Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.7% → 83.5%

Time: 26.8s

Alternatives: 19

Speedup: 2.0×

• 26.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 83.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := {\left(\sqrt[3]{\ell}\right)}^{2}\\ \mathbf{if}\;k\_m \leq 4.6 \cdot 10^{-92}:\\ \;\;\;\;{\left(\frac{\frac{t\_1}{t}}{{\left(\sqrt[3]{k\_m}\right)}^{2}}\right)}^{3}\\ \mathbf{elif}\;k\_m \leq 3.85 \cdot 10^{+28} \lor \neg \left(k\_m \leq 1.28 \cdot 10^{+90}\right):\\ \;\;\;\;\frac{2}{{\left(\frac{t}{t\_1} \cdot \sqrt[3]{\left(2 + {\left(\frac{k\_m}{t}\right)}^{2}\right) \cdot \left(\sin k\_m \cdot \tan k\_m\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(t \cdot {\sin k\_m}^{2}\right) \cdot {k\_m}^{2}}{\cos k\_m \cdot \ell}}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (pow (cbrt l) 2.0)))
   (if (<= k_m 4.6e-92)
     (pow (/ (/ t_1 t) (pow (cbrt k_m) 2.0)) 3.0)
     (if (or (<= k_m 3.85e+28) (not (<= k_m 1.28e+90)))
       (/
        2.0
        (pow
         (*
          (/ t t_1)
          (cbrt (* (+ 2.0 (pow (/ k_m t) 2.0)) (* (sin k_m) (tan k_m)))))
         3.0))
       (/
        2.0
        (/
         (/ (* (* t (pow (sin k_m) 2.0)) (pow k_m 2.0)) (* (cos k_m) l))
         l))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = pow(cbrt(l), 2.0);
	double tmp;
	if (k_m <= 4.6e-92) {
		tmp = pow(((t_1 / t) / pow(cbrt(k_m), 2.0)), 3.0);
	} else if ((k_m <= 3.85e+28) || !(k_m <= 1.28e+90)) {
		tmp = 2.0 / pow(((t / t_1) * cbrt(((2.0 + pow((k_m / t), 2.0)) * (sin(k_m) * tan(k_m))))), 3.0);
	} else {
		tmp = 2.0 / ((((t * pow(sin(k_m), 2.0)) * pow(k_m, 2.0)) / (cos(k_m) * l)) / l);
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = Math.pow(Math.cbrt(l), 2.0);
	double tmp;
	if (k_m <= 4.6e-92) {
		tmp = Math.pow(((t_1 / t) / Math.pow(Math.cbrt(k_m), 2.0)), 3.0);
	} else if ((k_m <= 3.85e+28) || !(k_m <= 1.28e+90)) {
		tmp = 2.0 / Math.pow(((t / t_1) * Math.cbrt(((2.0 + Math.pow((k_m / t), 2.0)) * (Math.sin(k_m) * Math.tan(k_m))))), 3.0);
	} else {
		tmp = 2.0 / ((((t * Math.pow(Math.sin(k_m), 2.0)) * Math.pow(k_m, 2.0)) / (Math.cos(k_m) * l)) / l);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = cbrt(l) ^ 2.0
	tmp = 0.0
	if (k_m <= 4.6e-92)
		tmp = Float64(Float64(t_1 / t) / (cbrt(k_m) ^ 2.0)) ^ 3.0;
	elseif ((k_m <= 3.85e+28) || !(k_m <= 1.28e+90))
		tmp = Float64(2.0 / (Float64(Float64(t / t_1) * cbrt(Float64(Float64(2.0 + (Float64(k_m / t) ^ 2.0)) * Float64(sin(k_m) * tan(k_m))))) ^ 3.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t * (sin(k_m) ^ 2.0)) * (k_m ^ 2.0)) / Float64(cos(k_m) * l)) / l));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[k$95$m, 4.6e-92], N[Power[N[(N[(t$95$1 / t), $MachinePrecision] / N[Power[N[Power[k$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], If[Or[LessEqual[k$95$m, 3.85e+28], N[Not[LessEqual[k$95$m, 1.28e+90]], $MachinePrecision]], N[(2.0 / N[Power[N[(N[(t / t$95$1), $MachinePrecision] * N[Power[N[(N[(2.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(t * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 4.60000000000000032e-92

   1. Initial program 52.9%

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

      1. associate-/r* 52.9%

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

      2. sqr-neg 52.9%

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

      3. associate-*l* 46.8%

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

      4. sqr-neg 46.8%

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

      5. associate-/r* 52.7%

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

      6. associate-+l+ 52.7%

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

      7. unpow2 52.7%

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

      8. times-frac 40.2%

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

      9. sqr-neg 40.2%

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

      10. times-frac 52.7%

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

      11. unpow2 52.7%

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

   3. Simplified 52.7%

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

   5. Taylor expanded in k around 0 45.6%

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

      1. *-commutative 45.6%

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

      2. associate-/r* 44.8%

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

   7. Simplified 44.8%

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

      1. unpow2 44.8%

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

   9. Applied egg-rr 44.8%

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

       1. expm1-log1p-u 34.7%

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

       2. expm1-udef 34.1%

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

       3. div-inv 34.1%

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

       4. pow-flip 34.1%

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

       5. metadata-eval 34.1%

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

   11. Applied egg-rr 34.1%

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

       1. expm1-def 34.7%

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

       2. expm1-log1p 44.8%

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

   13. Simplified 44.8%

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

       1. add-cube-cbrt 44.8%

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

       2. pow2 44.8%

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

       3. cbrt-div 44.8%

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

       4. metadata-eval 44.8%

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

       5. pow-flip 44.8%

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

       6. div-inv 44.8%

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

       7. cbrt-div 44.8%

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

       8. unpow2 44.8%

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

       9. cbrt-prod 44.8%

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

       10. unpow2 44.8%

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

       11. unpow3 44.8%

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

       12. add-cbrt-cube 44.8%

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

       13. cbrt-prod 44.7%

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

       14. pow2 44.7%

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

       15. cbrt-div 44.7%

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

   15. Applied egg-rr 78.3%

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

       1. pow-plus 78.3%

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

       2. metadata-eval 78.3%

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

   17. Simplified 78.3%

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

   if 4.60000000000000032e-92 < k < 3.8499999999999999e28 or 1.27999999999999995e90 < k

   1. Initial program 54.0%

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

      1. associate-/r* 60.3%

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

      2. +-commutative 60.3%

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

      3. associate-+r+ 60.3%

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

      4. metadata-eval 60.3%

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

      5. associate-*r* 60.2%

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

      6. add-cube-cbrt 60.1%

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

      7. pow3 60.1%

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

   4. Applied egg-rr 77.4%

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

   if 3.8499999999999999e28 < k < 1.27999999999999995e90

   1. Initial program 68.4%

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

      1. associate-/r* 68.4%

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

      2. +-commutative 68.4%

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

      3. associate-+r+ 68.4%

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

      4. metadata-eval 68.4%

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

      5. associate-*r* 68.4%

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

      6. *-commutative 68.4%

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

      7. associate-*l/ 68.4%

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

      8. associate-*r/ 68.4%

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

   4. Applied egg-rr 68.4%

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

   5. Taylor expanded in k around inf 95.5%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.6 \cdot 10^{-92}:\\ \;\;\;\;{\left(\frac{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}}{{\left(\sqrt[3]{k}\right)}^{2}}\right)}^{3}\\ \mathbf{elif}\;k \leq 3.85 \cdot 10^{+28} \lor \neg \left(k \leq 1.28 \cdot 10^{+90}\right):\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(t \cdot {\sin k}^{2}\right) \cdot {k}^{2}}{\cos k \cdot \ell}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := {\left(\frac{k\_m}{t}\right)}^{2}\\ \mathbf{if}\;t \leq 1.02 \cdot 10^{-76}:\\ \;\;\;\;\frac{2}{\frac{\frac{t \cdot {\sin k\_m}^{2}}{\cos k\_m} \cdot \frac{{k\_m}^{2}}{\ell}}{\ell}}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\frac{1}{\frac{\ell}{\tan k\_m \cdot \left(\frac{{t}^{3}}{\frac{\ell}{\sin k\_m}} \cdot \left(2 + t\_1\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k\_m \cdot {\left(\sqrt[3]{\sin k\_m} \cdot \left(\frac{t}{\sqrt[3]{\ell}} \cdot \sqrt[3]{\frac{1}{\ell}}\right)\right)}^{3}\right) \cdot \left(1 + \left(1 + t\_1\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (pow (/ k_m t) 2.0)))
   (if (<= t 1.02e-76)
     (/
      2.0
      (/ (* (/ (* t (pow (sin k_m) 2.0)) (cos k_m)) (/ (pow k_m 2.0) l)) l))
     (if (<= t 5.5e+102)
       (/
        2.0
        (/
         1.0
         (/ l (* (tan k_m) (* (/ (pow t 3.0) (/ l (sin k_m))) (+ 2.0 t_1))))))
       (/
        2.0
        (*
         (*
          (tan k_m)
          (pow (* (cbrt (sin k_m)) (* (/ t (cbrt l)) (cbrt (/ 1.0 l)))) 3.0))
         (+ 1.0 (+ 1.0 t_1))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = pow((k_m / t), 2.0);
	double tmp;
	if (t <= 1.02e-76) {
		tmp = 2.0 / ((((t * pow(sin(k_m), 2.0)) / cos(k_m)) * (pow(k_m, 2.0) / l)) / l);
	} else if (t <= 5.5e+102) {
		tmp = 2.0 / (1.0 / (l / (tan(k_m) * ((pow(t, 3.0) / (l / sin(k_m))) * (2.0 + t_1)))));
	} else {
		tmp = 2.0 / ((tan(k_m) * pow((cbrt(sin(k_m)) * ((t / cbrt(l)) * cbrt((1.0 / l)))), 3.0)) * (1.0 + (1.0 + t_1)));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = Math.pow((k_m / t), 2.0);
	double tmp;
	if (t <= 1.02e-76) {
		tmp = 2.0 / ((((t * Math.pow(Math.sin(k_m), 2.0)) / Math.cos(k_m)) * (Math.pow(k_m, 2.0) / l)) / l);
	} else if (t <= 5.5e+102) {
		tmp = 2.0 / (1.0 / (l / (Math.tan(k_m) * ((Math.pow(t, 3.0) / (l / Math.sin(k_m))) * (2.0 + t_1)))));
	} else {
		tmp = 2.0 / ((Math.tan(k_m) * Math.pow((Math.cbrt(Math.sin(k_m)) * ((t / Math.cbrt(l)) * Math.cbrt((1.0 / l)))), 3.0)) * (1.0 + (1.0 + t_1)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(k_m / t) ^ 2.0
	tmp = 0.0
	if (t <= 1.02e-76)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t * (sin(k_m) ^ 2.0)) / cos(k_m)) * Float64((k_m ^ 2.0) / l)) / l));
	elseif (t <= 5.5e+102)
		tmp = Float64(2.0 / Float64(1.0 / Float64(l / Float64(tan(k_m) * Float64(Float64((t ^ 3.0) / Float64(l / sin(k_m))) * Float64(2.0 + t_1))))));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k_m) * (Float64(cbrt(sin(k_m)) * Float64(Float64(t / cbrt(l)) * cbrt(Float64(1.0 / l)))) ^ 3.0)) * Float64(1.0 + Float64(1.0 + t_1))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t, 1.02e-76], N[(2.0 / N[(N[(N[(N[(t * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[Power[k$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.5e+102], N[(2.0 / N[(1.0 / N[(l / N[(N[Tan[k$95$m], $MachinePrecision] * N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k$95$m], $MachinePrecision] * N[Power[N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 1/3], $MachinePrecision] * N[(N[(t / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[(1.0 / l), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.02000000000000006e-76

   1. Initial program 47.6%

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

      1. associate-/r* 54.8%

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

      2. +-commutative 54.8%

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

      3. associate-+r+ 54.8%

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

      4. metadata-eval 54.8%

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

      5. associate-*r* 55.0%

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

      6. *-commutative 55.0%

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

      7. associate-*l/ 57.0%

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

      8. associate-*r/ 57.6%

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

   4. Applied egg-rr 57.6%

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

   5. Taylor expanded in k around inf 62.0%

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

      1. *-commutative 62.0%

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

      2. *-commutative 62.0%

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

      3. times-frac 62.5%

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

   7. Simplified 62.5%

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

   if 1.02000000000000006e-76 < t < 5.49999999999999981e102

   1. Initial program 79.2%

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

      1. associate-/r* 81.9%

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

      2. +-commutative 81.9%

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

      3. associate-+r+ 81.9%

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

      4. metadata-eval 81.9%

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

      5. associate-*r* 81.8%

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

      6. *-commutative 81.8%

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

      7. associate-*l/ 87.3%

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

      8. associate-*r/ 90.5%

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

   4. Applied egg-rr 90.5%

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

      1. clear-num 90.4%

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

      2. inv-pow 90.4%

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

      3. associate-*l* 90.5%

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

      4. associate-*l/ 92.4%

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

   6. Applied egg-rr 92.4%

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

      1. unpow-1 92.4%

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

      2. metadata-eval 92.4%

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

      3. associate-+r+ 92.4%

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

      4. *-commutative 92.4%

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

      5. associate-/l* 93.3%

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

      6. associate-+r+ 93.3%

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

      7. metadata-eval 93.3%

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

   8. Simplified 93.3%

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

   if 5.49999999999999981e102 < t

   1. Initial program 57.6%

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

      1. associate-/r* 60.8%

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

      2. div-inv 60.8%

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

      3. add-cube-cbrt 60.8%

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

      4. associate-*l* 60.8%

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

      5. pow2 60.8%

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

      6. cbrt-div 60.8%

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

      7. rem-cbrt-cube 60.8%

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

      8. cbrt-div 60.8%

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

      9. rem-cbrt-cube 84.7%

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

   4. Applied egg-rr 84.7%

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

      1. add-cube-cbrt 84.8%

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

      2. pow3 84.9%

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

   6. Applied egg-rr 93.8%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.02 \cdot 10^{-76}:\\ \;\;\;\;\frac{2}{\frac{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \frac{{k}^{2}}{\ell}}{\ell}}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\frac{1}{\frac{\ell}{\tan k \cdot \left(\frac{{t}^{3}}{\frac{\ell}{\sin k}} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot {\left(\sqrt[3]{\sin k} \cdot \left(\frac{t}{\sqrt[3]{\ell}} \cdot \sqrt[3]{\frac{1}{\ell}}\right)\right)}^{3}\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := {\left(\frac{k\_m}{t}\right)}^{2}\\ \mathbf{if}\;t \leq 1.45 \cdot 10^{-77}:\\ \;\;\;\;\frac{2}{\frac{\frac{t \cdot {\sin k\_m}^{2}}{\cos k\_m} \cdot \frac{{k\_m}^{2}}{\ell}}{\ell}}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\frac{1}{\frac{\ell}{\tan k\_m \cdot \left(\frac{{t}^{3}}{\frac{\ell}{\sin k\_m}} \cdot \left(2 + t\_1\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + t\_1\right)\right) \cdot \left(\tan k\_m \cdot {\left(\sqrt[3]{\sin k\_m} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (pow (/ k_m t) 2.0)))
   (if (<= t 1.45e-77)
     (/
      2.0
      (/ (* (/ (* t (pow (sin k_m) 2.0)) (cos k_m)) (/ (pow k_m 2.0) l)) l))
     (if (<= t 5.5e+102)
       (/
        2.0
        (/
         1.0
         (/ l (* (tan k_m) (* (/ (pow t 3.0) (/ l (sin k_m))) (+ 2.0 t_1))))))
       (/
        2.0
        (*
         (+ 1.0 (+ 1.0 t_1))
         (*
          (tan k_m)
          (pow (* (cbrt (sin k_m)) (/ t (pow (cbrt l) 2.0))) 3.0))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = pow((k_m / t), 2.0);
	double tmp;
	if (t <= 1.45e-77) {
		tmp = 2.0 / ((((t * pow(sin(k_m), 2.0)) / cos(k_m)) * (pow(k_m, 2.0) / l)) / l);
	} else if (t <= 5.5e+102) {
		tmp = 2.0 / (1.0 / (l / (tan(k_m) * ((pow(t, 3.0) / (l / sin(k_m))) * (2.0 + t_1)))));
	} else {
		tmp = 2.0 / ((1.0 + (1.0 + t_1)) * (tan(k_m) * pow((cbrt(sin(k_m)) * (t / pow(cbrt(l), 2.0))), 3.0)));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = Math.pow((k_m / t), 2.0);
	double tmp;
	if (t <= 1.45e-77) {
		tmp = 2.0 / ((((t * Math.pow(Math.sin(k_m), 2.0)) / Math.cos(k_m)) * (Math.pow(k_m, 2.0) / l)) / l);
	} else if (t <= 5.5e+102) {
		tmp = 2.0 / (1.0 / (l / (Math.tan(k_m) * ((Math.pow(t, 3.0) / (l / Math.sin(k_m))) * (2.0 + t_1)))));
	} else {
		tmp = 2.0 / ((1.0 + (1.0 + t_1)) * (Math.tan(k_m) * Math.pow((Math.cbrt(Math.sin(k_m)) * (t / Math.pow(Math.cbrt(l), 2.0))), 3.0)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(k_m / t) ^ 2.0
	tmp = 0.0
	if (t <= 1.45e-77)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t * (sin(k_m) ^ 2.0)) / cos(k_m)) * Float64((k_m ^ 2.0) / l)) / l));
	elseif (t <= 5.5e+102)
		tmp = Float64(2.0 / Float64(1.0 / Float64(l / Float64(tan(k_m) * Float64(Float64((t ^ 3.0) / Float64(l / sin(k_m))) * Float64(2.0 + t_1))))));
	else
		tmp = Float64(2.0 / Float64(Float64(1.0 + Float64(1.0 + t_1)) * Float64(tan(k_m) * (Float64(cbrt(sin(k_m)) * Float64(t / (cbrt(l) ^ 2.0))) ^ 3.0))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t, 1.45e-77], N[(2.0 / N[(N[(N[(N[(t * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[Power[k$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.5e+102], N[(2.0 / N[(1.0 / N[(l / N[(N[Tan[k$95$m], $MachinePrecision] * N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(1.0 + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[Power[N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 1/3], $MachinePrecision] * N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.4499999999999999e-77

   1. Initial program 47.6%

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

      1. associate-/r* 54.8%

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

      2. +-commutative 54.8%

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

      3. associate-+r+ 54.8%

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

      4. metadata-eval 54.8%

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

      5. associate-*r* 55.0%

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

      6. *-commutative 55.0%

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

      7. associate-*l/ 57.0%

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

      8. associate-*r/ 57.6%

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

   4. Applied egg-rr 57.6%

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

   5. Taylor expanded in k around inf 62.0%

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

      1. *-commutative 62.0%

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

      2. *-commutative 62.0%

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

      3. times-frac 62.5%

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

   7. Simplified 62.5%

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

   if 1.4499999999999999e-77 < t < 5.49999999999999981e102

   1. Initial program 79.2%

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

      1. associate-/r* 81.9%

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

      2. +-commutative 81.9%

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

      3. associate-+r+ 81.9%

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

      4. metadata-eval 81.9%

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

      5. associate-*r* 81.8%

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

      6. *-commutative 81.8%

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

      7. associate-*l/ 87.3%

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

      8. associate-*r/ 90.5%

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

   4. Applied egg-rr 90.5%

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

      1. clear-num 90.4%

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

      2. inv-pow 90.4%

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

      3. associate-*l* 90.5%

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

      4. associate-*l/ 92.4%

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

   6. Applied egg-rr 92.4%

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

      1. unpow-1 92.4%

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

      2. metadata-eval 92.4%

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

      3. associate-+r+ 92.4%

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

      4. *-commutative 92.4%

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

      5. associate-/l* 93.3%

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

      6. associate-+r+ 93.3%

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

      7. metadata-eval 93.3%

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

   8. Simplified 93.3%

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

   if 5.49999999999999981e102 < t

   1. Initial program 57.6%

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

      1. associate-/r* 60.8%

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

      2. add-cube-cbrt 60.8%

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

      3. pow3 60.8%

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

      4. *-commutative 60.8%

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

      5. cbrt-prod 60.8%

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

      6. associate-/r* 57.6%

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

      7. cbrt-div 57.6%

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

      8. rem-cbrt-cube 73.2%

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

      9. cbrt-prod 93.5%

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

      10. pow2 93.5%

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

   4. Applied egg-rr 93.5%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.45 \cdot 10^{-77}:\\ \;\;\;\;\frac{2}{\frac{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \frac{{k}^{2}}{\ell}}{\ell}}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\frac{1}{\frac{\ell}{\tan k \cdot \left(\frac{{t}^{3}}{\frac{\ell}{\sin k}} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot {\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := {\left(\frac{k\_m}{t}\right)}^{2}\\ \mathbf{if}\;t \leq 1.5 \cdot 10^{-77}:\\ \;\;\;\;\frac{2}{\frac{\frac{t \cdot {\sin k\_m}^{2}}{\cos k\_m} \cdot \frac{{k\_m}^{2}}{\ell}}{\ell}}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\frac{1}{\frac{\ell}{\tan k\_m \cdot \left(\frac{{t}^{3}}{\frac{\ell}{\sin k\_m}} \cdot \left(2 + t\_1\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + t\_1\right)\right) \cdot \left(\tan k\_m \cdot {\left(\left(t \cdot \sqrt[3]{\sin k\_m}\right) \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (pow (/ k_m t) 2.0)))
   (if (<= t 1.5e-77)
     (/
      2.0
      (/ (* (/ (* t (pow (sin k_m) 2.0)) (cos k_m)) (/ (pow k_m 2.0) l)) l))
     (if (<= t 5.5e+102)
       (/
        2.0
        (/
         1.0
         (/ l (* (tan k_m) (* (/ (pow t 3.0) (/ l (sin k_m))) (+ 2.0 t_1))))))
       (/
        2.0
        (*
         (+ 1.0 (+ 1.0 t_1))
         (*
          (tan k_m)
          (pow (* (* t (cbrt (sin k_m))) (pow (cbrt l) -2.0)) 3.0))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = pow((k_m / t), 2.0);
	double tmp;
	if (t <= 1.5e-77) {
		tmp = 2.0 / ((((t * pow(sin(k_m), 2.0)) / cos(k_m)) * (pow(k_m, 2.0) / l)) / l);
	} else if (t <= 5.5e+102) {
		tmp = 2.0 / (1.0 / (l / (tan(k_m) * ((pow(t, 3.0) / (l / sin(k_m))) * (2.0 + t_1)))));
	} else {
		tmp = 2.0 / ((1.0 + (1.0 + t_1)) * (tan(k_m) * pow(((t * cbrt(sin(k_m))) * pow(cbrt(l), -2.0)), 3.0)));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = Math.pow((k_m / t), 2.0);
	double tmp;
	if (t <= 1.5e-77) {
		tmp = 2.0 / ((((t * Math.pow(Math.sin(k_m), 2.0)) / Math.cos(k_m)) * (Math.pow(k_m, 2.0) / l)) / l);
	} else if (t <= 5.5e+102) {
		tmp = 2.0 / (1.0 / (l / (Math.tan(k_m) * ((Math.pow(t, 3.0) / (l / Math.sin(k_m))) * (2.0 + t_1)))));
	} else {
		tmp = 2.0 / ((1.0 + (1.0 + t_1)) * (Math.tan(k_m) * Math.pow(((t * Math.cbrt(Math.sin(k_m))) * Math.pow(Math.cbrt(l), -2.0)), 3.0)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(k_m / t) ^ 2.0
	tmp = 0.0
	if (t <= 1.5e-77)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t * (sin(k_m) ^ 2.0)) / cos(k_m)) * Float64((k_m ^ 2.0) / l)) / l));
	elseif (t <= 5.5e+102)
		tmp = Float64(2.0 / Float64(1.0 / Float64(l / Float64(tan(k_m) * Float64(Float64((t ^ 3.0) / Float64(l / sin(k_m))) * Float64(2.0 + t_1))))));
	else
		tmp = Float64(2.0 / Float64(Float64(1.0 + Float64(1.0 + t_1)) * Float64(tan(k_m) * (Float64(Float64(t * cbrt(sin(k_m))) * (cbrt(l) ^ -2.0)) ^ 3.0))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t, 1.5e-77], N[(2.0 / N[(N[(N[(N[(t * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[Power[k$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.5e+102], N[(2.0 / N[(1.0 / N[(l / N[(N[Tan[k$95$m], $MachinePrecision] * N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(1.0 + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[Power[N[(N[(t * N[Power[N[Sin[k$95$m], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.50000000000000008e-77

   1. Initial program 47.6%

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

      1. associate-/r* 54.8%

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

      2. +-commutative 54.8%

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

      3. associate-+r+ 54.8%

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

      4. metadata-eval 54.8%

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

      5. associate-*r* 55.0%

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

      6. *-commutative 55.0%

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

      7. associate-*l/ 57.0%

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

      8. associate-*r/ 57.6%

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

   4. Applied egg-rr 57.6%

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

   5. Taylor expanded in k around inf 62.0%

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

      1. *-commutative 62.0%

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

      2. *-commutative 62.0%

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

      3. times-frac 62.5%

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

   7. Simplified 62.5%

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

   if 1.50000000000000008e-77 < t < 5.49999999999999981e102

   1. Initial program 79.2%

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

      1. associate-/r* 81.9%

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

      2. +-commutative 81.9%

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

      3. associate-+r+ 81.9%

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

      4. metadata-eval 81.9%

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

      5. associate-*r* 81.8%

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

      6. *-commutative 81.8%

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

      7. associate-*l/ 87.3%

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

      8. associate-*r/ 90.5%

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

   4. Applied egg-rr 90.5%

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

      1. clear-num 90.4%

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

      2. inv-pow 90.4%

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

      3. associate-*l* 90.5%

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

      4. associate-*l/ 92.4%

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

   6. Applied egg-rr 92.4%

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

      1. unpow-1 92.4%

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

      2. metadata-eval 92.4%

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

      3. associate-+r+ 92.4%

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

      4. *-commutative 92.4%

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

      5. associate-/l* 93.3%

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

      6. associate-+r+ 93.3%

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

      7. metadata-eval 93.3%

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

   8. Simplified 93.3%

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

   if 5.49999999999999981e102 < t

   1. Initial program 57.6%

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

      1. associate-/r* 60.8%

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

      2. add-cube-cbrt 60.8%

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

      3. pow3 60.8%

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

      4. *-commutative 60.8%

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

      5. cbrt-prod 60.8%

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

      6. associate-/r* 57.6%

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

      7. cbrt-div 57.6%

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

      8. rem-cbrt-cube 73.2%

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

      9. cbrt-prod 93.5%

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

      10. pow2 93.5%

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

   4. Applied egg-rr 93.5%

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

      1. expm1-log1p-u 57.8%

         \[\leadsto \frac{2}{\left({\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)\right)\right)}}^{3} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. expm1-udef 48.2%

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

      3. div-inv 48.2%

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

      4. pow-flip 48.2%

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

      5. metadata-eval 48.2%

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

   6. Applied egg-rr 48.2%

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

      1. expm1-def 57.8%

         \[\leadsto \frac{2}{\left({\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin k} \cdot \left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)\right)\right)}}^{3} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. expm1-log1p 93.4%

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

      3. associate-*r* 93.5%

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

      4. *-commutative 93.5%

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

   8. Simplified 93.5%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.5 \cdot 10^{-77}:\\ \;\;\;\;\frac{2}{\frac{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \frac{{k}^{2}}{\ell}}{\ell}}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\frac{1}{\frac{\ell}{\tan k \cdot \left(\frac{{t}^{3}}{\frac{\ell}{\sin k}} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot {\left(\left(t \cdot \sqrt[3]{\sin k}\right) \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := {\left(\frac{k\_m}{t}\right)}^{2}\\ \mathbf{if}\;t \leq 1.26 \cdot 10^{-76}:\\ \;\;\;\;\frac{2}{\frac{\frac{t \cdot {\sin k\_m}^{2}}{\cos k\_m} \cdot \frac{{k\_m}^{2}}{\ell}}{\ell}}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\frac{1}{\frac{\ell}{\tan k\_m \cdot \left(\frac{{t}^{3}}{\frac{\ell}{\sin k\_m}} \cdot \left(2 + t\_1\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + t\_1\right)\right) \cdot \left(\tan k\_m \cdot {\left(\frac{t}{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{\sqrt[3]{\sin k\_m}}}\right)}^{3}\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (pow (/ k_m t) 2.0)))
   (if (<= t 1.26e-76)
     (/
      2.0
      (/ (* (/ (* t (pow (sin k_m) 2.0)) (cos k_m)) (/ (pow k_m 2.0) l)) l))
     (if (<= t 5.5e+102)
       (/
        2.0
        (/
         1.0
         (/ l (* (tan k_m) (* (/ (pow t 3.0) (/ l (sin k_m))) (+ 2.0 t_1))))))
       (/
        2.0
        (*
         (+ 1.0 (+ 1.0 t_1))
         (*
          (tan k_m)
          (pow (/ t (/ (pow (cbrt l) 2.0) (cbrt (sin k_m)))) 3.0))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = pow((k_m / t), 2.0);
	double tmp;
	if (t <= 1.26e-76) {
		tmp = 2.0 / ((((t * pow(sin(k_m), 2.0)) / cos(k_m)) * (pow(k_m, 2.0) / l)) / l);
	} else if (t <= 5.5e+102) {
		tmp = 2.0 / (1.0 / (l / (tan(k_m) * ((pow(t, 3.0) / (l / sin(k_m))) * (2.0 + t_1)))));
	} else {
		tmp = 2.0 / ((1.0 + (1.0 + t_1)) * (tan(k_m) * pow((t / (pow(cbrt(l), 2.0) / cbrt(sin(k_m)))), 3.0)));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = Math.pow((k_m / t), 2.0);
	double tmp;
	if (t <= 1.26e-76) {
		tmp = 2.0 / ((((t * Math.pow(Math.sin(k_m), 2.0)) / Math.cos(k_m)) * (Math.pow(k_m, 2.0) / l)) / l);
	} else if (t <= 5.5e+102) {
		tmp = 2.0 / (1.0 / (l / (Math.tan(k_m) * ((Math.pow(t, 3.0) / (l / Math.sin(k_m))) * (2.0 + t_1)))));
	} else {
		tmp = 2.0 / ((1.0 + (1.0 + t_1)) * (Math.tan(k_m) * Math.pow((t / (Math.pow(Math.cbrt(l), 2.0) / Math.cbrt(Math.sin(k_m)))), 3.0)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(k_m / t) ^ 2.0
	tmp = 0.0
	if (t <= 1.26e-76)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t * (sin(k_m) ^ 2.0)) / cos(k_m)) * Float64((k_m ^ 2.0) / l)) / l));
	elseif (t <= 5.5e+102)
		tmp = Float64(2.0 / Float64(1.0 / Float64(l / Float64(tan(k_m) * Float64(Float64((t ^ 3.0) / Float64(l / sin(k_m))) * Float64(2.0 + t_1))))));
	else
		tmp = Float64(2.0 / Float64(Float64(1.0 + Float64(1.0 + t_1)) * Float64(tan(k_m) * (Float64(t / Float64((cbrt(l) ^ 2.0) / cbrt(sin(k_m)))) ^ 3.0))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t, 1.26e-76], N[(2.0 / N[(N[(N[(N[(t * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[Power[k$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.5e+102], N[(2.0 / N[(1.0 / N[(l / N[(N[Tan[k$95$m], $MachinePrecision] * N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(1.0 + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[Power[N[(t / N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.26e-76

   1. Initial program 47.6%

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

      1. associate-/r* 54.8%

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

      2. +-commutative 54.8%

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

      3. associate-+r+ 54.8%

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

      4. metadata-eval 54.8%

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

      5. associate-*r* 55.0%

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

      6. *-commutative 55.0%

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

      7. associate-*l/ 57.0%

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

      8. associate-*r/ 57.6%

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

   4. Applied egg-rr 57.6%

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

   5. Taylor expanded in k around inf 62.0%

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

      1. *-commutative 62.0%

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

      2. *-commutative 62.0%

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

      3. times-frac 62.5%

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

   7. Simplified 62.5%

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

   if 1.26e-76 < t < 5.49999999999999981e102

   1. Initial program 79.2%

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

      1. associate-/r* 81.9%

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

      2. +-commutative 81.9%

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

      3. associate-+r+ 81.9%

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

      4. metadata-eval 81.9%

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

      5. associate-*r* 81.8%

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

      6. *-commutative 81.8%

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

      7. associate-*l/ 87.3%

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

      8. associate-*r/ 90.5%

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

   4. Applied egg-rr 90.5%

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

      1. clear-num 90.4%

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

      2. inv-pow 90.4%

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

      3. associate-*l* 90.5%

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

      4. associate-*l/ 92.4%

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

   6. Applied egg-rr 92.4%

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

      1. unpow-1 92.4%

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

      2. metadata-eval 92.4%

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

      3. associate-+r+ 92.4%

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

      4. *-commutative 92.4%

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

      5. associate-/l* 93.3%

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

      6. associate-+r+ 93.3%

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

      7. metadata-eval 93.3%

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

   8. Simplified 93.3%

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

   if 5.49999999999999981e102 < t

   1. Initial program 57.6%

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

      1. associate-/r* 60.8%

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

      2. add-cube-cbrt 60.8%

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

      3. pow3 60.8%

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

      4. *-commutative 60.8%

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

      5. cbrt-prod 60.8%

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

      6. associate-/r* 57.6%

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

      7. cbrt-div 57.6%

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

      8. rem-cbrt-cube 73.2%

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

      9. cbrt-prod 93.5%

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

      10. pow2 93.5%

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

   4. Applied egg-rr 93.5%

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

      1. associate-*r/ 93.6%

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

   6. Applied egg-rr 93.6%

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

      1. *-commutative 93.6%

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

      2. associate-/l* 93.5%

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

   8. Simplified 93.5%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.26 \cdot 10^{-76}:\\ \;\;\;\;\frac{2}{\frac{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \frac{{k}^{2}}{\ell}}{\ell}}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\frac{1}{\frac{\ell}{\tan k \cdot \left(\frac{{t}^{3}}{\frac{\ell}{\sin k}} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot {\left(\frac{t}{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{\sqrt[3]{\sin k}}}\right)}^{3}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := {\left(\frac{k\_m}{t}\right)}^{2}\\ \mathbf{if}\;t \leq 2.15 \cdot 10^{-76}:\\ \;\;\;\;\frac{2}{\frac{\frac{t \cdot {\sin k\_m}^{2}}{\cos k\_m} \cdot \frac{{k\_m}^{2}}{\ell}}{\ell}}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\frac{1}{\frac{\ell}{\tan k\_m \cdot \left(\frac{{t}^{3}}{\frac{\ell}{\sin k\_m}} \cdot \left(2 + t\_1\right)\right)}}}\\ \mathbf{elif}\;t \leq 9 \cdot 10^{+171}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + t\_1\right)\right) \cdot \left(\tan k\_m \cdot \left(\sin k\_m \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}}{{\left(\sqrt[3]{k\_m}\right)}^{2}}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (pow (/ k_m t) 2.0)))
   (if (<= t 2.15e-76)
     (/
      2.0
      (/ (* (/ (* t (pow (sin k_m) 2.0)) (cos k_m)) (/ (pow k_m 2.0) l)) l))
     (if (<= t 5.5e+102)
       (/
        2.0
        (/
         1.0
         (/ l (* (tan k_m) (* (/ (pow t 3.0) (/ l (sin k_m))) (+ 2.0 t_1))))))
       (if (<= t 9e+171)
         (/
          2.0
          (*
           (+ 1.0 (+ 1.0 t_1))
           (* (tan k_m) (* (sin k_m) (pow (/ (pow t 1.5) l) 2.0)))))
         (pow (/ (/ (pow (cbrt l) 2.0) t) (pow (cbrt k_m) 2.0)) 3.0))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = pow((k_m / t), 2.0);
	double tmp;
	if (t <= 2.15e-76) {
		tmp = 2.0 / ((((t * pow(sin(k_m), 2.0)) / cos(k_m)) * (pow(k_m, 2.0) / l)) / l);
	} else if (t <= 5.5e+102) {
		tmp = 2.0 / (1.0 / (l / (tan(k_m) * ((pow(t, 3.0) / (l / sin(k_m))) * (2.0 + t_1)))));
	} else if (t <= 9e+171) {
		tmp = 2.0 / ((1.0 + (1.0 + t_1)) * (tan(k_m) * (sin(k_m) * pow((pow(t, 1.5) / l), 2.0))));
	} else {
		tmp = pow(((pow(cbrt(l), 2.0) / t) / pow(cbrt(k_m), 2.0)), 3.0);
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = Math.pow((k_m / t), 2.0);
	double tmp;
	if (t <= 2.15e-76) {
		tmp = 2.0 / ((((t * Math.pow(Math.sin(k_m), 2.0)) / Math.cos(k_m)) * (Math.pow(k_m, 2.0) / l)) / l);
	} else if (t <= 5.5e+102) {
		tmp = 2.0 / (1.0 / (l / (Math.tan(k_m) * ((Math.pow(t, 3.0) / (l / Math.sin(k_m))) * (2.0 + t_1)))));
	} else if (t <= 9e+171) {
		tmp = 2.0 / ((1.0 + (1.0 + t_1)) * (Math.tan(k_m) * (Math.sin(k_m) * Math.pow((Math.pow(t, 1.5) / l), 2.0))));
	} else {
		tmp = Math.pow(((Math.pow(Math.cbrt(l), 2.0) / t) / Math.pow(Math.cbrt(k_m), 2.0)), 3.0);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(k_m / t) ^ 2.0
	tmp = 0.0
	if (t <= 2.15e-76)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t * (sin(k_m) ^ 2.0)) / cos(k_m)) * Float64((k_m ^ 2.0) / l)) / l));
	elseif (t <= 5.5e+102)
		tmp = Float64(2.0 / Float64(1.0 / Float64(l / Float64(tan(k_m) * Float64(Float64((t ^ 3.0) / Float64(l / sin(k_m))) * Float64(2.0 + t_1))))));
	elseif (t <= 9e+171)
		tmp = Float64(2.0 / Float64(Float64(1.0 + Float64(1.0 + t_1)) * Float64(tan(k_m) * Float64(sin(k_m) * (Float64((t ^ 1.5) / l) ^ 2.0)))));
	else
		tmp = Float64(Float64((cbrt(l) ^ 2.0) / t) / (cbrt(k_m) ^ 2.0)) ^ 3.0;
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t, 2.15e-76], N[(2.0 / N[(N[(N[(N[(t * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[Power[k$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.5e+102], N[(2.0 / N[(1.0 / N[(l / N[(N[Tan[k$95$m], $MachinePrecision] * N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 9e+171], N[(2.0 / N[(N[(1.0 + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[Power[N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision] / N[Power[N[Power[k$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if t < 2.15e-76

   1. Initial program 47.6%

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

      1. associate-/r* 54.8%

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

      2. +-commutative 54.8%

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

      3. associate-+r+ 54.8%

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

      4. metadata-eval 54.8%

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

      5. associate-*r* 55.0%

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

      6. *-commutative 55.0%

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

      7. associate-*l/ 57.0%

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

      8. associate-*r/ 57.6%

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

   4. Applied egg-rr 57.6%

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

   5. Taylor expanded in k around inf 62.0%

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

      1. *-commutative 62.0%

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

      2. *-commutative 62.0%

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

      3. times-frac 62.5%

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

   7. Simplified 62.5%

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

   if 2.15e-76 < t < 5.49999999999999981e102

   1. Initial program 79.2%

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

      1. associate-/r* 81.9%

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

      2. +-commutative 81.9%

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

      3. associate-+r+ 81.9%

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

      4. metadata-eval 81.9%

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

      5. associate-*r* 81.8%

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

      6. *-commutative 81.8%

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

      7. associate-*l/ 87.3%

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

      8. associate-*r/ 90.5%

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

   4. Applied egg-rr 90.5%

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

      1. clear-num 90.4%

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

      2. inv-pow 90.4%

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

      3. associate-*l* 90.5%

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

      4. associate-*l/ 92.4%

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

   6. Applied egg-rr 92.4%

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

      1. unpow-1 92.4%

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

      2. metadata-eval 92.4%

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

      3. associate-+r+ 92.4%

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

      4. *-commutative 92.4%

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

      5. associate-/l* 93.3%

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

      6. associate-+r+ 93.3%

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

      7. metadata-eval 93.3%

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

   8. Simplified 93.3%

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

   if 5.49999999999999981e102 < t < 8.99999999999999937e171

   1. Initial program 40.6%

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

      1. associate-/r* 42.5%

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

      2. add-sqr-sqrt 42.5%

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

      3. pow2 42.5%

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

      4. associate-/r* 40.6%

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

      5. sqrt-div 40.6%

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

      6. sqrt-pow1 66.0%

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

      7. metadata-eval 66.0%

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

      8. sqrt-prod 44.8%

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

      9. add-sqr-sqrt 94.9%

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

   4. Applied egg-rr 94.9%

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

   if 8.99999999999999937e171 < t

   1. Initial program 68.6%

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

      1. associate-/r* 68.6%

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

      2. sqr-neg 68.6%

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

      3. associate-*l* 58.1%

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

      4. sqr-neg 58.1%

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

      5. associate-/r* 62.0%

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

      6. associate-+l+ 62.0%

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

      7. unpow2 62.0%

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

      8. times-frac 42.5%

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

      9. sqr-neg 42.5%

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

      10. times-frac 62.0%

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

      11. unpow2 62.0%

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

   3. Simplified 62.0%

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

   5. Taylor expanded in k around 0 58.1%

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

      1. *-commutative 58.1%

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

      2. associate-/r* 58.1%

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

   7. Simplified 58.1%

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

      1. unpow2 58.1%

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

   9. Applied egg-rr 58.1%

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

       1. expm1-log1p-u 58.1%

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

       2. expm1-udef 58.1%

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

       3. div-inv 58.1%

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

       4. pow-flip 58.1%

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

       5. metadata-eval 58.1%

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

   11. Applied egg-rr 58.1%

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

       1. expm1-def 58.1%

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

       2. expm1-log1p 58.1%

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

   13. Simplified 58.1%

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

       1. add-cube-cbrt 58.1%

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

       2. pow2 58.1%

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

       3. cbrt-div 58.1%

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

       4. metadata-eval 58.1%

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

       5. pow-flip 58.1%

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

       6. div-inv 58.1%

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

       7. cbrt-div 58.1%

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

       8. unpow2 58.1%

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

       9. cbrt-prod 58.1%

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

       10. unpow2 58.1%

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

       11. unpow3 58.1%

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

       12. add-cbrt-cube 58.1%

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

       13. cbrt-prod 58.1%

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

       14. pow2 58.1%

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

       15. cbrt-div 58.1%

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

   15. Applied egg-rr 88.3%

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

       1. pow-plus 88.3%

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

       2. metadata-eval 88.3%

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

   17. Simplified 88.3%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.15 \cdot 10^{-76}:\\ \;\;\;\;\frac{2}{\frac{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \frac{{k}^{2}}{\ell}}{\ell}}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\frac{1}{\frac{\ell}{\tan k \cdot \left(\frac{{t}^{3}}{\frac{\ell}{\sin k}} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}}\\ \mathbf{elif}\;t \leq 9 \cdot 10^{+171}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}}{{\left(\sqrt[3]{k}\right)}^{2}}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := 2 + {\left(\frac{k\_m}{t}\right)}^{2}\\ \mathbf{if}\;t \leq 4 \cdot 10^{-76}:\\ \;\;\;\;\frac{2}{\frac{\frac{t \cdot {\sin k\_m}^{2}}{\cos k\_m} \cdot \frac{{k\_m}^{2}}{\ell}}{\ell}}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\frac{1}{\frac{\ell}{\tan k\_m \cdot \left(\frac{{t}^{3}}{\frac{\ell}{\sin k\_m}} \cdot t\_1\right)}}}\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{+160}:\\ \;\;\;\;\frac{2}{\left(\sin k\_m \cdot \frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}\right) \cdot \left(\tan k\_m \cdot t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}}{{\left(\sqrt[3]{k\_m}\right)}^{2}}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (+ 2.0 (pow (/ k_m t) 2.0))))
   (if (<= t 4e-76)
     (/
      2.0
      (/ (* (/ (* t (pow (sin k_m) 2.0)) (cos k_m)) (/ (pow k_m 2.0) l)) l))
     (if (<= t 5.5e+102)
       (/
        2.0
        (/ 1.0 (/ l (* (tan k_m) (* (/ (pow t 3.0) (/ l (sin k_m))) t_1)))))
       (if (<= t 7.2e+160)
         (/
          2.0
          (* (* (sin k_m) (/ (pow (/ t (cbrt l)) 3.0) l)) (* (tan k_m) t_1)))
         (pow (/ (/ (pow (cbrt l) 2.0) t) (pow (cbrt k_m) 2.0)) 3.0))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = 2.0 + pow((k_m / t), 2.0);
	double tmp;
	if (t <= 4e-76) {
		tmp = 2.0 / ((((t * pow(sin(k_m), 2.0)) / cos(k_m)) * (pow(k_m, 2.0) / l)) / l);
	} else if (t <= 5.5e+102) {
		tmp = 2.0 / (1.0 / (l / (tan(k_m) * ((pow(t, 3.0) / (l / sin(k_m))) * t_1))));
	} else if (t <= 7.2e+160) {
		tmp = 2.0 / ((sin(k_m) * (pow((t / cbrt(l)), 3.0) / l)) * (tan(k_m) * t_1));
	} else {
		tmp = pow(((pow(cbrt(l), 2.0) / t) / pow(cbrt(k_m), 2.0)), 3.0);
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = 2.0 + Math.pow((k_m / t), 2.0);
	double tmp;
	if (t <= 4e-76) {
		tmp = 2.0 / ((((t * Math.pow(Math.sin(k_m), 2.0)) / Math.cos(k_m)) * (Math.pow(k_m, 2.0) / l)) / l);
	} else if (t <= 5.5e+102) {
		tmp = 2.0 / (1.0 / (l / (Math.tan(k_m) * ((Math.pow(t, 3.0) / (l / Math.sin(k_m))) * t_1))));
	} else if (t <= 7.2e+160) {
		tmp = 2.0 / ((Math.sin(k_m) * (Math.pow((t / Math.cbrt(l)), 3.0) / l)) * (Math.tan(k_m) * t_1));
	} else {
		tmp = Math.pow(((Math.pow(Math.cbrt(l), 2.0) / t) / Math.pow(Math.cbrt(k_m), 2.0)), 3.0);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(2.0 + (Float64(k_m / t) ^ 2.0))
	tmp = 0.0
	if (t <= 4e-76)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t * (sin(k_m) ^ 2.0)) / cos(k_m)) * Float64((k_m ^ 2.0) / l)) / l));
	elseif (t <= 5.5e+102)
		tmp = Float64(2.0 / Float64(1.0 / Float64(l / Float64(tan(k_m) * Float64(Float64((t ^ 3.0) / Float64(l / sin(k_m))) * t_1)))));
	elseif (t <= 7.2e+160)
		tmp = Float64(2.0 / Float64(Float64(sin(k_m) * Float64((Float64(t / cbrt(l)) ^ 3.0) / l)) * Float64(tan(k_m) * t_1)));
	else
		tmp = Float64(Float64((cbrt(l) ^ 2.0) / t) / (cbrt(k_m) ^ 2.0)) ^ 3.0;
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(2.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 4e-76], N[(2.0 / N[(N[(N[(N[(t * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[Power[k$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.5e+102], N[(2.0 / N[(1.0 / N[(l / N[(N[Tan[k$95$m], $MachinePrecision] * N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 7.2e+160], N[(2.0 / N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Power[N[(t / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision] / N[Power[N[Power[k$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if t < 3.99999999999999971e-76

   1. Initial program 47.6%

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

      1. associate-/r* 54.8%

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

      2. +-commutative 54.8%

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

      3. associate-+r+ 54.8%

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

      4. metadata-eval 54.8%

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

      5. associate-*r* 55.0%

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

      6. *-commutative 55.0%

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

      7. associate-*l/ 57.0%

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

      8. associate-*r/ 57.6%

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

   4. Applied egg-rr 57.6%

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

   5. Taylor expanded in k around inf 62.0%

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

      1. *-commutative 62.0%

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

      2. *-commutative 62.0%

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

      3. times-frac 62.5%

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

   7. Simplified 62.5%

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

   if 3.99999999999999971e-76 < t < 5.49999999999999981e102

   1. Initial program 79.2%

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

      1. associate-/r* 81.9%

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

      2. +-commutative 81.9%

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

      3. associate-+r+ 81.9%

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

      4. metadata-eval 81.9%

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

      5. associate-*r* 81.8%

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

      6. *-commutative 81.8%

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

      7. associate-*l/ 87.3%

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

      8. associate-*r/ 90.5%

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

   4. Applied egg-rr 90.5%

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

      1. clear-num 90.4%

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

      2. inv-pow 90.4%

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

      3. associate-*l* 90.5%

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

      4. associate-*l/ 92.4%

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

   6. Applied egg-rr 92.4%

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

      1. unpow-1 92.4%

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

      2. metadata-eval 92.4%

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

      3. associate-+r+ 92.4%

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

      4. *-commutative 92.4%

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

      5. associate-/l* 93.3%

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

      6. associate-+r+ 93.3%

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

      7. metadata-eval 93.3%

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

   8. Simplified 93.3%

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

   if 5.49999999999999981e102 < t < 7.20000000000000042e160

   1. Initial program 39.6%

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

      1. associate-*l* 39.6%

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

      2. sqr-neg 39.6%

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

      3. sqr-neg 39.6%

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

      4. associate-/r* 41.6%

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

      5. distribute-rgt-in 41.6%

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

      6. unpow2 41.6%

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

      7. times-frac 41.1%

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

      8. sqr-neg 41.1%

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

      9. times-frac 41.6%

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

      10. unpow2 41.6%

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

      11. distribute-rgt-in 41.6%

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

   3. Simplified 41.6%

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

      1. add-cube-cbrt 41.6%

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

      2. pow3 41.6%

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

      3. cbrt-div 41.6%

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

      4. rem-cbrt-cube 88.8%

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

   6. Applied egg-rr 88.8%

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

   if 7.20000000000000042e160 < t

   1. Initial program 67.5%

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

      1. associate-/r* 67.5%

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

      2. sqr-neg 67.5%

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

      3. associate-*l* 57.6%

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

      4. sqr-neg 57.6%

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

      5. associate-/r* 61.3%

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

      6. associate-+l+ 61.3%

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

      7. unpow2 61.3%

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

      8. times-frac 39.9%

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

      9. sqr-neg 39.9%

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

      10. times-frac 61.3%

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

      11. unpow2 61.3%

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

   3. Simplified 61.3%

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

   5. Taylor expanded in k around 0 57.6%

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

      1. *-commutative 57.6%

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

      2. associate-/r* 57.6%

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

   7. Simplified 57.6%

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

      1. unpow2 57.6%

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

   9. Applied egg-rr 57.6%

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

       1. expm1-log1p-u 57.6%

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

       2. expm1-udef 57.6%

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

       3. div-inv 57.6%

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

       4. pow-flip 57.6%

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

       5. metadata-eval 57.6%

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

   11. Applied egg-rr 57.6%

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

       1. expm1-def 57.6%

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

       2. expm1-log1p 57.6%

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

   13. Simplified 57.6%

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

       1. add-cube-cbrt 57.6%

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

       2. pow2 57.6%

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

       3. cbrt-div 57.6%

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

       4. metadata-eval 57.6%

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

       5. pow-flip 57.6%

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

       6. div-inv 57.6%

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

       7. cbrt-div 57.6%

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

       8. unpow2 57.6%

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

       9. cbrt-prod 57.6%

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

       10. unpow2 57.6%

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

       11. unpow3 57.6%

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

       12. add-cbrt-cube 57.6%

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

       13. cbrt-prod 57.6%

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

       14. pow2 57.6%

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

       15. cbrt-div 57.6%

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

   15. Applied egg-rr 89.0%

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

       1. pow-plus 89.0%

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

       2. metadata-eval 89.0%

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

   17. Simplified 89.0%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4 \cdot 10^{-76}:\\ \;\;\;\;\frac{2}{\frac{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \frac{{k}^{2}}{\ell}}{\ell}}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\frac{1}{\frac{\ell}{\tan k \cdot \left(\frac{{t}^{3}}{\frac{\ell}{\sin k}} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}}\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{+160}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}}{{\left(\sqrt[3]{k}\right)}^{2}}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := {\left(\frac{k\_m}{t}\right)}^{2}\\ \mathbf{if}\;t \leq 4.9 \cdot 10^{-76}:\\ \;\;\;\;\frac{2}{\frac{\frac{t \cdot {\sin k\_m}^{2}}{\cos k\_m} \cdot \frac{{k\_m}^{2}}{\ell}}{\ell}}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+101}:\\ \;\;\;\;\frac{2}{\frac{1}{\frac{\ell}{\tan k\_m \cdot \left(\frac{{t}^{3}}{\frac{\ell}{\sin k\_m}} \cdot \left(2 + t\_1\right)\right)}}}\\ \mathbf{elif}\;t \leq 5.1 \cdot 10^{+160}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + t\_1\right)\right) \cdot \left(\tan k\_m \cdot \left(\sin k\_m \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}}{{\left(\sqrt[3]{k\_m}\right)}^{2}}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (pow (/ k_m t) 2.0)))
   (if (<= t 4.9e-76)
     (/
      2.0
      (/ (* (/ (* t (pow (sin k_m) 2.0)) (cos k_m)) (/ (pow k_m 2.0) l)) l))
     (if (<= t 6e+101)
       (/
        2.0
        (/
         1.0
         (/ l (* (tan k_m) (* (/ (pow t 3.0) (/ l (sin k_m))) (+ 2.0 t_1))))))
       (if (<= t 5.1e+160)
         (/
          2.0
          (*
           (+ 1.0 (+ 1.0 t_1))
           (* (tan k_m) (* (sin k_m) (* (/ (pow t 2.0) l) (/ t l))))))
         (pow (/ (/ (pow (cbrt l) 2.0) t) (pow (cbrt k_m) 2.0)) 3.0))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = pow((k_m / t), 2.0);
	double tmp;
	if (t <= 4.9e-76) {
		tmp = 2.0 / ((((t * pow(sin(k_m), 2.0)) / cos(k_m)) * (pow(k_m, 2.0) / l)) / l);
	} else if (t <= 6e+101) {
		tmp = 2.0 / (1.0 / (l / (tan(k_m) * ((pow(t, 3.0) / (l / sin(k_m))) * (2.0 + t_1)))));
	} else if (t <= 5.1e+160) {
		tmp = 2.0 / ((1.0 + (1.0 + t_1)) * (tan(k_m) * (sin(k_m) * ((pow(t, 2.0) / l) * (t / l)))));
	} else {
		tmp = pow(((pow(cbrt(l), 2.0) / t) / pow(cbrt(k_m), 2.0)), 3.0);
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = Math.pow((k_m / t), 2.0);
	double tmp;
	if (t <= 4.9e-76) {
		tmp = 2.0 / ((((t * Math.pow(Math.sin(k_m), 2.0)) / Math.cos(k_m)) * (Math.pow(k_m, 2.0) / l)) / l);
	} else if (t <= 6e+101) {
		tmp = 2.0 / (1.0 / (l / (Math.tan(k_m) * ((Math.pow(t, 3.0) / (l / Math.sin(k_m))) * (2.0 + t_1)))));
	} else if (t <= 5.1e+160) {
		tmp = 2.0 / ((1.0 + (1.0 + t_1)) * (Math.tan(k_m) * (Math.sin(k_m) * ((Math.pow(t, 2.0) / l) * (t / l)))));
	} else {
		tmp = Math.pow(((Math.pow(Math.cbrt(l), 2.0) / t) / Math.pow(Math.cbrt(k_m), 2.0)), 3.0);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(k_m / t) ^ 2.0
	tmp = 0.0
	if (t <= 4.9e-76)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t * (sin(k_m) ^ 2.0)) / cos(k_m)) * Float64((k_m ^ 2.0) / l)) / l));
	elseif (t <= 6e+101)
		tmp = Float64(2.0 / Float64(1.0 / Float64(l / Float64(tan(k_m) * Float64(Float64((t ^ 3.0) / Float64(l / sin(k_m))) * Float64(2.0 + t_1))))));
	elseif (t <= 5.1e+160)
		tmp = Float64(2.0 / Float64(Float64(1.0 + Float64(1.0 + t_1)) * Float64(tan(k_m) * Float64(sin(k_m) * Float64(Float64((t ^ 2.0) / l) * Float64(t / l))))));
	else
		tmp = Float64(Float64((cbrt(l) ^ 2.0) / t) / (cbrt(k_m) ^ 2.0)) ^ 3.0;
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t, 4.9e-76], N[(2.0 / N[(N[(N[(N[(t * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[Power[k$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6e+101], N[(2.0 / N[(1.0 / N[(l / N[(N[Tan[k$95$m], $MachinePrecision] * N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.1e+160], N[(2.0 / N[(N[(1.0 + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[(N[Power[t, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision] / N[Power[N[Power[k$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if t < 4.89999999999999972e-76

   1. Initial program 47.6%

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

      1. associate-/r* 54.8%

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

      2. +-commutative 54.8%

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

      3. associate-+r+ 54.8%

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

      4. metadata-eval 54.8%

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

      5. associate-*r* 55.0%

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

      6. *-commutative 55.0%

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

      7. associate-*l/ 57.0%

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

      8. associate-*r/ 57.6%

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

   4. Applied egg-rr 57.6%

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

   5. Taylor expanded in k around inf 62.0%

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

      1. *-commutative 62.0%

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

      2. *-commutative 62.0%

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

      3. times-frac 62.5%

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

   7. Simplified 62.5%

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

   if 4.89999999999999972e-76 < t < 5.99999999999999986e101

   1. Initial program 79.2%

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

      1. associate-/r* 81.9%

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

      2. +-commutative 81.9%

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

      3. associate-+r+ 81.9%

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

      4. metadata-eval 81.9%

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

      5. associate-*r* 81.8%

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

      6. *-commutative 81.8%

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

      7. associate-*l/ 87.3%

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

      8. associate-*r/ 90.5%

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

   4. Applied egg-rr 90.5%

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

      1. clear-num 90.4%

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

      2. inv-pow 90.4%

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

      3. associate-*l* 90.5%

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

      4. associate-*l/ 92.4%

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

   6. Applied egg-rr 92.4%

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

      1. unpow-1 92.4%

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

      2. metadata-eval 92.4%

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

      3. associate-+r+ 92.4%

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

      4. *-commutative 92.4%

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

      5. associate-/l* 93.3%

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

      6. associate-+r+ 93.3%

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

      7. metadata-eval 93.3%

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

   8. Simplified 93.3%

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

   if 5.99999999999999986e101 < t < 5.1000000000000001e160

   1. Initial program 39.6%

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

      1. unpow3 39.6%

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

      2. times-frac 78.5%

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

      3. pow2 78.5%

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

   4. Applied egg-rr 78.5%

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

   if 5.1000000000000001e160 < t

   1. Initial program 67.5%

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

      1. associate-/r* 67.5%

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

      2. sqr-neg 67.5%

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

      3. associate-*l* 57.6%

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

      4. sqr-neg 57.6%

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

      5. associate-/r* 61.3%

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

      6. associate-+l+ 61.3%

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

      7. unpow2 61.3%

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

      8. times-frac 39.9%

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

      9. sqr-neg 39.9%

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

      10. times-frac 61.3%

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

      11. unpow2 61.3%

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

   3. Simplified 61.3%

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

   5. Taylor expanded in k around 0 57.6%

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

      1. *-commutative 57.6%

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

      2. associate-/r* 57.6%

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

   7. Simplified 57.6%

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

      1. unpow2 57.6%

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

   9. Applied egg-rr 57.6%

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

       1. expm1-log1p-u 57.6%

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

       2. expm1-udef 57.6%

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

       3. div-inv 57.6%

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

       4. pow-flip 57.6%

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

       5. metadata-eval 57.6%

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

   11. Applied egg-rr 57.6%

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

       1. expm1-def 57.6%

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

       2. expm1-log1p 57.6%

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

   13. Simplified 57.6%

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

       1. add-cube-cbrt 57.6%

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

       2. pow2 57.6%

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

       3. cbrt-div 57.6%

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

       4. metadata-eval 57.6%

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

       5. pow-flip 57.6%

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

       6. div-inv 57.6%

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

       7. cbrt-div 57.6%

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

       8. unpow2 57.6%

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

       9. cbrt-prod 57.6%

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

       10. unpow2 57.6%

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

       11. unpow3 57.6%

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

       12. add-cbrt-cube 57.6%

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

       13. cbrt-prod 57.6%

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

       14. pow2 57.6%

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

       15. cbrt-div 57.6%

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

   15. Applied egg-rr 89.0%

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

       1. pow-plus 89.0%

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

       2. metadata-eval 89.0%

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

   17. Simplified 89.0%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.9 \cdot 10^{-76}:\\ \;\;\;\;\frac{2}{\frac{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \frac{{k}^{2}}{\ell}}{\ell}}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+101}:\\ \;\;\;\;\frac{2}{\frac{1}{\frac{\ell}{\tan k \cdot \left(\frac{{t}^{3}}{\frac{\ell}{\sin k}} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}}\\ \mathbf{elif}\;t \leq 5.1 \cdot 10^{+160}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}}{{\left(\sqrt[3]{k}\right)}^{2}}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := {\left(\frac{k\_m}{t}\right)}^{2}\\ \mathbf{if}\;t \leq 3.2 \cdot 10^{-76}:\\ \;\;\;\;\frac{2}{\frac{\frac{t \cdot {\sin k\_m}^{2}}{\cos k\_m} \cdot \frac{{k\_m}^{2}}{\ell}}{\ell}}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\frac{1}{\frac{\ell}{\tan k\_m \cdot \left(\frac{{t}^{3}}{\frac{\ell}{\sin k\_m}} \cdot \left(2 + t\_1\right)\right)}}}\\ \mathbf{elif}\;t \leq 5.1 \cdot 10^{+160}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + t\_1\right)\right) \cdot \left(\tan k\_m \cdot \left(\sin k\_m \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot {k\_m}^{-0.6666666666666666}}{t}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (pow (/ k_m t) 2.0)))
   (if (<= t 3.2e-76)
     (/
      2.0
      (/ (* (/ (* t (pow (sin k_m) 2.0)) (cos k_m)) (/ (pow k_m 2.0) l)) l))
     (if (<= t 5.5e+102)
       (/
        2.0
        (/
         1.0
         (/ l (* (tan k_m) (* (/ (pow t 3.0) (/ l (sin k_m))) (+ 2.0 t_1))))))
       (if (<= t 5.1e+160)
         (/
          2.0
          (*
           (+ 1.0 (+ 1.0 t_1))
           (* (tan k_m) (* (sin k_m) (* (/ (pow t 2.0) l) (/ t l))))))
         (pow
          (/ (* (pow (cbrt l) 2.0) (pow k_m -0.6666666666666666)) t)
          3.0))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = pow((k_m / t), 2.0);
	double tmp;
	if (t <= 3.2e-76) {
		tmp = 2.0 / ((((t * pow(sin(k_m), 2.0)) / cos(k_m)) * (pow(k_m, 2.0) / l)) / l);
	} else if (t <= 5.5e+102) {
		tmp = 2.0 / (1.0 / (l / (tan(k_m) * ((pow(t, 3.0) / (l / sin(k_m))) * (2.0 + t_1)))));
	} else if (t <= 5.1e+160) {
		tmp = 2.0 / ((1.0 + (1.0 + t_1)) * (tan(k_m) * (sin(k_m) * ((pow(t, 2.0) / l) * (t / l)))));
	} else {
		tmp = pow(((pow(cbrt(l), 2.0) * pow(k_m, -0.6666666666666666)) / t), 3.0);
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = Math.pow((k_m / t), 2.0);
	double tmp;
	if (t <= 3.2e-76) {
		tmp = 2.0 / ((((t * Math.pow(Math.sin(k_m), 2.0)) / Math.cos(k_m)) * (Math.pow(k_m, 2.0) / l)) / l);
	} else if (t <= 5.5e+102) {
		tmp = 2.0 / (1.0 / (l / (Math.tan(k_m) * ((Math.pow(t, 3.0) / (l / Math.sin(k_m))) * (2.0 + t_1)))));
	} else if (t <= 5.1e+160) {
		tmp = 2.0 / ((1.0 + (1.0 + t_1)) * (Math.tan(k_m) * (Math.sin(k_m) * ((Math.pow(t, 2.0) / l) * (t / l)))));
	} else {
		tmp = Math.pow(((Math.pow(Math.cbrt(l), 2.0) * Math.pow(k_m, -0.6666666666666666)) / t), 3.0);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(k_m / t) ^ 2.0
	tmp = 0.0
	if (t <= 3.2e-76)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t * (sin(k_m) ^ 2.0)) / cos(k_m)) * Float64((k_m ^ 2.0) / l)) / l));
	elseif (t <= 5.5e+102)
		tmp = Float64(2.0 / Float64(1.0 / Float64(l / Float64(tan(k_m) * Float64(Float64((t ^ 3.0) / Float64(l / sin(k_m))) * Float64(2.0 + t_1))))));
	elseif (t <= 5.1e+160)
		tmp = Float64(2.0 / Float64(Float64(1.0 + Float64(1.0 + t_1)) * Float64(tan(k_m) * Float64(sin(k_m) * Float64(Float64((t ^ 2.0) / l) * Float64(t / l))))));
	else
		tmp = Float64(Float64((cbrt(l) ^ 2.0) * (k_m ^ -0.6666666666666666)) / t) ^ 3.0;
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t, 3.2e-76], N[(2.0 / N[(N[(N[(N[(t * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[Power[k$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.5e+102], N[(2.0 / N[(1.0 / N[(l / N[(N[Tan[k$95$m], $MachinePrecision] * N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.1e+160], N[(2.0 / N[(N[(1.0 + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[(N[Power[t, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[k$95$m, -0.6666666666666666], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], 3.0], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if t < 3.1999999999999998e-76

   1. Initial program 47.6%

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

      1. associate-/r* 54.8%

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

      2. +-commutative 54.8%

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

      3. associate-+r+ 54.8%

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

      4. metadata-eval 54.8%

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

      5. associate-*r* 55.0%

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

      6. *-commutative 55.0%

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

      7. associate-*l/ 57.0%

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

      8. associate-*r/ 57.6%

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

   4. Applied egg-rr 57.6%

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

   5. Taylor expanded in k around inf 62.0%

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

      1. *-commutative 62.0%

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

      2. *-commutative 62.0%

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

      3. times-frac 62.5%

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

   7. Simplified 62.5%

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

   if 3.1999999999999998e-76 < t < 5.49999999999999981e102

   1. Initial program 79.2%

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

      1. associate-/r* 81.9%

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

      2. +-commutative 81.9%

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

      3. associate-+r+ 81.9%

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

      4. metadata-eval 81.9%

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

      5. associate-*r* 81.8%

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

      6. *-commutative 81.8%

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

      7. associate-*l/ 87.3%

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

      8. associate-*r/ 90.5%

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

   4. Applied egg-rr 90.5%

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

      1. clear-num 90.4%

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

      2. inv-pow 90.4%

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

      3. associate-*l* 90.5%

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

      4. associate-*l/ 92.4%

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

   6. Applied egg-rr 92.4%

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

      1. unpow-1 92.4%

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

      2. metadata-eval 92.4%

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

      3. associate-+r+ 92.4%

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

      4. *-commutative 92.4%

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

      5. associate-/l* 93.3%

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

      6. associate-+r+ 93.3%

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

      7. metadata-eval 93.3%

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

   8. Simplified 93.3%

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

   if 5.49999999999999981e102 < t < 5.1000000000000001e160

   1. Initial program 39.6%

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

      1. unpow3 39.6%

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

      2. times-frac 78.5%

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

      3. pow2 78.5%

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

   4. Applied egg-rr 78.5%

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

   if 5.1000000000000001e160 < t

   1. Initial program 67.5%

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

      1. associate-/r* 67.5%

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

      2. sqr-neg 67.5%

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

      3. associate-*l* 57.6%

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

      4. sqr-neg 57.6%

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

      5. associate-/r* 61.3%

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

      6. associate-+l+ 61.3%

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

      7. unpow2 61.3%

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

      8. times-frac 39.9%

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

      9. sqr-neg 39.9%

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

      10. times-frac 61.3%

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

      11. unpow2 61.3%

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

   3. Simplified 61.3%

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

   5. Taylor expanded in k around 0 57.6%

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

      1. *-commutative 57.6%

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

      2. associate-/r* 57.6%

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

   7. Simplified 57.6%

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

      1. add-cube-cbrt 57.6%

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

      2. pow2 57.6%

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

      3. div-inv 54.5%

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

      4. cbrt-prod 54.5%

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

      5. cbrt-div 54.5%

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

      6. unpow2 54.5%

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

      7. cbrt-prod 54.5%

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

      8. unpow2 54.5%

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

      9. unpow3 54.5%

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

      10. add-cbrt-cube 54.5%

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

      11. pow-flip 54.5%

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

      12. metadata-eval 54.5%

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

      13. div-inv 54.5%

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

   9. Applied egg-rr 62.9%

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

       1. pow-plus 62.9%

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

       2. metadata-eval 62.9%

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

       3. associate-*l/ 62.9%

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

   11. Simplified 62.9%

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

       1. pow1/3 62.9%

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

       2. pow-pow 50.1%

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

       3. metadata-eval 50.1%

          \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot {k}^{\color{blue}{-0.6666666666666666}}}{t}\right)}^{3} \]

   13. Applied egg-rr 50.1%

       \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \color{blue}{{k}^{-0.6666666666666666}}}{t}\right)}^{3} \]
3. Recombined 4 regimes into one program.

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.2 \cdot 10^{-76}:\\ \;\;\;\;\frac{2}{\frac{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \frac{{k}^{2}}{\ell}}{\ell}}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\frac{1}{\frac{\ell}{\tan k \cdot \left(\frac{{t}^{3}}{\frac{\ell}{\sin k}} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}}\\ \mathbf{elif}\;t \leq 5.1 \cdot 10^{+160}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot {k}^{-0.6666666666666666}}{t}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 1.65 \cdot 10^{-78}:\\ \;\;\;\;\frac{2}{\frac{\frac{t \cdot {\sin k\_m}^{2}}{\cos k\_m} \cdot \frac{{k\_m}^{2}}{\ell}}{\ell}}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\frac{1}{\frac{\ell}{\tan k\_m \cdot \left(\frac{{t}^{3}}{\frac{\ell}{\sin k\_m}} \cdot \left(2 + {\left(\frac{k\_m}{t}\right)}^{2}\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot {k\_m}^{-0.6666666666666666}}{t}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 1.65e-78)
   (/
    2.0
    (/ (* (/ (* t (pow (sin k_m) 2.0)) (cos k_m)) (/ (pow k_m 2.0) l)) l))
   (if (<= t 5.5e+102)
     (/
      2.0
      (/
       1.0
       (/
        l
        (*
         (tan k_m)
         (* (/ (pow t 3.0) (/ l (sin k_m))) (+ 2.0 (pow (/ k_m t) 2.0)))))))
     (pow (/ (* (pow (cbrt l) 2.0) (pow k_m -0.6666666666666666)) t) 3.0))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 1.65e-78) {
		tmp = 2.0 / ((((t * pow(sin(k_m), 2.0)) / cos(k_m)) * (pow(k_m, 2.0) / l)) / l);
	} else if (t <= 5.5e+102) {
		tmp = 2.0 / (1.0 / (l / (tan(k_m) * ((pow(t, 3.0) / (l / sin(k_m))) * (2.0 + pow((k_m / t), 2.0))))));
	} else {
		tmp = pow(((pow(cbrt(l), 2.0) * pow(k_m, -0.6666666666666666)) / t), 3.0);
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 1.65e-78) {
		tmp = 2.0 / ((((t * Math.pow(Math.sin(k_m), 2.0)) / Math.cos(k_m)) * (Math.pow(k_m, 2.0) / l)) / l);
	} else if (t <= 5.5e+102) {
		tmp = 2.0 / (1.0 / (l / (Math.tan(k_m) * ((Math.pow(t, 3.0) / (l / Math.sin(k_m))) * (2.0 + Math.pow((k_m / t), 2.0))))));
	} else {
		tmp = Math.pow(((Math.pow(Math.cbrt(l), 2.0) * Math.pow(k_m, -0.6666666666666666)) / t), 3.0);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 1.65e-78)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t * (sin(k_m) ^ 2.0)) / cos(k_m)) * Float64((k_m ^ 2.0) / l)) / l));
	elseif (t <= 5.5e+102)
		tmp = Float64(2.0 / Float64(1.0 / Float64(l / Float64(tan(k_m) * Float64(Float64((t ^ 3.0) / Float64(l / sin(k_m))) * Float64(2.0 + (Float64(k_m / t) ^ 2.0)))))));
	else
		tmp = Float64(Float64((cbrt(l) ^ 2.0) * (k_m ^ -0.6666666666666666)) / t) ^ 3.0;
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 1.65e-78], N[(2.0 / N[(N[(N[(N[(t * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[Power[k$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.5e+102], N[(2.0 / N[(1.0 / N[(l / N[(N[Tan[k$95$m], $MachinePrecision] * N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[k$95$m, -0.6666666666666666], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], 3.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.64999999999999991e-78

   1. Initial program 47.6%

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

      1. associate-/r* 54.8%

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

      2. +-commutative 54.8%

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

      3. associate-+r+ 54.8%

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

      4. metadata-eval 54.8%

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

      5. associate-*r* 55.0%

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

      6. *-commutative 55.0%

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

      7. associate-*l/ 57.0%

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

      8. associate-*r/ 57.6%

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

   4. Applied egg-rr 57.6%

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

   5. Taylor expanded in k around inf 62.0%

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

      1. *-commutative 62.0%

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

      2. *-commutative 62.0%

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

      3. times-frac 62.5%

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

   7. Simplified 62.5%

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

   if 1.64999999999999991e-78 < t < 5.49999999999999981e102

   1. Initial program 79.2%

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

      1. associate-/r* 81.9%

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

      2. +-commutative 81.9%

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

      3. associate-+r+ 81.9%

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

      4. metadata-eval 81.9%

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

      5. associate-*r* 81.8%

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

      6. *-commutative 81.8%

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

      7. associate-*l/ 87.3%

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

      8. associate-*r/ 90.5%

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

   4. Applied egg-rr 90.5%

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

      1. clear-num 90.4%

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

      2. inv-pow 90.4%

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

      3. associate-*l* 90.5%

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

      4. associate-*l/ 92.4%

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

   6. Applied egg-rr 92.4%

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

      1. unpow-1 92.4%

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

      2. metadata-eval 92.4%

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

      3. associate-+r+ 92.4%

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

      4. *-commutative 92.4%

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

      5. associate-/l* 93.3%

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

      6. associate-+r+ 93.3%

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

      7. metadata-eval 93.3%

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

   8. Simplified 93.3%

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

   if 5.49999999999999981e102 < t

   1. Initial program 57.6%

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

      1. associate-/r* 57.6%

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

      2. sqr-neg 57.6%

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

      3. associate-*l* 49.3%

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

      4. sqr-neg 49.3%

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

      5. associate-/r* 52.3%

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

      6. associate-+l+ 52.3%

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

      7. unpow2 52.3%

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

      8. times-frac 38.3%

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

      9. sqr-neg 38.3%

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

      10. times-frac 52.3%

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

      11. unpow2 52.3%

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

   3. Simplified 52.3%

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

   5. Taylor expanded in k around 0 49.3%

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

      1. *-commutative 49.3%

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

      2. associate-/r* 49.3%

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

   7. Simplified 49.3%

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

      1. add-cube-cbrt 49.3%

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

      2. pow2 49.3%

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

      3. div-inv 47.3%

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

      4. cbrt-prod 47.3%

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

      5. cbrt-div 47.3%

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

      6. unpow2 47.3%

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

      7. cbrt-prod 47.3%

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

      8. unpow2 47.3%

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

      9. unpow3 47.3%

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

      10. add-cbrt-cube 47.3%

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

      11. pow-flip 47.3%

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

      12. metadata-eval 47.3%

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

      13. div-inv 47.3%

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

   9. Applied egg-rr 61.0%

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

       1. pow-plus 61.0%

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

       2. metadata-eval 61.0%

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

       3. associate-*l/ 61.1%

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

   11. Simplified 61.1%

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

       1. pow1/3 60.9%

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

       2. pow-pow 48.2%

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

       3. metadata-eval 48.2%

          \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot {k}^{\color{blue}{-0.6666666666666666}}}{t}\right)}^{3} \]

   13. Applied egg-rr 48.2%

       \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \color{blue}{{k}^{-0.6666666666666666}}}{t}\right)}^{3} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.65 \cdot 10^{-78}:\\ \;\;\;\;\frac{2}{\frac{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \frac{{k}^{2}}{\ell}}{\ell}}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\frac{1}{\frac{\ell}{\tan k \cdot \left(\frac{{t}^{3}}{\frac{\ell}{\sin k}} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot {k}^{-0.6666666666666666}}{t}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 74.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 5.4 \cdot 10^{-78}:\\ \;\;\;\;\frac{2}{\frac{\frac{t \cdot {\sin k\_m}^{2}}{\cos k\_m} \cdot \frac{{k\_m}^{2}}{\ell}}{\ell}}\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{+95}:\\ \;\;\;\;\frac{2}{\frac{\left(\tan k\_m \cdot \left(2 + {\left(\frac{k\_m}{t}\right)}^{2}\right)\right) \cdot \left(\sin k\_m \cdot \frac{{t}^{3}}{\ell}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot {k\_m}^{-0.6666666666666666}}{t}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 5.4e-78)
   (/
    2.0
    (/ (* (/ (* t (pow (sin k_m) 2.0)) (cos k_m)) (/ (pow k_m 2.0) l)) l))
   (if (<= t 3.4e+95)
     (/
      2.0
      (/
       (*
        (* (tan k_m) (+ 2.0 (pow (/ k_m t) 2.0)))
        (* (sin k_m) (/ (pow t 3.0) l)))
       l))
     (pow (/ (* (pow (cbrt l) 2.0) (pow k_m -0.6666666666666666)) t) 3.0))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 5.4e-78) {
		tmp = 2.0 / ((((t * pow(sin(k_m), 2.0)) / cos(k_m)) * (pow(k_m, 2.0) / l)) / l);
	} else if (t <= 3.4e+95) {
		tmp = 2.0 / (((tan(k_m) * (2.0 + pow((k_m / t), 2.0))) * (sin(k_m) * (pow(t, 3.0) / l))) / l);
	} else {
		tmp = pow(((pow(cbrt(l), 2.0) * pow(k_m, -0.6666666666666666)) / t), 3.0);
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 5.4e-78) {
		tmp = 2.0 / ((((t * Math.pow(Math.sin(k_m), 2.0)) / Math.cos(k_m)) * (Math.pow(k_m, 2.0) / l)) / l);
	} else if (t <= 3.4e+95) {
		tmp = 2.0 / (((Math.tan(k_m) * (2.0 + Math.pow((k_m / t), 2.0))) * (Math.sin(k_m) * (Math.pow(t, 3.0) / l))) / l);
	} else {
		tmp = Math.pow(((Math.pow(Math.cbrt(l), 2.0) * Math.pow(k_m, -0.6666666666666666)) / t), 3.0);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 5.4e-78)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t * (sin(k_m) ^ 2.0)) / cos(k_m)) * Float64((k_m ^ 2.0) / l)) / l));
	elseif (t <= 3.4e+95)
		tmp = Float64(2.0 / Float64(Float64(Float64(tan(k_m) * Float64(2.0 + (Float64(k_m / t) ^ 2.0))) * Float64(sin(k_m) * Float64((t ^ 3.0) / l))) / l));
	else
		tmp = Float64(Float64((cbrt(l) ^ 2.0) * (k_m ^ -0.6666666666666666)) / t) ^ 3.0;
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 5.4e-78], N[(2.0 / N[(N[(N[(N[(t * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[Power[k$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.4e+95], N[(2.0 / N[(N[(N[(N[Tan[k$95$m], $MachinePrecision] * N[(2.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[k$95$m, -0.6666666666666666], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], 3.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 5.39999999999999987e-78

   1. Initial program 47.6%

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

      1. associate-/r* 54.8%

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

      2. +-commutative 54.8%

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

      3. associate-+r+ 54.8%

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

      4. metadata-eval 54.8%

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

      5. associate-*r* 55.0%

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

      6. *-commutative 55.0%

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

      7. associate-*l/ 57.0%

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

      8. associate-*r/ 57.6%

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

   4. Applied egg-rr 57.6%

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

   5. Taylor expanded in k around inf 62.0%

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

      1. *-commutative 62.0%

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

      2. *-commutative 62.0%

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

      3. times-frac 62.5%

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

   7. Simplified 62.5%

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

   if 5.39999999999999987e-78 < t < 3.40000000000000022e95

   1. Initial program 81.5%

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

      1. associate-/r* 84.2%

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

      2. +-commutative 84.2%

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

      3. associate-+r+ 84.2%

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

      4. metadata-eval 84.2%

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

      5. associate-*r* 84.1%

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

      6. *-commutative 84.1%

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

      7. associate-*l/ 89.8%

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

      8. associate-*r/ 93.0%

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

   4. Applied egg-rr 93.0%

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

   if 3.40000000000000022e95 < t

   1. Initial program 56.6%

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

      1. associate-/r* 56.6%

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

      2. sqr-neg 56.6%

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

      3. associate-*l* 48.3%

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

      4. sqr-neg 48.3%

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

      5. associate-/r* 51.3%

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

      6. associate-+l+ 51.3%

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

      7. unpow2 51.3%

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

      8. times-frac 37.6%

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

      9. sqr-neg 37.6%

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

      10. times-frac 51.3%

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

      11. unpow2 51.3%

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

   3. Simplified 51.3%

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

   5. Taylor expanded in k around 0 48.4%

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

      1. *-commutative 48.4%

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

      2. associate-/r* 48.3%

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

   7. Simplified 48.3%

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

      1. add-cube-cbrt 48.3%

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

      2. pow2 48.3%

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

      3. div-inv 46.4%

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

      4. cbrt-prod 46.4%

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

      5. cbrt-div 46.4%

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

      6. unpow2 46.4%

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

      7. cbrt-prod 46.4%

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

      8. unpow2 46.4%

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

      9. unpow3 46.4%

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

      10. add-cbrt-cube 46.4%

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

      11. pow-flip 46.4%

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

      12. metadata-eval 46.4%

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

      13. div-inv 46.4%

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

   9. Applied egg-rr 59.9%

      \[\leadsto \color{blue}{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t} \cdot \sqrt[3]{{k}^{-2}}\right)}^{2} \cdot \left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t} \cdot \sqrt[3]{{k}^{-2}}\right)} \]
   10. Step-by-step derivation

       1. pow-plus 59.9%

          \[\leadsto \color{blue}{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t} \cdot \sqrt[3]{{k}^{-2}}\right)}^{\left(2 + 1\right)}} \]

       2. metadata-eval 59.9%

          \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t} \cdot \sqrt[3]{{k}^{-2}}\right)}^{\color{blue}{3}} \]

       3. associate-*l/ 59.9%

          \[\leadsto {\color{blue}{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{{k}^{-2}}}{t}\right)}}^{3} \]

   11. Simplified 59.9%

       \[\leadsto \color{blue}{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{{k}^{-2}}}{t}\right)}^{3}} \]
   12. Step-by-step derivation

       1. pow1/3 59.8%

          \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \color{blue}{{\left({k}^{-2}\right)}^{0.3333333333333333}}}{t}\right)}^{3} \]

       2. pow-pow 48.9%

          \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \color{blue}{{k}^{\left(-2 \cdot 0.3333333333333333\right)}}}{t}\right)}^{3} \]

       3. metadata-eval 48.9%

          \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot {k}^{\color{blue}{-0.6666666666666666}}}{t}\right)}^{3} \]

   13. Applied egg-rr 48.9%

       \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \color{blue}{{k}^{-0.6666666666666666}}}{t}\right)}^{3} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.4 \cdot 10^{-78}:\\ \;\;\;\;\frac{2}{\frac{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \frac{{k}^{2}}{\ell}}{\ell}}\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{+95}:\\ \;\;\;\;\frac{2}{\frac{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot {k}^{-0.6666666666666666}}{t}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 72.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 7.8 \cdot 10^{+39}:\\ \;\;\;\;\frac{2}{\frac{\frac{t \cdot {\sin k\_m}^{2}}{\cos k\_m} \cdot \frac{{k\_m}^{2}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot {k\_m}^{-0.6666666666666666}}{t}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 7.8e+39)
   (/
    2.0
    (/ (* (/ (* t (pow (sin k_m) 2.0)) (cos k_m)) (/ (pow k_m 2.0) l)) l))
   (pow (/ (* (pow (cbrt l) 2.0) (pow k_m -0.6666666666666666)) t) 3.0)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 7.8e+39) {
		tmp = 2.0 / ((((t * pow(sin(k_m), 2.0)) / cos(k_m)) * (pow(k_m, 2.0) / l)) / l);
	} else {
		tmp = pow(((pow(cbrt(l), 2.0) * pow(k_m, -0.6666666666666666)) / t), 3.0);
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 7.8e+39) {
		tmp = 2.0 / ((((t * Math.pow(Math.sin(k_m), 2.0)) / Math.cos(k_m)) * (Math.pow(k_m, 2.0) / l)) / l);
	} else {
		tmp = Math.pow(((Math.pow(Math.cbrt(l), 2.0) * Math.pow(k_m, -0.6666666666666666)) / t), 3.0);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 7.8e+39)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t * (sin(k_m) ^ 2.0)) / cos(k_m)) * Float64((k_m ^ 2.0) / l)) / l));
	else
		tmp = Float64(Float64((cbrt(l) ^ 2.0) * (k_m ^ -0.6666666666666666)) / t) ^ 3.0;
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 7.8e+39], N[(2.0 / N[(N[(N[(N[(t * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[Power[k$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[k$95$m, -0.6666666666666666], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], 3.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 7.8000000000000002e39

   1. Initial program 51.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/r* 58.3%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. +-commutative 58.3%

         \[\leadsto \frac{2}{\left(\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

      3. associate-+r+ 58.3%

         \[\leadsto \frac{2}{\left(\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}} \]

      4. metadata-eval 58.3%

         \[\leadsto \frac{2}{\left(\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)} \]

      5. associate-*r* 58.5%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

      6. *-commutative 58.5%

         \[\leadsto \frac{2}{\color{blue}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right)}} \]

      7. associate-*l/ 61.2%

         \[\leadsto \frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell}}} \]

      8. associate-*r/ 61.8%

         \[\leadsto \frac{2}{\color{blue}{\frac{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\frac{{t}^{3}}{\ell} \cdot \sin k\right)}{\ell}}} \]

   4. Applied egg-rr 61.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\frac{{t}^{3}}{\ell} \cdot \sin k\right)}{\ell}}} \]

   5. Taylor expanded in k around inf 64.2%

      \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}}{\ell}} \]
   6. Step-by-step derivation

      1. *-commutative 64.2%

         \[\leadsto \frac{2}{\frac{\frac{\color{blue}{\left(t \cdot {\sin k}^{2}\right) \cdot {k}^{2}}}{\ell \cdot \cos k}}{\ell}} \]

      2. *-commutative 64.2%

         \[\leadsto \frac{2}{\frac{\frac{\left(t \cdot {\sin k}^{2}\right) \cdot {k}^{2}}{\color{blue}{\cos k \cdot \ell}}}{\ell}} \]

      3. times-frac 64.6%

         \[\leadsto \frac{2}{\frac{\color{blue}{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \frac{{k}^{2}}{\ell}}}{\ell}} \]

   7. Simplified 64.6%

      \[\leadsto \frac{2}{\frac{\color{blue}{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \frac{{k}^{2}}{\ell}}}{\ell}} \]

   if 7.8000000000000002e39 < t

   1. Initial program 61.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 61.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. sqr-neg 61.4%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. associate-*l* 52.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. sqr-neg 52.7%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. associate-/r* 55.3%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. associate-+l+ 55.3%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{\color{blue}{1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)}} \]

      7. unpow2 55.3%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right)} \]

      8. times-frac 43.5%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k \cdot k}{t \cdot t}} + 1\right)} \]

      9. sqr-neg 43.5%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}} + 1\right)} \]

      10. times-frac 55.3%

          \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}} + 1\right)} \]

      11. unpow2 55.3%

          \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{{\left(\frac{k}{-t}\right)}^{2}} + 1\right)} \]

   3. Simplified 55.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 51.1%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. *-commutative 51.1%

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      2. associate-/r* 52.7%

         \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]

   7. Simplified 52.7%

      \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]
   8. Step-by-step derivation

      1. add-cube-cbrt 52.6%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \cdot \sqrt[3]{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}}\right) \cdot \sqrt[3]{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}}} \]

      2. pow2 52.6%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}}\right)}^{2}} \cdot \sqrt[3]{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]

      3. div-inv 51.0%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\frac{{\ell}^{2}}{{t}^{3}} \cdot \frac{1}{{k}^{2}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]

      4. cbrt-prod 50.9%

         \[\leadsto {\color{blue}{\left(\sqrt[3]{\frac{{\ell}^{2}}{{t}^{3}}} \cdot \sqrt[3]{\frac{1}{{k}^{2}}}\right)}}^{2} \cdot \sqrt[3]{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]

      5. cbrt-div 50.9%

         \[\leadsto {\left(\color{blue}{\frac{\sqrt[3]{{\ell}^{2}}}{\sqrt[3]{{t}^{3}}}} \cdot \sqrt[3]{\frac{1}{{k}^{2}}}\right)}^{2} \cdot \sqrt[3]{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]

      6. unpow2 50.9%

         \[\leadsto {\left(\frac{\sqrt[3]{\color{blue}{\ell \cdot \ell}}}{\sqrt[3]{{t}^{3}}} \cdot \sqrt[3]{\frac{1}{{k}^{2}}}\right)}^{2} \cdot \sqrt[3]{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]

      7. cbrt-prod 50.9%

         \[\leadsto {\left(\frac{\color{blue}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}{\sqrt[3]{{t}^{3}}} \cdot \sqrt[3]{\frac{1}{{k}^{2}}}\right)}^{2} \cdot \sqrt[3]{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]

      8. unpow2 50.9%

         \[\leadsto {\left(\frac{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}}{\sqrt[3]{{t}^{3}}} \cdot \sqrt[3]{\frac{1}{{k}^{2}}}\right)}^{2} \cdot \sqrt[3]{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]

      9. unpow3 50.9%

         \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{\sqrt[3]{\color{blue}{\left(t \cdot t\right) \cdot t}}} \cdot \sqrt[3]{\frac{1}{{k}^{2}}}\right)}^{2} \cdot \sqrt[3]{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]

      10. add-cbrt-cube 50.9%

          \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{\color{blue}{t}} \cdot \sqrt[3]{\frac{1}{{k}^{2}}}\right)}^{2} \cdot \sqrt[3]{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]

      11. pow-flip 50.9%

          \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t} \cdot \sqrt[3]{\color{blue}{{k}^{\left(-2\right)}}}\right)}^{2} \cdot \sqrt[3]{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]

      12. metadata-eval 50.9%

          \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t} \cdot \sqrt[3]{{k}^{\color{blue}{-2}}}\right)}^{2} \cdot \sqrt[3]{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]

      13. div-inv 50.9%

          \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t} \cdot \sqrt[3]{{k}^{-2}}\right)}^{2} \cdot \sqrt[3]{\color{blue}{\frac{{\ell}^{2}}{{t}^{3}} \cdot \frac{1}{{k}^{2}}}} \]

   9. Applied egg-rr 60.9%

      \[\leadsto \color{blue}{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t} \cdot \sqrt[3]{{k}^{-2}}\right)}^{2} \cdot \left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t} \cdot \sqrt[3]{{k}^{-2}}\right)} \]
   10. Step-by-step derivation

       1. pow-plus 60.9%

          \[\leadsto \color{blue}{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t} \cdot \sqrt[3]{{k}^{-2}}\right)}^{\left(2 + 1\right)}} \]

       2. metadata-eval 60.9%

          \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t} \cdot \sqrt[3]{{k}^{-2}}\right)}^{\color{blue}{3}} \]

       3. associate-*l/ 60.9%

          \[\leadsto {\color{blue}{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{{k}^{-2}}}{t}\right)}}^{3} \]

   11. Simplified 60.9%

       \[\leadsto \color{blue}{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{{k}^{-2}}}{t}\right)}^{3}} \]
   12. Step-by-step derivation

       1. pow1/3 60.6%

          \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \color{blue}{{\left({k}^{-2}\right)}^{0.3333333333333333}}}{t}\right)}^{3} \]

       2. pow-pow 44.9%

          \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \color{blue}{{k}^{\left(-2 \cdot 0.3333333333333333\right)}}}{t}\right)}^{3} \]

       3. metadata-eval 44.9%

          \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot {k}^{\color{blue}{-0.6666666666666666}}}{t}\right)}^{3} \]

   13. Applied egg-rr 44.9%

       \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \color{blue}{{k}^{-0.6666666666666666}}}{t}\right)}^{3} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.8 \cdot 10^{+39}:\\ \;\;\;\;\frac{2}{\frac{\frac{t \cdot {\sin k}^{2}}{\cos k} \cdot \frac{{k}^{2}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot {k}^{-0.6666666666666666}}{t}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 81.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 9000000000000:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot {k\_m}^{-0.6666666666666666}}{t}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(t \cdot {\sin k\_m}^{2}\right) \cdot {k\_m}^{2}}{\cos k\_m \cdot \ell}}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 9000000000000.0)
   (pow (/ (* (pow (cbrt l) 2.0) (pow k_m -0.6666666666666666)) t) 3.0)
   (/
    2.0
    (/ (/ (* (* t (pow (sin k_m) 2.0)) (pow k_m 2.0)) (* (cos k_m) l)) l))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 9000000000000.0) {
		tmp = pow(((pow(cbrt(l), 2.0) * pow(k_m, -0.6666666666666666)) / t), 3.0);
	} else {
		tmp = 2.0 / ((((t * pow(sin(k_m), 2.0)) * pow(k_m, 2.0)) / (cos(k_m) * l)) / l);
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 9000000000000.0) {
		tmp = Math.pow(((Math.pow(Math.cbrt(l), 2.0) * Math.pow(k_m, -0.6666666666666666)) / t), 3.0);
	} else {
		tmp = 2.0 / ((((t * Math.pow(Math.sin(k_m), 2.0)) * Math.pow(k_m, 2.0)) / (Math.cos(k_m) * l)) / l);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 9000000000000.0)
		tmp = Float64(Float64((cbrt(l) ^ 2.0) * (k_m ^ -0.6666666666666666)) / t) ^ 3.0;
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t * (sin(k_m) ^ 2.0)) * (k_m ^ 2.0)) / Float64(cos(k_m) * l)) / l));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 9000000000000.0], N[Power[N[(N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[k$95$m, -0.6666666666666666], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], 3.0], $MachinePrecision], N[(2.0 / N[(N[(N[(N[(t * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 9e12

   1. Initial program 56.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 56.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. sqr-neg 56.9%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. associate-*l* 51.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. sqr-neg 51.5%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. associate-/r* 57.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. associate-+l+ 57.7%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{\color{blue}{1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)}} \]

      7. unpow2 57.7%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right)} \]

      8. times-frac 45.1%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k \cdot k}{t \cdot t}} + 1\right)} \]

      9. sqr-neg 45.1%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}} + 1\right)} \]

      10. times-frac 57.7%

          \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}} + 1\right)} \]

      11. unpow2 57.7%

          \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{{\left(\frac{k}{-t}\right)}^{2}} + 1\right)} \]

   3. Simplified 57.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 49.7%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. *-commutative 49.7%

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      2. associate-/r* 49.3%

         \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]

   7. Simplified 49.3%

      \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]
   8. Step-by-step derivation

      1. add-cube-cbrt 49.2%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \cdot \sqrt[3]{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}}\right) \cdot \sqrt[3]{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}}} \]

      2. pow2 49.2%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}}\right)}^{2}} \cdot \sqrt[3]{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]

      3. div-inv 48.7%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\frac{{\ell}^{2}}{{t}^{3}} \cdot \frac{1}{{k}^{2}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]

      4. cbrt-prod 48.7%

         \[\leadsto {\color{blue}{\left(\sqrt[3]{\frac{{\ell}^{2}}{{t}^{3}}} \cdot \sqrt[3]{\frac{1}{{k}^{2}}}\right)}}^{2} \cdot \sqrt[3]{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]

      5. cbrt-div 48.7%

         \[\leadsto {\left(\color{blue}{\frac{\sqrt[3]{{\ell}^{2}}}{\sqrt[3]{{t}^{3}}}} \cdot \sqrt[3]{\frac{1}{{k}^{2}}}\right)}^{2} \cdot \sqrt[3]{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]

      6. unpow2 48.7%

         \[\leadsto {\left(\frac{\sqrt[3]{\color{blue}{\ell \cdot \ell}}}{\sqrt[3]{{t}^{3}}} \cdot \sqrt[3]{\frac{1}{{k}^{2}}}\right)}^{2} \cdot \sqrt[3]{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]

      7. cbrt-prod 48.7%

         \[\leadsto {\left(\frac{\color{blue}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}{\sqrt[3]{{t}^{3}}} \cdot \sqrt[3]{\frac{1}{{k}^{2}}}\right)}^{2} \cdot \sqrt[3]{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]

      8. unpow2 48.7%

         \[\leadsto {\left(\frac{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}}{\sqrt[3]{{t}^{3}}} \cdot \sqrt[3]{\frac{1}{{k}^{2}}}\right)}^{2} \cdot \sqrt[3]{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]

      9. unpow3 48.7%

         \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{\sqrt[3]{\color{blue}{\left(t \cdot t\right) \cdot t}}} \cdot \sqrt[3]{\frac{1}{{k}^{2}}}\right)}^{2} \cdot \sqrt[3]{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]

      10. add-cbrt-cube 48.7%

          \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{\color{blue}{t}} \cdot \sqrt[3]{\frac{1}{{k}^{2}}}\right)}^{2} \cdot \sqrt[3]{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]

      11. pow-flip 48.7%

          \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t} \cdot \sqrt[3]{\color{blue}{{k}^{\left(-2\right)}}}\right)}^{2} \cdot \sqrt[3]{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]

      12. metadata-eval 48.7%

          \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t} \cdot \sqrt[3]{{k}^{\color{blue}{-2}}}\right)}^{2} \cdot \sqrt[3]{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]

      13. div-inv 48.7%

          \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t} \cdot \sqrt[3]{{k}^{-2}}\right)}^{2} \cdot \sqrt[3]{\color{blue}{\frac{{\ell}^{2}}{{t}^{3}} \cdot \frac{1}{{k}^{2}}}} \]

   9. Applied egg-rr 64.6%

      \[\leadsto \color{blue}{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t} \cdot \sqrt[3]{{k}^{-2}}\right)}^{2} \cdot \left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t} \cdot \sqrt[3]{{k}^{-2}}\right)} \]
   10. Step-by-step derivation

       1. pow-plus 64.6%

          \[\leadsto \color{blue}{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t} \cdot \sqrt[3]{{k}^{-2}}\right)}^{\left(2 + 1\right)}} \]

       2. metadata-eval 64.6%

          \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t} \cdot \sqrt[3]{{k}^{-2}}\right)}^{\color{blue}{3}} \]

       3. associate-*l/ 64.6%

          \[\leadsto {\color{blue}{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{{k}^{-2}}}{t}\right)}}^{3} \]

   11. Simplified 64.6%

       \[\leadsto \color{blue}{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{{k}^{-2}}}{t}\right)}^{3}} \]
   12. Step-by-step derivation

       1. pow1/3 64.1%

          \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \color{blue}{{\left({k}^{-2}\right)}^{0.3333333333333333}}}{t}\right)}^{3} \]

       2. pow-pow 34.2%

          \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \color{blue}{{k}^{\left(-2 \cdot 0.3333333333333333\right)}}}{t}\right)}^{3} \]

       3. metadata-eval 34.2%

          \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot {k}^{\color{blue}{-0.6666666666666666}}}{t}\right)}^{3} \]

   13. Applied egg-rr 34.2%

       \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \color{blue}{{k}^{-0.6666666666666666}}}{t}\right)}^{3} \]

   if 9e12 < k

   1. Initial program 43.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/r* 48.0%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. +-commutative 48.0%

         \[\leadsto \frac{2}{\left(\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

      3. associate-+r+ 48.0%

         \[\leadsto \frac{2}{\left(\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}} \]

      4. metadata-eval 48.0%

         \[\leadsto \frac{2}{\left(\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)} \]

      5. associate-*r* 48.0%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

      6. *-commutative 48.0%

         \[\leadsto \frac{2}{\color{blue}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right)}} \]

      7. associate-*l/ 48.0%

         \[\leadsto \frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell}}} \]

      8. associate-*r/ 49.7%

         \[\leadsto \frac{2}{\color{blue}{\frac{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\frac{{t}^{3}}{\ell} \cdot \sin k\right)}{\ell}}} \]

   4. Applied egg-rr 49.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\frac{{t}^{3}}{\ell} \cdot \sin k\right)}{\ell}}} \]

   5. Taylor expanded in k around inf 63.5%

      \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}}{\ell}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9000000000000:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot {k}^{-0.6666666666666666}}{t}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(t \cdot {\sin k}^{2}\right) \cdot {k}^{2}}{\cos k \cdot \ell}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 69.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot {k\_m}^{-0.6666666666666666}}{t}\right)}^{3} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (pow (/ (* (pow (cbrt l) 2.0) (pow k_m -0.6666666666666666)) t) 3.0))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return pow(((pow(cbrt(l), 2.0) * pow(k_m, -0.6666666666666666)) / t), 3.0);
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return Math.pow(((Math.pow(Math.cbrt(l), 2.0) * Math.pow(k_m, -0.6666666666666666)) / t), 3.0);
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64((cbrt(l) ^ 2.0) * (k_m ^ -0.6666666666666666)) / t) ^ 3.0
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[Power[N[(N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[k$95$m, -0.6666666666666666], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], 3.0], $MachinePrecision]

Derivation

1. Initial program 53.9%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/r* 54.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

   2. sqr-neg 54.1%

      \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. associate-*l* 49.9%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   4. sqr-neg 49.9%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   5. associate-/r* 55.6%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   6. associate-+l+ 55.6%

      \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{\color{blue}{1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)}} \]

   7. unpow2 55.6%

      \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right)} \]

   8. times-frac 43.9%

      \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k \cdot k}{t \cdot t}} + 1\right)} \]

   9. sqr-neg 43.9%

      \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}} + 1\right)} \]

   10. times-frac 55.6%

       \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}} + 1\right)} \]

   11. unpow2 55.6%

       \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{{\left(\frac{k}{-t}\right)}^{2}} + 1\right)} \]

3. Simplified 55.6%

   \[\leadsto \color{blue}{\frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
4. Add Preprocessing

5. Taylor expanded in k around 0 47.0%

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
6. Step-by-step derivation

   1. *-commutative 47.0%

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   2. associate-/r* 47.1%

      \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]

7. Simplified 47.1%

   \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]
8. Step-by-step derivation

   1. add-cube-cbrt 47.1%

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \cdot \sqrt[3]{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}}\right) \cdot \sqrt[3]{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}}} \]

   2. pow2 47.1%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}}\right)}^{2}} \cdot \sqrt[3]{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]

   3. div-inv 46.7%

      \[\leadsto {\left(\sqrt[3]{\color{blue}{\frac{{\ell}^{2}}{{t}^{3}} \cdot \frac{1}{{k}^{2}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]

   4. cbrt-prod 46.7%

      \[\leadsto {\color{blue}{\left(\sqrt[3]{\frac{{\ell}^{2}}{{t}^{3}}} \cdot \sqrt[3]{\frac{1}{{k}^{2}}}\right)}}^{2} \cdot \sqrt[3]{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]

   5. cbrt-div 46.7%

      \[\leadsto {\left(\color{blue}{\frac{\sqrt[3]{{\ell}^{2}}}{\sqrt[3]{{t}^{3}}}} \cdot \sqrt[3]{\frac{1}{{k}^{2}}}\right)}^{2} \cdot \sqrt[3]{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]

   6. unpow2 46.7%

      \[\leadsto {\left(\frac{\sqrt[3]{\color{blue}{\ell \cdot \ell}}}{\sqrt[3]{{t}^{3}}} \cdot \sqrt[3]{\frac{1}{{k}^{2}}}\right)}^{2} \cdot \sqrt[3]{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]

   7. cbrt-prod 46.7%

      \[\leadsto {\left(\frac{\color{blue}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}{\sqrt[3]{{t}^{3}}} \cdot \sqrt[3]{\frac{1}{{k}^{2}}}\right)}^{2} \cdot \sqrt[3]{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]

   8. unpow2 46.7%

      \[\leadsto {\left(\frac{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}}{\sqrt[3]{{t}^{3}}} \cdot \sqrt[3]{\frac{1}{{k}^{2}}}\right)}^{2} \cdot \sqrt[3]{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]

   9. unpow3 46.7%

      \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{\sqrt[3]{\color{blue}{\left(t \cdot t\right) \cdot t}}} \cdot \sqrt[3]{\frac{1}{{k}^{2}}}\right)}^{2} \cdot \sqrt[3]{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]

   10. add-cbrt-cube 46.7%

       \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{\color{blue}{t}} \cdot \sqrt[3]{\frac{1}{{k}^{2}}}\right)}^{2} \cdot \sqrt[3]{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]

   11. pow-flip 46.7%

       \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t} \cdot \sqrt[3]{\color{blue}{{k}^{\left(-2\right)}}}\right)}^{2} \cdot \sqrt[3]{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]

   12. metadata-eval 46.7%

       \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t} \cdot \sqrt[3]{{k}^{\color{blue}{-2}}}\right)}^{2} \cdot \sqrt[3]{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]

   13. div-inv 46.7%

       \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t} \cdot \sqrt[3]{{k}^{-2}}\right)}^{2} \cdot \sqrt[3]{\color{blue}{\frac{{\ell}^{2}}{{t}^{3}} \cdot \frac{1}{{k}^{2}}}} \]

9. Applied egg-rr 59.9%

   \[\leadsto \color{blue}{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t} \cdot \sqrt[3]{{k}^{-2}}\right)}^{2} \cdot \left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t} \cdot \sqrt[3]{{k}^{-2}}\right)} \]
10. Step-by-step derivation

    1. pow-plus 59.9%

       \[\leadsto \color{blue}{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t} \cdot \sqrt[3]{{k}^{-2}}\right)}^{\left(2 + 1\right)}} \]

    2. metadata-eval 59.9%

       \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t} \cdot \sqrt[3]{{k}^{-2}}\right)}^{\color{blue}{3}} \]

    3. associate-*l/ 60.0%

       \[\leadsto {\color{blue}{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{{k}^{-2}}}{t}\right)}}^{3} \]

11. Simplified 60.0%

    \[\leadsto \color{blue}{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{{k}^{-2}}}{t}\right)}^{3}} \]
12. Step-by-step derivation

    1. pow1/3 59.6%

       \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \color{blue}{{\left({k}^{-2}\right)}^{0.3333333333333333}}}{t}\right)}^{3} \]

    2. pow-pow 36.2%

       \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \color{blue}{{k}^{\left(-2 \cdot 0.3333333333333333\right)}}}{t}\right)}^{3} \]

    3. metadata-eval 36.2%

       \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot {k}^{\color{blue}{-0.6666666666666666}}}{t}\right)}^{3} \]

13. Applied egg-rr 36.2%

    \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \color{blue}{{k}^{-0.6666666666666666}}}{t}\right)}^{3} \]

14. Final simplification 36.2%

    \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot {k}^{-0.6666666666666666}}{t}\right)}^{3} \]
15. Add Preprocessing

Alternative 15: 61.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\ell \leq 9.5 \cdot 10^{-184} \lor \neg \left(\ell \leq 460000000000\right):\\ \;\;\;\;\frac{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3}}{k\_m \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{2}}{k\_m} \cdot \frac{{t}^{-3}}{k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (or (<= l 9.5e-184) (not (<= l 460000000000.0)))
   (/ (pow (/ (pow (cbrt l) 2.0) t) 3.0) (* k_m k_m))
   (* (/ (pow l 2.0) k_m) (/ (pow t -3.0) k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if ((l <= 9.5e-184) || !(l <= 460000000000.0)) {
		tmp = pow((pow(cbrt(l), 2.0) / t), 3.0) / (k_m * k_m);
	} else {
		tmp = (pow(l, 2.0) / k_m) * (pow(t, -3.0) / k_m);
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if ((l <= 9.5e-184) || !(l <= 460000000000.0)) {
		tmp = Math.pow((Math.pow(Math.cbrt(l), 2.0) / t), 3.0) / (k_m * k_m);
	} else {
		tmp = (Math.pow(l, 2.0) / k_m) * (Math.pow(t, -3.0) / k_m);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if ((l <= 9.5e-184) || !(l <= 460000000000.0))
		tmp = Float64((Float64((cbrt(l) ^ 2.0) / t) ^ 3.0) / Float64(k_m * k_m));
	else
		tmp = Float64(Float64((l ^ 2.0) / k_m) * Float64((t ^ -3.0) / k_m));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[Or[LessEqual[l, 9.5e-184], N[Not[LessEqual[l, 460000000000.0]], $MachinePrecision]], N[(N[Power[N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision], 3.0], $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[l, 2.0], $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[Power[t, -3.0], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 9.4999999999999991e-184 or 4.6e11 < l

   1. Initial program 52.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 53.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. sqr-neg 53.0%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. associate-*l* 49.1%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. sqr-neg 49.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. associate-/r* 55.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. associate-+l+ 55.7%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{\color{blue}{1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)}} \]

      7. unpow2 55.7%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right)} \]

      8. times-frac 43.1%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k \cdot k}{t \cdot t}} + 1\right)} \]

      9. sqr-neg 43.1%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}} + 1\right)} \]

      10. times-frac 55.7%

          \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}} + 1\right)} \]

      11. unpow2 55.7%

          \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{{\left(\frac{k}{-t}\right)}^{2}} + 1\right)} \]

   3. Simplified 55.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 46.0%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. *-commutative 46.0%

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      2. associate-/r* 46.6%

         \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]

   7. Simplified 46.6%

      \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 46.6%

         \[\leadsto \frac{\frac{{\ell}^{2}}{{t}^{3}}}{\color{blue}{k \cdot k}} \]

   9. Applied egg-rr 46.6%

      \[\leadsto \frac{\frac{{\ell}^{2}}{{t}^{3}}}{\color{blue}{k \cdot k}} \]
   10. Step-by-step derivation

       1. add-cube-cbrt 46.6%

          \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{{\ell}^{2}}{{t}^{3}}} \cdot \sqrt[3]{\frac{{\ell}^{2}}{{t}^{3}}}\right) \cdot \sqrt[3]{\frac{{\ell}^{2}}{{t}^{3}}}}}{k \cdot k} \]

       2. pow2 46.6%

          \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{\frac{{\ell}^{2}}{{t}^{3}}}\right)}^{2}} \cdot \sqrt[3]{\frac{{\ell}^{2}}{{t}^{3}}}}{k \cdot k} \]

       3. cbrt-div 46.6%

          \[\leadsto \frac{{\color{blue}{\left(\frac{\sqrt[3]{{\ell}^{2}}}{\sqrt[3]{{t}^{3}}}\right)}}^{2} \cdot \sqrt[3]{\frac{{\ell}^{2}}{{t}^{3}}}}{k \cdot k} \]

       4. unpow2 46.6%

          \[\leadsto \frac{{\left(\frac{\sqrt[3]{\color{blue}{\ell \cdot \ell}}}{\sqrt[3]{{t}^{3}}}\right)}^{2} \cdot \sqrt[3]{\frac{{\ell}^{2}}{{t}^{3}}}}{k \cdot k} \]

       5. cbrt-prod 46.6%

          \[\leadsto \frac{{\left(\frac{\color{blue}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}{\sqrt[3]{{t}^{3}}}\right)}^{2} \cdot \sqrt[3]{\frac{{\ell}^{2}}{{t}^{3}}}}{k \cdot k} \]

       6. unpow2 46.6%

          \[\leadsto \frac{{\left(\frac{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}}{\sqrt[3]{{t}^{3}}}\right)}^{2} \cdot \sqrt[3]{\frac{{\ell}^{2}}{{t}^{3}}}}{k \cdot k} \]

       7. unpow3 46.6%

          \[\leadsto \frac{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{\sqrt[3]{\color{blue}{\left(t \cdot t\right) \cdot t}}}\right)}^{2} \cdot \sqrt[3]{\frac{{\ell}^{2}}{{t}^{3}}}}{k \cdot k} \]

       8. add-cbrt-cube 46.6%

          \[\leadsto \frac{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{\color{blue}{t}}\right)}^{2} \cdot \sqrt[3]{\frac{{\ell}^{2}}{{t}^{3}}}}{k \cdot k} \]

       9. cbrt-div 46.6%

          \[\leadsto \frac{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{2} \cdot \color{blue}{\frac{\sqrt[3]{{\ell}^{2}}}{\sqrt[3]{{t}^{3}}}}}{k \cdot k} \]

       10. unpow2 46.6%

           \[\leadsto \frac{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{2} \cdot \frac{\sqrt[3]{\color{blue}{\ell \cdot \ell}}}{\sqrt[3]{{t}^{3}}}}{k \cdot k} \]

       11. cbrt-prod 51.9%

           \[\leadsto \frac{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{2} \cdot \frac{\color{blue}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}{\sqrt[3]{{t}^{3}}}}{k \cdot k} \]

       12. unpow2 51.9%

           \[\leadsto \frac{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{2} \cdot \frac{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}}{\sqrt[3]{{t}^{3}}}}{k \cdot k} \]

       13. unpow3 51.9%

           \[\leadsto \frac{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{2} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{\sqrt[3]{\color{blue}{\left(t \cdot t\right) \cdot t}}}}{k \cdot k} \]

       14. add-cbrt-cube 58.5%

           \[\leadsto \frac{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{2} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{\color{blue}{t}}}{k \cdot k} \]

   11. Applied egg-rr 58.5%

       \[\leadsto \frac{\color{blue}{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{2} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}}}{k \cdot k} \]
   12. Step-by-step derivation

       1. pow-plus 58.5%

          \[\leadsto \frac{\color{blue}{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{\left(2 + 1\right)}}}{k \cdot k} \]

       2. metadata-eval 58.5%

          \[\leadsto \frac{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{\color{blue}{3}}}{k \cdot k} \]

   13. Simplified 58.5%

       \[\leadsto \frac{\color{blue}{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3}}}{k \cdot k} \]

   if 9.4999999999999991e-184 < l < 4.6e11

   1. Initial program 61.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 61.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. sqr-neg 61.3%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. associate-*l* 55.2%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. sqr-neg 55.2%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. associate-/r* 55.2%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. associate-+l+ 55.2%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{\color{blue}{1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)}} \]

      7. unpow2 55.2%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right)} \]

      8. times-frac 49.3%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k \cdot k}{t \cdot t}} + 1\right)} \]

      9. sqr-neg 49.3%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}} + 1\right)} \]

      10. times-frac 55.2%

          \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}} + 1\right)} \]

      11. unpow2 55.2%

          \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{{\left(\frac{k}{-t}\right)}^{2}} + 1\right)} \]

   3. Simplified 55.2%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 54.1%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. *-commutative 54.1%

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      2. associate-/r* 50.1%

         \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]

   7. Simplified 50.1%

      \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 50.1%

         \[\leadsto \frac{\frac{{\ell}^{2}}{{t}^{3}}}{\color{blue}{k \cdot k}} \]

   9. Applied egg-rr 50.1%

      \[\leadsto \frac{\frac{{\ell}^{2}}{{t}^{3}}}{\color{blue}{k \cdot k}} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 49.6%

          \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{t}^{3}}\right)\right)}}{k \cdot k} \]

       2. expm1-udef 46.7%

          \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{t}^{3}}\right)} - 1}}{k \cdot k} \]

       3. div-inv 46.7%

          \[\leadsto \frac{e^{\mathsf{log1p}\left(\color{blue}{{\ell}^{2} \cdot \frac{1}{{t}^{3}}}\right)} - 1}{k \cdot k} \]

       4. pow-flip 46.7%

          \[\leadsto \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot \color{blue}{{t}^{\left(-3\right)}}\right)} - 1}{k \cdot k} \]

       5. metadata-eval 46.7%

          \[\leadsto \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {t}^{\color{blue}{-3}}\right)} - 1}{k \cdot k} \]

   11. Applied egg-rr 46.7%

       \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {t}^{-3}\right)} - 1}}{k \cdot k} \]
   12. Step-by-step derivation

       1. expm1-def 49.6%

          \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\ell}^{2} \cdot {t}^{-3}\right)\right)}}{k \cdot k} \]

       2. expm1-log1p 50.1%

          \[\leadsto \frac{\color{blue}{{\ell}^{2} \cdot {t}^{-3}}}{k \cdot k} \]

   13. Simplified 50.1%

       \[\leadsto \frac{\color{blue}{{\ell}^{2} \cdot {t}^{-3}}}{k \cdot k} \]
   14. Step-by-step derivation

       1. times-frac 60.8%

          \[\leadsto \color{blue}{\frac{{\ell}^{2}}{k} \cdot \frac{{t}^{-3}}{k}} \]

   15. Applied egg-rr 60.8%

       \[\leadsto \color{blue}{\frac{{\ell}^{2}}{k} \cdot \frac{{t}^{-3}}{k}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 9.5 \cdot 10^{-184} \lor \neg \left(\ell \leq 460000000000\right):\\ \;\;\;\;\frac{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{2}}{k} \cdot \frac{{t}^{-3}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 33.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{2}{\left(1 + \left(1 + {\left(\frac{k\_m}{t}\right)}^{2}\right)\right) \cdot {\left(k\_m \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (/
  2.0
  (* (+ 1.0 (+ 1.0 (pow (/ k_m t) 2.0))) (pow (* k_m (/ (pow t 1.5) l)) 2.0))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return 2.0 / ((1.0 + (1.0 + pow((k_m / t), 2.0))) * pow((k_m * (pow(t, 1.5) / l)), 2.0));
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = 2.0d0 / ((1.0d0 + (1.0d0 + ((k_m / t) ** 2.0d0))) * ((k_m * ((t ** 1.5d0) / l)) ** 2.0d0))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return 2.0 / ((1.0 + (1.0 + Math.pow((k_m / t), 2.0))) * Math.pow((k_m * (Math.pow(t, 1.5) / l)), 2.0));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return 2.0 / ((1.0 + (1.0 + math.pow((k_m / t), 2.0))) * math.pow((k_m * (math.pow(t, 1.5) / l)), 2.0))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(2.0 / Float64(Float64(1.0 + Float64(1.0 + (Float64(k_m / t) ^ 2.0))) * (Float64(k_m * Float64((t ^ 1.5) / l)) ^ 2.0)))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = 2.0 / ((1.0 + (1.0 + ((k_m / t) ^ 2.0))) * ((k_m * ((t ^ 1.5) / l)) ^ 2.0));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(2.0 / N[(N[(1.0 + N[(1.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(k$95$m * N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.9%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-*l* 49.8%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. associate-/r* 55.5%

      \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. add-sqr-sqrt 28.8%

      \[\leadsto \frac{2}{\color{blue}{\left(\sqrt{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. pow2 28.8%

      \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}\right)}^{2}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. sqrt-prod 21.5%

      \[\leadsto \frac{2}{{\color{blue}{\left(\sqrt{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}}^{2} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. associate-/r* 19.4%

      \[\leadsto \frac{2}{{\left(\sqrt{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   7. sqrt-div 19.4%

      \[\leadsto \frac{2}{{\left(\color{blue}{\frac{\sqrt{{t}^{3}}}{\sqrt{\ell \cdot \ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   8. sqrt-pow1 21.4%

      \[\leadsto \frac{2}{{\left(\frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   9. metadata-eval 21.4%

      \[\leadsto \frac{2}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   10. sqrt-prod 12.1%

       \[\leadsto \frac{2}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   11. add-sqr-sqrt 25.6%

       \[\leadsto \frac{2}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

4. Applied egg-rr 25.6%

   \[\leadsto \frac{2}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

5. Taylor expanded in k around 0 32.8%

   \[\leadsto \frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{k}\right)}^{2} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

6. Final simplification 32.8%

   \[\leadsto \frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(k \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}} \]
7. Add Preprocessing

Alternative 17: 57.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 5.2 \cdot 10^{-192}:\\ \;\;\;\;\frac{{\ell}^{2}}{k\_m} \cdot \frac{{t}^{-3}}{k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot \frac{{k\_m}^{2} \cdot {t}^{3}}{\ell}}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 5.2e-192)
   (* (/ (pow l 2.0) k_m) (/ (pow t -3.0) k_m))
   (/ 2.0 (/ (* 2.0 (/ (* (pow k_m 2.0) (pow t 3.0)) l)) l))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 5.2e-192) {
		tmp = (pow(l, 2.0) / k_m) * (pow(t, -3.0) / k_m);
	} else {
		tmp = 2.0 / ((2.0 * ((pow(k_m, 2.0) * pow(t, 3.0)) / l)) / l);
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 5.2d-192) then
        tmp = ((l ** 2.0d0) / k_m) * ((t ** (-3.0d0)) / k_m)
    else
        tmp = 2.0d0 / ((2.0d0 * (((k_m ** 2.0d0) * (t ** 3.0d0)) / l)) / l)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 5.2e-192) {
		tmp = (Math.pow(l, 2.0) / k_m) * (Math.pow(t, -3.0) / k_m);
	} else {
		tmp = 2.0 / ((2.0 * ((Math.pow(k_m, 2.0) * Math.pow(t, 3.0)) / l)) / l);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 5.2e-192:
		tmp = (math.pow(l, 2.0) / k_m) * (math.pow(t, -3.0) / k_m)
	else:
		tmp = 2.0 / ((2.0 * ((math.pow(k_m, 2.0) * math.pow(t, 3.0)) / l)) / l)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 5.2e-192)
		tmp = Float64(Float64((l ^ 2.0) / k_m) * Float64((t ^ -3.0) / k_m));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * Float64(Float64((k_m ^ 2.0) * (t ^ 3.0)) / l)) / l));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 5.2e-192)
		tmp = ((l ^ 2.0) / k_m) * ((t ^ -3.0) / k_m);
	else
		tmp = 2.0 / ((2.0 * (((k_m ^ 2.0) * (t ^ 3.0)) / l)) / l);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 5.2e-192], N[(N[(N[Power[l, 2.0], $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[Power[t, -3.0], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.2000000000000003e-192

   1. Initial program 53.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 53.6%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. sqr-neg 53.6%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. associate-*l* 46.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. sqr-neg 46.7%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. associate-/r* 51.3%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. associate-+l+ 51.3%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{\color{blue}{1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)}} \]

      7. unpow2 51.3%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right)} \]

      8. times-frac 39.7%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k \cdot k}{t \cdot t}} + 1\right)} \]

      9. sqr-neg 39.7%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}} + 1\right)} \]

      10. times-frac 51.3%

          \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}} + 1\right)} \]

      11. unpow2 51.3%

          \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{{\left(\frac{k}{-t}\right)}^{2}} + 1\right)} \]

   3. Simplified 51.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 45.3%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. *-commutative 45.3%

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      2. associate-/r* 44.3%

         \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]

   7. Simplified 44.3%

      \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 44.3%

         \[\leadsto \frac{\frac{{\ell}^{2}}{{t}^{3}}}{\color{blue}{k \cdot k}} \]

   9. Applied egg-rr 44.3%

      \[\leadsto \frac{\frac{{\ell}^{2}}{{t}^{3}}}{\color{blue}{k \cdot k}} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 35.5%

          \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{t}^{3}}\right)\right)}}{k \cdot k} \]

       2. expm1-udef 34.8%

          \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{t}^{3}}\right)} - 1}}{k \cdot k} \]

       3. div-inv 34.8%

          \[\leadsto \frac{e^{\mathsf{log1p}\left(\color{blue}{{\ell}^{2} \cdot \frac{1}{{t}^{3}}}\right)} - 1}{k \cdot k} \]

       4. pow-flip 34.8%

          \[\leadsto \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot \color{blue}{{t}^{\left(-3\right)}}\right)} - 1}{k \cdot k} \]

       5. metadata-eval 34.8%

          \[\leadsto \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {t}^{\color{blue}{-3}}\right)} - 1}{k \cdot k} \]

   11. Applied egg-rr 34.8%

       \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {t}^{-3}\right)} - 1}}{k \cdot k} \]
   12. Step-by-step derivation

       1. expm1-def 35.5%

          \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\ell}^{2} \cdot {t}^{-3}\right)\right)}}{k \cdot k} \]

       2. expm1-log1p 44.3%

          \[\leadsto \frac{\color{blue}{{\ell}^{2} \cdot {t}^{-3}}}{k \cdot k} \]

   13. Simplified 44.3%

       \[\leadsto \frac{\color{blue}{{\ell}^{2} \cdot {t}^{-3}}}{k \cdot k} \]
   14. Step-by-step derivation

       1. times-frac 53.4%

          \[\leadsto \color{blue}{\frac{{\ell}^{2}}{k} \cdot \frac{{t}^{-3}}{k}} \]

   15. Applied egg-rr 53.4%

       \[\leadsto \color{blue}{\frac{{\ell}^{2}}{k} \cdot \frac{{t}^{-3}}{k}} \]

   if 5.2000000000000003e-192 < k

   1. Initial program 54.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/r* 61.7%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. +-commutative 61.7%

         \[\leadsto \frac{2}{\left(\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

      3. associate-+r+ 61.7%

         \[\leadsto \frac{2}{\left(\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}} \]

      4. metadata-eval 61.7%

         \[\leadsto \frac{2}{\left(\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)} \]

      5. associate-*r* 61.7%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

      6. *-commutative 61.7%

         \[\leadsto \frac{2}{\color{blue}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right)}} \]

      7. associate-*l/ 61.7%

         \[\leadsto \frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell}}} \]

      8. associate-*r/ 62.6%

         \[\leadsto \frac{2}{\color{blue}{\frac{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\frac{{t}^{3}}{\ell} \cdot \sin k\right)}{\ell}}} \]

   4. Applied egg-rr 62.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\frac{{t}^{3}}{\ell} \cdot \sin k\right)}{\ell}}} \]

   5. Taylor expanded in k around 0 55.1%

      \[\leadsto \frac{2}{\frac{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{\ell}}}{\ell}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.2 \cdot 10^{-192}:\\ \;\;\;\;\frac{{\ell}^{2}}{k} \cdot \frac{{t}^{-3}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{\ell}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 57.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 5.2 \cdot 10^{-192}:\\ \;\;\;\;\frac{{\ell}^{2}}{k\_m} \cdot \frac{{t}^{-3}}{k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{2 \cdot {k\_m}^{2}}{\frac{\ell}{{t}^{3}}}}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 5.2e-192)
   (* (/ (pow l 2.0) k_m) (/ (pow t -3.0) k_m))
   (/ 2.0 (/ (/ (* 2.0 (pow k_m 2.0)) (/ l (pow t 3.0))) l))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 5.2e-192) {
		tmp = (pow(l, 2.0) / k_m) * (pow(t, -3.0) / k_m);
	} else {
		tmp = 2.0 / (((2.0 * pow(k_m, 2.0)) / (l / pow(t, 3.0))) / l);
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 5.2d-192) then
        tmp = ((l ** 2.0d0) / k_m) * ((t ** (-3.0d0)) / k_m)
    else
        tmp = 2.0d0 / (((2.0d0 * (k_m ** 2.0d0)) / (l / (t ** 3.0d0))) / l)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 5.2e-192) {
		tmp = (Math.pow(l, 2.0) / k_m) * (Math.pow(t, -3.0) / k_m);
	} else {
		tmp = 2.0 / (((2.0 * Math.pow(k_m, 2.0)) / (l / Math.pow(t, 3.0))) / l);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 5.2e-192:
		tmp = (math.pow(l, 2.0) / k_m) * (math.pow(t, -3.0) / k_m)
	else:
		tmp = 2.0 / (((2.0 * math.pow(k_m, 2.0)) / (l / math.pow(t, 3.0))) / l)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 5.2e-192)
		tmp = Float64(Float64((l ^ 2.0) / k_m) * Float64((t ^ -3.0) / k_m));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(2.0 * (k_m ^ 2.0)) / Float64(l / (t ^ 3.0))) / l));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 5.2e-192)
		tmp = ((l ^ 2.0) / k_m) * ((t ^ -3.0) / k_m);
	else
		tmp = 2.0 / (((2.0 * (k_m ^ 2.0)) / (l / (t ^ 3.0))) / l);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 5.2e-192], N[(N[(N[Power[l, 2.0], $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[Power[t, -3.0], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(2.0 * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.2000000000000003e-192

   1. Initial program 53.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 53.6%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. sqr-neg 53.6%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. associate-*l* 46.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. sqr-neg 46.7%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. associate-/r* 51.3%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. associate-+l+ 51.3%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{\color{blue}{1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)}} \]

      7. unpow2 51.3%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right)} \]

      8. times-frac 39.7%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k \cdot k}{t \cdot t}} + 1\right)} \]

      9. sqr-neg 39.7%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}} + 1\right)} \]

      10. times-frac 51.3%

          \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}} + 1\right)} \]

      11. unpow2 51.3%

          \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{{\left(\frac{k}{-t}\right)}^{2}} + 1\right)} \]

   3. Simplified 51.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 45.3%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. *-commutative 45.3%

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      2. associate-/r* 44.3%

         \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]

   7. Simplified 44.3%

      \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 44.3%

         \[\leadsto \frac{\frac{{\ell}^{2}}{{t}^{3}}}{\color{blue}{k \cdot k}} \]

   9. Applied egg-rr 44.3%

      \[\leadsto \frac{\frac{{\ell}^{2}}{{t}^{3}}}{\color{blue}{k \cdot k}} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 35.5%

          \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{t}^{3}}\right)\right)}}{k \cdot k} \]

       2. expm1-udef 34.8%

          \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{t}^{3}}\right)} - 1}}{k \cdot k} \]

       3. div-inv 34.8%

          \[\leadsto \frac{e^{\mathsf{log1p}\left(\color{blue}{{\ell}^{2} \cdot \frac{1}{{t}^{3}}}\right)} - 1}{k \cdot k} \]

       4. pow-flip 34.8%

          \[\leadsto \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot \color{blue}{{t}^{\left(-3\right)}}\right)} - 1}{k \cdot k} \]

       5. metadata-eval 34.8%

          \[\leadsto \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {t}^{\color{blue}{-3}}\right)} - 1}{k \cdot k} \]

   11. Applied egg-rr 34.8%

       \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {t}^{-3}\right)} - 1}}{k \cdot k} \]
   12. Step-by-step derivation

       1. expm1-def 35.5%

          \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\ell}^{2} \cdot {t}^{-3}\right)\right)}}{k \cdot k} \]

       2. expm1-log1p 44.3%

          \[\leadsto \frac{\color{blue}{{\ell}^{2} \cdot {t}^{-3}}}{k \cdot k} \]

   13. Simplified 44.3%

       \[\leadsto \frac{\color{blue}{{\ell}^{2} \cdot {t}^{-3}}}{k \cdot k} \]
   14. Step-by-step derivation

       1. times-frac 53.4%

          \[\leadsto \color{blue}{\frac{{\ell}^{2}}{k} \cdot \frac{{t}^{-3}}{k}} \]

   15. Applied egg-rr 53.4%

       \[\leadsto \color{blue}{\frac{{\ell}^{2}}{k} \cdot \frac{{t}^{-3}}{k}} \]

   if 5.2000000000000003e-192 < k

   1. Initial program 54.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/r* 61.7%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. +-commutative 61.7%

         \[\leadsto \frac{2}{\left(\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

      3. associate-+r+ 61.7%

         \[\leadsto \frac{2}{\left(\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}} \]

      4. metadata-eval 61.7%

         \[\leadsto \frac{2}{\left(\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)} \]

      5. associate-*r* 61.7%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

      6. *-commutative 61.7%

         \[\leadsto \frac{2}{\color{blue}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right)}} \]

      7. associate-*l/ 61.7%

         \[\leadsto \frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell}}} \]

      8. associate-*r/ 62.6%

         \[\leadsto \frac{2}{\color{blue}{\frac{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\frac{{t}^{3}}{\ell} \cdot \sin k\right)}{\ell}}} \]

   4. Applied egg-rr 62.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\frac{{t}^{3}}{\ell} \cdot \sin k\right)}{\ell}}} \]

   5. Taylor expanded in k around 0 55.1%

      \[\leadsto \frac{2}{\frac{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{\ell}}}{\ell}} \]
   6. Step-by-step derivation

      1. *-commutative 55.1%

         \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2} \cdot {t}^{3}}{\ell} \cdot 2}}{\ell}} \]

      2. associate-/l* 55.8%

         \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2}}{\frac{\ell}{{t}^{3}}}} \cdot 2}{\ell}} \]

      3. associate-*l/ 55.8%

         \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2} \cdot 2}{\frac{\ell}{{t}^{3}}}}}{\ell}} \]

   7. Simplified 55.8%

      \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2} \cdot 2}{\frac{\ell}{{t}^{3}}}}}{\ell}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.2 \cdot 10^{-192}:\\ \;\;\;\;\frac{{\ell}^{2}}{k} \cdot \frac{{t}^{-3}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{2 \cdot {k}^{2}}{\frac{\ell}{{t}^{3}}}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 55.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{{\ell}^{2}}{k\_m} \cdot \frac{{t}^{-3}}{k\_m} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (/ (pow l 2.0) k_m) (/ (pow t -3.0) k_m)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (pow(l, 2.0) / k_m) * (pow(t, -3.0) / k_m);
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = ((l ** 2.0d0) / k_m) * ((t ** (-3.0d0)) / k_m)
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return (Math.pow(l, 2.0) / k_m) * (Math.pow(t, -3.0) / k_m);
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (math.pow(l, 2.0) / k_m) * (math.pow(t, -3.0) / k_m)

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64((l ^ 2.0) / k_m) * Float64((t ^ -3.0) / k_m))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = ((l ^ 2.0) / k_m) * ((t ^ -3.0) / k_m);
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(N[Power[l, 2.0], $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[Power[t, -3.0], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.9%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/r* 54.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

   2. sqr-neg 54.1%

      \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. associate-*l* 49.9%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   4. sqr-neg 49.9%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   5. associate-/r* 55.6%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   6. associate-+l+ 55.6%

      \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{\color{blue}{1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)}} \]

   7. unpow2 55.6%

      \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right)} \]

   8. times-frac 43.9%

      \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k \cdot k}{t \cdot t}} + 1\right)} \]

   9. sqr-neg 43.9%

      \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}} + 1\right)} \]

   10. times-frac 55.6%

       \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}} + 1\right)} \]

   11. unpow2 55.6%

       \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{{\left(\frac{k}{-t}\right)}^{2}} + 1\right)} \]

3. Simplified 55.6%

   \[\leadsto \color{blue}{\frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
4. Add Preprocessing

5. Taylor expanded in k around 0 47.0%

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
6. Step-by-step derivation

   1. *-commutative 47.0%

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   2. associate-/r* 47.1%

      \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]

7. Simplified 47.1%

   \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]
8. Step-by-step derivation

   1. unpow2 47.1%

      \[\leadsto \frac{\frac{{\ell}^{2}}{{t}^{3}}}{\color{blue}{k \cdot k}} \]

9. Applied egg-rr 47.1%

   \[\leadsto \frac{\frac{{\ell}^{2}}{{t}^{3}}}{\color{blue}{k \cdot k}} \]
10. Step-by-step derivation

    1. expm1-log1p-u 39.8%

       \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{t}^{3}}\right)\right)}}{k \cdot k} \]

    2. expm1-udef 39.5%

       \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{t}^{3}}\right)} - 1}}{k \cdot k} \]

    3. div-inv 39.5%

       \[\leadsto \frac{e^{\mathsf{log1p}\left(\color{blue}{{\ell}^{2} \cdot \frac{1}{{t}^{3}}}\right)} - 1}{k \cdot k} \]

    4. pow-flip 39.5%

       \[\leadsto \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot \color{blue}{{t}^{\left(-3\right)}}\right)} - 1}{k \cdot k} \]

    5. metadata-eval 39.5%

       \[\leadsto \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {t}^{\color{blue}{-3}}\right)} - 1}{k \cdot k} \]

11. Applied egg-rr 39.5%

    \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {t}^{-3}\right)} - 1}}{k \cdot k} \]
12. Step-by-step derivation

    1. expm1-def 39.8%

       \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\ell}^{2} \cdot {t}^{-3}\right)\right)}}{k \cdot k} \]

    2. expm1-log1p 47.1%

       \[\leadsto \frac{\color{blue}{{\ell}^{2} \cdot {t}^{-3}}}{k \cdot k} \]

13. Simplified 47.1%

    \[\leadsto \frac{\color{blue}{{\ell}^{2} \cdot {t}^{-3}}}{k \cdot k} \]
14. Step-by-step derivation

    1. times-frac 52.2%

       \[\leadsto \color{blue}{\frac{{\ell}^{2}}{k} \cdot \frac{{t}^{-3}}{k}} \]

15. Applied egg-rr 52.2%

    \[\leadsto \color{blue}{\frac{{\ell}^{2}}{k} \cdot \frac{{t}^{-3}}{k}} \]

16. Final simplification 52.2%

    \[\leadsto \frac{{\ell}^{2}}{k} \cdot \frac{{t}^{-3}}{k} \]
17. Add Preprocessing

Reproduce

herbie shell --seed 2024026 
(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))))