Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.0% → 81.2%

Time: 19.4s

Alternatives: 11

Speedup: 2.0×

• 27.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 81.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2.25 \cdot 10^{+14}:\\ \;\;\;\;\frac{2}{{\left(\left(k\_m \cdot \sqrt{2}\right) \cdot {\left(\frac{\sqrt{t\_m}}{\sqrt[3]{\ell}}\right)}^{3}\right)}^{2}}\\ \mathbf{elif}\;k\_m \leq 6 \cdot 10^{+151}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(t\_m \cdot {k\_m}^{2}\right) \cdot {\ell}^{-2}\right) \cdot {\sin k\_m}^{2}}{\cos k\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\left(\sin k\_m \cdot \tan k\_m\right) \cdot \left(2 + {\left(\frac{k\_m}{t\_m}\right)}^{2}\right)}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 2.25e+14)
    (/ 2.0 (pow (* (* k_m (sqrt 2.0)) (pow (/ (sqrt t_m) (cbrt l)) 3.0)) 2.0))
    (if (<= k_m 6e+151)
      (/
       2.0
       (/
        (* (* (* t_m (pow k_m 2.0)) (pow l -2.0)) (pow (sin k_m) 2.0))
        (cos k_m)))
      (/
       2.0
       (pow
        (*
         (/ t_m (pow (cbrt l) 2.0))
         (cbrt (* (* (sin k_m) (tan k_m)) (+ 2.0 (pow (/ k_m t_m) 2.0)))))
        3.0))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.25e+14) {
		tmp = 2.0 / pow(((k_m * sqrt(2.0)) * pow((sqrt(t_m) / cbrt(l)), 3.0)), 2.0);
	} else if (k_m <= 6e+151) {
		tmp = 2.0 / ((((t_m * pow(k_m, 2.0)) * pow(l, -2.0)) * pow(sin(k_m), 2.0)) / cos(k_m));
	} else {
		tmp = 2.0 / pow(((t_m / pow(cbrt(l), 2.0)) * cbrt(((sin(k_m) * tan(k_m)) * (2.0 + pow((k_m / t_m), 2.0))))), 3.0);
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.25e+14) {
		tmp = 2.0 / Math.pow(((k_m * Math.sqrt(2.0)) * Math.pow((Math.sqrt(t_m) / Math.cbrt(l)), 3.0)), 2.0);
	} else if (k_m <= 6e+151) {
		tmp = 2.0 / ((((t_m * Math.pow(k_m, 2.0)) * Math.pow(l, -2.0)) * Math.pow(Math.sin(k_m), 2.0)) / Math.cos(k_m));
	} else {
		tmp = 2.0 / Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt(((Math.sin(k_m) * Math.tan(k_m)) * (2.0 + Math.pow((k_m / t_m), 2.0))))), 3.0);
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 2.25e+14)
		tmp = Float64(2.0 / (Float64(Float64(k_m * sqrt(2.0)) * (Float64(sqrt(t_m) / cbrt(l)) ^ 3.0)) ^ 2.0));
	elseif (k_m <= 6e+151)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m * (k_m ^ 2.0)) * (l ^ -2.0)) * (sin(k_m) ^ 2.0)) / cos(k_m)));
	else
		tmp = Float64(2.0 / (Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * cbrt(Float64(Float64(sin(k_m) * tan(k_m)) * Float64(2.0 + (Float64(k_m / t_m) ^ 2.0))))) ^ 3.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 2.25e+14], N[(2.0 / N[Power[N[(N[(k$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Sqrt[t$95$m], $MachinePrecision] / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 6e+151], N[(2.0 / N[(N[(N[(N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[l, -2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[Power[N[(k$95$m / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 2.25e14

   1. Initial program 58.2%

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

   3. Simplified 59.7%

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

   5. Taylor expanded in k around 0 60.4%

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

      1. add-cube-cbrt 60.4%

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

      2. pow3 60.4%

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

      3. associate-/l/ 55.0%

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

      4. unpow2 55.0%

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

      5. cbrt-div 55.0%

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

      6. unpow3 55.0%

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

      7. add-cbrt-cube 60.0%

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

      8. unpow2 60.0%

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

      9. cbrt-prod 66.0%

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

      10. unpow2 66.0%

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

      11. cube-mult 66.0%

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

      12. div-inv 66.0%

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

      13. pow-flip 66.0%

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

      14. metadata-eval 66.0%

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

      15. pow2 66.0%

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

      16. div-inv 66.0%

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

      17. pow-flip 66.0%

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

      18. metadata-eval 66.0%

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

   7. Applied egg-rr 66.0%

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

      1. unpow2 66.0%

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

      2. cube-mult 66.0%

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

   9. Simplified 66.0%

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

   11. Applied egg-rr 38.5%

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

       1. unpow1 38.5%

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

       2. *-commutative 38.5%

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

       3. associate-*l/ 38.5%

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

   13. Simplified 38.5%

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

       1. associate-*l/ 38.5%

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

       2. add-cube-cbrt 38.3%

          \[\leadsto \frac{2}{{\left(\color{blue}{\left(\left(\sqrt[3]{\frac{{t}^{1.5}}{\ell}} \cdot \sqrt[3]{\frac{{t}^{1.5}}{\ell}}\right) \cdot \sqrt[3]{\frac{{t}^{1.5}}{\ell}}\right)} \cdot \left(k \cdot \sqrt{2}\right)\right)}^{2}} \]

       3. associate-*l* 38.4%

          \[\leadsto \frac{2}{{\color{blue}{\left(\left(\sqrt[3]{\frac{{t}^{1.5}}{\ell}} \cdot \sqrt[3]{\frac{{t}^{1.5}}{\ell}}\right) \cdot \left(\sqrt[3]{\frac{{t}^{1.5}}{\ell}} \cdot \left(k \cdot \sqrt{2}\right)\right)\right)}}^{2}} \]

       4. pow2 38.4%

          \[\leadsto \frac{2}{{\left(\color{blue}{{\left(\sqrt[3]{\frac{{t}^{1.5}}{\ell}}\right)}^{2}} \cdot \left(\sqrt[3]{\frac{{t}^{1.5}}{\ell}} \cdot \left(k \cdot \sqrt{2}\right)\right)\right)}^{2}} \]

       5. cbrt-div 38.4%

          \[\leadsto \frac{2}{{\left({\color{blue}{\left(\frac{\sqrt[3]{{t}^{1.5}}}{\sqrt[3]{\ell}}\right)}}^{2} \cdot \left(\sqrt[3]{\frac{{t}^{1.5}}{\ell}} \cdot \left(k \cdot \sqrt{2}\right)\right)\right)}^{2}} \]

       6. metadata-eval 38.4%

          \[\leadsto \frac{2}{{\left({\left(\frac{\sqrt[3]{{t}^{\color{blue}{\left(\frac{3}{2}\right)}}}}{\sqrt[3]{\ell}}\right)}^{2} \cdot \left(\sqrt[3]{\frac{{t}^{1.5}}{\ell}} \cdot \left(k \cdot \sqrt{2}\right)\right)\right)}^{2}} \]

       7. sqrt-pow1 34.7%

          \[\leadsto \frac{2}{{\left({\left(\frac{\sqrt[3]{\color{blue}{\sqrt{{t}^{3}}}}}{\sqrt[3]{\ell}}\right)}^{2} \cdot \left(\sqrt[3]{\frac{{t}^{1.5}}{\ell}} \cdot \left(k \cdot \sqrt{2}\right)\right)\right)}^{2}} \]

       8. unpow3 34.7%

          \[\leadsto \frac{2}{{\left({\left(\frac{\sqrt[3]{\sqrt{\color{blue}{\left(t \cdot t\right) \cdot t}}}}{\sqrt[3]{\ell}}\right)}^{2} \cdot \left(\sqrt[3]{\frac{{t}^{1.5}}{\ell}} \cdot \left(k \cdot \sqrt{2}\right)\right)\right)}^{2}} \]

       9. sqrt-prod 37.0%

          \[\leadsto \frac{2}{{\left({\left(\frac{\sqrt[3]{\color{blue}{\sqrt{t \cdot t} \cdot \sqrt{t}}}}{\sqrt[3]{\ell}}\right)}^{2} \cdot \left(\sqrt[3]{\frac{{t}^{1.5}}{\ell}} \cdot \left(k \cdot \sqrt{2}\right)\right)\right)}^{2}} \]

       10. sqrt-unprod 38.4%

           \[\leadsto \frac{2}{{\left({\left(\frac{\sqrt[3]{\color{blue}{\left(\sqrt{t} \cdot \sqrt{t}\right)} \cdot \sqrt{t}}}{\sqrt[3]{\ell}}\right)}^{2} \cdot \left(\sqrt[3]{\frac{{t}^{1.5}}{\ell}} \cdot \left(k \cdot \sqrt{2}\right)\right)\right)}^{2}} \]

       11. add-cbrt-cube 38.4%

           \[\leadsto \frac{2}{{\left({\left(\frac{\color{blue}{\sqrt{t}}}{\sqrt[3]{\ell}}\right)}^{2} \cdot \left(\sqrt[3]{\frac{{t}^{1.5}}{\ell}} \cdot \left(k \cdot \sqrt{2}\right)\right)\right)}^{2}} \]

   15. Applied egg-rr 39.7%

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

       1. associate-*r* 39.7%

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

       2. *-commutative 39.7%

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

       3. pow-plus 39.7%

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

       4. metadata-eval 39.7%

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

   17. Simplified 39.7%

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

   if 2.25e14 < k < 5.9999999999999998e151

   1. Initial program 66.5%

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

   3. Simplified 66.7%

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

   5. Taylor expanded in t around 0 93.2%

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

      1. associate-*r* 93.2%

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

      2. times-frac 93.2%

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

   7. Simplified 93.2%

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

      1. associate-*r/ 93.2%

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

      2. div-inv 93.2%

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

      3. *-commutative 93.2%

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

      4. pow-flip 93.1%

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

      5. metadata-eval 93.1%

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

   9. Applied egg-rr 93.1%

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

   if 5.9999999999999998e151 < k

   1. Initial program 53.3%

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

   3. Simplified 53.4%

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

      1. associate-*l* 53.4%

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

      2. associate-/r* 66.6%

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

      3. associate-+r+ 66.6%

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

      4. metadata-eval 66.6%

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

      5. associate-*l* 66.6%

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

      6. add-cube-cbrt 66.5%

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

      7. pow3 66.6%

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

   6. Applied egg-rr 83.5%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.25 \cdot 10^{+14}:\\ \;\;\;\;\frac{2}{{\left(\left(k \cdot \sqrt{2}\right) \cdot {\left(\frac{\sqrt{t}}{\sqrt[3]{\ell}}\right)}^{3}\right)}^{2}}\\ \mathbf{elif}\;k \leq 6 \cdot 10^{+151}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(t \cdot {k}^{2}\right) \cdot {\ell}^{-2}\right) \cdot {\sin k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := k\_m \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 4.3 \cdot 10^{+14}:\\ \;\;\;\;\frac{2}{{\left(t\_2 \cdot {\left(\frac{\sqrt{t\_m}}{\sqrt[3]{\ell}}\right)}^{3}\right)}^{2}}\\ \mathbf{elif}\;k\_m \leq 5.8 \cdot 10^{+152}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot {k\_m}^{2}}{{\ell}^{2}} \cdot \frac{{\sin k\_m}^{2}}{\cos k\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t\_m \cdot \left(\sqrt{t\_m} \cdot \frac{t\_2}{\ell}\right)\right)}^{2}}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (* k_m (sqrt 2.0))))
   (*
    t_s
    (if (<= k_m 4.3e+14)
      (/ 2.0 (pow (* t_2 (pow (/ (sqrt t_m) (cbrt l)) 3.0)) 2.0))
      (if (<= k_m 5.8e+152)
        (/
         2.0
         (*
          (/ (* t_m (pow k_m 2.0)) (pow l 2.0))
          (/ (pow (sin k_m) 2.0) (cos k_m))))
        (/ 2.0 (pow (* t_m (* (sqrt t_m) (/ t_2 l))) 2.0)))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = k_m * sqrt(2.0);
	double tmp;
	if (k_m <= 4.3e+14) {
		tmp = 2.0 / pow((t_2 * pow((sqrt(t_m) / cbrt(l)), 3.0)), 2.0);
	} else if (k_m <= 5.8e+152) {
		tmp = 2.0 / (((t_m * pow(k_m, 2.0)) / pow(l, 2.0)) * (pow(sin(k_m), 2.0) / cos(k_m)));
	} else {
		tmp = 2.0 / pow((t_m * (sqrt(t_m) * (t_2 / l))), 2.0);
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = k_m * Math.sqrt(2.0);
	double tmp;
	if (k_m <= 4.3e+14) {
		tmp = 2.0 / Math.pow((t_2 * Math.pow((Math.sqrt(t_m) / Math.cbrt(l)), 3.0)), 2.0);
	} else if (k_m <= 5.8e+152) {
		tmp = 2.0 / (((t_m * Math.pow(k_m, 2.0)) / Math.pow(l, 2.0)) * (Math.pow(Math.sin(k_m), 2.0) / Math.cos(k_m)));
	} else {
		tmp = 2.0 / Math.pow((t_m * (Math.sqrt(t_m) * (t_2 / l))), 2.0);
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = Float64(k_m * sqrt(2.0))
	tmp = 0.0
	if (k_m <= 4.3e+14)
		tmp = Float64(2.0 / (Float64(t_2 * (Float64(sqrt(t_m) / cbrt(l)) ^ 3.0)) ^ 2.0));
	elseif (k_m <= 5.8e+152)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k_m ^ 2.0)) / (l ^ 2.0)) * Float64((sin(k_m) ^ 2.0) / cos(k_m))));
	else
		tmp = Float64(2.0 / (Float64(t_m * Float64(sqrt(t_m) * Float64(t_2 / l))) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := Block[{t$95$2 = N[(k$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 4.3e+14], N[(2.0 / N[Power[N[(t$95$2 * N[Power[N[(N[Sqrt[t$95$m], $MachinePrecision] / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 5.8e+152], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(t$95$m * N[(N[Sqrt[t$95$m], $MachinePrecision] * N[(t$95$2 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 4.3e14

   1. Initial program 58.2%

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

   3. Simplified 59.7%

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

   5. Taylor expanded in k around 0 60.4%

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

      1. add-cube-cbrt 60.4%

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

      2. pow3 60.4%

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

      3. associate-/l/ 55.0%

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

      4. unpow2 55.0%

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

      5. cbrt-div 55.0%

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

      6. unpow3 55.0%

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

      7. add-cbrt-cube 60.0%

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

      8. unpow2 60.0%

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

      9. cbrt-prod 66.0%

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

      10. unpow2 66.0%

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

      11. cube-mult 66.0%

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

      12. div-inv 66.0%

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

      13. pow-flip 66.0%

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

      14. metadata-eval 66.0%

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

      15. pow2 66.0%

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

      16. div-inv 66.0%

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

      17. pow-flip 66.0%

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

      18. metadata-eval 66.0%

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

   7. Applied egg-rr 66.0%

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

      1. unpow2 66.0%

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

      2. cube-mult 66.0%

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

   9. Simplified 66.0%

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

   11. Applied egg-rr 38.5%

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

       1. unpow1 38.5%

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

       2. *-commutative 38.5%

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

       3. associate-*l/ 38.5%

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

   13. Simplified 38.5%

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

       1. associate-*l/ 38.5%

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

       2. add-cube-cbrt 38.3%

          \[\leadsto \frac{2}{{\left(\color{blue}{\left(\left(\sqrt[3]{\frac{{t}^{1.5}}{\ell}} \cdot \sqrt[3]{\frac{{t}^{1.5}}{\ell}}\right) \cdot \sqrt[3]{\frac{{t}^{1.5}}{\ell}}\right)} \cdot \left(k \cdot \sqrt{2}\right)\right)}^{2}} \]

       3. associate-*l* 38.4%

          \[\leadsto \frac{2}{{\color{blue}{\left(\left(\sqrt[3]{\frac{{t}^{1.5}}{\ell}} \cdot \sqrt[3]{\frac{{t}^{1.5}}{\ell}}\right) \cdot \left(\sqrt[3]{\frac{{t}^{1.5}}{\ell}} \cdot \left(k \cdot \sqrt{2}\right)\right)\right)}}^{2}} \]

       4. pow2 38.4%

          \[\leadsto \frac{2}{{\left(\color{blue}{{\left(\sqrt[3]{\frac{{t}^{1.5}}{\ell}}\right)}^{2}} \cdot \left(\sqrt[3]{\frac{{t}^{1.5}}{\ell}} \cdot \left(k \cdot \sqrt{2}\right)\right)\right)}^{2}} \]

       5. cbrt-div 38.4%

          \[\leadsto \frac{2}{{\left({\color{blue}{\left(\frac{\sqrt[3]{{t}^{1.5}}}{\sqrt[3]{\ell}}\right)}}^{2} \cdot \left(\sqrt[3]{\frac{{t}^{1.5}}{\ell}} \cdot \left(k \cdot \sqrt{2}\right)\right)\right)}^{2}} \]

       6. metadata-eval 38.4%

          \[\leadsto \frac{2}{{\left({\left(\frac{\sqrt[3]{{t}^{\color{blue}{\left(\frac{3}{2}\right)}}}}{\sqrt[3]{\ell}}\right)}^{2} \cdot \left(\sqrt[3]{\frac{{t}^{1.5}}{\ell}} \cdot \left(k \cdot \sqrt{2}\right)\right)\right)}^{2}} \]

       7. sqrt-pow1 34.7%

          \[\leadsto \frac{2}{{\left({\left(\frac{\sqrt[3]{\color{blue}{\sqrt{{t}^{3}}}}}{\sqrt[3]{\ell}}\right)}^{2} \cdot \left(\sqrt[3]{\frac{{t}^{1.5}}{\ell}} \cdot \left(k \cdot \sqrt{2}\right)\right)\right)}^{2}} \]

       8. unpow3 34.7%

          \[\leadsto \frac{2}{{\left({\left(\frac{\sqrt[3]{\sqrt{\color{blue}{\left(t \cdot t\right) \cdot t}}}}{\sqrt[3]{\ell}}\right)}^{2} \cdot \left(\sqrt[3]{\frac{{t}^{1.5}}{\ell}} \cdot \left(k \cdot \sqrt{2}\right)\right)\right)}^{2}} \]

       9. sqrt-prod 37.0%

          \[\leadsto \frac{2}{{\left({\left(\frac{\sqrt[3]{\color{blue}{\sqrt{t \cdot t} \cdot \sqrt{t}}}}{\sqrt[3]{\ell}}\right)}^{2} \cdot \left(\sqrt[3]{\frac{{t}^{1.5}}{\ell}} \cdot \left(k \cdot \sqrt{2}\right)\right)\right)}^{2}} \]

       10. sqrt-unprod 38.4%

           \[\leadsto \frac{2}{{\left({\left(\frac{\sqrt[3]{\color{blue}{\left(\sqrt{t} \cdot \sqrt{t}\right)} \cdot \sqrt{t}}}{\sqrt[3]{\ell}}\right)}^{2} \cdot \left(\sqrt[3]{\frac{{t}^{1.5}}{\ell}} \cdot \left(k \cdot \sqrt{2}\right)\right)\right)}^{2}} \]

       11. add-cbrt-cube 38.4%

           \[\leadsto \frac{2}{{\left({\left(\frac{\color{blue}{\sqrt{t}}}{\sqrt[3]{\ell}}\right)}^{2} \cdot \left(\sqrt[3]{\frac{{t}^{1.5}}{\ell}} \cdot \left(k \cdot \sqrt{2}\right)\right)\right)}^{2}} \]

   15. Applied egg-rr 39.7%

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

       1. associate-*r* 39.7%

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

       2. *-commutative 39.7%

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

       3. pow-plus 39.7%

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

       4. metadata-eval 39.7%

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

   17. Simplified 39.7%

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

   if 4.3e14 < k < 5.7999999999999997e152

   1. Initial program 66.5%

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

   3. Simplified 66.7%

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

   5. Taylor expanded in t around 0 93.2%

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

      1. associate-*r* 93.2%

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

      2. times-frac 93.2%

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

   7. Simplified 93.2%

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

   if 5.7999999999999997e152 < k

   1. Initial program 53.3%

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

   3. Simplified 66.6%

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

   5. Taylor expanded in k around 0 60.4%

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

      1. add-cube-cbrt 60.4%

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

      2. pow3 60.4%

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

      3. associate-/l/ 50.3%

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

      4. unpow2 50.3%

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

      5. cbrt-div 50.3%

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

      6. unpow3 50.3%

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

      7. add-cbrt-cube 56.9%

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

      8. unpow2 56.9%

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

      9. cbrt-prod 67.1%

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

      10. unpow2 67.1%

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

      11. cube-mult 67.1%

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

      12. div-inv 67.1%

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

      13. pow-flip 67.1%

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

      14. metadata-eval 67.1%

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

      15. pow2 67.1%

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

      16. div-inv 67.1%

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

      17. pow-flip 67.1%

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

      18. metadata-eval 67.1%

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

   7. Applied egg-rr 67.1%

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

      1. unpow2 67.1%

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

      2. cube-mult 67.1%

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

   9. Simplified 67.1%

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

   11. Applied egg-rr 36.8%

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

       1. unpow1 36.8%

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

       2. *-commutative 36.8%

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

       3. associate-*l/ 36.8%

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

   13. Simplified 36.8%

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

       1. associate-/l* 36.7%

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

       2. add-cbrt-cube 30.0%

          \[\leadsto \frac{2}{{\left(\color{blue}{\sqrt[3]{\left({t}^{1.5} \cdot {t}^{1.5}\right) \cdot {t}^{1.5}}} \cdot \frac{k \cdot \sqrt{2}}{\ell}\right)}^{2}} \]

       3. pow-prod-up 30.0%

          \[\leadsto \frac{2}{{\left(\sqrt[3]{\color{blue}{{t}^{\left(1.5 + 1.5\right)}} \cdot {t}^{1.5}} \cdot \frac{k \cdot \sqrt{2}}{\ell}\right)}^{2}} \]

       4. metadata-eval 30.0%

          \[\leadsto \frac{2}{{\left(\sqrt[3]{{t}^{\color{blue}{3}} \cdot {t}^{1.5}} \cdot \frac{k \cdot \sqrt{2}}{\ell}\right)}^{2}} \]

       5. cbrt-prod 36.7%

          \[\leadsto \frac{2}{{\left(\color{blue}{\left(\sqrt[3]{{t}^{3}} \cdot \sqrt[3]{{t}^{1.5}}\right)} \cdot \frac{k \cdot \sqrt{2}}{\ell}\right)}^{2}} \]

       6. rem-cbrt-cube 36.7%

          \[\leadsto \frac{2}{{\left(\left(\color{blue}{t} \cdot \sqrt[3]{{t}^{1.5}}\right) \cdot \frac{k \cdot \sqrt{2}}{\ell}\right)}^{2}} \]

       7. associate-*l* 36.7%

          \[\leadsto \frac{2}{{\color{blue}{\left(t \cdot \left(\sqrt[3]{{t}^{1.5}} \cdot \frac{k \cdot \sqrt{2}}{\ell}\right)\right)}}^{2}} \]

       8. metadata-eval 36.7%

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

       9. sqrt-pow1 36.9%

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

       10. unpow3 36.9%

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

       11. sqrt-prod 36.7%

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

       12. sqrt-unprod 36.7%

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

       13. add-cbrt-cube 43.4%

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

   15. Applied egg-rr 43.4%

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

4. Final simplification 43.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.3 \cdot 10^{+14}:\\ \;\;\;\;\frac{2}{{\left(\left(k \cdot \sqrt{2}\right) \cdot {\left(\frac{\sqrt{t}}{\sqrt[3]{\ell}}\right)}^{3}\right)}^{2}}\\ \mathbf{elif}\;k \leq 5.8 \cdot 10^{+152}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{2}}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t \cdot \left(\sqrt{t} \cdot \frac{k \cdot \sqrt{2}}{\ell}\right)\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := k\_m \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 3.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{2}{{\left(t\_2 \cdot {\left(\frac{\sqrt{t\_m}}{\sqrt[3]{\ell}}\right)}^{3}\right)}^{2}}\\ \mathbf{elif}\;k\_m \leq 5.3 \cdot 10^{+153}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(t\_m \cdot {k\_m}^{2}\right) \cdot {\ell}^{-2}\right) \cdot {\sin k\_m}^{2}}{\cos k\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t\_m \cdot \left(\sqrt{t\_m} \cdot \frac{t\_2}{\ell}\right)\right)}^{2}}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (* k_m (sqrt 2.0))))
   (*
    t_s
    (if (<= k_m 3.2e+14)
      (/ 2.0 (pow (* t_2 (pow (/ (sqrt t_m) (cbrt l)) 3.0)) 2.0))
      (if (<= k_m 5.3e+153)
        (/
         2.0
         (/
          (* (* (* t_m (pow k_m 2.0)) (pow l -2.0)) (pow (sin k_m) 2.0))
          (cos k_m)))
        (/ 2.0 (pow (* t_m (* (sqrt t_m) (/ t_2 l))) 2.0)))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = k_m * sqrt(2.0);
	double tmp;
	if (k_m <= 3.2e+14) {
		tmp = 2.0 / pow((t_2 * pow((sqrt(t_m) / cbrt(l)), 3.0)), 2.0);
	} else if (k_m <= 5.3e+153) {
		tmp = 2.0 / ((((t_m * pow(k_m, 2.0)) * pow(l, -2.0)) * pow(sin(k_m), 2.0)) / cos(k_m));
	} else {
		tmp = 2.0 / pow((t_m * (sqrt(t_m) * (t_2 / l))), 2.0);
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = k_m * Math.sqrt(2.0);
	double tmp;
	if (k_m <= 3.2e+14) {
		tmp = 2.0 / Math.pow((t_2 * Math.pow((Math.sqrt(t_m) / Math.cbrt(l)), 3.0)), 2.0);
	} else if (k_m <= 5.3e+153) {
		tmp = 2.0 / ((((t_m * Math.pow(k_m, 2.0)) * Math.pow(l, -2.0)) * Math.pow(Math.sin(k_m), 2.0)) / Math.cos(k_m));
	} else {
		tmp = 2.0 / Math.pow((t_m * (Math.sqrt(t_m) * (t_2 / l))), 2.0);
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = Float64(k_m * sqrt(2.0))
	tmp = 0.0
	if (k_m <= 3.2e+14)
		tmp = Float64(2.0 / (Float64(t_2 * (Float64(sqrt(t_m) / cbrt(l)) ^ 3.0)) ^ 2.0));
	elseif (k_m <= 5.3e+153)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m * (k_m ^ 2.0)) * (l ^ -2.0)) * (sin(k_m) ^ 2.0)) / cos(k_m)));
	else
		tmp = Float64(2.0 / (Float64(t_m * Float64(sqrt(t_m) * Float64(t_2 / l))) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := Block[{t$95$2 = N[(k$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 3.2e+14], N[(2.0 / N[Power[N[(t$95$2 * N[Power[N[(N[Sqrt[t$95$m], $MachinePrecision] / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 5.3e+153], N[(2.0 / N[(N[(N[(N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[l, -2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(t$95$m * N[(N[Sqrt[t$95$m], $MachinePrecision] * N[(t$95$2 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 3.2e14

   1. Initial program 58.2%

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

   3. Simplified 59.7%

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

   5. Taylor expanded in k around 0 60.4%

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

      1. add-cube-cbrt 60.4%

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

      2. pow3 60.4%

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

      3. associate-/l/ 55.0%

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

      4. unpow2 55.0%

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

      5. cbrt-div 55.0%

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

      6. unpow3 55.0%

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

      7. add-cbrt-cube 60.0%

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

      8. unpow2 60.0%

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

      9. cbrt-prod 66.0%

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

      10. unpow2 66.0%

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

      11. cube-mult 66.0%

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

      12. div-inv 66.0%

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

      13. pow-flip 66.0%

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

      14. metadata-eval 66.0%

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

      15. pow2 66.0%

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

      16. div-inv 66.0%

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

      17. pow-flip 66.0%

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

      18. metadata-eval 66.0%

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

   7. Applied egg-rr 66.0%

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

      1. unpow2 66.0%

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

      2. cube-mult 66.0%

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

   9. Simplified 66.0%

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

   11. Applied egg-rr 38.5%

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

       1. unpow1 38.5%

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

       2. *-commutative 38.5%

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

       3. associate-*l/ 38.5%

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

   13. Simplified 38.5%

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

       1. associate-*l/ 38.5%

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

       2. add-cube-cbrt 38.3%

          \[\leadsto \frac{2}{{\left(\color{blue}{\left(\left(\sqrt[3]{\frac{{t}^{1.5}}{\ell}} \cdot \sqrt[3]{\frac{{t}^{1.5}}{\ell}}\right) \cdot \sqrt[3]{\frac{{t}^{1.5}}{\ell}}\right)} \cdot \left(k \cdot \sqrt{2}\right)\right)}^{2}} \]

       3. associate-*l* 38.4%

          \[\leadsto \frac{2}{{\color{blue}{\left(\left(\sqrt[3]{\frac{{t}^{1.5}}{\ell}} \cdot \sqrt[3]{\frac{{t}^{1.5}}{\ell}}\right) \cdot \left(\sqrt[3]{\frac{{t}^{1.5}}{\ell}} \cdot \left(k \cdot \sqrt{2}\right)\right)\right)}}^{2}} \]

       4. pow2 38.4%

          \[\leadsto \frac{2}{{\left(\color{blue}{{\left(\sqrt[3]{\frac{{t}^{1.5}}{\ell}}\right)}^{2}} \cdot \left(\sqrt[3]{\frac{{t}^{1.5}}{\ell}} \cdot \left(k \cdot \sqrt{2}\right)\right)\right)}^{2}} \]

       5. cbrt-div 38.4%

          \[\leadsto \frac{2}{{\left({\color{blue}{\left(\frac{\sqrt[3]{{t}^{1.5}}}{\sqrt[3]{\ell}}\right)}}^{2} \cdot \left(\sqrt[3]{\frac{{t}^{1.5}}{\ell}} \cdot \left(k \cdot \sqrt{2}\right)\right)\right)}^{2}} \]

       6. metadata-eval 38.4%

          \[\leadsto \frac{2}{{\left({\left(\frac{\sqrt[3]{{t}^{\color{blue}{\left(\frac{3}{2}\right)}}}}{\sqrt[3]{\ell}}\right)}^{2} \cdot \left(\sqrt[3]{\frac{{t}^{1.5}}{\ell}} \cdot \left(k \cdot \sqrt{2}\right)\right)\right)}^{2}} \]

       7. sqrt-pow1 34.7%

          \[\leadsto \frac{2}{{\left({\left(\frac{\sqrt[3]{\color{blue}{\sqrt{{t}^{3}}}}}{\sqrt[3]{\ell}}\right)}^{2} \cdot \left(\sqrt[3]{\frac{{t}^{1.5}}{\ell}} \cdot \left(k \cdot \sqrt{2}\right)\right)\right)}^{2}} \]

       8. unpow3 34.7%

          \[\leadsto \frac{2}{{\left({\left(\frac{\sqrt[3]{\sqrt{\color{blue}{\left(t \cdot t\right) \cdot t}}}}{\sqrt[3]{\ell}}\right)}^{2} \cdot \left(\sqrt[3]{\frac{{t}^{1.5}}{\ell}} \cdot \left(k \cdot \sqrt{2}\right)\right)\right)}^{2}} \]

       9. sqrt-prod 37.0%

          \[\leadsto \frac{2}{{\left({\left(\frac{\sqrt[3]{\color{blue}{\sqrt{t \cdot t} \cdot \sqrt{t}}}}{\sqrt[3]{\ell}}\right)}^{2} \cdot \left(\sqrt[3]{\frac{{t}^{1.5}}{\ell}} \cdot \left(k \cdot \sqrt{2}\right)\right)\right)}^{2}} \]

       10. sqrt-unprod 38.4%

           \[\leadsto \frac{2}{{\left({\left(\frac{\sqrt[3]{\color{blue}{\left(\sqrt{t} \cdot \sqrt{t}\right)} \cdot \sqrt{t}}}{\sqrt[3]{\ell}}\right)}^{2} \cdot \left(\sqrt[3]{\frac{{t}^{1.5}}{\ell}} \cdot \left(k \cdot \sqrt{2}\right)\right)\right)}^{2}} \]

       11. add-cbrt-cube 38.4%

           \[\leadsto \frac{2}{{\left({\left(\frac{\color{blue}{\sqrt{t}}}{\sqrt[3]{\ell}}\right)}^{2} \cdot \left(\sqrt[3]{\frac{{t}^{1.5}}{\ell}} \cdot \left(k \cdot \sqrt{2}\right)\right)\right)}^{2}} \]

   15. Applied egg-rr 39.7%

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

       1. associate-*r* 39.7%

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

       2. *-commutative 39.7%

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

       3. pow-plus 39.7%

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

       4. metadata-eval 39.7%

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

   17. Simplified 39.7%

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

   if 3.2e14 < k < 5.2999999999999999e153

   1. Initial program 66.5%

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

   3. Simplified 66.7%

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

   5. Taylor expanded in t around 0 93.2%

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

      1. associate-*r* 93.2%

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

      2. times-frac 93.2%

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

   7. Simplified 93.2%

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

      1. associate-*r/ 93.2%

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

      2. div-inv 93.2%

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

      3. *-commutative 93.2%

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

      4. pow-flip 93.1%

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

      5. metadata-eval 93.1%

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

   9. Applied egg-rr 93.1%

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

   if 5.2999999999999999e153 < k

   1. Initial program 53.3%

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

   3. Simplified 66.6%

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

   5. Taylor expanded in k around 0 60.4%

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

      1. add-cube-cbrt 60.4%

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

      2. pow3 60.4%

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

      3. associate-/l/ 50.3%

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

      4. unpow2 50.3%

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

      5. cbrt-div 50.3%

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

      6. unpow3 50.3%

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

      7. add-cbrt-cube 56.9%

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

      8. unpow2 56.9%

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

      9. cbrt-prod 67.1%

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

      10. unpow2 67.1%

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

      11. cube-mult 67.1%

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

      12. div-inv 67.1%

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

      13. pow-flip 67.1%

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

      14. metadata-eval 67.1%

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

      15. pow2 67.1%

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

      16. div-inv 67.1%

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

      17. pow-flip 67.1%

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

      18. metadata-eval 67.1%

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

   7. Applied egg-rr 67.1%

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

      1. unpow2 67.1%

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

      2. cube-mult 67.1%

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

   9. Simplified 67.1%

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

   11. Applied egg-rr 36.8%

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

       1. unpow1 36.8%

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

       2. *-commutative 36.8%

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

       3. associate-*l/ 36.8%

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

   13. Simplified 36.8%

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

       1. associate-/l* 36.7%

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

       2. add-cbrt-cube 30.0%

          \[\leadsto \frac{2}{{\left(\color{blue}{\sqrt[3]{\left({t}^{1.5} \cdot {t}^{1.5}\right) \cdot {t}^{1.5}}} \cdot \frac{k \cdot \sqrt{2}}{\ell}\right)}^{2}} \]

       3. pow-prod-up 30.0%

          \[\leadsto \frac{2}{{\left(\sqrt[3]{\color{blue}{{t}^{\left(1.5 + 1.5\right)}} \cdot {t}^{1.5}} \cdot \frac{k \cdot \sqrt{2}}{\ell}\right)}^{2}} \]

       4. metadata-eval 30.0%

          \[\leadsto \frac{2}{{\left(\sqrt[3]{{t}^{\color{blue}{3}} \cdot {t}^{1.5}} \cdot \frac{k \cdot \sqrt{2}}{\ell}\right)}^{2}} \]

       5. cbrt-prod 36.7%

          \[\leadsto \frac{2}{{\left(\color{blue}{\left(\sqrt[3]{{t}^{3}} \cdot \sqrt[3]{{t}^{1.5}}\right)} \cdot \frac{k \cdot \sqrt{2}}{\ell}\right)}^{2}} \]

       6. rem-cbrt-cube 36.7%

          \[\leadsto \frac{2}{{\left(\left(\color{blue}{t} \cdot \sqrt[3]{{t}^{1.5}}\right) \cdot \frac{k \cdot \sqrt{2}}{\ell}\right)}^{2}} \]

       7. associate-*l* 36.7%

          \[\leadsto \frac{2}{{\color{blue}{\left(t \cdot \left(\sqrt[3]{{t}^{1.5}} \cdot \frac{k \cdot \sqrt{2}}{\ell}\right)\right)}}^{2}} \]

       8. metadata-eval 36.7%

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

       9. sqrt-pow1 36.9%

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

       10. unpow3 36.9%

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

       11. sqrt-prod 36.7%

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

       12. sqrt-unprod 36.7%

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

       13. add-cbrt-cube 43.4%

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

   15. Applied egg-rr 43.4%

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

4. Final simplification 43.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{2}{{\left(\left(k \cdot \sqrt{2}\right) \cdot {\left(\frac{\sqrt{t}}{\sqrt[3]{\ell}}\right)}^{3}\right)}^{2}}\\ \mathbf{elif}\;k \leq 5.3 \cdot 10^{+153}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(t \cdot {k}^{2}\right) \cdot {\ell}^{-2}\right) \cdot {\sin k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t \cdot \left(\sqrt{t} \cdot \frac{k \cdot \sqrt{2}}{\ell}\right)\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := k\_m \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2.1 \cdot 10^{+14}:\\ \;\;\;\;\frac{2}{{\left(t\_2 \cdot {\left(\frac{\sqrt{t\_m}}{\sqrt[3]{\ell}}\right)}^{3}\right)}^{2}}\\ \mathbf{elif}\;k\_m \leq 5.3 \cdot 10^{+153}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot {k\_m}^{2}}{{\ell}^{2}} \cdot \frac{0.5 - \frac{\cos \left(k\_m \cdot 2\right)}{2}}{\cos k\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t\_m \cdot \left(\sqrt{t\_m} \cdot \frac{t\_2}{\ell}\right)\right)}^{2}}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (* k_m (sqrt 2.0))))
   (*
    t_s
    (if (<= k_m 2.1e+14)
      (/ 2.0 (pow (* t_2 (pow (/ (sqrt t_m) (cbrt l)) 3.0)) 2.0))
      (if (<= k_m 5.3e+153)
        (/
         2.0
         (*
          (/ (* t_m (pow k_m 2.0)) (pow l 2.0))
          (/ (- 0.5 (/ (cos (* k_m 2.0)) 2.0)) (cos k_m))))
        (/ 2.0 (pow (* t_m (* (sqrt t_m) (/ t_2 l))) 2.0)))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = k_m * sqrt(2.0);
	double tmp;
	if (k_m <= 2.1e+14) {
		tmp = 2.0 / pow((t_2 * pow((sqrt(t_m) / cbrt(l)), 3.0)), 2.0);
	} else if (k_m <= 5.3e+153) {
		tmp = 2.0 / (((t_m * pow(k_m, 2.0)) / pow(l, 2.0)) * ((0.5 - (cos((k_m * 2.0)) / 2.0)) / cos(k_m)));
	} else {
		tmp = 2.0 / pow((t_m * (sqrt(t_m) * (t_2 / l))), 2.0);
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = k_m * Math.sqrt(2.0);
	double tmp;
	if (k_m <= 2.1e+14) {
		tmp = 2.0 / Math.pow((t_2 * Math.pow((Math.sqrt(t_m) / Math.cbrt(l)), 3.0)), 2.0);
	} else if (k_m <= 5.3e+153) {
		tmp = 2.0 / (((t_m * Math.pow(k_m, 2.0)) / Math.pow(l, 2.0)) * ((0.5 - (Math.cos((k_m * 2.0)) / 2.0)) / Math.cos(k_m)));
	} else {
		tmp = 2.0 / Math.pow((t_m * (Math.sqrt(t_m) * (t_2 / l))), 2.0);
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = Float64(k_m * sqrt(2.0))
	tmp = 0.0
	if (k_m <= 2.1e+14)
		tmp = Float64(2.0 / (Float64(t_2 * (Float64(sqrt(t_m) / cbrt(l)) ^ 3.0)) ^ 2.0));
	elseif (k_m <= 5.3e+153)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k_m ^ 2.0)) / (l ^ 2.0)) * Float64(Float64(0.5 - Float64(cos(Float64(k_m * 2.0)) / 2.0)) / cos(k_m))));
	else
		tmp = Float64(2.0 / (Float64(t_m * Float64(sqrt(t_m) * Float64(t_2 / l))) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := Block[{t$95$2 = N[(k$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 2.1e+14], N[(2.0 / N[Power[N[(t$95$2 * N[Power[N[(N[Sqrt[t$95$m], $MachinePrecision] / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 5.3e+153], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 - N[(N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(t$95$m * N[(N[Sqrt[t$95$m], $MachinePrecision] * N[(t$95$2 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 2.1e14

   1. Initial program 58.2%

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

   3. Simplified 59.7%

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

   5. Taylor expanded in k around 0 60.4%

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

      1. add-cube-cbrt 60.4%

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

      2. pow3 60.4%

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

      3. associate-/l/ 55.0%

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

      4. unpow2 55.0%

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

      5. cbrt-div 55.0%

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

      6. unpow3 55.0%

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

      7. add-cbrt-cube 60.0%

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

      8. unpow2 60.0%

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

      9. cbrt-prod 66.0%

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

      10. unpow2 66.0%

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

      11. cube-mult 66.0%

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

      12. div-inv 66.0%

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

      13. pow-flip 66.0%

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

      14. metadata-eval 66.0%

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

      15. pow2 66.0%

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

      16. div-inv 66.0%

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

      17. pow-flip 66.0%

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

      18. metadata-eval 66.0%

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

   7. Applied egg-rr 66.0%

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

      1. unpow2 66.0%

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

      2. cube-mult 66.0%

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

   9. Simplified 66.0%

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

   11. Applied egg-rr 38.5%

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

       1. unpow1 38.5%

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

       2. *-commutative 38.5%

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

       3. associate-*l/ 38.5%

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

   13. Simplified 38.5%

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

       1. associate-*l/ 38.5%

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

       2. add-cube-cbrt 38.3%

          \[\leadsto \frac{2}{{\left(\color{blue}{\left(\left(\sqrt[3]{\frac{{t}^{1.5}}{\ell}} \cdot \sqrt[3]{\frac{{t}^{1.5}}{\ell}}\right) \cdot \sqrt[3]{\frac{{t}^{1.5}}{\ell}}\right)} \cdot \left(k \cdot \sqrt{2}\right)\right)}^{2}} \]

       3. associate-*l* 38.4%

          \[\leadsto \frac{2}{{\color{blue}{\left(\left(\sqrt[3]{\frac{{t}^{1.5}}{\ell}} \cdot \sqrt[3]{\frac{{t}^{1.5}}{\ell}}\right) \cdot \left(\sqrt[3]{\frac{{t}^{1.5}}{\ell}} \cdot \left(k \cdot \sqrt{2}\right)\right)\right)}}^{2}} \]

       4. pow2 38.4%

          \[\leadsto \frac{2}{{\left(\color{blue}{{\left(\sqrt[3]{\frac{{t}^{1.5}}{\ell}}\right)}^{2}} \cdot \left(\sqrt[3]{\frac{{t}^{1.5}}{\ell}} \cdot \left(k \cdot \sqrt{2}\right)\right)\right)}^{2}} \]

       5. cbrt-div 38.4%

          \[\leadsto \frac{2}{{\left({\color{blue}{\left(\frac{\sqrt[3]{{t}^{1.5}}}{\sqrt[3]{\ell}}\right)}}^{2} \cdot \left(\sqrt[3]{\frac{{t}^{1.5}}{\ell}} \cdot \left(k \cdot \sqrt{2}\right)\right)\right)}^{2}} \]

       6. metadata-eval 38.4%

          \[\leadsto \frac{2}{{\left({\left(\frac{\sqrt[3]{{t}^{\color{blue}{\left(\frac{3}{2}\right)}}}}{\sqrt[3]{\ell}}\right)}^{2} \cdot \left(\sqrt[3]{\frac{{t}^{1.5}}{\ell}} \cdot \left(k \cdot \sqrt{2}\right)\right)\right)}^{2}} \]

       7. sqrt-pow1 34.7%

          \[\leadsto \frac{2}{{\left({\left(\frac{\sqrt[3]{\color{blue}{\sqrt{{t}^{3}}}}}{\sqrt[3]{\ell}}\right)}^{2} \cdot \left(\sqrt[3]{\frac{{t}^{1.5}}{\ell}} \cdot \left(k \cdot \sqrt{2}\right)\right)\right)}^{2}} \]

       8. unpow3 34.7%

          \[\leadsto \frac{2}{{\left({\left(\frac{\sqrt[3]{\sqrt{\color{blue}{\left(t \cdot t\right) \cdot t}}}}{\sqrt[3]{\ell}}\right)}^{2} \cdot \left(\sqrt[3]{\frac{{t}^{1.5}}{\ell}} \cdot \left(k \cdot \sqrt{2}\right)\right)\right)}^{2}} \]

       9. sqrt-prod 37.0%

          \[\leadsto \frac{2}{{\left({\left(\frac{\sqrt[3]{\color{blue}{\sqrt{t \cdot t} \cdot \sqrt{t}}}}{\sqrt[3]{\ell}}\right)}^{2} \cdot \left(\sqrt[3]{\frac{{t}^{1.5}}{\ell}} \cdot \left(k \cdot \sqrt{2}\right)\right)\right)}^{2}} \]

       10. sqrt-unprod 38.4%

           \[\leadsto \frac{2}{{\left({\left(\frac{\sqrt[3]{\color{blue}{\left(\sqrt{t} \cdot \sqrt{t}\right)} \cdot \sqrt{t}}}{\sqrt[3]{\ell}}\right)}^{2} \cdot \left(\sqrt[3]{\frac{{t}^{1.5}}{\ell}} \cdot \left(k \cdot \sqrt{2}\right)\right)\right)}^{2}} \]

       11. add-cbrt-cube 38.4%

           \[\leadsto \frac{2}{{\left({\left(\frac{\color{blue}{\sqrt{t}}}{\sqrt[3]{\ell}}\right)}^{2} \cdot \left(\sqrt[3]{\frac{{t}^{1.5}}{\ell}} \cdot \left(k \cdot \sqrt{2}\right)\right)\right)}^{2}} \]

   15. Applied egg-rr 39.7%

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

       1. associate-*r* 39.7%

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

       2. *-commutative 39.7%

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

       3. pow-plus 39.7%

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

       4. metadata-eval 39.7%

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

   17. Simplified 39.7%

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

   if 2.1e14 < k < 5.2999999999999999e153

   1. Initial program 66.5%

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

   3. Simplified 66.7%

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

   5. Taylor expanded in t around 0 93.2%

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

      1. associate-*r* 93.2%

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

      2. times-frac 93.2%

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

   7. Simplified 93.2%

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

      1. unpow2 93.2%

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

      2. sin-mult 93.3%

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

   9. Applied egg-rr 93.3%

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

       1. div-sub 93.3%

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

       2. +-inverses 93.3%

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

       3. cos-0 93.3%

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

       4. metadata-eval 93.3%

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

       5. count-2 93.3%

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

       6. *-commutative 93.3%

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

   11. Simplified 93.3%

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

   if 5.2999999999999999e153 < k

   1. Initial program 53.3%

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

   3. Simplified 66.6%

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

   5. Taylor expanded in k around 0 60.4%

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

      1. add-cube-cbrt 60.4%

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

      2. pow3 60.4%

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

      3. associate-/l/ 50.3%

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

      4. unpow2 50.3%

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

      5. cbrt-div 50.3%

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

      6. unpow3 50.3%

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

      7. add-cbrt-cube 56.9%

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

      8. unpow2 56.9%

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

      9. cbrt-prod 67.1%

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

      10. unpow2 67.1%

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

      11. cube-mult 67.1%

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

      12. div-inv 67.1%

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

      13. pow-flip 67.1%

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

      14. metadata-eval 67.1%

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

      15. pow2 67.1%

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

      16. div-inv 67.1%

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

      17. pow-flip 67.1%

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

      18. metadata-eval 67.1%

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

   7. Applied egg-rr 67.1%

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

      1. unpow2 67.1%

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

      2. cube-mult 67.1%

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

   9. Simplified 67.1%

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

   11. Applied egg-rr 36.8%

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

       1. unpow1 36.8%

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

       2. *-commutative 36.8%

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

       3. associate-*l/ 36.8%

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

   13. Simplified 36.8%

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

       1. associate-/l* 36.7%

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

       2. add-cbrt-cube 30.0%

          \[\leadsto \frac{2}{{\left(\color{blue}{\sqrt[3]{\left({t}^{1.5} \cdot {t}^{1.5}\right) \cdot {t}^{1.5}}} \cdot \frac{k \cdot \sqrt{2}}{\ell}\right)}^{2}} \]

       3. pow-prod-up 30.0%

          \[\leadsto \frac{2}{{\left(\sqrt[3]{\color{blue}{{t}^{\left(1.5 + 1.5\right)}} \cdot {t}^{1.5}} \cdot \frac{k \cdot \sqrt{2}}{\ell}\right)}^{2}} \]

       4. metadata-eval 30.0%

          \[\leadsto \frac{2}{{\left(\sqrt[3]{{t}^{\color{blue}{3}} \cdot {t}^{1.5}} \cdot \frac{k \cdot \sqrt{2}}{\ell}\right)}^{2}} \]

       5. cbrt-prod 36.7%

          \[\leadsto \frac{2}{{\left(\color{blue}{\left(\sqrt[3]{{t}^{3}} \cdot \sqrt[3]{{t}^{1.5}}\right)} \cdot \frac{k \cdot \sqrt{2}}{\ell}\right)}^{2}} \]

       6. rem-cbrt-cube 36.7%

          \[\leadsto \frac{2}{{\left(\left(\color{blue}{t} \cdot \sqrt[3]{{t}^{1.5}}\right) \cdot \frac{k \cdot \sqrt{2}}{\ell}\right)}^{2}} \]

       7. associate-*l* 36.7%

          \[\leadsto \frac{2}{{\color{blue}{\left(t \cdot \left(\sqrt[3]{{t}^{1.5}} \cdot \frac{k \cdot \sqrt{2}}{\ell}\right)\right)}}^{2}} \]

       8. metadata-eval 36.7%

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

       9. sqrt-pow1 36.9%

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

       10. unpow3 36.9%

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

       11. sqrt-prod 36.7%

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

       12. sqrt-unprod 36.7%

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

       13. add-cbrt-cube 43.4%

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

   15. Applied egg-rr 43.4%

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

4. Final simplification 43.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.1 \cdot 10^{+14}:\\ \;\;\;\;\frac{2}{{\left(\left(k \cdot \sqrt{2}\right) \cdot {\left(\frac{\sqrt{t}}{\sqrt[3]{\ell}}\right)}^{3}\right)}^{2}}\\ \mathbf{elif}\;k \leq 5.3 \cdot 10^{+153}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{2}}{{\ell}^{2}} \cdot \frac{0.5 - \frac{\cos \left(k \cdot 2\right)}{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t \cdot \left(\sqrt{t} \cdot \frac{k \cdot \sqrt{2}}{\ell}\right)\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 9.5 \cdot 10^{+14} \lor \neg \left(k\_m \leq 5.3 \cdot 10^{+153}\right):\\ \;\;\;\;\frac{2}{{\left(t\_m \cdot \left(\sqrt{t\_m} \cdot \frac{k\_m \cdot \sqrt{2}}{\ell}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot {k\_m}^{2}}{{\ell}^{2}} \cdot \frac{0.5 - \frac{\cos \left(k\_m \cdot 2\right)}{2}}{\cos k\_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (or (<= k_m 9.5e+14) (not (<= k_m 5.3e+153)))
    (/ 2.0 (pow (* t_m (* (sqrt t_m) (/ (* k_m (sqrt 2.0)) l))) 2.0))
    (/
     2.0
     (*
      (/ (* t_m (pow k_m 2.0)) (pow l 2.0))
      (/ (- 0.5 (/ (cos (* k_m 2.0)) 2.0)) (cos k_m)))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if ((k_m <= 9.5e+14) || !(k_m <= 5.3e+153)) {
		tmp = 2.0 / pow((t_m * (sqrt(t_m) * ((k_m * sqrt(2.0)) / l))), 2.0);
	} else {
		tmp = 2.0 / (((t_m * pow(k_m, 2.0)) / pow(l, 2.0)) * ((0.5 - (cos((k_m * 2.0)) / 2.0)) / cos(k_m)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if ((k_m <= 9.5d+14) .or. (.not. (k_m <= 5.3d+153))) then
        tmp = 2.0d0 / ((t_m * (sqrt(t_m) * ((k_m * sqrt(2.0d0)) / l))) ** 2.0d0)
    else
        tmp = 2.0d0 / (((t_m * (k_m ** 2.0d0)) / (l ** 2.0d0)) * ((0.5d0 - (cos((k_m * 2.0d0)) / 2.0d0)) / cos(k_m)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if ((k_m <= 9.5e+14) || !(k_m <= 5.3e+153)) {
		tmp = 2.0 / Math.pow((t_m * (Math.sqrt(t_m) * ((k_m * Math.sqrt(2.0)) / l))), 2.0);
	} else {
		tmp = 2.0 / (((t_m * Math.pow(k_m, 2.0)) / Math.pow(l, 2.0)) * ((0.5 - (Math.cos((k_m * 2.0)) / 2.0)) / Math.cos(k_m)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if (k_m <= 9.5e+14) or not (k_m <= 5.3e+153):
		tmp = 2.0 / math.pow((t_m * (math.sqrt(t_m) * ((k_m * math.sqrt(2.0)) / l))), 2.0)
	else:
		tmp = 2.0 / (((t_m * math.pow(k_m, 2.0)) / math.pow(l, 2.0)) * ((0.5 - (math.cos((k_m * 2.0)) / 2.0)) / math.cos(k_m)))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if ((k_m <= 9.5e+14) || !(k_m <= 5.3e+153))
		tmp = Float64(2.0 / (Float64(t_m * Float64(sqrt(t_m) * Float64(Float64(k_m * sqrt(2.0)) / l))) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k_m ^ 2.0)) / (l ^ 2.0)) * Float64(Float64(0.5 - Float64(cos(Float64(k_m * 2.0)) / 2.0)) / cos(k_m))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if ((k_m <= 9.5e+14) || ~((k_m <= 5.3e+153)))
		tmp = 2.0 / ((t_m * (sqrt(t_m) * ((k_m * sqrt(2.0)) / l))) ^ 2.0);
	else
		tmp = 2.0 / (((t_m * (k_m ^ 2.0)) / (l ^ 2.0)) * ((0.5 - (cos((k_m * 2.0)) / 2.0)) / cos(k_m)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[Or[LessEqual[k$95$m, 9.5e+14], N[Not[LessEqual[k$95$m, 5.3e+153]], $MachinePrecision]], N[(2.0 / N[Power[N[(t$95$m * N[(N[Sqrt[t$95$m], $MachinePrecision] * N[(N[(k$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 - N[(N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 9.5e14 or 5.2999999999999999e153 < k

   1. Initial program 57.6%

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

   3. Simplified 60.5%

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

   5. Taylor expanded in k around 0 60.4%

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

      1. add-cube-cbrt 60.4%

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

      2. pow3 60.4%

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

      3. associate-/l/ 54.4%

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

      4. unpow2 54.4%

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

      5. cbrt-div 54.4%

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

      6. unpow3 54.4%

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

      7. add-cbrt-cube 59.7%

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

      8. unpow2 59.7%

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

      9. cbrt-prod 66.1%

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

      10. unpow2 66.1%

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

      11. cube-mult 66.1%

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

      12. div-inv 66.1%

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

      13. pow-flip 66.1%

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

      14. metadata-eval 66.1%

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

      15. pow2 66.1%

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

      16. div-inv 66.1%

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

      17. pow-flip 66.1%

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

      18. metadata-eval 66.1%

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

   7. Applied egg-rr 66.1%

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

      1. unpow2 66.1%

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

      2. cube-mult 66.1%

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

   9. Simplified 66.1%

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

   11. Applied egg-rr 38.3%

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

       1. unpow1 38.3%

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

       2. *-commutative 38.3%

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

       3. associate-*l/ 38.3%

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

   13. Simplified 38.3%

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

       1. associate-/l* 37.9%

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

       2. add-cbrt-cube 33.7%

          \[\leadsto \frac{2}{{\left(\color{blue}{\sqrt[3]{\left({t}^{1.5} \cdot {t}^{1.5}\right) \cdot {t}^{1.5}}} \cdot \frac{k \cdot \sqrt{2}}{\ell}\right)}^{2}} \]

       3. pow-prod-up 33.7%

          \[\leadsto \frac{2}{{\left(\sqrt[3]{\color{blue}{{t}^{\left(1.5 + 1.5\right)}} \cdot {t}^{1.5}} \cdot \frac{k \cdot \sqrt{2}}{\ell}\right)}^{2}} \]

       4. metadata-eval 33.7%

          \[\leadsto \frac{2}{{\left(\sqrt[3]{{t}^{\color{blue}{3}} \cdot {t}^{1.5}} \cdot \frac{k \cdot \sqrt{2}}{\ell}\right)}^{2}} \]

       5. cbrt-prod 34.9%

          \[\leadsto \frac{2}{{\left(\color{blue}{\left(\sqrt[3]{{t}^{3}} \cdot \sqrt[3]{{t}^{1.5}}\right)} \cdot \frac{k \cdot \sqrt{2}}{\ell}\right)}^{2}} \]

       6. rem-cbrt-cube 37.9%

          \[\leadsto \frac{2}{{\left(\left(\color{blue}{t} \cdot \sqrt[3]{{t}^{1.5}}\right) \cdot \frac{k \cdot \sqrt{2}}{\ell}\right)}^{2}} \]

       7. associate-*l* 37.9%

          \[\leadsto \frac{2}{{\color{blue}{\left(t \cdot \left(\sqrt[3]{{t}^{1.5}} \cdot \frac{k \cdot \sqrt{2}}{\ell}\right)\right)}}^{2}} \]

       8. metadata-eval 37.9%

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

       9. sqrt-pow1 35.5%

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

       10. unpow3 35.5%

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

       11. sqrt-prod 37.4%

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

       12. sqrt-unprod 37.8%

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

       13. add-cbrt-cube 39.6%

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

   15. Applied egg-rr 39.6%

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

   if 9.5e14 < k < 5.2999999999999999e153

   1. Initial program 66.5%

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

   3. Simplified 66.7%

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

   5. Taylor expanded in t around 0 93.2%

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

      1. associate-*r* 93.2%

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

      2. times-frac 93.2%

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

   7. Simplified 93.2%

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

      1. unpow2 93.2%

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

      2. sin-mult 93.3%

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

   9. Applied egg-rr 93.3%

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

       1. div-sub 93.3%

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

       2. +-inverses 93.3%

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

       3. cos-0 93.3%

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

       4. metadata-eval 93.3%

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

       5. count-2 93.3%

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

       6. *-commutative 93.3%

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

   11. Simplified 93.3%

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

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.5 \cdot 10^{+14} \lor \neg \left(k \leq 5.3 \cdot 10^{+153}\right):\\ \;\;\;\;\frac{2}{{\left(t \cdot \left(\sqrt{t} \cdot \frac{k \cdot \sqrt{2}}{\ell}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{2}}{{\ell}^{2}} \cdot \frac{0.5 - \frac{\cos \left(k \cdot 2\right)}{2}}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.4 \cdot 10^{-96}:\\ \;\;\;\;2 \cdot {\left(\ell \cdot \frac{{k\_m}^{-2}}{\sqrt{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t\_m \cdot \left(\sqrt{t\_m} \cdot \frac{k\_m \cdot \sqrt{2}}{\ell}\right)\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.4e-96)
    (* 2.0 (pow (* l (/ (pow k_m -2.0) (sqrt t_m))) 2.0))
    (/ 2.0 (pow (* t_m (* (sqrt t_m) (/ (* k_m (sqrt 2.0)) l))) 2.0)))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (t_m <= 3.4e-96) {
		tmp = 2.0 * pow((l * (pow(k_m, -2.0) / sqrt(t_m))), 2.0);
	} else {
		tmp = 2.0 / pow((t_m * (sqrt(t_m) * ((k_m * sqrt(2.0)) / l))), 2.0);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t_m <= 3.4d-96) then
        tmp = 2.0d0 * ((l * ((k_m ** (-2.0d0)) / sqrt(t_m))) ** 2.0d0)
    else
        tmp = 2.0d0 / ((t_m * (sqrt(t_m) * ((k_m * sqrt(2.0d0)) / l))) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (t_m <= 3.4e-96) {
		tmp = 2.0 * Math.pow((l * (Math.pow(k_m, -2.0) / Math.sqrt(t_m))), 2.0);
	} else {
		tmp = 2.0 / Math.pow((t_m * (Math.sqrt(t_m) * ((k_m * Math.sqrt(2.0)) / l))), 2.0);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if t_m <= 3.4e-96:
		tmp = 2.0 * math.pow((l * (math.pow(k_m, -2.0) / math.sqrt(t_m))), 2.0)
	else:
		tmp = 2.0 / math.pow((t_m * (math.sqrt(t_m) * ((k_m * math.sqrt(2.0)) / l))), 2.0)
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (t_m <= 3.4e-96)
		tmp = Float64(2.0 * (Float64(l * Float64((k_m ^ -2.0) / sqrt(t_m))) ^ 2.0));
	else
		tmp = Float64(2.0 / (Float64(t_m * Float64(sqrt(t_m) * Float64(Float64(k_m * sqrt(2.0)) / l))) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (t_m <= 3.4e-96)
		tmp = 2.0 * ((l * ((k_m ^ -2.0) / sqrt(t_m))) ^ 2.0);
	else
		tmp = 2.0 / ((t_m * (sqrt(t_m) * ((k_m * sqrt(2.0)) / l))) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 3.4e-96], N[(2.0 * N[Power[N[(l * N[(N[Power[k$95$m, -2.0], $MachinePrecision] / N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(t$95$m * N[(N[Sqrt[t$95$m], $MachinePrecision] * N[(N[(k$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.4000000000000001e-96

   1. Initial program 52.2%

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

   3. Simplified 50.0%

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

   5. Taylor expanded in k around 0 46.9%

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

      1. associate-/r* 47.9%

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

   7. Simplified 47.9%

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

   8. Taylor expanded in k around inf 57.0%

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

      1. *-commutative 57.0%

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

      2. associate-/r* 58.3%

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

   10. Simplified 58.3%

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

       1. *-un-lft-identity 58.3%

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

       2. div-inv 58.3%

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

       3. pow-flip 58.3%

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

       4. metadata-eval 58.3%

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

   12. Applied egg-rr 58.3%

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

       1. *-lft-identity 58.3%

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

       2. associate-*l/ 57.9%

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

       3. associate-/l* 57.0%

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

   14. Simplified 57.0%

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

       1. add-sqr-sqrt 38.7%

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

       2. pow2 38.7%

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

       3. sqrt-prod 35.3%

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

       4. sqrt-pow1 39.4%

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

       5. metadata-eval 39.4%

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

       6. pow1 39.4%

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

       7. sqrt-div 18.2%

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

       8. sqrt-pow1 18.2%

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

       9. metadata-eval 18.2%

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

   16. Applied egg-rr 18.2%

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

   if 3.4000000000000001e-96 < t

   1. Initial program 71.7%

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

   3. Simplified 68.0%

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

   5. Taylor expanded in k around 0 65.6%

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

      1. add-cube-cbrt 65.5%

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

      2. pow3 65.5%

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

      3. associate-/l/ 61.5%

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

      4. unpow2 61.5%

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

      5. cbrt-div 61.5%

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

      6. unpow3 61.5%

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

      7. add-cbrt-cube 64.6%

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

      8. unpow2 64.6%

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

      9. cbrt-prod 71.8%

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

      10. unpow2 71.8%

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

      11. cube-mult 71.7%

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

      12. div-inv 71.8%

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

      13. pow-flip 71.8%

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

      14. metadata-eval 71.8%

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

      15. pow2 71.8%

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

      16. div-inv 71.7%

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

      17. pow-flip 71.7%

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

      18. metadata-eval 71.7%

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

   7. Applied egg-rr 71.7%

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

      1. unpow2 71.7%

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

      2. cube-mult 71.7%

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

   9. Simplified 71.7%

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

   11. Applied egg-rr 90.6%

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

       1. unpow1 90.6%

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

       2. *-commutative 90.6%

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

       3. associate-*l/ 90.6%

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

   13. Simplified 90.6%

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

       1. associate-/l* 88.3%

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

       2. add-cbrt-cube 79.1%

          \[\leadsto \frac{2}{{\left(\color{blue}{\sqrt[3]{\left({t}^{1.5} \cdot {t}^{1.5}\right) \cdot {t}^{1.5}}} \cdot \frac{k \cdot \sqrt{2}}{\ell}\right)}^{2}} \]

       3. pow-prod-up 79.1%

          \[\leadsto \frac{2}{{\left(\sqrt[3]{\color{blue}{{t}^{\left(1.5 + 1.5\right)}} \cdot {t}^{1.5}} \cdot \frac{k \cdot \sqrt{2}}{\ell}\right)}^{2}} \]

       4. metadata-eval 79.1%

          \[\leadsto \frac{2}{{\left(\sqrt[3]{{t}^{\color{blue}{3}} \cdot {t}^{1.5}} \cdot \frac{k \cdot \sqrt{2}}{\ell}\right)}^{2}} \]

       5. cbrt-prod 81.7%

          \[\leadsto \frac{2}{{\left(\color{blue}{\left(\sqrt[3]{{t}^{3}} \cdot \sqrt[3]{{t}^{1.5}}\right)} \cdot \frac{k \cdot \sqrt{2}}{\ell}\right)}^{2}} \]

       6. rem-cbrt-cube 88.3%

          \[\leadsto \frac{2}{{\left(\left(\color{blue}{t} \cdot \sqrt[3]{{t}^{1.5}}\right) \cdot \frac{k \cdot \sqrt{2}}{\ell}\right)}^{2}} \]

       7. associate-*l* 88.3%

          \[\leadsto \frac{2}{{\color{blue}{\left(t \cdot \left(\sqrt[3]{{t}^{1.5}} \cdot \frac{k \cdot \sqrt{2}}{\ell}\right)\right)}}^{2}} \]

       8. metadata-eval 88.3%

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

       9. sqrt-pow1 81.7%

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

       10. unpow3 81.6%

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

       11. sqrt-prod 86.7%

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

       12. sqrt-unprod 88.2%

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

       13. add-cbrt-cube 89.7%

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

   15. Applied egg-rr 89.7%

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

4. Final simplification 39.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.4 \cdot 10^{-96}:\\ \;\;\;\;2 \cdot {\left(\ell \cdot \frac{{k}^{-2}}{\sqrt{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t \cdot \left(\sqrt{t} \cdot \frac{k \cdot \sqrt{2}}{\ell}\right)\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.1 \cdot 10^{-105}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\ell}{\sqrt{t\_m}}\right)}^{2}}{{k\_m}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(k\_m \cdot \frac{{t\_m}^{1.5}}{\ell}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.1e-105)
    (* 2.0 (/ (pow (/ l (sqrt t_m)) 2.0) (pow k_m 4.0)))
    (/ 1.0 (pow (* k_m (/ (pow t_m 1.5) l)) 2.0)))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (t_m <= 3.1e-105) {
		tmp = 2.0 * (pow((l / sqrt(t_m)), 2.0) / pow(k_m, 4.0));
	} else {
		tmp = 1.0 / pow((k_m * (pow(t_m, 1.5) / l)), 2.0);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t_m <= 3.1d-105) then
        tmp = 2.0d0 * (((l / sqrt(t_m)) ** 2.0d0) / (k_m ** 4.0d0))
    else
        tmp = 1.0d0 / ((k_m * ((t_m ** 1.5d0) / l)) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (t_m <= 3.1e-105) {
		tmp = 2.0 * (Math.pow((l / Math.sqrt(t_m)), 2.0) / Math.pow(k_m, 4.0));
	} else {
		tmp = 1.0 / Math.pow((k_m * (Math.pow(t_m, 1.5) / l)), 2.0);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if t_m <= 3.1e-105:
		tmp = 2.0 * (math.pow((l / math.sqrt(t_m)), 2.0) / math.pow(k_m, 4.0))
	else:
		tmp = 1.0 / math.pow((k_m * (math.pow(t_m, 1.5) / l)), 2.0)
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (t_m <= 3.1e-105)
		tmp = Float64(2.0 * Float64((Float64(l / sqrt(t_m)) ^ 2.0) / (k_m ^ 4.0)));
	else
		tmp = Float64(1.0 / (Float64(k_m * Float64((t_m ^ 1.5) / l)) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (t_m <= 3.1e-105)
		tmp = 2.0 * (((l / sqrt(t_m)) ^ 2.0) / (k_m ^ 4.0));
	else
		tmp = 1.0 / ((k_m * ((t_m ^ 1.5) / l)) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 3.1e-105], N[(2.0 * N[(N[Power[N[(l / N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Power[N[(k$95$m * N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.10000000000000014e-105

   1. Initial program 52.0%

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

   3. Simplified 49.8%

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

   5. Taylor expanded in k around 0 46.5%

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

      1. associate-/r* 47.6%

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

   7. Simplified 47.6%

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

   8. Taylor expanded in k around inf 56.8%

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

      1. *-commutative 56.8%

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

      2. associate-/r* 58.2%

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

   10. Simplified 58.2%

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

       1. add-sqr-sqrt 30.7%

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

       2. sqrt-div 15.3%

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

       3. sqrt-pow1 9.2%

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

       4. metadata-eval 9.2%

          \[\leadsto 2 \cdot \frac{\frac{{\ell}^{\color{blue}{1}}}{\sqrt{t}} \cdot \sqrt{\frac{{\ell}^{2}}{t}}}{{k}^{4}} \]

       5. pow1 9.2%

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

       6. sqrt-div 9.2%

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

       7. sqrt-pow1 17.2%

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

       8. metadata-eval 17.2%

          \[\leadsto 2 \cdot \frac{\frac{\ell}{\sqrt{t}} \cdot \frac{{\ell}^{\color{blue}{1}}}{\sqrt{t}}}{{k}^{4}} \]

       9. pow1 17.2%

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

   12. Applied egg-rr 17.2%

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

       1. unpow2 17.2%

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

   14. Simplified 17.2%

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

   if 3.10000000000000014e-105 < t

   1. Initial program 71.5%

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

   3. Simplified 69.2%

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

   5. Taylor expanded in k around 0 66.9%

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

      1. add-cube-cbrt 66.8%

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

      2. pow3 66.8%

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

      3. associate-/l/ 61.7%

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

      4. unpow2 61.7%

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

      5. cbrt-div 61.7%

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

      6. unpow3 61.7%

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

      7. add-cbrt-cube 64.7%

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

      8. unpow2 64.7%

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

      9. cbrt-prod 72.8%

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

      10. unpow2 72.8%

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

      11. cube-mult 72.8%

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

      12. div-inv 72.8%

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

      13. pow-flip 72.8%

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

      14. metadata-eval 72.8%

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

      15. pow2 72.8%

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

      16. div-inv 72.8%

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

      17. pow-flip 72.8%

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

      18. metadata-eval 72.8%

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

   7. Applied egg-rr 72.8%

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

      1. unpow2 72.8%

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

      2. cube-mult 72.8%

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

   9. Simplified 72.8%

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

   11. Applied egg-rr 91.0%

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

       1. unpow1 91.0%

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

       2. *-commutative 91.0%

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

   13. Simplified 91.0%

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

       1. frac-2neg 91.0%

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

       2. metadata-eval 91.0%

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

       3. div-inv 91.0%

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

       4. associate-*l/ 91.0%

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

       5. associate-*l/ 91.0%

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

       6. associate-*r* 90.9%

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

       7. unpow-prod-down 91.0%

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

       8. pow2 91.0%

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

       9. rem-square-sqrt 91.1%

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

   15. Applied egg-rr 91.1%

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

       1. associate-*r/ 91.1%

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

       2. metadata-eval 91.1%

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

       3. neg-mul-1 91.1%

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

       4. associate-/r* 91.1%

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

       5. metadata-eval 91.1%

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

       6. *-commutative 91.1%

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

       7. associate-/r* 91.1%

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

       8. metadata-eval 91.1%

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

       9. *-commutative 91.1%

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

   17. Simplified 91.1%

       \[\leadsto \color{blue}{\frac{1}{{\left(k \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.1 \cdot 10^{-105}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\ell}{\sqrt{t}}\right)}^{2}}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(k \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.42 \cdot 10^{-95}:\\ \;\;\;\;2 \cdot {\left(\ell \cdot \frac{{k\_m}^{-2}}{\sqrt{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(k\_m \cdot \frac{{t\_m}^{1.5}}{\ell}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.42e-95)
    (* 2.0 (pow (* l (/ (pow k_m -2.0) (sqrt t_m))) 2.0))
    (/ 1.0 (pow (* k_m (/ (pow t_m 1.5) l)) 2.0)))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (t_m <= 1.42e-95) {
		tmp = 2.0 * pow((l * (pow(k_m, -2.0) / sqrt(t_m))), 2.0);
	} else {
		tmp = 1.0 / pow((k_m * (pow(t_m, 1.5) / l)), 2.0);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t_m <= 1.42d-95) then
        tmp = 2.0d0 * ((l * ((k_m ** (-2.0d0)) / sqrt(t_m))) ** 2.0d0)
    else
        tmp = 1.0d0 / ((k_m * ((t_m ** 1.5d0) / l)) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (t_m <= 1.42e-95) {
		tmp = 2.0 * Math.pow((l * (Math.pow(k_m, -2.0) / Math.sqrt(t_m))), 2.0);
	} else {
		tmp = 1.0 / Math.pow((k_m * (Math.pow(t_m, 1.5) / l)), 2.0);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if t_m <= 1.42e-95:
		tmp = 2.0 * math.pow((l * (math.pow(k_m, -2.0) / math.sqrt(t_m))), 2.0)
	else:
		tmp = 1.0 / math.pow((k_m * (math.pow(t_m, 1.5) / l)), 2.0)
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (t_m <= 1.42e-95)
		tmp = Float64(2.0 * (Float64(l * Float64((k_m ^ -2.0) / sqrt(t_m))) ^ 2.0));
	else
		tmp = Float64(1.0 / (Float64(k_m * Float64((t_m ^ 1.5) / l)) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (t_m <= 1.42e-95)
		tmp = 2.0 * ((l * ((k_m ^ -2.0) / sqrt(t_m))) ^ 2.0);
	else
		tmp = 1.0 / ((k_m * ((t_m ^ 1.5) / l)) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 1.42e-95], N[(2.0 * N[Power[N[(l * N[(N[Power[k$95$m, -2.0], $MachinePrecision] / N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Power[N[(k$95$m * N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.42000000000000007e-95

   1. Initial program 52.2%

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

   3. Simplified 50.0%

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

   5. Taylor expanded in k around 0 46.9%

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

      1. associate-/r* 47.9%

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

   7. Simplified 47.9%

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

   8. Taylor expanded in k around inf 57.0%

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

      1. *-commutative 57.0%

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

      2. associate-/r* 58.3%

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

   10. Simplified 58.3%

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

       1. *-un-lft-identity 58.3%

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

       2. div-inv 58.3%

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

       3. pow-flip 58.3%

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

       4. metadata-eval 58.3%

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

   12. Applied egg-rr 58.3%

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

       1. *-lft-identity 58.3%

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

       2. associate-*l/ 57.9%

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

       3. associate-/l* 57.0%

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

   14. Simplified 57.0%

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

       1. add-sqr-sqrt 38.7%

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

       2. pow2 38.7%

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

       3. sqrt-prod 35.3%

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

       4. sqrt-pow1 39.4%

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

       5. metadata-eval 39.4%

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

       6. pow1 39.4%

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

       7. sqrt-div 18.2%

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

       8. sqrt-pow1 18.2%

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

       9. metadata-eval 18.2%

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

   16. Applied egg-rr 18.2%

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

   if 1.42000000000000007e-95 < t

   1. Initial program 71.7%

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

   3. Simplified 68.0%

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

   5. Taylor expanded in k around 0 65.6%

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

      1. add-cube-cbrt 65.5%

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

      2. pow3 65.5%

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

      3. associate-/l/ 61.5%

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

      4. unpow2 61.5%

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

      5. cbrt-div 61.5%

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

      6. unpow3 61.5%

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

      7. add-cbrt-cube 64.6%

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

      8. unpow2 64.6%

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

      9. cbrt-prod 71.8%

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

      10. unpow2 71.8%

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

      11. cube-mult 71.7%

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

      12. div-inv 71.8%

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

      13. pow-flip 71.8%

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

      14. metadata-eval 71.8%

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

      15. pow2 71.8%

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

      16. div-inv 71.7%

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

      17. pow-flip 71.7%

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

      18. metadata-eval 71.7%

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

   7. Applied egg-rr 71.7%

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

      1. unpow2 71.7%

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

      2. cube-mult 71.7%

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

   9. Simplified 71.7%

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

   11. Applied egg-rr 90.6%

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

       1. unpow1 90.6%

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

       2. *-commutative 90.6%

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

   13. Simplified 90.6%

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

       1. frac-2neg 90.6%

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

       2. metadata-eval 90.6%

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

       3. div-inv 90.6%

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

       4. associate-*l/ 90.6%

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

       5. associate-*l/ 90.6%

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

       6. associate-*r* 90.6%

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

       7. unpow-prod-down 90.6%

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

       8. pow2 90.6%

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

       9. rem-square-sqrt 90.7%

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

   15. Applied egg-rr 90.7%

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

       1. associate-*r/ 90.7%

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

       2. metadata-eval 90.7%

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

       3. neg-mul-1 90.7%

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

       4. associate-/r* 90.7%

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

       5. metadata-eval 90.7%

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

       6. *-commutative 90.7%

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

       7. associate-/r* 90.7%

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

       8. metadata-eval 90.7%

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

       9. *-commutative 90.7%

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

   17. Simplified 90.7%

       \[\leadsto \color{blue}{\frac{1}{{\left(k \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.42 \cdot 10^{-95}:\\ \;\;\;\;2 \cdot {\left(\ell \cdot \frac{{k}^{-2}}{\sqrt{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(k \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.1 \cdot 10^{-111}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{t\_m}}{{k\_m}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(k\_m \cdot \frac{{t\_m}^{1.5}}{\ell}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.1e-111)
    (* 2.0 (/ (/ (pow l 2.0) t_m) (pow k_m 4.0)))
    (/ 1.0 (pow (* k_m (/ (pow t_m 1.5) l)) 2.0)))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (t_m <= 3.1e-111) {
		tmp = 2.0 * ((pow(l, 2.0) / t_m) / pow(k_m, 4.0));
	} else {
		tmp = 1.0 / pow((k_m * (pow(t_m, 1.5) / l)), 2.0);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t_m <= 3.1d-111) then
        tmp = 2.0d0 * (((l ** 2.0d0) / t_m) / (k_m ** 4.0d0))
    else
        tmp = 1.0d0 / ((k_m * ((t_m ** 1.5d0) / l)) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (t_m <= 3.1e-111) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / t_m) / Math.pow(k_m, 4.0));
	} else {
		tmp = 1.0 / Math.pow((k_m * (Math.pow(t_m, 1.5) / l)), 2.0);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if t_m <= 3.1e-111:
		tmp = 2.0 * ((math.pow(l, 2.0) / t_m) / math.pow(k_m, 4.0))
	else:
		tmp = 1.0 / math.pow((k_m * (math.pow(t_m, 1.5) / l)), 2.0)
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (t_m <= 3.1e-111)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / t_m) / (k_m ^ 4.0)));
	else
		tmp = Float64(1.0 / (Float64(k_m * Float64((t_m ^ 1.5) / l)) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (t_m <= 3.1e-111)
		tmp = 2.0 * (((l ^ 2.0) / t_m) / (k_m ^ 4.0));
	else
		tmp = 1.0 / ((k_m * ((t_m ^ 1.5) / l)) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 3.1e-111], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Power[N[(k$95$m * N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.10000000000000014e-111

   1. Initial program 52.0%

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

   3. Simplified 49.8%

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

   5. Taylor expanded in k around 0 46.5%

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

      1. associate-/r* 47.6%

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

   7. Simplified 47.6%

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

   8. Taylor expanded in k around inf 56.8%

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

      1. *-commutative 56.8%

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

      2. associate-/r* 58.2%

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

   10. Simplified 58.2%

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

   if 3.10000000000000014e-111 < t

   1. Initial program 71.5%

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

   3. Simplified 69.2%

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

   5. Taylor expanded in k around 0 66.9%

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

      1. add-cube-cbrt 66.8%

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

      2. pow3 66.8%

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

      3. associate-/l/ 61.7%

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

      4. unpow2 61.7%

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

      5. cbrt-div 61.7%

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

      6. unpow3 61.7%

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

      7. add-cbrt-cube 64.7%

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

      8. unpow2 64.7%

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

      9. cbrt-prod 72.8%

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

      10. unpow2 72.8%

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

      11. cube-mult 72.8%

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

      12. div-inv 72.8%

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

      13. pow-flip 72.8%

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

      14. metadata-eval 72.8%

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

      15. pow2 72.8%

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

      16. div-inv 72.8%

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

      17. pow-flip 72.8%

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

      18. metadata-eval 72.8%

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

   7. Applied egg-rr 72.8%

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

      1. unpow2 72.8%

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

      2. cube-mult 72.8%

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

   9. Simplified 72.8%

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

   11. Applied egg-rr 91.0%

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

       1. unpow1 91.0%

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

       2. *-commutative 91.0%

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

   13. Simplified 91.0%

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

       1. frac-2neg 91.0%

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

       2. metadata-eval 91.0%

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

       3. div-inv 91.0%

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

       4. associate-*l/ 91.0%

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

       5. associate-*l/ 91.0%

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

       6. associate-*r* 90.9%

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

       7. unpow-prod-down 91.0%

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

       8. pow2 91.0%

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

       9. rem-square-sqrt 91.1%

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

   15. Applied egg-rr 91.1%

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

       1. associate-*r/ 91.1%

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

       2. metadata-eval 91.1%

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

       3. neg-mul-1 91.1%

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

       4. associate-/r* 91.1%

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

       5. metadata-eval 91.1%

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

       6. *-commutative 91.1%

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

       7. associate-/r* 91.1%

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

       8. metadata-eval 91.1%

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

       9. *-commutative 91.1%

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

   17. Simplified 91.1%

       \[\leadsto \color{blue}{\frac{1}{{\left(k \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.1 \cdot 10^{-111}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(k \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 51.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(2 \cdot \left({\ell}^{2} \cdot \frac{{k\_m}^{-4}}{t\_m}\right)\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* 2.0 (* (pow l 2.0) (/ (pow k_m -4.0) t_m)))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (2.0 * (pow(l, 2.0) * (pow(k_m, -4.0) / t_m)));
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = t_s * (2.0d0 * ((l ** 2.0d0) * ((k_m ** (-4.0d0)) / t_m)))
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (2.0 * (Math.pow(l, 2.0) * (Math.pow(k_m, -4.0) / t_m)));
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	return t_s * (2.0 * (math.pow(l, 2.0) * (math.pow(k_m, -4.0) / t_m)))

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	return Float64(t_s * Float64(2.0 * Float64((l ^ 2.0) * Float64((k_m ^ -4.0) / t_m))))
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * (2.0 * ((l ^ 2.0) * ((k_m ^ -4.0) / t_m)));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[Power[k$95$m, -4.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.1%

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

3. Simplified 54.6%

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

5. Taylor expanded in k around 0 52.0%

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

   1. associate-/r* 52.0%

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

7. Simplified 52.0%

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

8. Taylor expanded in k around inf 55.2%

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

   1. *-commutative 55.2%

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

   2. associate-/r* 56.4%

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

10. Simplified 56.4%

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

    1. *-un-lft-identity 56.4%

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

    2. div-inv 56.4%

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

    3. pow-flip 56.4%

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

    4. metadata-eval 56.4%

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

12. Applied egg-rr 56.4%

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

    1. *-lft-identity 56.4%

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

    2. associate-*l/ 55.8%

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

    3. associate-/l* 55.2%

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

14. Simplified 55.2%

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

15. Final simplification 55.2%

    \[\leadsto 2 \cdot \left({\ell}^{2} \cdot \frac{{k}^{-4}}{t}\right) \]
16. Add Preprocessing

Alternative 11: 52.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(2 \cdot \frac{\frac{{\ell}^{2}}{t\_m}}{{k\_m}^{4}}\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* 2.0 (/ (/ (pow l 2.0) t_m) (pow k_m 4.0)))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (2.0 * ((pow(l, 2.0) / t_m) / pow(k_m, 4.0)));
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = t_s * (2.0d0 * (((l ** 2.0d0) / t_m) / (k_m ** 4.0d0)))
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (2.0 * ((Math.pow(l, 2.0) / t_m) / Math.pow(k_m, 4.0)));
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	return t_s * (2.0 * ((math.pow(l, 2.0) / t_m) / math.pow(k_m, 4.0)))

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	return Float64(t_s * Float64(2.0 * Float64(Float64((l ^ 2.0) / t_m) / (k_m ^ 4.0))))
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * (2.0 * (((l ^ 2.0) / t_m) / (k_m ^ 4.0)));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.1%

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

3. Simplified 54.6%

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

5. Taylor expanded in k around 0 52.0%

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

   1. associate-/r* 52.0%

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

7. Simplified 52.0%

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

8. Taylor expanded in k around inf 55.2%

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

   1. *-commutative 55.2%

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

   2. associate-/r* 56.4%

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

10. Simplified 56.4%

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

11. Final simplification 56.4%

    \[\leadsto 2 \cdot \frac{\frac{{\ell}^{2}}{t}}{{k}^{4}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024077 
(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))))