

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

      2. *-commutative 70.5%

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

      3. associate-/l* 70.0%

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

      4. *-commutative 70.0%

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

      5. associate-/l* 70.0%

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

   4. Applied egg-rr 70.0%

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

      1. distribute-lft-out 70.0%

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

      2. +-commutative 70.0%

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

      3. *-commutative 70.0%

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

      4. associate-+r+ 70.0%

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

      5. metadata-eval 70.0%

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

      6. associate-*r/ 60.7%

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

      7. associate-/l* 70.0%

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

      8. associate-/r* 70.5%

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

      9. *-commutative 70.5%

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

   6. Simplified 70.5%

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

   if -0.0105000000000000007 < t < 7.9000000000000001e-268

   1. Initial program 43.2%

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

   2. Taylor expanded in t around 0 79.3%

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

      1. associate-*r* 79.3%

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

      2. *-commutative 79.3%

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

      3. times-frac 77.5%

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

   4. Simplified 77.5%

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

   5. Taylor expanded in k around 0 59.4%

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

      1. associate-/r* 59.5%

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

   7. Simplified 59.5%

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

      1. expm1-log1p-u 59.5%

         \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)\right)}}{t} \]

      2. expm1-udef 57.9%

         \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)} - 1}}{t} \]

      3. div-inv 57.9%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left(\color{blue}{{\ell}^{2} \cdot \frac{1}{{k}^{4}}}\right)} - 1}{t} \]

      4. pow-flip 57.9%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-4\right)}}\right)} - 1}{t} \]

      5. metadata-eval 57.9%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{\color{blue}{-4}}\right)} - 1}{t} \]

   9. Applied egg-rr 57.9%

      \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)} - 1}}{t} \]
   10. Step-by-step derivation

       1. expm1-def 59.5%

          \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)\right)}}{t} \]

       2. expm1-log1p 59.5%

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

   11. Simplified 59.5%

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

       1. add-sqr-sqrt 59.5%

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

       2. pow2 59.5%

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

       3. sqrt-prod 59.5%

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

       4. unpow2 59.5%

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

       5. sqrt-prod 37.2%

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

       6. add-sqr-sqrt 65.4%

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

       7. sqrt-pow1 66.7%

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

       8. metadata-eval 66.7%

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

   13. Applied egg-rr 66.7%

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

   if 7.9000000000000001e-268 < t < 1.1499999999999999e-84

   1. Initial program 38.1%

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

      1. add-sqr-sqrt 33.4%

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

      2. pow2 33.4%

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

   3. Applied egg-rr 56.0%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}} \]

   4. Taylor expanded in k around inf 76.6%

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

   if 1.1499999999999999e-84 < t

   1. Initial program 63.0%

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

      1. add-sqr-sqrt 39.7%

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

      2. pow2 39.7%

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

   3. Applied egg-rr 53.3%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}} \]

   4. Taylor expanded in k around 0 68.4%

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

      1. associate-*l/ 68.4%

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

      2. sqrt-pow1 78.1%

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

      3. metadata-eval 78.1%

         \[\leadsto \frac{2}{{\left(\frac{\left(k \cdot \sqrt{2}\right) \cdot {t}^{\color{blue}{1.5}}}{\ell}\right)}^{2}} \]

   6. Applied egg-rr 78.1%

      \[\leadsto \frac{2}{{\color{blue}{\left(\frac{\left(k \cdot \sqrt{2}\right) \cdot {t}^{1.5}}{\ell}\right)}}^{2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.0105:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{\tan k}{\frac{{\ell}^{2}}{k \cdot {t}^{3}}}}\\ \mathbf{elif}\;t \leq 7.9 \cdot 10^{-268}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-84}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{\left(k \cdot \sqrt{2}\right) \cdot {t}^{1.5}}{\ell}\right)}^{2}}\\ \end{array} \]

Alternative 13: 70.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.0105:\\ \;\;\;\;\frac{2}{2 \cdot \frac{1}{\frac{{\ell}^{2}}{k} \cdot \frac{\cos k}{\sin k \cdot {t}^{3}}}}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-267}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{-83}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{\left(k \cdot \sqrt{2}\right) \cdot {t}^{1.5}}{\ell}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -0.0105)
   (/
    2.0
    (* 2.0 (/ 1.0 (* (/ (pow l 2.0) k) (/ (cos k) (* (sin k) (pow t 3.0)))))))
   (if (<= t 4.2e-267)
     (* 2.0 (/ (pow (* l (pow k -2.0)) 2.0) t))
     (if (<= t 9.2e-83)
       (/ 2.0 (pow (* (/ (* k (sin k)) l) (sqrt (/ t (cos k)))) 2.0))
       (/ 2.0 (pow (/ (* (* k (sqrt 2.0)) (pow t 1.5)) l) 2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -0.0105) {
		tmp = 2.0 / (2.0 * (1.0 / ((pow(l, 2.0) / k) * (cos(k) / (sin(k) * pow(t, 3.0))))));
	} else if (t <= 4.2e-267) {
		tmp = 2.0 * (pow((l * pow(k, -2.0)), 2.0) / t);
	} else if (t <= 9.2e-83) {
		tmp = 2.0 / pow((((k * sin(k)) / l) * sqrt((t / cos(k)))), 2.0);
	} else {
		tmp = 2.0 / pow((((k * sqrt(2.0)) * pow(t, 1.5)) / 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 <= (-0.0105d0)) then
        tmp = 2.0d0 / (2.0d0 * (1.0d0 / (((l ** 2.0d0) / k) * (cos(k) / (sin(k) * (t ** 3.0d0))))))
    else if (t <= 4.2d-267) then
        tmp = 2.0d0 * (((l * (k ** (-2.0d0))) ** 2.0d0) / t)
    else if (t <= 9.2d-83) then
        tmp = 2.0d0 / ((((k * sin(k)) / l) * sqrt((t / cos(k)))) ** 2.0d0)
    else
        tmp = 2.0d0 / ((((k * sqrt(2.0d0)) * (t ** 1.5d0)) / l) ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -0.0105) {
		tmp = 2.0 / (2.0 * (1.0 / ((Math.pow(l, 2.0) / k) * (Math.cos(k) / (Math.sin(k) * Math.pow(t, 3.0))))));
	} else if (t <= 4.2e-267) {
		tmp = 2.0 * (Math.pow((l * Math.pow(k, -2.0)), 2.0) / t);
	} else if (t <= 9.2e-83) {
		tmp = 2.0 / Math.pow((((k * Math.sin(k)) / l) * Math.sqrt((t / Math.cos(k)))), 2.0);
	} else {
		tmp = 2.0 / Math.pow((((k * Math.sqrt(2.0)) * Math.pow(t, 1.5)) / l), 2.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -0.0105:
		tmp = 2.0 / (2.0 * (1.0 / ((math.pow(l, 2.0) / k) * (math.cos(k) / (math.sin(k) * math.pow(t, 3.0))))))
	elif t <= 4.2e-267:
		tmp = 2.0 * (math.pow((l * math.pow(k, -2.0)), 2.0) / t)
	elif t <= 9.2e-83:
		tmp = 2.0 / math.pow((((k * math.sin(k)) / l) * math.sqrt((t / math.cos(k)))), 2.0)
	else:
		tmp = 2.0 / math.pow((((k * math.sqrt(2.0)) * math.pow(t, 1.5)) / l), 2.0)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -0.0105)
		tmp = Float64(2.0 / Float64(2.0 * Float64(1.0 / Float64(Float64((l ^ 2.0) / k) * Float64(cos(k) / Float64(sin(k) * (t ^ 3.0)))))));
	elseif (t <= 4.2e-267)
		tmp = Float64(2.0 * Float64((Float64(l * (k ^ -2.0)) ^ 2.0) / t));
	elseif (t <= 9.2e-83)
		tmp = Float64(2.0 / (Float64(Float64(Float64(k * sin(k)) / l) * sqrt(Float64(t / cos(k)))) ^ 2.0));
	else
		tmp = Float64(2.0 / (Float64(Float64(Float64(k * sqrt(2.0)) * (t ^ 1.5)) / l) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -0.0105)
		tmp = 2.0 / (2.0 * (1.0 / (((l ^ 2.0) / k) * (cos(k) / (sin(k) * (t ^ 3.0))))));
	elseif (t <= 4.2e-267)
		tmp = 2.0 * (((l * (k ^ -2.0)) ^ 2.0) / t);
	elseif (t <= 9.2e-83)
		tmp = 2.0 / ((((k * sin(k)) / l) * sqrt((t / cos(k)))) ^ 2.0);
	else
		tmp = 2.0 / ((((k * sqrt(2.0)) * (t ^ 1.5)) / l) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -0.0105], N[(2.0 / N[(2.0 * N[(1.0 / N[(N[(N[Power[l, 2.0], $MachinePrecision] / k), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.2e-267], N[(2.0 * N[(N[Power[N[(l * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 9.2e-83], N[(2.0 / N[Power[N[(N[(N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(N[(k * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[t, 1.5], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < -0.0105000000000000007

   1. Initial program 75.8%

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

   2. Taylor expanded in k around 0 70.5%

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

   3. Taylor expanded in t around inf 67.2%

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

      1. clear-num 67.2%

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

      2. inv-pow 67.2%

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

      3. *-commutative 67.2%

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

   5. Applied egg-rr 67.2%

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

      1. unpow-1 67.2%

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

      2. times-frac 70.4%

         \[\leadsto \frac{2}{2 \cdot \frac{1}{\color{blue}{\frac{{\ell}^{2}}{k} \cdot \frac{\cos k}{\sin k \cdot {t}^{3}}}}} \]

   7. Simplified 70.4%

      \[\leadsto \frac{2}{2 \cdot \color{blue}{\frac{1}{\frac{{\ell}^{2}}{k} \cdot \frac{\cos k}{\sin k \cdot {t}^{3}}}}} \]

   if -0.0105000000000000007 < t < 4.2000000000000003e-267

   1. Initial program 43.2%

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

   2. Taylor expanded in t around 0 79.3%

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

      1. associate-*r* 79.3%

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

      2. *-commutative 79.3%

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

      3. times-frac 77.5%

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

   4. Simplified 77.5%

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

   5. Taylor expanded in k around 0 59.4%

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

      1. associate-/r* 59.5%

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

   7. Simplified 59.5%

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

      1. expm1-log1p-u 59.5%

         \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)\right)}}{t} \]

      2. expm1-udef 57.9%

         \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)} - 1}}{t} \]

      3. div-inv 57.9%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left(\color{blue}{{\ell}^{2} \cdot \frac{1}{{k}^{4}}}\right)} - 1}{t} \]

      4. pow-flip 57.9%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-4\right)}}\right)} - 1}{t} \]

      5. metadata-eval 57.9%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{\color{blue}{-4}}\right)} - 1}{t} \]

   9. Applied egg-rr 57.9%

      \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)} - 1}}{t} \]
   10. Step-by-step derivation

       1. expm1-def 59.5%

          \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)\right)}}{t} \]

       2. expm1-log1p 59.5%

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

   11. Simplified 59.5%

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

       1. add-sqr-sqrt 59.5%

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

       2. pow2 59.5%

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

       3. sqrt-prod 59.5%

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

       4. unpow2 59.5%

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

       5. sqrt-prod 37.2%

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

       6. add-sqr-sqrt 65.4%

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

       7. sqrt-pow1 66.7%

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

       8. metadata-eval 66.7%

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

   13. Applied egg-rr 66.7%

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

   if 4.2000000000000003e-267 < t < 9.19999999999999959e-83

   1. Initial program 38.1%

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

      1. add-sqr-sqrt 33.4%

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

      2. pow2 33.4%

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

   3. Applied egg-rr 56.0%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}} \]

   4. Taylor expanded in k around inf 76.6%

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

   if 9.19999999999999959e-83 < t

   1. Initial program 63.0%

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

      1. add-sqr-sqrt 39.7%

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

      2. pow2 39.7%

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

   3. Applied egg-rr 53.3%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}} \]

   4. Taylor expanded in k around 0 68.4%

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

      1. associate-*l/ 68.4%

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

      2. sqrt-pow1 78.1%

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

      3. metadata-eval 78.1%

         \[\leadsto \frac{2}{{\left(\frac{\left(k \cdot \sqrt{2}\right) \cdot {t}^{\color{blue}{1.5}}}{\ell}\right)}^{2}} \]

   6. Applied egg-rr 78.1%

      \[\leadsto \frac{2}{{\color{blue}{\left(\frac{\left(k \cdot \sqrt{2}\right) \cdot {t}^{1.5}}{\ell}\right)}}^{2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.0105:\\ \;\;\;\;\frac{2}{2 \cdot \frac{1}{\frac{{\ell}^{2}}{k} \cdot \frac{\cos k}{\sin k \cdot {t}^{3}}}}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-267}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{-83}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{\left(k \cdot \sqrt{2}\right) \cdot {t}^{1.5}}{\ell}\right)}^{2}}\\ \end{array} \]

Alternative 14: 70.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -860000000:\\ \;\;\;\;\frac{2}{2 \cdot \frac{k \cdot \left(\sin k \cdot {t}^{3}\right)}{{\ell}^{2}}}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{-267}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{elif}\;t \leq 8.4 \cdot 10^{-83}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{\left(k \cdot \sqrt{2}\right) \cdot {t}^{1.5}}{\ell}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -860000000.0)
   (/ 2.0 (* 2.0 (/ (* k (* (sin k) (pow t 3.0))) (pow l 2.0))))
   (if (<= t 1.1e-267)
     (* 2.0 (/ (pow (* l (pow k -2.0)) 2.0) t))
     (if (<= t 8.4e-83)
       (/ 2.0 (pow (* (/ (* k (sin k)) l) (sqrt (/ t (cos k)))) 2.0))
       (/ 2.0 (pow (/ (* (* k (sqrt 2.0)) (pow t 1.5)) l) 2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -860000000.0) {
		tmp = 2.0 / (2.0 * ((k * (sin(k) * pow(t, 3.0))) / pow(l, 2.0)));
	} else if (t <= 1.1e-267) {
		tmp = 2.0 * (pow((l * pow(k, -2.0)), 2.0) / t);
	} else if (t <= 8.4e-83) {
		tmp = 2.0 / pow((((k * sin(k)) / l) * sqrt((t / cos(k)))), 2.0);
	} else {
		tmp = 2.0 / pow((((k * sqrt(2.0)) * pow(t, 1.5)) / 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 <= (-860000000.0d0)) then
        tmp = 2.0d0 / (2.0d0 * ((k * (sin(k) * (t ** 3.0d0))) / (l ** 2.0d0)))
    else if (t <= 1.1d-267) then
        tmp = 2.0d0 * (((l * (k ** (-2.0d0))) ** 2.0d0) / t)
    else if (t <= 8.4d-83) then
        tmp = 2.0d0 / ((((k * sin(k)) / l) * sqrt((t / cos(k)))) ** 2.0d0)
    else
        tmp = 2.0d0 / ((((k * sqrt(2.0d0)) * (t ** 1.5d0)) / l) ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -860000000.0) {
		tmp = 2.0 / (2.0 * ((k * (Math.sin(k) * Math.pow(t, 3.0))) / Math.pow(l, 2.0)));
	} else if (t <= 1.1e-267) {
		tmp = 2.0 * (Math.pow((l * Math.pow(k, -2.0)), 2.0) / t);
	} else if (t <= 8.4e-83) {
		tmp = 2.0 / Math.pow((((k * Math.sin(k)) / l) * Math.sqrt((t / Math.cos(k)))), 2.0);
	} else {
		tmp = 2.0 / Math.pow((((k * Math.sqrt(2.0)) * Math.pow(t, 1.5)) / l), 2.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -860000000.0:
		tmp = 2.0 / (2.0 * ((k * (math.sin(k) * math.pow(t, 3.0))) / math.pow(l, 2.0)))
	elif t <= 1.1e-267:
		tmp = 2.0 * (math.pow((l * math.pow(k, -2.0)), 2.0) / t)
	elif t <= 8.4e-83:
		tmp = 2.0 / math.pow((((k * math.sin(k)) / l) * math.sqrt((t / math.cos(k)))), 2.0)
	else:
		tmp = 2.0 / math.pow((((k * math.sqrt(2.0)) * math.pow(t, 1.5)) / l), 2.0)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -860000000.0)
		tmp = Float64(2.0 / Float64(2.0 * Float64(Float64(k * Float64(sin(k) * (t ^ 3.0))) / (l ^ 2.0))));
	elseif (t <= 1.1e-267)
		tmp = Float64(2.0 * Float64((Float64(l * (k ^ -2.0)) ^ 2.0) / t));
	elseif (t <= 8.4e-83)
		tmp = Float64(2.0 / (Float64(Float64(Float64(k * sin(k)) / l) * sqrt(Float64(t / cos(k)))) ^ 2.0));
	else
		tmp = Float64(2.0 / (Float64(Float64(Float64(k * sqrt(2.0)) * (t ^ 1.5)) / l) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -860000000.0)
		tmp = 2.0 / (2.0 * ((k * (sin(k) * (t ^ 3.0))) / (l ^ 2.0)));
	elseif (t <= 1.1e-267)
		tmp = 2.0 * (((l * (k ^ -2.0)) ^ 2.0) / t);
	elseif (t <= 8.4e-83)
		tmp = 2.0 / ((((k * sin(k)) / l) * sqrt((t / cos(k)))) ^ 2.0);
	else
		tmp = 2.0 / ((((k * sqrt(2.0)) * (t ^ 1.5)) / l) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -860000000.0], N[(2.0 / N[(2.0 * N[(N[(k * N[(N[Sin[k], $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.1e-267], N[(2.0 * N[(N[Power[N[(l * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 8.4e-83], N[(2.0 / N[Power[N[(N[(N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(N[(k * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[t, 1.5], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < -8.6e8

   1. Initial program 76.2%

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

   2. Taylor expanded in k around 0 70.6%

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

   3. Taylor expanded in t around inf 67.2%

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

   4. Taylor expanded in k around 0 67.5%

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

   if -8.6e8 < t < 1.09999999999999994e-267

   1. Initial program 44.3%

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

   2. Taylor expanded in t around 0 78.8%

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

      1. associate-*r* 78.8%

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

      2. *-commutative 78.8%

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

      3. times-frac 77.0%

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

   4. Simplified 77.0%

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

   5. Taylor expanded in k around 0 59.7%

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

      1. associate-/r* 59.8%

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

   7. Simplified 59.8%

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

      1. expm1-log1p-u 59.8%

         \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)\right)}}{t} \]

      2. expm1-udef 58.3%

         \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)} - 1}}{t} \]

      3. div-inv 58.3%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left(\color{blue}{{\ell}^{2} \cdot \frac{1}{{k}^{4}}}\right)} - 1}{t} \]

      4. pow-flip 58.3%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-4\right)}}\right)} - 1}{t} \]

      5. metadata-eval 58.3%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{\color{blue}{-4}}\right)} - 1}{t} \]

   9. Applied egg-rr 58.3%

      \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)} - 1}}{t} \]
   10. Step-by-step derivation

       1. expm1-def 59.8%

          \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)\right)}}{t} \]

       2. expm1-log1p 59.8%

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

   11. Simplified 59.8%

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

       1. add-sqr-sqrt 59.8%

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

       2. pow2 59.8%

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

       3. sqrt-prod 59.8%

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

       4. unpow2 59.8%

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

       5. sqrt-prod 38.5%

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

       6. add-sqr-sqrt 65.5%

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

       7. sqrt-pow1 66.8%

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

       8. metadata-eval 66.8%

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

   13. Applied egg-rr 66.8%

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

   if 1.09999999999999994e-267 < t < 8.3999999999999996e-83

   1. Initial program 38.1%

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

      1. add-sqr-sqrt 33.4%

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

      2. pow2 33.4%

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

   3. Applied egg-rr 56.0%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}} \]

   4. Taylor expanded in k around inf 76.6%

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

   if 8.3999999999999996e-83 < t

   1. Initial program 63.0%

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

      1. add-sqr-sqrt 39.7%

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

      2. pow2 39.7%

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

   3. Applied egg-rr 53.3%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}} \]

   4. Taylor expanded in k around 0 68.4%

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

      1. associate-*l/ 68.4%

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

      2. sqrt-pow1 78.1%

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

      3. metadata-eval 78.1%

         \[\leadsto \frac{2}{{\left(\frac{\left(k \cdot \sqrt{2}\right) \cdot {t}^{\color{blue}{1.5}}}{\ell}\right)}^{2}} \]

   6. Applied egg-rr 78.1%

      \[\leadsto \frac{2}{{\color{blue}{\left(\frac{\left(k \cdot \sqrt{2}\right) \cdot {t}^{1.5}}{\ell}\right)}}^{2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -860000000:\\ \;\;\;\;\frac{2}{2 \cdot \frac{k \cdot \left(\sin k \cdot {t}^{3}\right)}{{\ell}^{2}}}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{-267}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{elif}\;t \leq 8.4 \cdot 10^{-83}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{\left(k \cdot \sqrt{2}\right) \cdot {t}^{1.5}}{\ell}\right)}^{2}}\\ \end{array} \]

Alternative 15: 70.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.0106:\\ \;\;\;\;\frac{2}{2 \cdot \left(\frac{k \cdot {t}^{3}}{\cos k} \cdot \frac{\sin k}{{\ell}^{2}}\right)}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{-268}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{-83}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{\left(k \cdot \sqrt{2}\right) \cdot {t}^{1.5}}{\ell}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -0.0106)
   (/ 2.0 (* 2.0 (* (/ (* k (pow t 3.0)) (cos k)) (/ (sin k) (pow l 2.0)))))
   (if (<= t 5.5e-268)
     (* 2.0 (/ (pow (* l (pow k -2.0)) 2.0) t))
     (if (<= t 9.2e-83)
       (/ 2.0 (pow (* (/ (* k (sin k)) l) (sqrt (/ t (cos k)))) 2.0))
       (/ 2.0 (pow (/ (* (* k (sqrt 2.0)) (pow t 1.5)) l) 2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -0.0106) {
		tmp = 2.0 / (2.0 * (((k * pow(t, 3.0)) / cos(k)) * (sin(k) / pow(l, 2.0))));
	} else if (t <= 5.5e-268) {
		tmp = 2.0 * (pow((l * pow(k, -2.0)), 2.0) / t);
	} else if (t <= 9.2e-83) {
		tmp = 2.0 / pow((((k * sin(k)) / l) * sqrt((t / cos(k)))), 2.0);
	} else {
		tmp = 2.0 / pow((((k * sqrt(2.0)) * pow(t, 1.5)) / 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 <= (-0.0106d0)) then
        tmp = 2.0d0 / (2.0d0 * (((k * (t ** 3.0d0)) / cos(k)) * (sin(k) / (l ** 2.0d0))))
    else if (t <= 5.5d-268) then
        tmp = 2.0d0 * (((l * (k ** (-2.0d0))) ** 2.0d0) / t)
    else if (t <= 9.2d-83) then
        tmp = 2.0d0 / ((((k * sin(k)) / l) * sqrt((t / cos(k)))) ** 2.0d0)
    else
        tmp = 2.0d0 / ((((k * sqrt(2.0d0)) * (t ** 1.5d0)) / l) ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -0.0106) {
		tmp = 2.0 / (2.0 * (((k * Math.pow(t, 3.0)) / Math.cos(k)) * (Math.sin(k) / Math.pow(l, 2.0))));
	} else if (t <= 5.5e-268) {
		tmp = 2.0 * (Math.pow((l * Math.pow(k, -2.0)), 2.0) / t);
	} else if (t <= 9.2e-83) {
		tmp = 2.0 / Math.pow((((k * Math.sin(k)) / l) * Math.sqrt((t / Math.cos(k)))), 2.0);
	} else {
		tmp = 2.0 / Math.pow((((k * Math.sqrt(2.0)) * Math.pow(t, 1.5)) / l), 2.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -0.0106:
		tmp = 2.0 / (2.0 * (((k * math.pow(t, 3.0)) / math.cos(k)) * (math.sin(k) / math.pow(l, 2.0))))
	elif t <= 5.5e-268:
		tmp = 2.0 * (math.pow((l * math.pow(k, -2.0)), 2.0) / t)
	elif t <= 9.2e-83:
		tmp = 2.0 / math.pow((((k * math.sin(k)) / l) * math.sqrt((t / math.cos(k)))), 2.0)
	else:
		tmp = 2.0 / math.pow((((k * math.sqrt(2.0)) * math.pow(t, 1.5)) / l), 2.0)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -0.0106)
		tmp = Float64(2.0 / Float64(2.0 * Float64(Float64(Float64(k * (t ^ 3.0)) / cos(k)) * Float64(sin(k) / (l ^ 2.0)))));
	elseif (t <= 5.5e-268)
		tmp = Float64(2.0 * Float64((Float64(l * (k ^ -2.0)) ^ 2.0) / t));
	elseif (t <= 9.2e-83)
		tmp = Float64(2.0 / (Float64(Float64(Float64(k * sin(k)) / l) * sqrt(Float64(t / cos(k)))) ^ 2.0));
	else
		tmp = Float64(2.0 / (Float64(Float64(Float64(k * sqrt(2.0)) * (t ^ 1.5)) / l) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -0.0106)
		tmp = 2.0 / (2.0 * (((k * (t ^ 3.0)) / cos(k)) * (sin(k) / (l ^ 2.0))));
	elseif (t <= 5.5e-268)
		tmp = 2.0 * (((l * (k ^ -2.0)) ^ 2.0) / t);
	elseif (t <= 9.2e-83)
		tmp = 2.0 / ((((k * sin(k)) / l) * sqrt((t / cos(k)))) ^ 2.0);
	else
		tmp = 2.0 / ((((k * sqrt(2.0)) * (t ^ 1.5)) / l) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -0.0106], N[(2.0 / N[(2.0 * N[(N[(N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.5e-268], N[(2.0 * N[(N[Power[N[(l * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 9.2e-83], N[(2.0 / N[Power[N[(N[(N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(N[(k * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[t, 1.5], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < -0.0106

   1. Initial program 75.8%

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

   2. Taylor expanded in k around 0 70.5%

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

   3. Taylor expanded in t around inf 67.2%

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

      1. associate-*r* 67.2%

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

      2. *-commutative 67.2%

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

      3. times-frac 70.4%

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

   5. Simplified 70.4%

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

   if -0.0106 < t < 5.4999999999999997e-268

   1. Initial program 43.2%

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

   2. Taylor expanded in t around 0 79.3%

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

      1. associate-*r* 79.3%

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

      2. *-commutative 79.3%

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

      3. times-frac 77.5%

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

   4. Simplified 77.5%

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

   5. Taylor expanded in k around 0 59.4%

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

      1. associate-/r* 59.5%

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

   7. Simplified 59.5%

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

      1. expm1-log1p-u 59.5%

         \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)\right)}}{t} \]

      2. expm1-udef 57.9%

         \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)} - 1}}{t} \]

      3. div-inv 57.9%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left(\color{blue}{{\ell}^{2} \cdot \frac{1}{{k}^{4}}}\right)} - 1}{t} \]

      4. pow-flip 57.9%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-4\right)}}\right)} - 1}{t} \]

      5. metadata-eval 57.9%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{\color{blue}{-4}}\right)} - 1}{t} \]

   9. Applied egg-rr 57.9%

      \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)} - 1}}{t} \]
   10. Step-by-step derivation

       1. expm1-def 59.5%

          \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)\right)}}{t} \]

       2. expm1-log1p 59.5%

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

   11. Simplified 59.5%

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

       1. add-sqr-sqrt 59.5%

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

       2. pow2 59.5%

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

       3. sqrt-prod 59.5%

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

       4. unpow2 59.5%

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

       5. sqrt-prod 37.2%

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

       6. add-sqr-sqrt 65.4%

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

       7. sqrt-pow1 66.7%

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

       8. metadata-eval 66.7%

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

   13. Applied egg-rr 66.7%

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

   if 5.4999999999999997e-268 < t < 9.19999999999999959e-83

   1. Initial program 38.1%

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

      1. add-sqr-sqrt 33.4%

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

      2. pow2 33.4%

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

   3. Applied egg-rr 56.0%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}} \]

   4. Taylor expanded in k around inf 76.6%

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

   if 9.19999999999999959e-83 < t

   1. Initial program 63.0%

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

      1. add-sqr-sqrt 39.7%

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

      2. pow2 39.7%

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

   3. Applied egg-rr 53.3%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}} \]

   4. Taylor expanded in k around 0 68.4%

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

      1. associate-*l/ 68.4%

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

      2. sqrt-pow1 78.1%

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

      3. metadata-eval 78.1%

         \[\leadsto \frac{2}{{\left(\frac{\left(k \cdot \sqrt{2}\right) \cdot {t}^{\color{blue}{1.5}}}{\ell}\right)}^{2}} \]

   6. Applied egg-rr 78.1%

      \[\leadsto \frac{2}{{\color{blue}{\left(\frac{\left(k \cdot \sqrt{2}\right) \cdot {t}^{1.5}}{\ell}\right)}}^{2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.0106:\\ \;\;\;\;\frac{2}{2 \cdot \left(\frac{k \cdot {t}^{3}}{\cos k} \cdot \frac{\sin k}{{\ell}^{2}}\right)}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{-268}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{-83}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{\left(k \cdot \sqrt{2}\right) \cdot {t}^{1.5}}{\ell}\right)}^{2}}\\ \end{array} \]

Alternative 16: 67.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -860000000:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\\ \mathbf{elif}\;t \leq 1.02 \cdot 10^{-52}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left({t}^{1.5} \cdot \frac{k}{\frac{\ell}{\sqrt{2}}}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -860000000.0)
   (/ (pow l 2.0) (* (pow k 2.0) (pow t 3.0)))
   (if (<= t 1.02e-52)
     (* 2.0 (/ (pow (* l (pow k -2.0)) 2.0) t))
     (/ 2.0 (pow (* (pow t 1.5) (/ k (/ l (sqrt 2.0)))) 2.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -860000000.0) {
		tmp = pow(l, 2.0) / (pow(k, 2.0) * pow(t, 3.0));
	} else if (t <= 1.02e-52) {
		tmp = 2.0 * (pow((l * pow(k, -2.0)), 2.0) / t);
	} else {
		tmp = 2.0 / pow((pow(t, 1.5) * (k / (l / sqrt(2.0)))), 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 <= (-860000000.0d0)) then
        tmp = (l ** 2.0d0) / ((k ** 2.0d0) * (t ** 3.0d0))
    else if (t <= 1.02d-52) then
        tmp = 2.0d0 * (((l * (k ** (-2.0d0))) ** 2.0d0) / t)
    else
        tmp = 2.0d0 / (((t ** 1.5d0) * (k / (l / sqrt(2.0d0)))) ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -860000000.0) {
		tmp = Math.pow(l, 2.0) / (Math.pow(k, 2.0) * Math.pow(t, 3.0));
	} else if (t <= 1.02e-52) {
		tmp = 2.0 * (Math.pow((l * Math.pow(k, -2.0)), 2.0) / t);
	} else {
		tmp = 2.0 / Math.pow((Math.pow(t, 1.5) * (k / (l / Math.sqrt(2.0)))), 2.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -860000000.0:
		tmp = math.pow(l, 2.0) / (math.pow(k, 2.0) * math.pow(t, 3.0))
	elif t <= 1.02e-52:
		tmp = 2.0 * (math.pow((l * math.pow(k, -2.0)), 2.0) / t)
	else:
		tmp = 2.0 / math.pow((math.pow(t, 1.5) * (k / (l / math.sqrt(2.0)))), 2.0)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -860000000.0)
		tmp = Float64((l ^ 2.0) / Float64((k ^ 2.0) * (t ^ 3.0)));
	elseif (t <= 1.02e-52)
		tmp = Float64(2.0 * Float64((Float64(l * (k ^ -2.0)) ^ 2.0) / t));
	else
		tmp = Float64(2.0 / (Float64((t ^ 1.5) * Float64(k / Float64(l / sqrt(2.0)))) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -860000000.0)
		tmp = (l ^ 2.0) / ((k ^ 2.0) * (t ^ 3.0));
	elseif (t <= 1.02e-52)
		tmp = 2.0 * (((l * (k ^ -2.0)) ^ 2.0) / t);
	else
		tmp = 2.0 / (((t ^ 1.5) * (k / (l / sqrt(2.0)))) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -860000000.0], N[(N[Power[l, 2.0], $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.02e-52], N[(2.0 * N[(N[Power[N[(l * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[Power[t, 1.5], $MachinePrecision] * N[(k / N[(l / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -8.6e8

   1. Initial program 76.2%

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

   3. Simplified 66.5%

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

   4. Taylor expanded in k around 0 60.5%

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

   if -8.6e8 < t < 1.02000000000000009e-52

   1. Initial program 44.0%

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

   2. Taylor expanded in t around 0 70.8%

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

      1. associate-*r* 70.8%

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

      2. *-commutative 70.8%

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

      3. times-frac 70.6%

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

   4. Simplified 70.6%

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

   5. Taylor expanded in k around 0 55.4%

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

      1. associate-/r* 56.2%

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

   7. Simplified 56.2%

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

      1. expm1-log1p-u 56.2%

         \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)\right)}}{t} \]

      2. expm1-udef 54.5%

         \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)} - 1}}{t} \]

      3. div-inv 54.5%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left(\color{blue}{{\ell}^{2} \cdot \frac{1}{{k}^{4}}}\right)} - 1}{t} \]

      4. pow-flip 54.5%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-4\right)}}\right)} - 1}{t} \]

      5. metadata-eval 54.5%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{\color{blue}{-4}}\right)} - 1}{t} \]

   9. Applied egg-rr 54.5%

      \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)} - 1}}{t} \]
   10. Step-by-step derivation

       1. expm1-def 56.2%

          \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)\right)}}{t} \]

       2. expm1-log1p 56.2%

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

   11. Simplified 56.2%

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

       1. add-sqr-sqrt 56.2%

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

       2. pow2 56.2%

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

       3. sqrt-prod 56.2%

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

       4. unpow2 56.2%

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

       5. sqrt-prod 32.5%

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

       6. add-sqr-sqrt 62.4%

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

       7. sqrt-pow1 64.7%

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

       8. metadata-eval 64.7%

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

   13. Applied egg-rr 64.7%

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

   if 1.02000000000000009e-52 < t

   1. Initial program 62.4%

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

      1. add-sqr-sqrt 37.7%

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

      2. pow2 37.7%

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

   3. Applied egg-rr 53.0%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}} \]

   4. Taylor expanded in k around 0 68.5%

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

      1. expm1-log1p-u 39.0%

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

      2. expm1-udef 37.5%

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

      3. associate-*l/ 37.5%

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

      4. sqrt-pow1 43.9%

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

      5. metadata-eval 43.9%

         \[\leadsto \frac{2}{{\left(e^{\mathsf{log1p}\left(\frac{\left(k \cdot \sqrt{2}\right) \cdot {t}^{\color{blue}{1.5}}}{\ell}\right)} - 1\right)}^{2}} \]

   6. Applied egg-rr 43.9%

      \[\leadsto \frac{2}{{\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\left(k \cdot \sqrt{2}\right) \cdot {t}^{1.5}}{\ell}\right)} - 1\right)}}^{2}} \]
   7. Step-by-step derivation

      1. expm1-def 49.2%

         \[\leadsto \frac{2}{{\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(k \cdot \sqrt{2}\right) \cdot {t}^{1.5}}{\ell}\right)\right)\right)}}^{2}} \]

      2. expm1-log1p 79.4%

         \[\leadsto \frac{2}{{\color{blue}{\left(\frac{\left(k \cdot \sqrt{2}\right) \cdot {t}^{1.5}}{\ell}\right)}}^{2}} \]

      3. associate-/l* 76.8%

         \[\leadsto \frac{2}{{\color{blue}{\left(\frac{k \cdot \sqrt{2}}{\frac{\ell}{{t}^{1.5}}}\right)}}^{2}} \]

      4. associate-/r/ 79.1%

         \[\leadsto \frac{2}{{\color{blue}{\left(\frac{k \cdot \sqrt{2}}{\ell} \cdot {t}^{1.5}\right)}}^{2}} \]

      5. associate-*l/ 79.1%

         \[\leadsto \frac{2}{{\left(\color{blue}{\left(\frac{k}{\ell} \cdot \sqrt{2}\right)} \cdot {t}^{1.5}\right)}^{2}} \]

      6. associate-/r/ 79.0%

         \[\leadsto \frac{2}{{\left(\color{blue}{\frac{k}{\frac{\ell}{\sqrt{2}}}} \cdot {t}^{1.5}\right)}^{2}} \]

   8. Simplified 79.0%

      \[\leadsto \frac{2}{{\color{blue}{\left(\frac{k}{\frac{\ell}{\sqrt{2}}} \cdot {t}^{1.5}\right)}}^{2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -860000000:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\\ \mathbf{elif}\;t \leq 1.02 \cdot 10^{-52}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left({t}^{1.5} \cdot \frac{k}{\frac{\ell}{\sqrt{2}}}\right)}^{2}}\\ \end{array} \]

Alternative 17: 67.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -940000000:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{-45}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left({t}^{1.5} \cdot \frac{k \cdot \sqrt{2}}{\ell}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -940000000.0)
   (/ (pow l 2.0) (* (pow k 2.0) (pow t 3.0)))
   (if (<= t 2.8e-45)
     (* 2.0 (/ (pow (* l (pow k -2.0)) 2.0) t))
     (/ 2.0 (pow (* (pow t 1.5) (/ (* k (sqrt 2.0)) l)) 2.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -940000000.0) {
		tmp = pow(l, 2.0) / (pow(k, 2.0) * pow(t, 3.0));
	} else if (t <= 2.8e-45) {
		tmp = 2.0 * (pow((l * pow(k, -2.0)), 2.0) / t);
	} else {
		tmp = 2.0 / pow((pow(t, 1.5) * ((k * sqrt(2.0)) / 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 <= (-940000000.0d0)) then
        tmp = (l ** 2.0d0) / ((k ** 2.0d0) * (t ** 3.0d0))
    else if (t <= 2.8d-45) then
        tmp = 2.0d0 * (((l * (k ** (-2.0d0))) ** 2.0d0) / t)
    else
        tmp = 2.0d0 / (((t ** 1.5d0) * ((k * sqrt(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 <= -940000000.0) {
		tmp = Math.pow(l, 2.0) / (Math.pow(k, 2.0) * Math.pow(t, 3.0));
	} else if (t <= 2.8e-45) {
		tmp = 2.0 * (Math.pow((l * Math.pow(k, -2.0)), 2.0) / t);
	} else {
		tmp = 2.0 / Math.pow((Math.pow(t, 1.5) * ((k * Math.sqrt(2.0)) / l)), 2.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -940000000.0:
		tmp = math.pow(l, 2.0) / (math.pow(k, 2.0) * math.pow(t, 3.0))
	elif t <= 2.8e-45:
		tmp = 2.0 * (math.pow((l * math.pow(k, -2.0)), 2.0) / t)
	else:
		tmp = 2.0 / math.pow((math.pow(t, 1.5) * ((k * math.sqrt(2.0)) / l)), 2.0)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -940000000.0)
		tmp = Float64((l ^ 2.0) / Float64((k ^ 2.0) * (t ^ 3.0)));
	elseif (t <= 2.8e-45)
		tmp = Float64(2.0 * Float64((Float64(l * (k ^ -2.0)) ^ 2.0) / t));
	else
		tmp = Float64(2.0 / (Float64((t ^ 1.5) * Float64(Float64(k * sqrt(2.0)) / l)) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -940000000.0)
		tmp = (l ^ 2.0) / ((k ^ 2.0) * (t ^ 3.0));
	elseif (t <= 2.8e-45)
		tmp = 2.0 * (((l * (k ^ -2.0)) ^ 2.0) / t);
	else
		tmp = 2.0 / (((t ^ 1.5) * ((k * sqrt(2.0)) / l)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -940000000.0], N[(N[Power[l, 2.0], $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.8e-45], N[(2.0 * N[(N[Power[N[(l * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[Power[t, 1.5], $MachinePrecision] * N[(N[(k * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -9.4e8

   1. Initial program 76.2%

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

   3. Simplified 66.5%

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

   4. Taylor expanded in k around 0 60.5%

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

   if -9.4e8 < t < 2.8000000000000001e-45

   1. Initial program 44.0%

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

   2. Taylor expanded in t around 0 70.8%

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

      1. associate-*r* 70.8%

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

      2. *-commutative 70.8%

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

      3. times-frac 70.6%

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

   4. Simplified 70.6%

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

   5. Taylor expanded in k around 0 55.4%

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

      1. associate-/r* 56.2%

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

   7. Simplified 56.2%

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

      1. expm1-log1p-u 56.2%

         \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)\right)}}{t} \]

      2. expm1-udef 54.5%

         \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)} - 1}}{t} \]

      3. div-inv 54.5%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left(\color{blue}{{\ell}^{2} \cdot \frac{1}{{k}^{4}}}\right)} - 1}{t} \]

      4. pow-flip 54.5%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-4\right)}}\right)} - 1}{t} \]

      5. metadata-eval 54.5%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{\color{blue}{-4}}\right)} - 1}{t} \]

   9. Applied egg-rr 54.5%

      \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)} - 1}}{t} \]
   10. Step-by-step derivation

       1. expm1-def 56.2%

          \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)\right)}}{t} \]

       2. expm1-log1p 56.2%

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

   11. Simplified 56.2%

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

       1. add-sqr-sqrt 56.2%

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

       2. pow2 56.2%

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

       3. sqrt-prod 56.2%

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

       4. unpow2 56.2%

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

       5. sqrt-prod 32.5%

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

       6. add-sqr-sqrt 62.4%

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

       7. sqrt-pow1 64.7%

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

       8. metadata-eval 64.7%

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

   13. Applied egg-rr 64.7%

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

   if 2.8000000000000001e-45 < t

   1. Initial program 62.4%

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

      1. add-sqr-sqrt 37.7%

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

      2. pow2 37.7%

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

   3. Applied egg-rr 53.0%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}} \]

   4. Taylor expanded in k around 0 68.5%

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

      1. expm1-log1p-u 68.1%

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

      2. expm1-udef 60.0%

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

      3. sqrt-pow1 69.9%

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

      4. metadata-eval 69.9%

         \[\leadsto \frac{2}{{\left(\frac{k \cdot \sqrt{2}}{\ell} \cdot \left(e^{\mathsf{log1p}\left({t}^{\color{blue}{1.5}}\right)} - 1\right)\right)}^{2}} \]

   6. Applied egg-rr 69.9%

      \[\leadsto \frac{2}{{\left(\frac{k \cdot \sqrt{2}}{\ell} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({t}^{1.5}\right)} - 1\right)}\right)}^{2}} \]
   7. Step-by-step derivation

      1. expm1-def 78.0%

         \[\leadsto \frac{2}{{\left(\frac{k \cdot \sqrt{2}}{\ell} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({t}^{1.5}\right)\right)}\right)}^{2}} \]

      2. expm1-log1p 79.1%

         \[\leadsto \frac{2}{{\left(\frac{k \cdot \sqrt{2}}{\ell} \cdot \color{blue}{{t}^{1.5}}\right)}^{2}} \]

   8. Simplified 79.1%

      \[\leadsto \frac{2}{{\left(\frac{k \cdot \sqrt{2}}{\ell} \cdot \color{blue}{{t}^{1.5}}\right)}^{2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -940000000:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{-45}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left({t}^{1.5} \cdot \frac{k \cdot \sqrt{2}}{\ell}\right)}^{2}}\\ \end{array} \]

Alternative 18: 67.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -880000000:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-51}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{\left(k \cdot \sqrt{2}\right) \cdot {t}^{1.5}}{\ell}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -880000000.0)
   (/ (pow l 2.0) (* (pow k 2.0) (pow t 3.0)))
   (if (<= t 1.05e-51)
     (* 2.0 (/ (pow (* l (pow k -2.0)) 2.0) t))
     (/ 2.0 (pow (/ (* (* k (sqrt 2.0)) (pow t 1.5)) l) 2.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -880000000.0) {
		tmp = pow(l, 2.0) / (pow(k, 2.0) * pow(t, 3.0));
	} else if (t <= 1.05e-51) {
		tmp = 2.0 * (pow((l * pow(k, -2.0)), 2.0) / t);
	} else {
		tmp = 2.0 / pow((((k * sqrt(2.0)) * pow(t, 1.5)) / 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 <= (-880000000.0d0)) then
        tmp = (l ** 2.0d0) / ((k ** 2.0d0) * (t ** 3.0d0))
    else if (t <= 1.05d-51) then
        tmp = 2.0d0 * (((l * (k ** (-2.0d0))) ** 2.0d0) / t)
    else
        tmp = 2.0d0 / ((((k * sqrt(2.0d0)) * (t ** 1.5d0)) / l) ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -880000000.0) {
		tmp = Math.pow(l, 2.0) / (Math.pow(k, 2.0) * Math.pow(t, 3.0));
	} else if (t <= 1.05e-51) {
		tmp = 2.0 * (Math.pow((l * Math.pow(k, -2.0)), 2.0) / t);
	} else {
		tmp = 2.0 / Math.pow((((k * Math.sqrt(2.0)) * Math.pow(t, 1.5)) / l), 2.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -880000000.0:
		tmp = math.pow(l, 2.0) / (math.pow(k, 2.0) * math.pow(t, 3.0))
	elif t <= 1.05e-51:
		tmp = 2.0 * (math.pow((l * math.pow(k, -2.0)), 2.0) / t)
	else:
		tmp = 2.0 / math.pow((((k * math.sqrt(2.0)) * math.pow(t, 1.5)) / l), 2.0)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -880000000.0)
		tmp = Float64((l ^ 2.0) / Float64((k ^ 2.0) * (t ^ 3.0)));
	elseif (t <= 1.05e-51)
		tmp = Float64(2.0 * Float64((Float64(l * (k ^ -2.0)) ^ 2.0) / t));
	else
		tmp = Float64(2.0 / (Float64(Float64(Float64(k * sqrt(2.0)) * (t ^ 1.5)) / l) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -880000000.0)
		tmp = (l ^ 2.0) / ((k ^ 2.0) * (t ^ 3.0));
	elseif (t <= 1.05e-51)
		tmp = 2.0 * (((l * (k ^ -2.0)) ^ 2.0) / t);
	else
		tmp = 2.0 / ((((k * sqrt(2.0)) * (t ^ 1.5)) / l) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -880000000.0], N[(N[Power[l, 2.0], $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.05e-51], N[(2.0 * N[(N[Power[N[(l * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(N[(k * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[t, 1.5], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -8.8e8

   1. Initial program 76.2%

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

   3. Simplified 66.5%

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

   4. Taylor expanded in k around 0 60.5%

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

   if -8.8e8 < t < 1.05000000000000001e-51

   1. Initial program 44.0%

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

   2. Taylor expanded in t around 0 70.8%

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

      1. associate-*r* 70.8%

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

      2. *-commutative 70.8%

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

      3. times-frac 70.6%

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

   4. Simplified 70.6%

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

   5. Taylor expanded in k around 0 55.4%

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

      1. associate-/r* 56.2%

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

   7. Simplified 56.2%

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

      1. expm1-log1p-u 56.2%

         \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)\right)}}{t} \]

      2. expm1-udef 54.5%

         \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)} - 1}}{t} \]

      3. div-inv 54.5%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left(\color{blue}{{\ell}^{2} \cdot \frac{1}{{k}^{4}}}\right)} - 1}{t} \]

      4. pow-flip 54.5%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-4\right)}}\right)} - 1}{t} \]

      5. metadata-eval 54.5%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{\color{blue}{-4}}\right)} - 1}{t} \]

   9. Applied egg-rr 54.5%

      \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)} - 1}}{t} \]
   10. Step-by-step derivation

       1. expm1-def 56.2%

          \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)\right)}}{t} \]

       2. expm1-log1p 56.2%

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

   11. Simplified 56.2%

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

       1. add-sqr-sqrt 56.2%

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

       2. pow2 56.2%

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

       3. sqrt-prod 56.2%

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

       4. unpow2 56.2%

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

       5. sqrt-prod 32.5%

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

       6. add-sqr-sqrt 62.4%

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

       7. sqrt-pow1 64.7%

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

       8. metadata-eval 64.7%

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

   13. Applied egg-rr 64.7%

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

   if 1.05000000000000001e-51 < t

   1. Initial program 62.4%

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

      1. add-sqr-sqrt 37.7%

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

      2. pow2 37.7%

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

   3. Applied egg-rr 53.0%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}} \]

   4. Taylor expanded in k around 0 68.5%

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

      1. associate-*l/ 68.6%

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

      2. sqrt-pow1 79.4%

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

      3. metadata-eval 79.4%

         \[\leadsto \frac{2}{{\left(\frac{\left(k \cdot \sqrt{2}\right) \cdot {t}^{\color{blue}{1.5}}}{\ell}\right)}^{2}} \]

   6. Applied egg-rr 79.4%

      \[\leadsto \frac{2}{{\color{blue}{\left(\frac{\left(k \cdot \sqrt{2}\right) \cdot {t}^{1.5}}{\ell}\right)}}^{2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -880000000:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-51}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{\left(k \cdot \sqrt{2}\right) \cdot {t}^{1.5}}{\ell}\right)}^{2}}\\ \end{array} \]

Alternative 19: 69.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -920000000:\\ \;\;\;\;\frac{2}{2 \cdot \frac{k \cdot \left(\sin k \cdot {t}^{3}\right)}{{\ell}^{2}}}\\ \mathbf{elif}\;t \leq 10^{-46}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{\left(k \cdot \sqrt{2}\right) \cdot {t}^{1.5}}{\ell}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -920000000.0)
   (/ 2.0 (* 2.0 (/ (* k (* (sin k) (pow t 3.0))) (pow l 2.0))))
   (if (<= t 1e-46)
     (* 2.0 (/ (pow (* l (pow k -2.0)) 2.0) t))
     (/ 2.0 (pow (/ (* (* k (sqrt 2.0)) (pow t 1.5)) l) 2.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -920000000.0) {
		tmp = 2.0 / (2.0 * ((k * (sin(k) * pow(t, 3.0))) / pow(l, 2.0)));
	} else if (t <= 1e-46) {
		tmp = 2.0 * (pow((l * pow(k, -2.0)), 2.0) / t);
	} else {
		tmp = 2.0 / pow((((k * sqrt(2.0)) * pow(t, 1.5)) / 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 <= (-920000000.0d0)) then
        tmp = 2.0d0 / (2.0d0 * ((k * (sin(k) * (t ** 3.0d0))) / (l ** 2.0d0)))
    else if (t <= 1d-46) then
        tmp = 2.0d0 * (((l * (k ** (-2.0d0))) ** 2.0d0) / t)
    else
        tmp = 2.0d0 / ((((k * sqrt(2.0d0)) * (t ** 1.5d0)) / l) ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -920000000.0) {
		tmp = 2.0 / (2.0 * ((k * (Math.sin(k) * Math.pow(t, 3.0))) / Math.pow(l, 2.0)));
	} else if (t <= 1e-46) {
		tmp = 2.0 * (Math.pow((l * Math.pow(k, -2.0)), 2.0) / t);
	} else {
		tmp = 2.0 / Math.pow((((k * Math.sqrt(2.0)) * Math.pow(t, 1.5)) / l), 2.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -920000000.0:
		tmp = 2.0 / (2.0 * ((k * (math.sin(k) * math.pow(t, 3.0))) / math.pow(l, 2.0)))
	elif t <= 1e-46:
		tmp = 2.0 * (math.pow((l * math.pow(k, -2.0)), 2.0) / t)
	else:
		tmp = 2.0 / math.pow((((k * math.sqrt(2.0)) * math.pow(t, 1.5)) / l), 2.0)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -920000000.0)
		tmp = Float64(2.0 / Float64(2.0 * Float64(Float64(k * Float64(sin(k) * (t ^ 3.0))) / (l ^ 2.0))));
	elseif (t <= 1e-46)
		tmp = Float64(2.0 * Float64((Float64(l * (k ^ -2.0)) ^ 2.0) / t));
	else
		tmp = Float64(2.0 / (Float64(Float64(Float64(k * sqrt(2.0)) * (t ^ 1.5)) / l) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -920000000.0)
		tmp = 2.0 / (2.0 * ((k * (sin(k) * (t ^ 3.0))) / (l ^ 2.0)));
	elseif (t <= 1e-46)
		tmp = 2.0 * (((l * (k ^ -2.0)) ^ 2.0) / t);
	else
		tmp = 2.0 / ((((k * sqrt(2.0)) * (t ^ 1.5)) / l) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -920000000.0], N[(2.0 / N[(2.0 * N[(N[(k * N[(N[Sin[k], $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1e-46], N[(2.0 * N[(N[Power[N[(l * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(N[(k * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[t, 1.5], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -9.2e8

   1. Initial program 76.2%

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

   2. Taylor expanded in k around 0 70.6%

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

   3. Taylor expanded in t around inf 67.2%

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

   4. Taylor expanded in k around 0 67.5%

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

   if -9.2e8 < t < 1.00000000000000002e-46

   1. Initial program 44.0%

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

   2. Taylor expanded in t around 0 70.8%

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

      1. associate-*r* 70.8%

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

      2. *-commutative 70.8%

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

      3. times-frac 70.6%

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

   4. Simplified 70.6%

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

   5. Taylor expanded in k around 0 55.4%

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

      1. associate-/r* 56.2%

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

   7. Simplified 56.2%

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

      1. expm1-log1p-u 56.2%

         \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)\right)}}{t} \]

      2. expm1-udef 54.5%

         \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)} - 1}}{t} \]

      3. div-inv 54.5%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left(\color{blue}{{\ell}^{2} \cdot \frac{1}{{k}^{4}}}\right)} - 1}{t} \]

      4. pow-flip 54.5%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-4\right)}}\right)} - 1}{t} \]

      5. metadata-eval 54.5%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{\color{blue}{-4}}\right)} - 1}{t} \]

   9. Applied egg-rr 54.5%

      \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)} - 1}}{t} \]
   10. Step-by-step derivation

       1. expm1-def 56.2%

          \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)\right)}}{t} \]

       2. expm1-log1p 56.2%

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

   11. Simplified 56.2%

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

       1. add-sqr-sqrt 56.2%

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

       2. pow2 56.2%

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

       3. sqrt-prod 56.2%

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

       4. unpow2 56.2%

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

       5. sqrt-prod 32.5%

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

       6. add-sqr-sqrt 62.4%

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

       7. sqrt-pow1 64.7%

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

       8. metadata-eval 64.7%

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

   13. Applied egg-rr 64.7%

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

   if 1.00000000000000002e-46 < t

   1. Initial program 62.4%

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

      1. add-sqr-sqrt 37.7%

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

      2. pow2 37.7%

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

   3. Applied egg-rr 53.0%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right) \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}}} \]

   4. Taylor expanded in k around 0 68.5%

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

      1. associate-*l/ 68.6%

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

      2. sqrt-pow1 79.4%

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

      3. metadata-eval 79.4%

         \[\leadsto \frac{2}{{\left(\frac{\left(k \cdot \sqrt{2}\right) \cdot {t}^{\color{blue}{1.5}}}{\ell}\right)}^{2}} \]

   6. Applied egg-rr 79.4%

      \[\leadsto \frac{2}{{\color{blue}{\left(\frac{\left(k \cdot \sqrt{2}\right) \cdot {t}^{1.5}}{\ell}\right)}}^{2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -920000000:\\ \;\;\;\;\frac{2}{2 \cdot \frac{k \cdot \left(\sin k \cdot {t}^{3}\right)}{{\ell}^{2}}}\\ \mathbf{elif}\;t \leq 10^{-46}:\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{\left(k \cdot \sqrt{2}\right) \cdot {t}^{1.5}}{\ell}\right)}^{2}}\\ \end{array} \]

Alternative 20: 57.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 6 \cdot 10^{-139} \lor \neg \left(k \leq 2.8 \cdot 10^{+80}\right):\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\ell}^{2}}{{k}^{2}}}{{t}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= k 6e-139) (not (<= k 2.8e+80)))
   (* 2.0 (/ (pow (* l (pow k -2.0)) 2.0) t))
   (/ (/ (pow l 2.0) (pow k 2.0)) (pow t 3.0))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((k <= 6e-139) || !(k <= 2.8e+80)) {
		tmp = 2.0 * (pow((l * pow(k, -2.0)), 2.0) / t);
	} else {
		tmp = (pow(l, 2.0) / pow(k, 2.0)) / pow(t, 3.0);
	}
	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 <= 6d-139) .or. (.not. (k <= 2.8d+80))) then
        tmp = 2.0d0 * (((l * (k ** (-2.0d0))) ** 2.0d0) / t)
    else
        tmp = ((l ** 2.0d0) / (k ** 2.0d0)) / (t ** 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((k <= 6e-139) || !(k <= 2.8e+80)) {
		tmp = 2.0 * (Math.pow((l * Math.pow(k, -2.0)), 2.0) / t);
	} else {
		tmp = (Math.pow(l, 2.0) / Math.pow(k, 2.0)) / Math.pow(t, 3.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (k <= 6e-139) or not (k <= 2.8e+80):
		tmp = 2.0 * (math.pow((l * math.pow(k, -2.0)), 2.0) / t)
	else:
		tmp = (math.pow(l, 2.0) / math.pow(k, 2.0)) / math.pow(t, 3.0)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((k <= 6e-139) || !(k <= 2.8e+80))
		tmp = Float64(2.0 * Float64((Float64(l * (k ^ -2.0)) ^ 2.0) / t));
	else
		tmp = Float64(Float64((l ^ 2.0) / (k ^ 2.0)) / (t ^ 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((k <= 6e-139) || ~((k <= 2.8e+80)))
		tmp = 2.0 * (((l * (k ^ -2.0)) ^ 2.0) / t);
	else
		tmp = ((l ^ 2.0) / (k ^ 2.0)) / (t ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[k, 6e-139], N[Not[LessEqual[k, 2.8e+80]], $MachinePrecision]], N[(2.0 * N[(N[Power[N[(l * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.9999999999999998e-139 or 2.79999999999999984e80 < k

   1. Initial program 56.8%

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

   2. Taylor expanded in t around 0 62.1%

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

      1. associate-*r* 62.1%

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

      2. *-commutative 62.1%

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

      3. times-frac 62.6%

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

   4. Simplified 62.6%

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

   5. Taylor expanded in k around 0 55.8%

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

      1. associate-/r* 55.8%

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

   7. Simplified 55.8%

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

      1. expm1-log1p-u 55.8%

         \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)\right)}}{t} \]

      2. expm1-udef 54.8%

         \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)} - 1}}{t} \]

      3. div-inv 54.8%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left(\color{blue}{{\ell}^{2} \cdot \frac{1}{{k}^{4}}}\right)} - 1}{t} \]

      4. pow-flip 54.8%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-4\right)}}\right)} - 1}{t} \]

      5. metadata-eval 54.8%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{\color{blue}{-4}}\right)} - 1}{t} \]

   9. Applied egg-rr 54.8%

      \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)} - 1}}{t} \]
   10. Step-by-step derivation

       1. expm1-def 55.8%

          \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)\right)}}{t} \]

       2. expm1-log1p 55.8%

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

   11. Simplified 55.8%

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

       1. add-sqr-sqrt 55.8%

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

       2. pow2 55.8%

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

       3. sqrt-prod 55.8%

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

       4. unpow2 55.8%

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

       5. sqrt-prod 29.2%

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

       6. add-sqr-sqrt 60.3%

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

       7. sqrt-pow1 60.8%

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

       8. metadata-eval 60.8%

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

   13. Applied egg-rr 60.8%

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

   if 5.9999999999999998e-139 < k < 2.79999999999999984e80

   1. Initial program 59.1%

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

   3. Simplified 57.6%

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

   4. Taylor expanded in k around 0 53.2%

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

   5. Taylor expanded in t around 0 57.6%

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

      1. associate-/r* 57.6%

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

   7. Simplified 57.6%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6 \cdot 10^{-139} \lor \neg \left(k \leq 2.8 \cdot 10^{+80}\right):\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\ell}^{2}}{{k}^{2}}}{{t}^{3}}\\ \end{array} \]

Alternative 21: 58.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 8 \cdot 10^{-139} \lor \neg \left(k \leq 2.8 \cdot 10^{+80}\right):\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= k 8e-139) (not (<= k 2.8e+80)))
   (* 2.0 (/ (pow (* l (pow k -2.0)) 2.0) t))
   (/ (pow l 2.0) (* (pow k 2.0) (pow t 3.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((k <= 8e-139) || !(k <= 2.8e+80)) {
		tmp = 2.0 * (pow((l * pow(k, -2.0)), 2.0) / t);
	} else {
		tmp = pow(l, 2.0) / (pow(k, 2.0) * pow(t, 3.0));
	}
	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 <= 8d-139) .or. (.not. (k <= 2.8d+80))) then
        tmp = 2.0d0 * (((l * (k ** (-2.0d0))) ** 2.0d0) / t)
    else
        tmp = (l ** 2.0d0) / ((k ** 2.0d0) * (t ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((k <= 8e-139) || !(k <= 2.8e+80)) {
		tmp = 2.0 * (Math.pow((l * Math.pow(k, -2.0)), 2.0) / t);
	} else {
		tmp = Math.pow(l, 2.0) / (Math.pow(k, 2.0) * Math.pow(t, 3.0));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (k <= 8e-139) or not (k <= 2.8e+80):
		tmp = 2.0 * (math.pow((l * math.pow(k, -2.0)), 2.0) / t)
	else:
		tmp = math.pow(l, 2.0) / (math.pow(k, 2.0) * math.pow(t, 3.0))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((k <= 8e-139) || !(k <= 2.8e+80))
		tmp = Float64(2.0 * Float64((Float64(l * (k ^ -2.0)) ^ 2.0) / t));
	else
		tmp = Float64((l ^ 2.0) / Float64((k ^ 2.0) * (t ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((k <= 8e-139) || ~((k <= 2.8e+80)))
		tmp = 2.0 * (((l * (k ^ -2.0)) ^ 2.0) / t);
	else
		tmp = (l ^ 2.0) / ((k ^ 2.0) * (t ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[k, 8e-139], N[Not[LessEqual[k, 2.8e+80]], $MachinePrecision]], N[(2.0 * N[(N[Power[N[(l * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[Power[l, 2.0], $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 8.00000000000000024e-139 or 2.79999999999999984e80 < k

   1. Initial program 56.8%

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

   2. Taylor expanded in t around 0 62.1%

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

      1. associate-*r* 62.1%

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

      2. *-commutative 62.1%

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

      3. times-frac 62.6%

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

   4. Simplified 62.6%

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

   5. Taylor expanded in k around 0 55.8%

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

      1. associate-/r* 55.8%

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

   7. Simplified 55.8%

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

      1. expm1-log1p-u 55.8%

         \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)\right)}}{t} \]

      2. expm1-udef 54.8%

         \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)} - 1}}{t} \]

      3. div-inv 54.8%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left(\color{blue}{{\ell}^{2} \cdot \frac{1}{{k}^{4}}}\right)} - 1}{t} \]

      4. pow-flip 54.8%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-4\right)}}\right)} - 1}{t} \]

      5. metadata-eval 54.8%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{\color{blue}{-4}}\right)} - 1}{t} \]

   9. Applied egg-rr 54.8%

      \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)} - 1}}{t} \]
   10. Step-by-step derivation

       1. expm1-def 55.8%

          \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)\right)}}{t} \]

       2. expm1-log1p 55.8%

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

   11. Simplified 55.8%

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

       1. add-sqr-sqrt 55.8%

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

       2. pow2 55.8%

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

       3. sqrt-prod 55.8%

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

       4. unpow2 55.8%

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

       5. sqrt-prod 29.2%

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

       6. add-sqr-sqrt 60.3%

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

       7. sqrt-pow1 60.8%

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

       8. metadata-eval 60.8%

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

   13. Applied egg-rr 60.8%

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

   if 8.00000000000000024e-139 < k < 2.79999999999999984e80

   1. Initial program 59.1%

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

   3. Simplified 57.6%

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

   4. Taylor expanded in k around 0 57.6%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8 \cdot 10^{-139} \lor \neg \left(k \leq 2.8 \cdot 10^{+80}\right):\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\\ \end{array} \]

Alternative 22: 58.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 6.5 \cdot 10^{-139} \lor \neg \left(k \leq 2.8 \cdot 10^{+80}\right):\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot \left(2 \cdot {t}^{-3}\right)}{2 \cdot {k}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= k 6.5e-139) (not (<= k 2.8e+80)))
   (* 2.0 (/ (pow (* l (pow k -2.0)) 2.0) t))
   (/ (* (* l l) (* 2.0 (pow t -3.0))) (* 2.0 (pow k 2.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((k <= 6.5e-139) || !(k <= 2.8e+80)) {
		tmp = 2.0 * (pow((l * pow(k, -2.0)), 2.0) / t);
	} else {
		tmp = ((l * l) * (2.0 * pow(t, -3.0))) / (2.0 * pow(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 <= 6.5d-139) .or. (.not. (k <= 2.8d+80))) then
        tmp = 2.0d0 * (((l * (k ** (-2.0d0))) ** 2.0d0) / t)
    else
        tmp = ((l * l) * (2.0d0 * (t ** (-3.0d0)))) / (2.0d0 * (k ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((k <= 6.5e-139) || !(k <= 2.8e+80)) {
		tmp = 2.0 * (Math.pow((l * Math.pow(k, -2.0)), 2.0) / t);
	} else {
		tmp = ((l * l) * (2.0 * Math.pow(t, -3.0))) / (2.0 * Math.pow(k, 2.0));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (k <= 6.5e-139) or not (k <= 2.8e+80):
		tmp = 2.0 * (math.pow((l * math.pow(k, -2.0)), 2.0) / t)
	else:
		tmp = ((l * l) * (2.0 * math.pow(t, -3.0))) / (2.0 * math.pow(k, 2.0))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((k <= 6.5e-139) || !(k <= 2.8e+80))
		tmp = Float64(2.0 * Float64((Float64(l * (k ^ -2.0)) ^ 2.0) / t));
	else
		tmp = Float64(Float64(Float64(l * l) * Float64(2.0 * (t ^ -3.0))) / Float64(2.0 * (k ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((k <= 6.5e-139) || ~((k <= 2.8e+80)))
		tmp = 2.0 * (((l * (k ^ -2.0)) ^ 2.0) / t);
	else
		tmp = ((l * l) * (2.0 * (t ^ -3.0))) / (2.0 * (k ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[k, 6.5e-139], N[Not[LessEqual[k, 2.8e+80]], $MachinePrecision]], N[(2.0 * N[(N[Power[N[(l * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * l), $MachinePrecision] * N[(2.0 * N[Power[t, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.5e-139 or 2.79999999999999984e80 < k

   1. Initial program 56.8%

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

   2. Taylor expanded in t around 0 62.1%

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

      1. associate-*r* 62.1%

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

      2. *-commutative 62.1%

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

      3. times-frac 62.6%

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

   4. Simplified 62.6%

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

   5. Taylor expanded in k around 0 55.8%

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

      1. associate-/r* 55.8%

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

   7. Simplified 55.8%

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

      1. expm1-log1p-u 55.8%

         \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)\right)}}{t} \]

      2. expm1-udef 54.8%

         \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)} - 1}}{t} \]

      3. div-inv 54.8%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left(\color{blue}{{\ell}^{2} \cdot \frac{1}{{k}^{4}}}\right)} - 1}{t} \]

      4. pow-flip 54.8%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-4\right)}}\right)} - 1}{t} \]

      5. metadata-eval 54.8%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{\color{blue}{-4}}\right)} - 1}{t} \]

   9. Applied egg-rr 54.8%

      \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)} - 1}}{t} \]
   10. Step-by-step derivation

       1. expm1-def 55.8%

          \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)\right)}}{t} \]

       2. expm1-log1p 55.8%

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

   11. Simplified 55.8%

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

       1. add-sqr-sqrt 55.8%

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

       2. pow2 55.8%

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

       3. sqrt-prod 55.8%

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

       4. unpow2 55.8%

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

       5. sqrt-prod 29.2%

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

       6. add-sqr-sqrt 60.3%

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

       7. sqrt-pow1 60.8%

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

       8. metadata-eval 60.8%

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

   13. Applied egg-rr 60.8%

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

   if 6.5e-139 < k < 2.79999999999999984e80

   1. Initial program 59.1%

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

   3. Simplified 57.6%

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

   4. Taylor expanded in k around 0 53.2%

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

      1. div-inv 53.2%

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

      2. pow-flip 53.2%

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

      3. metadata-eval 53.2%

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

   6. Applied egg-rr 53.2%

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

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.5 \cdot 10^{-139} \lor \neg \left(k \leq 2.8 \cdot 10^{+80}\right):\\ \;\;\;\;2 \cdot \frac{{\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot \left(2 \cdot {t}^{-3}\right)}{2 \cdot {k}^{2}}\\ \end{array} \]

Alternative 23: 52.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \frac{{\ell}^{2} \cdot {k}^{-4}}{t} \end{array} \]

FPCore

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

C

double code(double t, double l, double k) {
	return 2.0 * ((pow(l, 2.0) * pow(k, -4.0)) / 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 * (((l ** 2.0d0) * (k ** (-4.0d0))) / t)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[t_, l_, k_] := N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Power[k, -4.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.2%

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

2. Taylor expanded in t around 0 61.3%

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

   1. associate-*r* 61.3%

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

   2. *-commutative 61.3%

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

   3. times-frac 61.2%

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

4. Simplified 61.2%

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

5. Taylor expanded in k around 0 52.8%

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

   1. associate-/r* 52.8%

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

7. Simplified 52.8%

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

   1. expm1-log1p-u 52.8%

      \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)\right)}}{t} \]

   2. expm1-udef 52.0%

      \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)} - 1}}{t} \]

   3. div-inv 52.0%

      \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left(\color{blue}{{\ell}^{2} \cdot \frac{1}{{k}^{4}}}\right)} - 1}{t} \]

   4. pow-flip 52.0%

      \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-4\right)}}\right)} - 1}{t} \]

   5. metadata-eval 52.0%

      \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{\color{blue}{-4}}\right)} - 1}{t} \]

9. Applied egg-rr 52.0%

   \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)} - 1}}{t} \]
10. Step-by-step derivation

    1. expm1-def 52.7%

       \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)\right)}}{t} \]

    2. expm1-log1p 52.7%

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

11. Simplified 52.7%

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

12. Final simplification 52.7%

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

Alternative 24: 52.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double t, double l, double k) {
	return 2.0 * ((pow(l, 2.0) / pow(k, 4.0)) / 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 * (((l ** 2.0d0) / (k ** 4.0d0)) / t)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[t_, l_, k_] := N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.2%

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

2. Taylor expanded in t around 0 61.3%

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

   1. associate-*r* 61.3%

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

   2. *-commutative 61.3%

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

   3. times-frac 61.2%

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

4. Simplified 61.2%

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

5. Taylor expanded in k around 0 52.8%

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

   1. associate-/r* 52.8%

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

7. Simplified 52.8%

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

8. Final simplification 52.8%

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

Alternative 25: 57.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double t, double l, double k) {
	return 2.0 * (pow((l * pow(k, -2.0)), 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 * (((l * (k ** (-2.0d0))) ** 2.0d0) / t)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[t_, l_, k_] := N[(2.0 * N[(N[Power[N[(l * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.2%

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

2. Taylor expanded in t around 0 61.3%

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

   1. associate-*r* 61.3%

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

   2. *-commutative 61.3%

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

   3. times-frac 61.2%

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

4. Simplified 61.2%

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

5. Taylor expanded in k around 0 52.8%

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

   1. associate-/r* 52.8%

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

7. Simplified 52.8%

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

   1. expm1-log1p-u 52.8%

      \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)\right)}}{t} \]

   2. expm1-udef 52.0%

      \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)} - 1}}{t} \]

   3. div-inv 52.0%

      \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left(\color{blue}{{\ell}^{2} \cdot \frac{1}{{k}^{4}}}\right)} - 1}{t} \]

   4. pow-flip 52.0%

      \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-4\right)}}\right)} - 1}{t} \]

   5. metadata-eval 52.0%

      \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{\color{blue}{-4}}\right)} - 1}{t} \]

9. Applied egg-rr 52.0%

   \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)} - 1}}{t} \]
10. Step-by-step derivation

    1. expm1-def 52.7%

       \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)\right)}}{t} \]

    2. expm1-log1p 52.7%

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

11. Simplified 52.7%

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

    1. add-sqr-sqrt 52.7%

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

    2. pow2 52.7%

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

    3. sqrt-prod 52.7%

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

    4. unpow2 52.7%

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

    5. sqrt-prod 28.2%

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

    6. add-sqr-sqrt 56.2%

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

    7. sqrt-pow1 57.0%

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

    8. metadata-eval 57.0%

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

13. Applied egg-rr 57.0%

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

14. Final simplification 57.0%

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

Reproduce

herbie shell --seed 2023305 
(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))))