Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.4% → 92.9%

Time: 22.7s

Alternatives: 27

Speedup: 2.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 92.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 4.2 \cdot 10^{-42}:\\ \;\;\;\;\frac{\cos k}{t_m} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(\left(t_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t_m}\right)}^{2}\right)}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4.2e-42)
    (* (/ (cos k) t_m) (pow (* l (/ (sqrt 2.0) (* k (sin k)))) 2.0))
    (/
     2.0
     (pow
      (*
       (* (* t_m (pow (cbrt l) -2.0)) (cbrt (sin k)))
       (cbrt (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))))
      3.0)))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.2e-42) {
		tmp = (cos(k) / t_m) * pow((l * (sqrt(2.0) / (k * sin(k)))), 2.0);
	} else {
		tmp = 2.0 / pow((((t_m * pow(cbrt(l), -2.0)) * cbrt(sin(k))) * cbrt((tan(k) * (2.0 + pow((k / t_m), 2.0))))), 3.0);
	}
	return t_s * tmp;
}

Java

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) {
	double tmp;
	if (t_m <= 4.2e-42) {
		tmp = (Math.cos(k) / t_m) * Math.pow((l * (Math.sqrt(2.0) / (k * Math.sin(k)))), 2.0);
	} else {
		tmp = 2.0 / Math.pow((((t_m * Math.pow(Math.cbrt(l), -2.0)) * Math.cbrt(Math.sin(k))) * Math.cbrt((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0))))), 3.0);
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 4.2e-42)
		tmp = Float64(Float64(cos(k) / t_m) * (Float64(l * Float64(sqrt(2.0) / Float64(k * sin(k)))) ^ 2.0));
	else
		tmp = Float64(2.0 / (Float64(Float64(Float64(t_m * (cbrt(l) ^ -2.0)) * cbrt(sin(k))) * cbrt(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))))) ^ 3.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[t$95$m, 4.2e-42], N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.20000000000000013e-42

   1. Initial program 49.0%

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

   3. Simplified 49.0%

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

      1. add-sqr-sqrt 38.3%

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

   5. Applied egg-rr 43.8%

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

      1. unpow2 43.8%

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

      2. associate-/l* 43.8%

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

      3. associate-*r* 43.8%

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

      4. *-commutative 43.8%

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

   7. Simplified 43.8%

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

   8. Taylor expanded in k around inf 39.7%

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

      1. expm1-log1p-u 39.4%

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

      2. expm1-udef 34.7%

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

      3. *-commutative 34.7%

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

      4. unpow-prod-down 34.6%

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

      5. pow2 34.6%

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

      6. add-sqr-sqrt 54.2%

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

      7. times-frac 54.7%

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

   10. Applied egg-rr 54.7%

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

       1. expm1-def 58.1%

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

       2. expm1-log1p 75.1%

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

       3. associate-*r/ 75.2%

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

       4. associate-*l/ 75.2%

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

       5. associate-/r* 75.2%

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

       6. *-rgt-identity 75.2%

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

       7. associate-*r/ 75.2%

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

       8. associate-*l* 75.2%

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

       9. associate-*r/ 75.2%

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

       10. *-commutative 75.2%

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

       11. *-lft-identity 75.2%

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

   12. Simplified 75.2%

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

   if 4.20000000000000013e-42 < t

   1. Initial program 63.2%

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

      1. associate-*l* 63.2%

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

      2. *-commutative 63.2%

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

      3. *-commutative 63.2%

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

      4. associate-/r* 69.1%

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

      5. distribute-rgt-in 69.1%

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

      6. unpow2 69.1%

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

      7. times-frac 52.5%

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

      8. sqr-neg 52.5%

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

      9. times-frac 69.1%

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

      10. unpow2 69.1%

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

      11. distribute-rgt-in 69.1%

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

      12. +-commutative 69.1%

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

   3. Simplified 69.1%

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

      1. associate-/r* 63.2%

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

      2. unpow3 63.2%

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

      3. times-frac 78.0%

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

      4. pow2 78.0%

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

   5. Applied egg-rr 78.0%

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

      1. add-cube-cbrt 77.9%

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

      2. pow3 77.9%

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

      3. cbrt-prod 77.8%

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

      4. frac-times 63.2%

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

      5. unpow2 63.2%

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

      6. unpow3 63.1%

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

      7. unpow2 63.1%

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

      8. cbrt-div 66.0%

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

      9. unpow3 66.0%

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

      10. add-cbrt-cube 78.2%

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

      11. unpow2 78.2%

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

      12. cbrt-prod 90.7%

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

      13. pow2 90.7%

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

   7. Applied egg-rr 90.7%

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

      1. add-cube-cbrt 90.6%

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

      2. pow3 90.6%

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

   9. Applied egg-rr 90.6%

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

       1. add-cube-cbrt 90.5%

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

       2. pow3 90.5%

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

   11. Applied egg-rr 97.5%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{-42}:\\ \;\;\;\;\frac{\cos k}{t} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)}^{3}}\\ \end{array} \]

Alternative 2: 89.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 4.5 \cdot 10^{-42}:\\ \;\;\;\;\frac{\cos k}{t_m} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t_m}\right)}^{2}\right)\right) \cdot {\left(\sqrt[3]{\sin k} \cdot \frac{t_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4.5e-42)
    (* (/ (cos k) t_m) (pow (* l (/ (sqrt 2.0) (* k (sin k)))) 2.0))
    (/
     2.0
     (*
      (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))
      (pow (* (cbrt (sin k)) (/ t_m (pow (cbrt l) 2.0))) 3.0))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.5e-42) {
		tmp = (cos(k) / t_m) * pow((l * (sqrt(2.0) / (k * sin(k)))), 2.0);
	} else {
		tmp = 2.0 / ((tan(k) * (2.0 + pow((k / t_m), 2.0))) * pow((cbrt(sin(k)) * (t_m / pow(cbrt(l), 2.0))), 3.0));
	}
	return t_s * tmp;
}

Java

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) {
	double tmp;
	if (t_m <= 4.5e-42) {
		tmp = (Math.cos(k) / t_m) * Math.pow((l * (Math.sqrt(2.0) / (k * Math.sin(k)))), 2.0);
	} else {
		tmp = 2.0 / ((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0))) * Math.pow((Math.cbrt(Math.sin(k)) * (t_m / Math.pow(Math.cbrt(l), 2.0))), 3.0));
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 4.5e-42)
		tmp = Float64(Float64(cos(k) / t_m) * (Float64(l * Float64(sqrt(2.0) / Float64(k * sin(k)))) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * (Float64(cbrt(sin(k)) * Float64(t_m / (cbrt(l) ^ 2.0))) ^ 3.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[t$95$m, 4.5e-42], N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.5e-42

   1. Initial program 49.0%

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

   3. Simplified 49.0%

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

      1. add-sqr-sqrt 38.3%

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

   5. Applied egg-rr 43.8%

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

      1. unpow2 43.8%

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

      2. associate-/l* 43.8%

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

      3. associate-*r* 43.8%

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

      4. *-commutative 43.8%

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

   7. Simplified 43.8%

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

   8. Taylor expanded in k around inf 39.7%

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

      1. expm1-log1p-u 39.4%

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

      2. expm1-udef 34.7%

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

      3. *-commutative 34.7%

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

      4. unpow-prod-down 34.6%

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

      5. pow2 34.6%

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

      6. add-sqr-sqrt 54.2%

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

      7. times-frac 54.7%

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

   10. Applied egg-rr 54.7%

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

       1. expm1-def 58.1%

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

       2. expm1-log1p 75.1%

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

       3. associate-*r/ 75.2%

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

       4. associate-*l/ 75.2%

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

       5. associate-/r* 75.2%

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

       6. *-rgt-identity 75.2%

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

       7. associate-*r/ 75.2%

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

       8. associate-*l* 75.2%

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

       9. associate-*r/ 75.2%

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

       10. *-commutative 75.2%

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

       11. *-lft-identity 75.2%

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

   12. Simplified 75.2%

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

   if 4.5e-42 < t

   1. Initial program 63.2%

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

      1. associate-*l* 63.2%

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

      2. *-commutative 63.2%

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

      3. *-commutative 63.2%

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

      4. associate-/r* 69.1%

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

      5. distribute-rgt-in 69.1%

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

      6. unpow2 69.1%

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

      7. times-frac 52.5%

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

      8. sqr-neg 52.5%

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

      9. times-frac 69.1%

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

      10. unpow2 69.1%

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

      11. distribute-rgt-in 69.1%

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

      12. +-commutative 69.1%

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

   3. Simplified 69.1%

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

      1. associate-/r* 63.2%

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

      2. unpow3 63.2%

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

      3. times-frac 78.0%

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

      4. pow2 78.0%

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

   5. Applied egg-rr 78.0%

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

      1. add-cube-cbrt 77.9%

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

      2. pow3 77.9%

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

      3. cbrt-prod 77.8%

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

      4. frac-times 63.2%

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

      5. unpow2 63.2%

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

      6. unpow3 63.1%

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

      7. unpow2 63.1%

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

      8. cbrt-div 66.0%

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

      9. unpow3 66.0%

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

      10. add-cbrt-cube 78.2%

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

      11. unpow2 78.2%

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

      12. cbrt-prod 90.7%

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

      13. pow2 90.7%

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

   7. Applied egg-rr 90.7%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.5 \cdot 10^{-42}:\\ \;\;\;\;\frac{\cos k}{t} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \end{array} \]

Alternative 3: 87.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \tan k \cdot \left(2 + {\left(\frac{k}{t_m}\right)}^{2}\right)\\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 4.4 \cdot 10^{-42}:\\ \;\;\;\;\frac{\cos k}{t_m} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \mathbf{elif}\;t_m \leq 4.2 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{t_2 \cdot \left(\frac{1}{\ell} \cdot \left(\sin k \cdot \frac{{t_m}^{3}}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(t_2 \cdot {\left(t_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))))
   (*
    t_s
    (if (<= t_m 4.4e-42)
      (* (/ (cos k) t_m) (pow (* l (/ (sqrt 2.0) (* k (sin k)))) 2.0))
      (if (<= t_m 4.2e+102)
        (/ 2.0 (* t_2 (* (/ 1.0 l) (* (sin k) (/ (pow t_m 3.0) l)))))
        (/ 2.0 (* (sin k) (* t_2 (pow (* t_m (pow (cbrt l) -2.0)) 3.0)))))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = tan(k) * (2.0 + pow((k / t_m), 2.0));
	double tmp;
	if (t_m <= 4.4e-42) {
		tmp = (cos(k) / t_m) * pow((l * (sqrt(2.0) / (k * sin(k)))), 2.0);
	} else if (t_m <= 4.2e+102) {
		tmp = 2.0 / (t_2 * ((1.0 / l) * (sin(k) * (pow(t_m, 3.0) / l))));
	} else {
		tmp = 2.0 / (sin(k) * (t_2 * pow((t_m * pow(cbrt(l), -2.0)), 3.0)));
	}
	return t_s * tmp;
}

Java

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) {
	double t_2 = Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0));
	double tmp;
	if (t_m <= 4.4e-42) {
		tmp = (Math.cos(k) / t_m) * Math.pow((l * (Math.sqrt(2.0) / (k * Math.sin(k)))), 2.0);
	} else if (t_m <= 4.2e+102) {
		tmp = 2.0 / (t_2 * ((1.0 / l) * (Math.sin(k) * (Math.pow(t_m, 3.0) / l))));
	} else {
		tmp = 2.0 / (Math.sin(k) * (t_2 * Math.pow((t_m * Math.pow(Math.cbrt(l), -2.0)), 3.0)));
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0)))
	tmp = 0.0
	if (t_m <= 4.4e-42)
		tmp = Float64(Float64(cos(k) / t_m) * (Float64(l * Float64(sqrt(2.0) / Float64(k * sin(k)))) ^ 2.0));
	elseif (t_m <= 4.2e+102)
		tmp = Float64(2.0 / Float64(t_2 * Float64(Float64(1.0 / l) * Float64(sin(k) * Float64((t_m ^ 3.0) / l)))));
	else
		tmp = Float64(2.0 / Float64(sin(k) * Float64(t_2 * (Float64(t_m * (cbrt(l) ^ -2.0)) ^ 3.0))));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_] := Block[{t$95$2 = N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4.4e-42], N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.2e+102], N[(2.0 / N[(t$95$2 * N[(N[(1.0 / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(t$95$2 * N[Power[N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 4.4000000000000001e-42

   1. Initial program 49.0%

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

   3. Simplified 49.0%

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

      1. add-sqr-sqrt 38.3%

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

   5. Applied egg-rr 43.8%

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

      1. unpow2 43.8%

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

      2. associate-/l* 43.8%

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

      3. associate-*r* 43.8%

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

      4. *-commutative 43.8%

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

   7. Simplified 43.8%

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

   8. Taylor expanded in k around inf 39.7%

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

      1. expm1-log1p-u 39.4%

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

      2. expm1-udef 34.7%

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

      3. *-commutative 34.7%

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

      4. unpow-prod-down 34.6%

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

      5. pow2 34.6%

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

      6. add-sqr-sqrt 54.2%

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

      7. times-frac 54.7%

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

   10. Applied egg-rr 54.7%

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

       1. expm1-def 58.1%

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

       2. expm1-log1p 75.1%

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

       3. associate-*r/ 75.2%

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

       4. associate-*l/ 75.2%

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

       5. associate-/r* 75.2%

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

       6. *-rgt-identity 75.2%

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

       7. associate-*r/ 75.2%

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

       8. associate-*l* 75.2%

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

       9. associate-*r/ 75.2%

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

       10. *-commutative 75.2%

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

       11. *-lft-identity 75.2%

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

   12. Simplified 75.2%

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

   if 4.4000000000000001e-42 < t < 4.20000000000000003e102

   1. Initial program 76.5%

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

      1. associate-*l* 76.6%

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

      2. *-commutative 76.6%

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

      3. *-commutative 76.6%

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

      4. associate-/r* 81.4%

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

      5. distribute-rgt-in 81.4%

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

      6. unpow2 81.4%

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

      7. times-frac 72.8%

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

      8. sqr-neg 72.8%

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

      9. times-frac 81.4%

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

      10. unpow2 81.4%

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

      11. distribute-rgt-in 81.4%

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

      12. +-commutative 81.4%

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

   3. Simplified 81.4%

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

      1. associate-*l/ 95.0%

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

      2. clear-num 95.0%

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

   5. Applied egg-rr 95.0%

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

      1. associate-/r/ 94.9%

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

      2. *-commutative 94.9%

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

   7. Simplified 94.9%

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

   if 4.20000000000000003e102 < t

   1. Initial program 56.8%

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

      1. associate-*l* 56.8%

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

      2. *-commutative 56.8%

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

      3. *-commutative 56.8%

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

      4. associate-/r* 63.3%

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

      5. distribute-rgt-in 63.3%

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

      6. unpow2 63.3%

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

      7. times-frac 42.9%

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

      8. sqr-neg 42.9%

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

      9. times-frac 63.3%

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

      10. unpow2 63.3%

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

      11. distribute-rgt-in 63.3%

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

      12. +-commutative 63.3%

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

   3. Simplified 63.3%

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

      1. associate-/r* 56.8%

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

      2. unpow3 56.8%

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

      3. times-frac 76.4%

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

      4. pow2 76.4%

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

   5. Applied egg-rr 76.4%

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

      1. add-cube-cbrt 76.4%

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

      2. pow3 76.4%

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

      3. cbrt-prod 76.3%

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

      4. frac-times 56.8%

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

      5. unpow2 56.8%

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

      6. unpow3 56.8%

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

      7. unpow2 56.8%

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

      8. cbrt-div 56.8%

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

      9. unpow3 56.8%

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

      10. add-cbrt-cube 74.8%

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

      11. unpow2 74.8%

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

      12. cbrt-prod 89.1%

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

      13. pow2 89.1%

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

   7. Applied egg-rr 89.1%

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

      1. add-cube-cbrt 89.0%

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

      2. pow3 89.0%

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

   9. Applied egg-rr 89.0%

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

       1. expm1-log1p-u 65.9%

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

       2. expm1-udef 53.3%

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

   11. Applied egg-rr 51.3%

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

       1. expm1-def 62.1%

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

       2. expm1-log1p 85.1%

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

       3. associate-*l* 85.1%

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

   13. Simplified 85.1%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.4 \cdot 10^{-42}:\\ \;\;\;\;\frac{\cos k}{t} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\frac{1}{\ell} \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}\right)}\\ \end{array} \]

Alternative 4: 87.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \tan k \cdot \left(2 + {\left(\frac{k}{t_m}\right)}^{2}\right)\\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 4.2 \cdot 10^{-42}:\\ \;\;\;\;\frac{\cos k}{t_m} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \mathbf{elif}\;t_m \leq 6.5 \cdot 10^{+69}:\\ \;\;\;\;\frac{2}{t_2 \cdot \left(\frac{1}{\ell} \cdot \left(\sin k \cdot \frac{{t_m}^{3}}{\ell}\right)\right)}\\ \mathbf{elif}\;t_m \leq 1.25 \cdot 10^{+183}:\\ \;\;\;\;\frac{2}{t_2 \cdot \left(\sin k \cdot {\left(\frac{{t_m}^{1.5}}{\ell}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(k \cdot 2\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))))
   (*
    t_s
    (if (<= t_m 4.2e-42)
      (* (/ (cos k) t_m) (pow (* l (/ (sqrt 2.0) (* k (sin k)))) 2.0))
      (if (<= t_m 6.5e+69)
        (/ 2.0 (* t_2 (* (/ 1.0 l) (* (sin k) (/ (pow t_m 3.0) l)))))
        (if (<= t_m 1.25e+183)
          (/ 2.0 (* t_2 (* (sin k) (pow (/ (pow t_m 1.5) l) 2.0))))
          (/
           2.0
           (*
            (pow (* (cbrt (sin k)) (/ t_m (pow (cbrt l) 2.0))) 3.0)
            (* k 2.0)))))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = tan(k) * (2.0 + pow((k / t_m), 2.0));
	double tmp;
	if (t_m <= 4.2e-42) {
		tmp = (cos(k) / t_m) * pow((l * (sqrt(2.0) / (k * sin(k)))), 2.0);
	} else if (t_m <= 6.5e+69) {
		tmp = 2.0 / (t_2 * ((1.0 / l) * (sin(k) * (pow(t_m, 3.0) / l))));
	} else if (t_m <= 1.25e+183) {
		tmp = 2.0 / (t_2 * (sin(k) * pow((pow(t_m, 1.5) / l), 2.0)));
	} else {
		tmp = 2.0 / (pow((cbrt(sin(k)) * (t_m / pow(cbrt(l), 2.0))), 3.0) * (k * 2.0));
	}
	return t_s * tmp;
}

Java

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) {
	double t_2 = Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0));
	double tmp;
	if (t_m <= 4.2e-42) {
		tmp = (Math.cos(k) / t_m) * Math.pow((l * (Math.sqrt(2.0) / (k * Math.sin(k)))), 2.0);
	} else if (t_m <= 6.5e+69) {
		tmp = 2.0 / (t_2 * ((1.0 / l) * (Math.sin(k) * (Math.pow(t_m, 3.0) / l))));
	} else if (t_m <= 1.25e+183) {
		tmp = 2.0 / (t_2 * (Math.sin(k) * Math.pow((Math.pow(t_m, 1.5) / l), 2.0)));
	} else {
		tmp = 2.0 / (Math.pow((Math.cbrt(Math.sin(k)) * (t_m / Math.pow(Math.cbrt(l), 2.0))), 3.0) * (k * 2.0));
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0)))
	tmp = 0.0
	if (t_m <= 4.2e-42)
		tmp = Float64(Float64(cos(k) / t_m) * (Float64(l * Float64(sqrt(2.0) / Float64(k * sin(k)))) ^ 2.0));
	elseif (t_m <= 6.5e+69)
		tmp = Float64(2.0 / Float64(t_2 * Float64(Float64(1.0 / l) * Float64(sin(k) * Float64((t_m ^ 3.0) / l)))));
	elseif (t_m <= 1.25e+183)
		tmp = Float64(2.0 / Float64(t_2 * Float64(sin(k) * (Float64((t_m ^ 1.5) / l) ^ 2.0))));
	else
		tmp = Float64(2.0 / Float64((Float64(cbrt(sin(k)) * Float64(t_m / (cbrt(l) ^ 2.0))) ^ 3.0) * Float64(k * 2.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_] := Block[{t$95$2 = N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4.2e-42], N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6.5e+69], N[(2.0 / N[(t$95$2 * N[(N[(1.0 / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.25e+183], N[(2.0 / N[(t$95$2 * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(k * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 4.20000000000000013e-42

   1. Initial program 49.0%

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

   3. Simplified 49.0%

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

      1. add-sqr-sqrt 38.3%

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

   5. Applied egg-rr 43.8%

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

      1. unpow2 43.8%

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

      2. associate-/l* 43.8%

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

      3. associate-*r* 43.8%

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

      4. *-commutative 43.8%

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

   7. Simplified 43.8%

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

   8. Taylor expanded in k around inf 39.7%

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

      1. expm1-log1p-u 39.4%

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

      2. expm1-udef 34.7%

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

      3. *-commutative 34.7%

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

      4. unpow-prod-down 34.6%

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

      5. pow2 34.6%

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

      6. add-sqr-sqrt 54.2%

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

      7. times-frac 54.7%

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

   10. Applied egg-rr 54.7%

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

       1. expm1-def 58.1%

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

       2. expm1-log1p 75.1%

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

       3. associate-*r/ 75.2%

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

       4. associate-*l/ 75.2%

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

       5. associate-/r* 75.2%

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

       6. *-rgt-identity 75.2%

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

       7. associate-*r/ 75.2%

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

       8. associate-*l* 75.2%

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

       9. associate-*r/ 75.2%

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

       10. *-commutative 75.2%

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

       11. *-lft-identity 75.2%

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

   12. Simplified 75.2%

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

   if 4.20000000000000013e-42 < t < 6.5000000000000001e69

   1. Initial program 79.2%

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

      1. associate-*l* 79.2%

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

      2. *-commutative 79.2%

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

      3. *-commutative 79.2%

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

      4. associate-/r* 84.5%

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

      5. distribute-rgt-in 84.5%

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

      6. unpow2 84.5%

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

      7. times-frac 75.0%

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

      8. sqr-neg 75.0%

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

      9. times-frac 84.5%

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

      10. unpow2 84.5%

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

      11. distribute-rgt-in 84.5%

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

      12. +-commutative 84.5%

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

   3. Simplified 84.5%

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

      1. associate-*l/ 99.5%

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

      2. clear-num 99.5%

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

   5. Applied egg-rr 99.5%

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

      1. associate-/r/ 99.5%

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

      2. *-commutative 99.5%

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

   7. Simplified 99.5%

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

   if 6.5000000000000001e69 < t < 1.25000000000000002e183

   1. Initial program 46.1%

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

      1. associate-*l* 46.1%

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

      2. *-commutative 46.1%

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

      3. *-commutative 46.1%

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

      4. associate-/r* 46.7%

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

      5. distribute-rgt-in 46.7%

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

      6. unpow2 46.7%

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

      7. times-frac 36.1%

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

      8. sqr-neg 36.1%

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

      9. times-frac 46.7%

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

      10. unpow2 46.7%

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

      11. distribute-rgt-in 46.7%

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

      12. +-commutative 46.7%

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

   3. Simplified 46.7%

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

      1. associate-/r* 46.1%

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

      2. add-sqr-sqrt 46.1%

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

      3. pow2 46.1%

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

      4. sqrt-div 46.1%

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

      5. sqrt-pow1 72.3%

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

      6. metadata-eval 72.3%

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

      7. sqrt-prod 41.9%

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

      8. add-sqr-sqrt 87.2%

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

   5. Applied egg-rr 87.2%

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

   if 1.25000000000000002e183 < t

   1. Initial program 64.0%

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

      1. associate-*l* 64.0%

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

      2. *-commutative 64.0%

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

      3. *-commutative 64.0%

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

      4. associate-/r* 74.2%

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

      5. distribute-rgt-in 74.2%

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

      6. unpow2 74.2%

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

      7. times-frac 48.2%

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

      8. sqr-neg 48.2%

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

      9. times-frac 74.2%

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

      10. unpow2 74.2%

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

      11. distribute-rgt-in 74.2%

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

      12. +-commutative 74.2%

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

   3. Simplified 74.2%

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

      1. associate-/r* 64.0%

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

      2. unpow3 64.0%

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

      3. times-frac 74.2%

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

      4. pow2 74.2%

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

   5. Applied egg-rr 74.2%

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

      1. add-cube-cbrt 74.2%

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

      2. pow3 74.2%

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

      3. cbrt-prod 74.2%

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

      4. frac-times 64.0%

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

      5. unpow2 64.0%

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

      6. unpow3 64.0%

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

      7. unpow2 64.0%

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

      8. cbrt-div 64.0%

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

      9. unpow3 64.0%

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

      10. add-cbrt-cube 75.1%

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

      11. unpow2 75.1%

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

      12. cbrt-prod 88.2%

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

      13. pow2 88.2%

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

   7. Applied egg-rr 88.2%

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

   8. Taylor expanded in k around 0 88.2%

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

      1. *-commutative 81.4%

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

   10. Simplified 88.2%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{-42}:\\ \;\;\;\;\frac{\cos k}{t} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{+69}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\frac{1}{\ell} \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell}\right)\right)}\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{+183}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(k \cdot 2\right)}\\ \end{array} \]

Alternative 5: 86.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \tan k \cdot \left(2 + {\left(\frac{k}{t_m}\right)}^{2}\right)\\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 4.2 \cdot 10^{-42}:\\ \;\;\;\;\frac{\cos k}{t_m} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \mathbf{elif}\;t_m \leq 9.5 \cdot 10^{+68}:\\ \;\;\;\;\frac{2}{t_2 \cdot \left(\frac{1}{\ell} \cdot \left(\sin k \cdot \frac{{t_m}^{3}}{\ell}\right)\right)}\\ \mathbf{elif}\;t_m \leq 1.34 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{t_2 \cdot \left(\sin k \cdot \left(\frac{{t_m}^{2}}{\ell} \cdot \frac{t_m}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(k \cdot 2\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))))
   (*
    t_s
    (if (<= t_m 4.2e-42)
      (* (/ (cos k) t_m) (pow (* l (/ (sqrt 2.0) (* k (sin k)))) 2.0))
      (if (<= t_m 9.5e+68)
        (/ 2.0 (* t_2 (* (/ 1.0 l) (* (sin k) (/ (pow t_m 3.0) l)))))
        (if (<= t_m 1.34e+154)
          (/ 2.0 (* t_2 (* (sin k) (* (/ (pow t_m 2.0) l) (/ t_m l)))))
          (/
           2.0
           (*
            (pow (* (cbrt (sin k)) (/ t_m (pow (cbrt l) 2.0))) 3.0)
            (* k 2.0)))))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = tan(k) * (2.0 + pow((k / t_m), 2.0));
	double tmp;
	if (t_m <= 4.2e-42) {
		tmp = (cos(k) / t_m) * pow((l * (sqrt(2.0) / (k * sin(k)))), 2.0);
	} else if (t_m <= 9.5e+68) {
		tmp = 2.0 / (t_2 * ((1.0 / l) * (sin(k) * (pow(t_m, 3.0) / l))));
	} else if (t_m <= 1.34e+154) {
		tmp = 2.0 / (t_2 * (sin(k) * ((pow(t_m, 2.0) / l) * (t_m / l))));
	} else {
		tmp = 2.0 / (pow((cbrt(sin(k)) * (t_m / pow(cbrt(l), 2.0))), 3.0) * (k * 2.0));
	}
	return t_s * tmp;
}

Java

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) {
	double t_2 = Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0));
	double tmp;
	if (t_m <= 4.2e-42) {
		tmp = (Math.cos(k) / t_m) * Math.pow((l * (Math.sqrt(2.0) / (k * Math.sin(k)))), 2.0);
	} else if (t_m <= 9.5e+68) {
		tmp = 2.0 / (t_2 * ((1.0 / l) * (Math.sin(k) * (Math.pow(t_m, 3.0) / l))));
	} else if (t_m <= 1.34e+154) {
		tmp = 2.0 / (t_2 * (Math.sin(k) * ((Math.pow(t_m, 2.0) / l) * (t_m / l))));
	} else {
		tmp = 2.0 / (Math.pow((Math.cbrt(Math.sin(k)) * (t_m / Math.pow(Math.cbrt(l), 2.0))), 3.0) * (k * 2.0));
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0)))
	tmp = 0.0
	if (t_m <= 4.2e-42)
		tmp = Float64(Float64(cos(k) / t_m) * (Float64(l * Float64(sqrt(2.0) / Float64(k * sin(k)))) ^ 2.0));
	elseif (t_m <= 9.5e+68)
		tmp = Float64(2.0 / Float64(t_2 * Float64(Float64(1.0 / l) * Float64(sin(k) * Float64((t_m ^ 3.0) / l)))));
	elseif (t_m <= 1.34e+154)
		tmp = Float64(2.0 / Float64(t_2 * Float64(sin(k) * Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l)))));
	else
		tmp = Float64(2.0 / Float64((Float64(cbrt(sin(k)) * Float64(t_m / (cbrt(l) ^ 2.0))) ^ 3.0) * Float64(k * 2.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_] := Block[{t$95$2 = N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4.2e-42], N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 9.5e+68], N[(2.0 / N[(t$95$2 * N[(N[(1.0 / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.34e+154], N[(2.0 / N[(t$95$2 * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(k * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 4.20000000000000013e-42

   1. Initial program 49.0%

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

   3. Simplified 49.0%

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

      1. add-sqr-sqrt 38.3%

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

   5. Applied egg-rr 43.8%

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

      1. unpow2 43.8%

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

      2. associate-/l* 43.8%

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

      3. associate-*r* 43.8%

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

      4. *-commutative 43.8%

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

   7. Simplified 43.8%

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

   8. Taylor expanded in k around inf 39.7%

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

      1. expm1-log1p-u 39.4%

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

      2. expm1-udef 34.7%

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

      3. *-commutative 34.7%

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

      4. unpow-prod-down 34.6%

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

      5. pow2 34.6%

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

      6. add-sqr-sqrt 54.2%

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

      7. times-frac 54.7%

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

   10. Applied egg-rr 54.7%

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

       1. expm1-def 58.1%

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

       2. expm1-log1p 75.1%

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

       3. associate-*r/ 75.2%

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

       4. associate-*l/ 75.2%

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

       5. associate-/r* 75.2%

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

       6. *-rgt-identity 75.2%

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

       7. associate-*r/ 75.2%

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

       8. associate-*l* 75.2%

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

       9. associate-*r/ 75.2%

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

       10. *-commutative 75.2%

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

       11. *-lft-identity 75.2%

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

   12. Simplified 75.2%

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

   if 4.20000000000000013e-42 < t < 9.50000000000000069e68

   1. Initial program 79.2%

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

      1. associate-*l* 79.2%

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

      2. *-commutative 79.2%

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

      3. *-commutative 79.2%

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

      4. associate-/r* 84.5%

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

      5. distribute-rgt-in 84.5%

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

      6. unpow2 84.5%

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

      7. times-frac 75.0%

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

      8. sqr-neg 75.0%

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

      9. times-frac 84.5%

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

      10. unpow2 84.5%

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

      11. distribute-rgt-in 84.5%

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

      12. +-commutative 84.5%

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

   3. Simplified 84.5%

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

      1. associate-*l/ 99.5%

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

      2. clear-num 99.5%

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

   5. Applied egg-rr 99.5%

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

      1. associate-/r/ 99.5%

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

      2. *-commutative 99.5%

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

   7. Simplified 99.5%

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

   if 9.50000000000000069e68 < t < 1.34000000000000001e154

   1. Initial program 38.4%

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

      1. associate-*l* 38.4%

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

      2. *-commutative 38.4%

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

      3. *-commutative 38.4%

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

      4. associate-/r* 39.1%

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

      5. distribute-rgt-in 39.1%

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

      6. unpow2 39.1%

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

      7. times-frac 39.1%

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

      8. sqr-neg 39.1%

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

      9. times-frac 39.1%

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

      10. unpow2 39.1%

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

      11. distribute-rgt-in 39.1%

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

      12. +-commutative 39.1%

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

   3. Simplified 39.1%

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

      1. associate-/r* 38.4%

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

      2. unpow3 38.4%

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

      3. times-frac 87.2%

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

      4. pow2 87.2%

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

   5. Applied egg-rr 87.2%

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

   if 1.34000000000000001e154 < t

   1. Initial program 63.0%

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

      1. associate-*l* 63.0%

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

      2. *-commutative 63.0%

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

      3. *-commutative 63.0%

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

      4. associate-/r* 71.2%

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

      5. distribute-rgt-in 71.2%

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

      6. unpow2 71.2%

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

      7. times-frac 44.7%

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

      8. sqr-neg 44.7%

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

      9. times-frac 71.2%

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

      10. unpow2 71.2%

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

      11. distribute-rgt-in 71.2%

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

      12. +-commutative 71.2%

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

   3. Simplified 71.2%

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

      1. associate-/r* 63.0%

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

      2. unpow3 63.0%

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

      3. times-frac 71.2%

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

      4. pow2 71.2%

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

   5. Applied egg-rr 71.2%

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

      1. add-cube-cbrt 71.2%

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

      2. pow3 71.2%

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

      3. cbrt-prod 71.2%

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

      4. frac-times 63.0%

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

      5. unpow2 63.0%

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

      6. unpow3 63.0%

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

      7. unpow2 63.0%

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

      8. cbrt-div 63.0%

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

      9. unpow3 63.0%

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

      10. add-cbrt-cube 74.7%

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

      11. unpow2 74.7%

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

      12. cbrt-prod 87.8%

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

      13. pow2 87.8%

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

   7. Applied egg-rr 87.8%

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

   8. Taylor expanded in k around 0 87.8%

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

      1. *-commutative 82.5%

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

   10. Simplified 87.8%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{-42}:\\ \;\;\;\;\frac{\cos k}{t} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{+68}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\frac{1}{\ell} \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell}\right)\right)}\\ \mathbf{elif}\;t \leq 1.34 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(k \cdot 2\right)}\\ \end{array} \]

Alternative 6: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \tan k \cdot \left(2 + {\left(\frac{k}{t_m}\right)}^{2}\right)\\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 4 \cdot 10^{-42}:\\ \;\;\;\;\frac{\cos k}{t_m} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \mathbf{elif}\;t_m \leq 6.5 \cdot 10^{+69}:\\ \;\;\;\;\frac{2}{t_2 \cdot \frac{\sin k \cdot \frac{{t_m}^{3}}{\ell}}{\ell}}\\ \mathbf{elif}\;t_m \leq 1.45 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{t_2 \cdot \left(\sin k \cdot \left(\frac{{t_m}^{2}}{\ell} \cdot \frac{t_m}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot {\left(\frac{\frac{t_m}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{3}\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))))
   (*
    t_s
    (if (<= t_m 4e-42)
      (* (/ (cos k) t_m) (pow (* l (/ (sqrt 2.0) (* k (sin k)))) 2.0))
      (if (<= t_m 6.5e+69)
        (/ 2.0 (* t_2 (/ (* (sin k) (/ (pow t_m 3.0) l)) l)))
        (if (<= t_m 1.45e+154)
          (/ 2.0 (* t_2 (* (sin k) (* (/ (pow t_m 2.0) l) (/ t_m l)))))
          (/
           2.0
           (*
            (* k 2.0)
            (* (sin k) (pow (/ (/ t_m (cbrt l)) (cbrt l)) 3.0))))))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = tan(k) * (2.0 + pow((k / t_m), 2.0));
	double tmp;
	if (t_m <= 4e-42) {
		tmp = (cos(k) / t_m) * pow((l * (sqrt(2.0) / (k * sin(k)))), 2.0);
	} else if (t_m <= 6.5e+69) {
		tmp = 2.0 / (t_2 * ((sin(k) * (pow(t_m, 3.0) / l)) / l));
	} else if (t_m <= 1.45e+154) {
		tmp = 2.0 / (t_2 * (sin(k) * ((pow(t_m, 2.0) / l) * (t_m / l))));
	} else {
		tmp = 2.0 / ((k * 2.0) * (sin(k) * pow(((t_m / cbrt(l)) / cbrt(l)), 3.0)));
	}
	return t_s * tmp;
}

Java

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) {
	double t_2 = Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0));
	double tmp;
	if (t_m <= 4e-42) {
		tmp = (Math.cos(k) / t_m) * Math.pow((l * (Math.sqrt(2.0) / (k * Math.sin(k)))), 2.0);
	} else if (t_m <= 6.5e+69) {
		tmp = 2.0 / (t_2 * ((Math.sin(k) * (Math.pow(t_m, 3.0) / l)) / l));
	} else if (t_m <= 1.45e+154) {
		tmp = 2.0 / (t_2 * (Math.sin(k) * ((Math.pow(t_m, 2.0) / l) * (t_m / l))));
	} else {
		tmp = 2.0 / ((k * 2.0) * (Math.sin(k) * Math.pow(((t_m / Math.cbrt(l)) / Math.cbrt(l)), 3.0)));
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0)))
	tmp = 0.0
	if (t_m <= 4e-42)
		tmp = Float64(Float64(cos(k) / t_m) * (Float64(l * Float64(sqrt(2.0) / Float64(k * sin(k)))) ^ 2.0));
	elseif (t_m <= 6.5e+69)
		tmp = Float64(2.0 / Float64(t_2 * Float64(Float64(sin(k) * Float64((t_m ^ 3.0) / l)) / l)));
	elseif (t_m <= 1.45e+154)
		tmp = Float64(2.0 / Float64(t_2 * Float64(sin(k) * Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l)))));
	else
		tmp = Float64(2.0 / Float64(Float64(k * 2.0) * Float64(sin(k) * (Float64(Float64(t_m / cbrt(l)) / cbrt(l)) ^ 3.0))));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_] := Block[{t$95$2 = N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4e-42], N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6.5e+69], N[(2.0 / N[(t$95$2 * N[(N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.45e+154], N[(2.0 / N[(t$95$2 * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * 2.0), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 4.00000000000000015e-42

   1. Initial program 49.0%

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

   3. Simplified 49.0%

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

      1. add-sqr-sqrt 38.3%

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

   5. Applied egg-rr 43.8%

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

      1. unpow2 43.8%

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

      2. associate-/l* 43.8%

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

      3. associate-*r* 43.8%

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

      4. *-commutative 43.8%

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

   7. Simplified 43.8%

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

   8. Taylor expanded in k around inf 39.7%

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

      1. expm1-log1p-u 39.4%

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

      2. expm1-udef 34.7%

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

      3. *-commutative 34.7%

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

      4. unpow-prod-down 34.6%

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

      5. pow2 34.6%

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

      6. add-sqr-sqrt 54.2%

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

      7. times-frac 54.7%

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

   10. Applied egg-rr 54.7%

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

       1. expm1-def 58.1%

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

       2. expm1-log1p 75.1%

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

       3. associate-*r/ 75.2%

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

       4. associate-*l/ 75.2%

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

       5. associate-/r* 75.2%

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

       6. *-rgt-identity 75.2%

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

       7. associate-*r/ 75.2%

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

       8. associate-*l* 75.2%

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

       9. associate-*r/ 75.2%

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

       10. *-commutative 75.2%

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

       11. *-lft-identity 75.2%

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

   12. Simplified 75.2%

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

   if 4.00000000000000015e-42 < t < 6.5000000000000001e69

   1. Initial program 79.2%

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

      1. associate-*l* 79.2%

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

      2. *-commutative 79.2%

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

      3. *-commutative 79.2%

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

      4. associate-/r* 84.5%

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

      5. distribute-rgt-in 84.5%

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

      6. unpow2 84.5%

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

      7. times-frac 75.0%

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

      8. sqr-neg 75.0%

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

      9. times-frac 84.5%

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

      10. unpow2 84.5%

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

      11. distribute-rgt-in 84.5%

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

      12. +-commutative 84.5%

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

   3. Simplified 84.5%

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

      1. associate-*l/ 99.5%

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

   5. Applied egg-rr 99.5%

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

   if 6.5000000000000001e69 < t < 1.4499999999999999e154

   1. Initial program 38.4%

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

      1. associate-*l* 38.4%

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

      2. *-commutative 38.4%

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

      3. *-commutative 38.4%

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

      4. associate-/r* 39.1%

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

      5. distribute-rgt-in 39.1%

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

      6. unpow2 39.1%

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

      7. times-frac 39.1%

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

      8. sqr-neg 39.1%

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

      9. times-frac 39.1%

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

      10. unpow2 39.1%

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

      11. distribute-rgt-in 39.1%

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

      12. +-commutative 39.1%

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

   3. Simplified 39.1%

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

      1. associate-/r* 38.4%

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

      2. unpow3 38.4%

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

      3. times-frac 87.2%

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

      4. pow2 87.2%

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

   5. Applied egg-rr 87.2%

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

   if 1.4499999999999999e154 < t

   1. Initial program 63.0%

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

      1. associate-*l* 63.0%

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

      2. *-commutative 63.0%

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

      3. *-commutative 63.0%

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

      4. associate-/r* 71.2%

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

      5. distribute-rgt-in 71.2%

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

      6. unpow2 71.2%

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

      7. times-frac 44.7%

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

      8. sqr-neg 44.7%

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

      9. times-frac 71.2%

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

      10. unpow2 71.2%

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

      11. distribute-rgt-in 71.2%

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

      12. +-commutative 71.2%

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

   3. Simplified 71.2%

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

      1. add-cube-cbrt 71.2%

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

      2. *-un-lft-identity 71.2%

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

      3. times-frac 71.2%

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

      4. pow2 71.2%

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

      5. cbrt-div 71.2%

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

      6. rem-cbrt-cube 71.2%

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

      7. cbrt-div 71.2%

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

      8. rem-cbrt-cube 82.5%

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

   5. Applied egg-rr 82.5%

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

   6. Taylor expanded in k around 0 82.5%

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

      1. *-commutative 82.5%

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

   8. Simplified 82.5%

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

      1. add-cube-cbrt 82.5%

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

      2. pow3 82.5%

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

      3. frac-times 73.9%

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

      4. unpow2 73.9%

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

      5. *-un-lft-identity 73.9%

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

      6. cbrt-div 73.9%

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

      7. add-cbrt-cube 82.4%

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

   10. Applied egg-rr 82.4%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4 \cdot 10^{-42}:\\ \;\;\;\;\frac{\cos k}{t} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{+69}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \frac{\sin k \cdot \frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot {\left(\frac{\frac{t}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{3}\right)}\\ \end{array} \]

Alternative 7: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \tan k \cdot \left(2 + {\left(\frac{k}{t_m}\right)}^{2}\right)\\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 4 \cdot 10^{-42}:\\ \;\;\;\;\frac{\cos k}{t_m} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \mathbf{elif}\;t_m \leq 2 \cdot 10^{+68}:\\ \;\;\;\;\frac{2}{t_2 \cdot \left(\frac{1}{\ell} \cdot \left(\sin k \cdot \frac{{t_m}^{3}}{\ell}\right)\right)}\\ \mathbf{elif}\;t_m \leq 1.34 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{t_2 \cdot \left(\sin k \cdot \left(\frac{{t_m}^{2}}{\ell} \cdot \frac{t_m}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot {\left(\frac{\frac{t_m}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{3}\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))))
   (*
    t_s
    (if (<= t_m 4e-42)
      (* (/ (cos k) t_m) (pow (* l (/ (sqrt 2.0) (* k (sin k)))) 2.0))
      (if (<= t_m 2e+68)
        (/ 2.0 (* t_2 (* (/ 1.0 l) (* (sin k) (/ (pow t_m 3.0) l)))))
        (if (<= t_m 1.34e+154)
          (/ 2.0 (* t_2 (* (sin k) (* (/ (pow t_m 2.0) l) (/ t_m l)))))
          (/
           2.0
           (*
            (* k 2.0)
            (* (sin k) (pow (/ (/ t_m (cbrt l)) (cbrt l)) 3.0))))))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = tan(k) * (2.0 + pow((k / t_m), 2.0));
	double tmp;
	if (t_m <= 4e-42) {
		tmp = (cos(k) / t_m) * pow((l * (sqrt(2.0) / (k * sin(k)))), 2.0);
	} else if (t_m <= 2e+68) {
		tmp = 2.0 / (t_2 * ((1.0 / l) * (sin(k) * (pow(t_m, 3.0) / l))));
	} else if (t_m <= 1.34e+154) {
		tmp = 2.0 / (t_2 * (sin(k) * ((pow(t_m, 2.0) / l) * (t_m / l))));
	} else {
		tmp = 2.0 / ((k * 2.0) * (sin(k) * pow(((t_m / cbrt(l)) / cbrt(l)), 3.0)));
	}
	return t_s * tmp;
}

Java

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) {
	double t_2 = Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0));
	double tmp;
	if (t_m <= 4e-42) {
		tmp = (Math.cos(k) / t_m) * Math.pow((l * (Math.sqrt(2.0) / (k * Math.sin(k)))), 2.0);
	} else if (t_m <= 2e+68) {
		tmp = 2.0 / (t_2 * ((1.0 / l) * (Math.sin(k) * (Math.pow(t_m, 3.0) / l))));
	} else if (t_m <= 1.34e+154) {
		tmp = 2.0 / (t_2 * (Math.sin(k) * ((Math.pow(t_m, 2.0) / l) * (t_m / l))));
	} else {
		tmp = 2.0 / ((k * 2.0) * (Math.sin(k) * Math.pow(((t_m / Math.cbrt(l)) / Math.cbrt(l)), 3.0)));
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0)))
	tmp = 0.0
	if (t_m <= 4e-42)
		tmp = Float64(Float64(cos(k) / t_m) * (Float64(l * Float64(sqrt(2.0) / Float64(k * sin(k)))) ^ 2.0));
	elseif (t_m <= 2e+68)
		tmp = Float64(2.0 / Float64(t_2 * Float64(Float64(1.0 / l) * Float64(sin(k) * Float64((t_m ^ 3.0) / l)))));
	elseif (t_m <= 1.34e+154)
		tmp = Float64(2.0 / Float64(t_2 * Float64(sin(k) * Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l)))));
	else
		tmp = Float64(2.0 / Float64(Float64(k * 2.0) * Float64(sin(k) * (Float64(Float64(t_m / cbrt(l)) / cbrt(l)) ^ 3.0))));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_] := Block[{t$95$2 = N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4e-42], N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2e+68], N[(2.0 / N[(t$95$2 * N[(N[(1.0 / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.34e+154], N[(2.0 / N[(t$95$2 * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * 2.0), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 4.00000000000000015e-42

   1. Initial program 49.0%

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

   3. Simplified 49.0%

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

      1. add-sqr-sqrt 38.3%

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

   5. Applied egg-rr 43.8%

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

      1. unpow2 43.8%

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

      2. associate-/l* 43.8%

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

      3. associate-*r* 43.8%

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

      4. *-commutative 43.8%

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

   7. Simplified 43.8%

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

   8. Taylor expanded in k around inf 39.7%

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

      1. expm1-log1p-u 39.4%

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

      2. expm1-udef 34.7%

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

      3. *-commutative 34.7%

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

      4. unpow-prod-down 34.6%

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

      5. pow2 34.6%

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

      6. add-sqr-sqrt 54.2%

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

      7. times-frac 54.7%

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

   10. Applied egg-rr 54.7%

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

       1. expm1-def 58.1%

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

       2. expm1-log1p 75.1%

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

       3. associate-*r/ 75.2%

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

       4. associate-*l/ 75.2%

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

       5. associate-/r* 75.2%

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

       6. *-rgt-identity 75.2%

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

       7. associate-*r/ 75.2%

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

       8. associate-*l* 75.2%

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

       9. associate-*r/ 75.2%

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

       10. *-commutative 75.2%

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

       11. *-lft-identity 75.2%

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

   12. Simplified 75.2%

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

   if 4.00000000000000015e-42 < t < 1.99999999999999991e68

   1. Initial program 79.2%

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

      1. associate-*l* 79.2%

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

      2. *-commutative 79.2%

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

      3. *-commutative 79.2%

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

      4. associate-/r* 84.5%

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

      5. distribute-rgt-in 84.5%

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

      6. unpow2 84.5%

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

      7. times-frac 75.0%

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

      8. sqr-neg 75.0%

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

      9. times-frac 84.5%

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

      10. unpow2 84.5%

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

      11. distribute-rgt-in 84.5%

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

      12. +-commutative 84.5%

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

   3. Simplified 84.5%

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

      1. associate-*l/ 99.5%

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

      2. clear-num 99.5%

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

   5. Applied egg-rr 99.5%

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

      1. associate-/r/ 99.5%

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

      2. *-commutative 99.5%

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

   7. Simplified 99.5%

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

   if 1.99999999999999991e68 < t < 1.34000000000000001e154

   1. Initial program 38.4%

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

      1. associate-*l* 38.4%

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

      2. *-commutative 38.4%

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

      3. *-commutative 38.4%

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

      4. associate-/r* 39.1%

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

      5. distribute-rgt-in 39.1%

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

      6. unpow2 39.1%

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

      7. times-frac 39.1%

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

      8. sqr-neg 39.1%

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

      9. times-frac 39.1%

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

      10. unpow2 39.1%

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

      11. distribute-rgt-in 39.1%

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

      12. +-commutative 39.1%

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

   3. Simplified 39.1%

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

      1. associate-/r* 38.4%

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

      2. unpow3 38.4%

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

      3. times-frac 87.2%

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

      4. pow2 87.2%

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

   5. Applied egg-rr 87.2%

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

   if 1.34000000000000001e154 < t

   1. Initial program 63.0%

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

      1. associate-*l* 63.0%

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

      2. *-commutative 63.0%

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

      3. *-commutative 63.0%

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

      4. associate-/r* 71.2%

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

      5. distribute-rgt-in 71.2%

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

      6. unpow2 71.2%

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

      7. times-frac 44.7%

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

      8. sqr-neg 44.7%

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

      9. times-frac 71.2%

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

      10. unpow2 71.2%

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

      11. distribute-rgt-in 71.2%

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

      12. +-commutative 71.2%

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

   3. Simplified 71.2%

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

      1. add-cube-cbrt 71.2%

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

      2. *-un-lft-identity 71.2%

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

      3. times-frac 71.2%

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

      4. pow2 71.2%

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

      5. cbrt-div 71.2%

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

      6. rem-cbrt-cube 71.2%

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

      7. cbrt-div 71.2%

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

      8. rem-cbrt-cube 82.5%

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

   5. Applied egg-rr 82.5%

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

   6. Taylor expanded in k around 0 82.5%

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

      1. *-commutative 82.5%

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

   8. Simplified 82.5%

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

      1. add-cube-cbrt 82.5%

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

      2. pow3 82.5%

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

      3. frac-times 73.9%

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

      4. unpow2 73.9%

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

      5. *-un-lft-identity 73.9%

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

      6. cbrt-div 73.9%

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

      7. add-cbrt-cube 82.4%

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

   10. Applied egg-rr 82.4%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4 \cdot 10^{-42}:\\ \;\;\;\;\frac{\cos k}{t} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{+68}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\frac{1}{\ell} \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell}\right)\right)}\\ \mathbf{elif}\;t \leq 1.34 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot {\left(\frac{\frac{t}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{3}\right)}\\ \end{array} \]

Alternative 8: 82.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 4.5 \cdot 10^{-42}:\\ \;\;\;\;\frac{\cos k}{t_m} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \mathbf{elif}\;t_m \leq 2.4 \cdot 10^{+102}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot \sqrt{{t_m}^{3}}}\right)}^{2}\\ \mathbf{elif}\;t_m \leq 1.34 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{{t_m}^{2}}{\ell} \cdot \frac{t_m}{\ell}\right)\right) \cdot \left(2 \cdot \frac{\sin k}{\cos k}\right)}\\ \mathbf{elif}\;t_m \leq 9.2 \cdot 10^{+235}:\\ \;\;\;\;\frac{\ell \cdot \ell}{{\left(t_m \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot {\left(\frac{\frac{t_m}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{3}\right)}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4.5e-42)
    (* (/ (cos k) t_m) (pow (* l (/ (sqrt 2.0) (* k (sin k)))) 2.0))
    (if (<= t_m 2.4e+102)
      (pow (/ l (* k (sqrt (pow t_m 3.0)))) 2.0)
      (if (<= t_m 1.34e+154)
        (/
         2.0
         (*
          (* (sin k) (* (/ (pow t_m 2.0) l) (/ t_m l)))
          (* 2.0 (/ (sin k) (cos k)))))
        (if (<= t_m 9.2e+235)
          (/ (* l l) (pow (* t_m (pow (cbrt k) 2.0)) 3.0))
          (/
           2.0
           (*
            (* k 2.0)
            (* (sin k) (pow (/ (/ t_m (cbrt l)) (cbrt l)) 3.0))))))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.5e-42) {
		tmp = (cos(k) / t_m) * pow((l * (sqrt(2.0) / (k * sin(k)))), 2.0);
	} else if (t_m <= 2.4e+102) {
		tmp = pow((l / (k * sqrt(pow(t_m, 3.0)))), 2.0);
	} else if (t_m <= 1.34e+154) {
		tmp = 2.0 / ((sin(k) * ((pow(t_m, 2.0) / l) * (t_m / l))) * (2.0 * (sin(k) / cos(k))));
	} else if (t_m <= 9.2e+235) {
		tmp = (l * l) / pow((t_m * pow(cbrt(k), 2.0)), 3.0);
	} else {
		tmp = 2.0 / ((k * 2.0) * (sin(k) * pow(((t_m / cbrt(l)) / cbrt(l)), 3.0)));
	}
	return t_s * tmp;
}

Java

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) {
	double tmp;
	if (t_m <= 4.5e-42) {
		tmp = (Math.cos(k) / t_m) * Math.pow((l * (Math.sqrt(2.0) / (k * Math.sin(k)))), 2.0);
	} else if (t_m <= 2.4e+102) {
		tmp = Math.pow((l / (k * Math.sqrt(Math.pow(t_m, 3.0)))), 2.0);
	} else if (t_m <= 1.34e+154) {
		tmp = 2.0 / ((Math.sin(k) * ((Math.pow(t_m, 2.0) / l) * (t_m / l))) * (2.0 * (Math.sin(k) / Math.cos(k))));
	} else if (t_m <= 9.2e+235) {
		tmp = (l * l) / Math.pow((t_m * Math.pow(Math.cbrt(k), 2.0)), 3.0);
	} else {
		tmp = 2.0 / ((k * 2.0) * (Math.sin(k) * Math.pow(((t_m / Math.cbrt(l)) / Math.cbrt(l)), 3.0)));
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 4.5e-42)
		tmp = Float64(Float64(cos(k) / t_m) * (Float64(l * Float64(sqrt(2.0) / Float64(k * sin(k)))) ^ 2.0));
	elseif (t_m <= 2.4e+102)
		tmp = Float64(l / Float64(k * sqrt((t_m ^ 3.0)))) ^ 2.0;
	elseif (t_m <= 1.34e+154)
		tmp = Float64(2.0 / Float64(Float64(sin(k) * Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l))) * Float64(2.0 * Float64(sin(k) / cos(k)))));
	elseif (t_m <= 9.2e+235)
		tmp = Float64(Float64(l * l) / (Float64(t_m * (cbrt(k) ^ 2.0)) ^ 3.0));
	else
		tmp = Float64(2.0 / Float64(Float64(k * 2.0) * Float64(sin(k) * (Float64(Float64(t_m / cbrt(l)) / cbrt(l)) ^ 3.0))));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[t$95$m, 4.5e-42], N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.4e+102], N[Power[N[(l / N[(k * N[Sqrt[N[Power[t$95$m, 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[t$95$m, 1.34e+154], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[Sin[k], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 9.2e+235], N[(N[(l * l), $MachinePrecision] / N[Power[N[(t$95$m * N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * 2.0), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]

Derivation

1. Split input into 5 regimes

2. if t < 4.5e-42

   1. Initial program 49.0%

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

   3. Simplified 49.0%

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

      1. add-sqr-sqrt 38.3%

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

   5. Applied egg-rr 43.8%

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

      1. unpow2 43.8%

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

      2. associate-/l* 43.8%

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

      3. associate-*r* 43.8%

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

      4. *-commutative 43.8%

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

   7. Simplified 43.8%

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

   8. Taylor expanded in k around inf 39.7%

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

      1. expm1-log1p-u 39.4%

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

      2. expm1-udef 34.7%

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

      3. *-commutative 34.7%

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

      4. unpow-prod-down 34.6%

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

      5. pow2 34.6%

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

      6. add-sqr-sqrt 54.2%

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

      7. times-frac 54.7%

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

   10. Applied egg-rr 54.7%

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

       1. expm1-def 58.1%

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

       2. expm1-log1p 75.1%

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

       3. associate-*r/ 75.2%

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

       4. associate-*l/ 75.2%

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

       5. associate-/r* 75.2%

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

       6. *-rgt-identity 75.2%

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

       7. associate-*r/ 75.2%

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

       8. associate-*l* 75.2%

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

       9. associate-*r/ 75.2%

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

       10. *-commutative 75.2%

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

       11. *-lft-identity 75.2%

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

   12. Simplified 75.2%

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

   if 4.5e-42 < t < 2.39999999999999994e102

   1. Initial program 76.5%

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

   3. Simplified 81.0%

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

      1. add-sqr-sqrt 71.5%

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

   5. Applied egg-rr 71.6%

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

      1. unpow2 71.6%

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

      2. associate-/l* 71.8%

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

      3. associate-*r* 71.8%

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

      4. *-commutative 71.8%

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

   7. Simplified 71.8%

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

   8. Taylor expanded in k around 0 81.6%

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

   if 2.39999999999999994e102 < t < 1.34000000000000001e154

   1. Initial program 35.8%

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

      1. associate-*l* 35.8%

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

      2. *-commutative 35.8%

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

      3. *-commutative 35.8%

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

      4. associate-/r* 36.6%

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

      5. distribute-rgt-in 36.6%

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

      6. unpow2 36.6%

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

      7. times-frac 36.6%

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

      8. sqr-neg 36.6%

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

      9. times-frac 36.6%

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

      10. unpow2 36.6%

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

      11. distribute-rgt-in 36.6%

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

      12. +-commutative 36.6%

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

   3. Simplified 36.6%

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

      1. associate-/r* 35.8%

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

      2. unpow3 35.8%

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

      3. times-frac 94.3%

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

      4. pow2 94.3%

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

   5. Applied egg-rr 94.3%

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

   6. Taylor expanded in t around inf 94.4%

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

   if 1.34000000000000001e154 < t < 9.2e235

   1. Initial program 58.7%

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

   3. Simplified 38.6%

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

   4. Taylor expanded in k around 0 38.6%

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

      1. unpow2 38.6%

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

   6. Applied egg-rr 38.6%

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

      1. add-cube-cbrt 38.6%

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

      2. pow3 38.6%

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

      3. *-commutative 38.6%

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

      4. cbrt-prod 38.6%

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

      5. unpow3 38.6%

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

      6. add-cbrt-cube 52.9%

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

      7. unpow2 52.9%

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

      8. cbrt-prod 85.6%

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

      9. pow2 85.6%

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

   8. Applied egg-rr 85.6%

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

   if 9.2e235 < t

   1. Initial program 70.1%

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

      1. associate-*l* 70.1%

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

      2. *-commutative 70.1%

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

      3. *-commutative 70.1%

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

      4. associate-/r* 85.6%

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

      5. distribute-rgt-in 85.6%

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

      6. unpow2 85.6%

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

      7. times-frac 54.8%

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

      8. sqr-neg 54.8%

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

      9. times-frac 85.6%

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

      10. unpow2 85.6%

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

      11. distribute-rgt-in 85.6%

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

      12. +-commutative 85.6%

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

   3. Simplified 85.6%

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

      1. add-cube-cbrt 85.6%

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

      2. *-un-lft-identity 85.6%

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

      3. times-frac 85.6%

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

      4. pow2 85.6%

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

      5. cbrt-div 85.6%

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

      6. rem-cbrt-cube 85.6%

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

      7. cbrt-div 85.6%

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

      8. rem-cbrt-cube 85.6%

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

   5. Applied egg-rr 85.6%

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

   6. Taylor expanded in k around 0 85.6%

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

      1. *-commutative 85.6%

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

   8. Simplified 85.6%

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

      1. add-cube-cbrt 85.6%

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

      2. pow3 85.6%

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

      3. frac-times 85.6%

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

      4. unpow2 85.6%

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

      5. *-un-lft-identity 85.6%

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

      6. cbrt-div 85.6%

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

      7. add-cbrt-cube 85.6%

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

   10. Applied egg-rr 85.6%

       \[\leadsto \frac{2}{\left(\color{blue}{{\left(\frac{\frac{t}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{3}} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.5 \cdot 10^{-42}:\\ \;\;\;\;\frac{\cos k}{t} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{+102}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot \sqrt{{t}^{3}}}\right)}^{2}\\ \mathbf{elif}\;t \leq 1.34 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \left(2 \cdot \frac{\sin k}{\cos k}\right)}\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{+235}:\\ \;\;\;\;\frac{\ell \cdot \ell}{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot {\left(\frac{\frac{t}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{3}\right)}\\ \end{array} \]

Alternative 9: 83.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 4.4 \cdot 10^{-42}:\\ \;\;\;\;\frac{\cos k}{t_m} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \mathbf{elif}\;t_m \leq 3.2 \cdot 10^{+88}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t_m}\right)}^{2}\right)\right) \cdot \frac{\sin k \cdot \frac{{t_m}^{3}}{\ell}}{\ell}}\\ \mathbf{elif}\;t_m \leq 1.34 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{{t_m}^{2}}{\ell} \cdot \frac{t_m}{\ell}\right)\right) \cdot \left(2 \cdot \frac{\sin k}{\cos k}\right)}\\ \mathbf{elif}\;t_m \leq 2.4 \cdot 10^{+236}:\\ \;\;\;\;\frac{\ell \cdot \ell}{{\left(t_m \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot {\left(\frac{\frac{t_m}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{3}\right)}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4.4e-42)
    (* (/ (cos k) t_m) (pow (* l (/ (sqrt 2.0) (* k (sin k)))) 2.0))
    (if (<= t_m 3.2e+88)
      (/
       2.0
       (*
        (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))
        (/ (* (sin k) (/ (pow t_m 3.0) l)) l)))
      (if (<= t_m 1.34e+154)
        (/
         2.0
         (*
          (* (sin k) (* (/ (pow t_m 2.0) l) (/ t_m l)))
          (* 2.0 (/ (sin k) (cos k)))))
        (if (<= t_m 2.4e+236)
          (/ (* l l) (pow (* t_m (pow (cbrt k) 2.0)) 3.0))
          (/
           2.0
           (*
            (* k 2.0)
            (* (sin k) (pow (/ (/ t_m (cbrt l)) (cbrt l)) 3.0))))))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.4e-42) {
		tmp = (cos(k) / t_m) * pow((l * (sqrt(2.0) / (k * sin(k)))), 2.0);
	} else if (t_m <= 3.2e+88) {
		tmp = 2.0 / ((tan(k) * (2.0 + pow((k / t_m), 2.0))) * ((sin(k) * (pow(t_m, 3.0) / l)) / l));
	} else if (t_m <= 1.34e+154) {
		tmp = 2.0 / ((sin(k) * ((pow(t_m, 2.0) / l) * (t_m / l))) * (2.0 * (sin(k) / cos(k))));
	} else if (t_m <= 2.4e+236) {
		tmp = (l * l) / pow((t_m * pow(cbrt(k), 2.0)), 3.0);
	} else {
		tmp = 2.0 / ((k * 2.0) * (sin(k) * pow(((t_m / cbrt(l)) / cbrt(l)), 3.0)));
	}
	return t_s * tmp;
}

Java

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) {
	double tmp;
	if (t_m <= 4.4e-42) {
		tmp = (Math.cos(k) / t_m) * Math.pow((l * (Math.sqrt(2.0) / (k * Math.sin(k)))), 2.0);
	} else if (t_m <= 3.2e+88) {
		tmp = 2.0 / ((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0))) * ((Math.sin(k) * (Math.pow(t_m, 3.0) / l)) / l));
	} else if (t_m <= 1.34e+154) {
		tmp = 2.0 / ((Math.sin(k) * ((Math.pow(t_m, 2.0) / l) * (t_m / l))) * (2.0 * (Math.sin(k) / Math.cos(k))));
	} else if (t_m <= 2.4e+236) {
		tmp = (l * l) / Math.pow((t_m * Math.pow(Math.cbrt(k), 2.0)), 3.0);
	} else {
		tmp = 2.0 / ((k * 2.0) * (Math.sin(k) * Math.pow(((t_m / Math.cbrt(l)) / Math.cbrt(l)), 3.0)));
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 4.4e-42)
		tmp = Float64(Float64(cos(k) / t_m) * (Float64(l * Float64(sqrt(2.0) / Float64(k * sin(k)))) ^ 2.0));
	elseif (t_m <= 3.2e+88)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * Float64(Float64(sin(k) * Float64((t_m ^ 3.0) / l)) / l)));
	elseif (t_m <= 1.34e+154)
		tmp = Float64(2.0 / Float64(Float64(sin(k) * Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l))) * Float64(2.0 * Float64(sin(k) / cos(k)))));
	elseif (t_m <= 2.4e+236)
		tmp = Float64(Float64(l * l) / (Float64(t_m * (cbrt(k) ^ 2.0)) ^ 3.0));
	else
		tmp = Float64(2.0 / Float64(Float64(k * 2.0) * Float64(sin(k) * (Float64(Float64(t_m / cbrt(l)) / cbrt(l)) ^ 3.0))));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[t$95$m, 4.4e-42], N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.2e+88], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.34e+154], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[Sin[k], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.4e+236], N[(N[(l * l), $MachinePrecision] / N[Power[N[(t$95$m * N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * 2.0), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]

Derivation

1. Split input into 5 regimes

2. if t < 4.4000000000000001e-42

   1. Initial program 49.0%

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

   3. Simplified 49.0%

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

      1. add-sqr-sqrt 38.3%

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

   5. Applied egg-rr 43.8%

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

      1. unpow2 43.8%

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

      2. associate-/l* 43.8%

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

      3. associate-*r* 43.8%

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

      4. *-commutative 43.8%

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

   7. Simplified 43.8%

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

   8. Taylor expanded in k around inf 39.7%

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

      1. expm1-log1p-u 39.4%

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

      2. expm1-udef 34.7%

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

      3. *-commutative 34.7%

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

      4. unpow-prod-down 34.6%

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

      5. pow2 34.6%

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

      6. add-sqr-sqrt 54.2%

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

      7. times-frac 54.7%

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

   10. Applied egg-rr 54.7%

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

       1. expm1-def 58.1%

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

       2. expm1-log1p 75.1%

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

       3. associate-*r/ 75.2%

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

       4. associate-*l/ 75.2%

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

       5. associate-/r* 75.2%

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

       6. *-rgt-identity 75.2%

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

       7. associate-*r/ 75.2%

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

       8. associate-*l* 75.2%

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

       9. associate-*r/ 75.2%

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

       10. *-commutative 75.2%

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

       11. *-lft-identity 75.2%

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

   12. Simplified 75.2%

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

   if 4.4000000000000001e-42 < t < 3.1999999999999999e88

   1. Initial program 80.2%

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

      1. associate-*l* 80.3%

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

      2. *-commutative 80.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. *-commutative 80.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-/r* 85.3%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. distribute-rgt-in 85.3%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

      6. unpow2 85.3%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      7. times-frac 76.3%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      8. sqr-neg 76.3%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      9. times-frac 85.3%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      10. unpow2 85.3%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      11. distribute-rgt-in 85.3%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

      12. +-commutative 85.3%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)\right)}\right)} \]

   3. Simplified 85.3%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*l/ 99.5%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell}} \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   5. Applied egg-rr 99.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell}} \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   if 3.1999999999999999e88 < t < 1.34000000000000001e154

   1. Initial program 32.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 32.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. *-commutative 32.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. *-commutative 32.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-/r* 33.6%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. distribute-rgt-in 33.6%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

      6. unpow2 33.6%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      7. times-frac 33.6%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      8. sqr-neg 33.6%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      9. times-frac 33.6%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      10. unpow2 33.6%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      11. distribute-rgt-in 33.6%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

      12. +-commutative 33.6%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)\right)}\right)} \]

   3. Simplified 33.6%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-/r* 32.8%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      2. unpow3 32.8%

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      3. times-frac 86.0%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      4. pow2 86.0%

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{2}}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   5. Applied egg-rr 86.0%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   6. Taylor expanded in t around inf 86.2%

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot \frac{\sin k}{\cos k}\right)}} \]

   if 1.34000000000000001e154 < t < 2.40000000000000013e236

   1. Initial program 58.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 38.6%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

   4. Taylor expanded in k around 0 38.6%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 38.6%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   6. Applied egg-rr 38.6%

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]
   7. Step-by-step derivation

      1. add-cube-cbrt 38.6%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\sqrt[3]{{k}^{2} \cdot {t}^{3}} \cdot \sqrt[3]{{k}^{2} \cdot {t}^{3}}\right) \cdot \sqrt[3]{{k}^{2} \cdot {t}^{3}}}} \]

      2. pow3 38.6%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{\left(\sqrt[3]{{k}^{2} \cdot {t}^{3}}\right)}^{3}}} \]

      3. *-commutative 38.6%

         \[\leadsto \frac{\ell \cdot \ell}{{\left(\sqrt[3]{\color{blue}{{t}^{3} \cdot {k}^{2}}}\right)}^{3}} \]

      4. cbrt-prod 38.6%

         \[\leadsto \frac{\ell \cdot \ell}{{\color{blue}{\left(\sqrt[3]{{t}^{3}} \cdot \sqrt[3]{{k}^{2}}\right)}}^{3}} \]

      5. unpow3 38.6%

         \[\leadsto \frac{\ell \cdot \ell}{{\left(\sqrt[3]{\color{blue}{\left(t \cdot t\right) \cdot t}} \cdot \sqrt[3]{{k}^{2}}\right)}^{3}} \]

      6. add-cbrt-cube 52.9%

         \[\leadsto \frac{\ell \cdot \ell}{{\left(\color{blue}{t} \cdot \sqrt[3]{{k}^{2}}\right)}^{3}} \]

      7. unpow2 52.9%

         \[\leadsto \frac{\ell \cdot \ell}{{\left(t \cdot \sqrt[3]{\color{blue}{k \cdot k}}\right)}^{3}} \]

      8. cbrt-prod 85.6%

         \[\leadsto \frac{\ell \cdot \ell}{{\left(t \cdot \color{blue}{\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)}\right)}^{3}} \]

      9. pow2 85.6%

         \[\leadsto \frac{\ell \cdot \ell}{{\left(t \cdot \color{blue}{{\left(\sqrt[3]{k}\right)}^{2}}\right)}^{3}} \]

   8. Applied egg-rr 85.6%

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}} \]

   if 2.40000000000000013e236 < t

   1. Initial program 70.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 70.1%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. *-commutative 70.1%

         \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. *-commutative 70.1%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-/r* 85.6%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. distribute-rgt-in 85.6%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

      6. unpow2 85.6%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      7. times-frac 54.8%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      8. sqr-neg 54.8%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      9. times-frac 85.6%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      10. unpow2 85.6%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      11. distribute-rgt-in 85.6%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

      12. +-commutative 85.6%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)\right)}\right)} \]

   3. Simplified 85.6%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 85.6%

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      2. *-un-lft-identity 85.6%

         \[\leadsto \frac{2}{\left(\frac{\left(\sqrt[3]{\frac{{t}^{3}}{\ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}}{\color{blue}{1 \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      3. times-frac 85.6%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{\sqrt[3]{\frac{{t}^{3}}{\ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}}{1} \cdot \frac{\sqrt[3]{\frac{{t}^{3}}{\ell}}}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      4. pow2 85.6%

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{\left(\sqrt[3]{\frac{{t}^{3}}{\ell}}\right)}^{2}}}{1} \cdot \frac{\sqrt[3]{\frac{{t}^{3}}{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      5. cbrt-div 85.6%

         \[\leadsto \frac{2}{\left(\left(\frac{{\color{blue}{\left(\frac{\sqrt[3]{{t}^{3}}}{\sqrt[3]{\ell}}\right)}}^{2}}{1} \cdot \frac{\sqrt[3]{\frac{{t}^{3}}{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      6. rem-cbrt-cube 85.6%

         \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{\color{blue}{t}}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{\frac{{t}^{3}}{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      7. cbrt-div 85.6%

         \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\color{blue}{\frac{\sqrt[3]{{t}^{3}}}{\sqrt[3]{\ell}}}}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      8. rem-cbrt-cube 85.6%

         \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{\color{blue}{t}}{\sqrt[3]{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   5. Applied egg-rr 85.6%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   6. Taylor expanded in k around 0 85.6%

      \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]
   7. Step-by-step derivation

      1. *-commutative 85.6%

         \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   8. Simplified 85.6%

      \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]
   9. Step-by-step derivation

      1. add-cube-cbrt 85.6%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\sqrt[3]{\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}} \cdot \sqrt[3]{\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}}\right) \cdot \sqrt[3]{\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}}\right)} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      2. pow3 85.6%

         \[\leadsto \frac{2}{\left(\color{blue}{{\left(\sqrt[3]{\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}}\right)}^{3}} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      3. frac-times 85.6%

         \[\leadsto \frac{2}{\left({\left(\sqrt[3]{\color{blue}{\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2} \cdot \frac{t}{\sqrt[3]{\ell}}}{1 \cdot \ell}}}\right)}^{3} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      4. unpow2 85.6%

         \[\leadsto \frac{2}{\left({\left(\sqrt[3]{\frac{\color{blue}{\left(\frac{t}{\sqrt[3]{\ell}} \cdot \frac{t}{\sqrt[3]{\ell}}\right)} \cdot \frac{t}{\sqrt[3]{\ell}}}{1 \cdot \ell}}\right)}^{3} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      5. *-un-lft-identity 85.6%

         \[\leadsto \frac{2}{\left({\left(\sqrt[3]{\frac{\left(\frac{t}{\sqrt[3]{\ell}} \cdot \frac{t}{\sqrt[3]{\ell}}\right) \cdot \frac{t}{\sqrt[3]{\ell}}}{\color{blue}{\ell}}}\right)}^{3} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      6. cbrt-div 85.6%

         \[\leadsto \frac{2}{\left({\color{blue}{\left(\frac{\sqrt[3]{\left(\frac{t}{\sqrt[3]{\ell}} \cdot \frac{t}{\sqrt[3]{\ell}}\right) \cdot \frac{t}{\sqrt[3]{\ell}}}}{\sqrt[3]{\ell}}\right)}}^{3} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      7. add-cbrt-cube 85.6%

         \[\leadsto \frac{2}{\left({\left(\frac{\color{blue}{\frac{t}{\sqrt[3]{\ell}}}}{\sqrt[3]{\ell}}\right)}^{3} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

   10. Applied egg-rr 85.6%

       \[\leadsto \frac{2}{\left(\color{blue}{{\left(\frac{\frac{t}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{3}} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.4 \cdot 10^{-42}:\\ \;\;\;\;\frac{\cos k}{t} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{+88}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \frac{\sin k \cdot \frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{elif}\;t \leq 1.34 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \left(2 \cdot \frac{\sin k}{\cos k}\right)}\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{+236}:\\ \;\;\;\;\frac{\ell \cdot \ell}{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot {\left(\frac{\frac{t}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{3}\right)}\\ \end{array} \]

Alternative 10: 82.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 4.5 \cdot 10^{-42}:\\ \;\;\;\;\frac{\cos k}{t_m} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \mathbf{elif}\;t_m \leq 4.2 \cdot 10^{+102}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot \sqrt{{t_m}^{3}}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot {\left(\frac{\frac{t_m}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{3}\right)}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4.5e-42)
    (* (/ (cos k) t_m) (pow (* l (/ (sqrt 2.0) (* k (sin k)))) 2.0))
    (if (<= t_m 4.2e+102)
      (pow (/ l (* k (sqrt (pow t_m 3.0)))) 2.0)
      (/
       2.0
       (* (* k 2.0) (* (sin k) (pow (/ (/ t_m (cbrt l)) (cbrt l)) 3.0))))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.5e-42) {
		tmp = (cos(k) / t_m) * pow((l * (sqrt(2.0) / (k * sin(k)))), 2.0);
	} else if (t_m <= 4.2e+102) {
		tmp = pow((l / (k * sqrt(pow(t_m, 3.0)))), 2.0);
	} else {
		tmp = 2.0 / ((k * 2.0) * (sin(k) * pow(((t_m / cbrt(l)) / cbrt(l)), 3.0)));
	}
	return t_s * tmp;
}

Java

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) {
	double tmp;
	if (t_m <= 4.5e-42) {
		tmp = (Math.cos(k) / t_m) * Math.pow((l * (Math.sqrt(2.0) / (k * Math.sin(k)))), 2.0);
	} else if (t_m <= 4.2e+102) {
		tmp = Math.pow((l / (k * Math.sqrt(Math.pow(t_m, 3.0)))), 2.0);
	} else {
		tmp = 2.0 / ((k * 2.0) * (Math.sin(k) * Math.pow(((t_m / Math.cbrt(l)) / Math.cbrt(l)), 3.0)));
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 4.5e-42)
		tmp = Float64(Float64(cos(k) / t_m) * (Float64(l * Float64(sqrt(2.0) / Float64(k * sin(k)))) ^ 2.0));
	elseif (t_m <= 4.2e+102)
		tmp = Float64(l / Float64(k * sqrt((t_m ^ 3.0)))) ^ 2.0;
	else
		tmp = Float64(2.0 / Float64(Float64(k * 2.0) * Float64(sin(k) * (Float64(Float64(t_m / cbrt(l)) / cbrt(l)) ^ 3.0))));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[t$95$m, 4.5e-42], N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.2e+102], N[Power[N[(l / N[(k * N[Sqrt[N[Power[t$95$m, 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 / N[(N[(k * 2.0), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 4.5e-42

   1. Initial program 49.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 49.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 38.3%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \cdot \sqrt{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}}} \]

   5. Applied egg-rr 43.8%

      \[\leadsto \color{blue}{\frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}} \]
   6. Step-by-step derivation

      1. unpow2 43.8%

         \[\leadsto \color{blue}{{\left(\frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right)}^{2}} \]

      2. associate-/l* 43.8%

         \[\leadsto {\color{blue}{\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}}\right)}}^{2} \]

      3. associate-*r* 43.8%

         \[\leadsto {\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\color{blue}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}}}}\right)}^{2} \]

      4. *-commutative 43.8%

         \[\leadsto {\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\color{blue}{\left({t}^{3} \cdot \tan k\right)} \cdot \sin k}}}}\right)}^{2} \]

   7. Simplified 43.8%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\left({t}^{3} \cdot \tan k\right) \cdot \sin k}}}}\right)}^{2}} \]

   8. Taylor expanded in k around inf 39.7%

      \[\leadsto {\color{blue}{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t}}\right)}}^{2} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 39.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t}}\right)}^{2}\right)\right)} \]

      2. expm1-udef 34.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t}}\right)}^{2}\right)} - 1} \]

      3. *-commutative 34.7%

         \[\leadsto e^{\mathsf{log1p}\left({\color{blue}{\left(\sqrt{\frac{\cos k}{t}} \cdot \frac{\ell \cdot \sqrt{2}}{k \cdot \sin k}\right)}}^{2}\right)} - 1 \]

      4. unpow-prod-down 34.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left(\sqrt{\frac{\cos k}{t}}\right)}^{2} \cdot {\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k}\right)}^{2}}\right)} - 1 \]

      5. pow2 34.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{\frac{\cos k}{t}} \cdot \sqrt{\frac{\cos k}{t}}\right)} \cdot {\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k}\right)}^{2}\right)} - 1 \]

      6. add-sqr-sqrt 54.2%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\cos k}{t}} \cdot {\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k}\right)}^{2}\right)} - 1 \]

      7. times-frac 54.7%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\cos k}{t} \cdot {\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\sqrt{2}}{\sin k}\right)}}^{2}\right)} - 1 \]

   10. Applied egg-rr 54.7%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\cos k}{t} \cdot {\left(\frac{\ell}{k} \cdot \frac{\sqrt{2}}{\sin k}\right)}^{2}\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 58.1%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\cos k}{t} \cdot {\left(\frac{\ell}{k} \cdot \frac{\sqrt{2}}{\sin k}\right)}^{2}\right)\right)} \]

       2. expm1-log1p 75.1%

          \[\leadsto \color{blue}{\frac{\cos k}{t} \cdot {\left(\frac{\ell}{k} \cdot \frac{\sqrt{2}}{\sin k}\right)}^{2}} \]

       3. associate-*r/ 75.2%

          \[\leadsto \frac{\cos k}{t} \cdot {\color{blue}{\left(\frac{\frac{\ell}{k} \cdot \sqrt{2}}{\sin k}\right)}}^{2} \]

       4. associate-*l/ 75.2%

          \[\leadsto \frac{\cos k}{t} \cdot {\left(\frac{\color{blue}{\frac{\ell \cdot \sqrt{2}}{k}}}{\sin k}\right)}^{2} \]

       5. associate-/r* 75.2%

          \[\leadsto \frac{\cos k}{t} \cdot {\color{blue}{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k}\right)}}^{2} \]

       6. *-rgt-identity 75.2%

          \[\leadsto \frac{\cos k}{t} \cdot {\left(\frac{\color{blue}{\left(\ell \cdot \sqrt{2}\right) \cdot 1}}{k \cdot \sin k}\right)}^{2} \]

       7. associate-*r/ 75.2%

          \[\leadsto \frac{\cos k}{t} \cdot {\color{blue}{\left(\left(\ell \cdot \sqrt{2}\right) \cdot \frac{1}{k \cdot \sin k}\right)}}^{2} \]

       8. associate-*l* 75.2%

          \[\leadsto \frac{\cos k}{t} \cdot {\color{blue}{\left(\ell \cdot \left(\sqrt{2} \cdot \frac{1}{k \cdot \sin k}\right)\right)}}^{2} \]

       9. associate-*r/ 75.2%

          \[\leadsto \frac{\cos k}{t} \cdot {\left(\ell \cdot \color{blue}{\frac{\sqrt{2} \cdot 1}{k \cdot \sin k}}\right)}^{2} \]

       10. *-commutative 75.2%

           \[\leadsto \frac{\cos k}{t} \cdot {\left(\ell \cdot \frac{\color{blue}{1 \cdot \sqrt{2}}}{k \cdot \sin k}\right)}^{2} \]

       11. *-lft-identity 75.2%

           \[\leadsto \frac{\cos k}{t} \cdot {\left(\ell \cdot \frac{\color{blue}{\sqrt{2}}}{k \cdot \sin k}\right)}^{2} \]

   12. Simplified 75.2%

       \[\leadsto \color{blue}{\frac{\cos k}{t} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2}} \]

   if 4.5e-42 < t < 4.20000000000000003e102

   1. Initial program 76.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 81.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 71.5%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \cdot \sqrt{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}}} \]

   5. Applied egg-rr 71.6%

      \[\leadsto \color{blue}{\frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}} \]
   6. Step-by-step derivation

      1. unpow2 71.6%

         \[\leadsto \color{blue}{{\left(\frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right)}^{2}} \]

      2. associate-/l* 71.8%

         \[\leadsto {\color{blue}{\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}}\right)}}^{2} \]

      3. associate-*r* 71.8%

         \[\leadsto {\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\color{blue}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}}}}\right)}^{2} \]

      4. *-commutative 71.8%

         \[\leadsto {\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\color{blue}{\left({t}^{3} \cdot \tan k\right)} \cdot \sin k}}}}\right)}^{2} \]

   7. Simplified 71.8%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\left({t}^{3} \cdot \tan k\right) \cdot \sin k}}}}\right)}^{2}} \]

   8. Taylor expanded in k around 0 81.6%

      \[\leadsto {\left(\frac{\ell}{\color{blue}{k \cdot \sqrt{{t}^{3}}}}\right)}^{2} \]

   if 4.20000000000000003e102 < t

   1. Initial program 56.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 56.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. *-commutative 56.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. *-commutative 56.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-/r* 63.3%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. distribute-rgt-in 63.3%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

      6. unpow2 63.3%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      7. times-frac 42.9%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      8. sqr-neg 42.9%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      9. times-frac 63.3%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      10. unpow2 63.3%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      11. distribute-rgt-in 63.3%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

      12. +-commutative 63.3%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)\right)}\right)} \]

   3. Simplified 63.3%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 63.3%

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      2. *-un-lft-identity 63.3%

         \[\leadsto \frac{2}{\left(\frac{\left(\sqrt[3]{\frac{{t}^{3}}{\ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}}{\color{blue}{1 \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      3. times-frac 63.3%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{\sqrt[3]{\frac{{t}^{3}}{\ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}}{1} \cdot \frac{\sqrt[3]{\frac{{t}^{3}}{\ell}}}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      4. pow2 63.3%

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{\left(\sqrt[3]{\frac{{t}^{3}}{\ell}}\right)}^{2}}}{1} \cdot \frac{\sqrt[3]{\frac{{t}^{3}}{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      5. cbrt-div 63.3%

         \[\leadsto \frac{2}{\left(\left(\frac{{\color{blue}{\left(\frac{\sqrt[3]{{t}^{3}}}{\sqrt[3]{\ell}}\right)}}^{2}}{1} \cdot \frac{\sqrt[3]{\frac{{t}^{3}}{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      6. rem-cbrt-cube 63.3%

         \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{\color{blue}{t}}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{\frac{{t}^{3}}{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      7. cbrt-div 63.3%

         \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\color{blue}{\frac{\sqrt[3]{{t}^{3}}}{\sqrt[3]{\ell}}}}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      8. rem-cbrt-cube 85.1%

         \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{\color{blue}{t}}{\sqrt[3]{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   5. Applied egg-rr 85.1%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   6. Taylor expanded in k around 0 80.8%

      \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]
   7. Step-by-step derivation

      1. *-commutative 80.8%

         \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   8. Simplified 80.8%

      \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]
   9. Step-by-step derivation

      1. add-cube-cbrt 80.8%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(\sqrt[3]{\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}} \cdot \sqrt[3]{\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}}\right) \cdot \sqrt[3]{\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}}\right)} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      2. pow3 80.8%

         \[\leadsto \frac{2}{\left(\color{blue}{{\left(\sqrt[3]{\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}}\right)}^{3}} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      3. frac-times 74.1%

         \[\leadsto \frac{2}{\left({\left(\sqrt[3]{\color{blue}{\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2} \cdot \frac{t}{\sqrt[3]{\ell}}}{1 \cdot \ell}}}\right)}^{3} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      4. unpow2 74.1%

         \[\leadsto \frac{2}{\left({\left(\sqrt[3]{\frac{\color{blue}{\left(\frac{t}{\sqrt[3]{\ell}} \cdot \frac{t}{\sqrt[3]{\ell}}\right)} \cdot \frac{t}{\sqrt[3]{\ell}}}{1 \cdot \ell}}\right)}^{3} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      5. *-un-lft-identity 74.1%

         \[\leadsto \frac{2}{\left({\left(\sqrt[3]{\frac{\left(\frac{t}{\sqrt[3]{\ell}} \cdot \frac{t}{\sqrt[3]{\ell}}\right) \cdot \frac{t}{\sqrt[3]{\ell}}}{\color{blue}{\ell}}}\right)}^{3} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      6. cbrt-div 74.1%

         \[\leadsto \frac{2}{\left({\color{blue}{\left(\frac{\sqrt[3]{\left(\frac{t}{\sqrt[3]{\ell}} \cdot \frac{t}{\sqrt[3]{\ell}}\right) \cdot \frac{t}{\sqrt[3]{\ell}}}}{\sqrt[3]{\ell}}\right)}}^{3} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      7. add-cbrt-cube 80.7%

         \[\leadsto \frac{2}{\left({\left(\frac{\color{blue}{\frac{t}{\sqrt[3]{\ell}}}}{\sqrt[3]{\ell}}\right)}^{3} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

   10. Applied egg-rr 80.7%

       \[\leadsto \frac{2}{\left(\color{blue}{{\left(\frac{\frac{t}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{3}} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.5 \cdot 10^{-42}:\\ \;\;\;\;\frac{\cos k}{t} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{+102}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot \sqrt{{t}^{3}}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot {\left(\frac{\frac{t}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{3}\right)}\\ \end{array} \]

Alternative 11: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 4.5 \cdot 10^{-42}:\\ \;\;\;\;\frac{\cos k}{t_m} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \mathbf{elif}\;t_m \leq 4.2 \cdot 10^{+102}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot \sqrt{{t_m}^{3}}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot \frac{{\left(\frac{t_m}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4.5e-42)
    (* (/ (cos k) t_m) (pow (* l (/ (sqrt 2.0) (* k (sin k)))) 2.0))
    (if (<= t_m 4.2e+102)
      (pow (/ l (* k (sqrt (pow t_m 3.0)))) 2.0)
      (/ 2.0 (* (* k 2.0) (* (sin k) (/ (pow (/ t_m (cbrt l)) 3.0) l))))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.5e-42) {
		tmp = (cos(k) / t_m) * pow((l * (sqrt(2.0) / (k * sin(k)))), 2.0);
	} else if (t_m <= 4.2e+102) {
		tmp = pow((l / (k * sqrt(pow(t_m, 3.0)))), 2.0);
	} else {
		tmp = 2.0 / ((k * 2.0) * (sin(k) * (pow((t_m / cbrt(l)), 3.0) / l)));
	}
	return t_s * tmp;
}

Java

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) {
	double tmp;
	if (t_m <= 4.5e-42) {
		tmp = (Math.cos(k) / t_m) * Math.pow((l * (Math.sqrt(2.0) / (k * Math.sin(k)))), 2.0);
	} else if (t_m <= 4.2e+102) {
		tmp = Math.pow((l / (k * Math.sqrt(Math.pow(t_m, 3.0)))), 2.0);
	} else {
		tmp = 2.0 / ((k * 2.0) * (Math.sin(k) * (Math.pow((t_m / Math.cbrt(l)), 3.0) / l)));
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 4.5e-42)
		tmp = Float64(Float64(cos(k) / t_m) * (Float64(l * Float64(sqrt(2.0) / Float64(k * sin(k)))) ^ 2.0));
	elseif (t_m <= 4.2e+102)
		tmp = Float64(l / Float64(k * sqrt((t_m ^ 3.0)))) ^ 2.0;
	else
		tmp = Float64(2.0 / Float64(Float64(k * 2.0) * Float64(sin(k) * Float64((Float64(t_m / cbrt(l)) ^ 3.0) / l))));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[t$95$m, 4.5e-42], N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.2e+102], N[Power[N[(l / N[(k * N[Sqrt[N[Power[t$95$m, 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 / N[(N[(k * 2.0), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 4.5e-42

   1. Initial program 49.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 49.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 38.3%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \cdot \sqrt{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}}} \]

   5. Applied egg-rr 43.8%

      \[\leadsto \color{blue}{\frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}} \]
   6. Step-by-step derivation

      1. unpow2 43.8%

         \[\leadsto \color{blue}{{\left(\frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right)}^{2}} \]

      2. associate-/l* 43.8%

         \[\leadsto {\color{blue}{\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}}\right)}}^{2} \]

      3. associate-*r* 43.8%

         \[\leadsto {\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\color{blue}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}}}}\right)}^{2} \]

      4. *-commutative 43.8%

         \[\leadsto {\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\color{blue}{\left({t}^{3} \cdot \tan k\right)} \cdot \sin k}}}}\right)}^{2} \]

   7. Simplified 43.8%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\left({t}^{3} \cdot \tan k\right) \cdot \sin k}}}}\right)}^{2}} \]

   8. Taylor expanded in k around inf 39.7%

      \[\leadsto {\color{blue}{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t}}\right)}}^{2} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 39.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t}}\right)}^{2}\right)\right)} \]

      2. expm1-udef 34.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t}}\right)}^{2}\right)} - 1} \]

      3. *-commutative 34.7%

         \[\leadsto e^{\mathsf{log1p}\left({\color{blue}{\left(\sqrt{\frac{\cos k}{t}} \cdot \frac{\ell \cdot \sqrt{2}}{k \cdot \sin k}\right)}}^{2}\right)} - 1 \]

      4. unpow-prod-down 34.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left(\sqrt{\frac{\cos k}{t}}\right)}^{2} \cdot {\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k}\right)}^{2}}\right)} - 1 \]

      5. pow2 34.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{\frac{\cos k}{t}} \cdot \sqrt{\frac{\cos k}{t}}\right)} \cdot {\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k}\right)}^{2}\right)} - 1 \]

      6. add-sqr-sqrt 54.2%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\cos k}{t}} \cdot {\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k}\right)}^{2}\right)} - 1 \]

      7. times-frac 54.7%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\cos k}{t} \cdot {\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\sqrt{2}}{\sin k}\right)}}^{2}\right)} - 1 \]

   10. Applied egg-rr 54.7%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\cos k}{t} \cdot {\left(\frac{\ell}{k} \cdot \frac{\sqrt{2}}{\sin k}\right)}^{2}\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 58.1%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\cos k}{t} \cdot {\left(\frac{\ell}{k} \cdot \frac{\sqrt{2}}{\sin k}\right)}^{2}\right)\right)} \]

       2. expm1-log1p 75.1%

          \[\leadsto \color{blue}{\frac{\cos k}{t} \cdot {\left(\frac{\ell}{k} \cdot \frac{\sqrt{2}}{\sin k}\right)}^{2}} \]

       3. associate-*r/ 75.2%

          \[\leadsto \frac{\cos k}{t} \cdot {\color{blue}{\left(\frac{\frac{\ell}{k} \cdot \sqrt{2}}{\sin k}\right)}}^{2} \]

       4. associate-*l/ 75.2%

          \[\leadsto \frac{\cos k}{t} \cdot {\left(\frac{\color{blue}{\frac{\ell \cdot \sqrt{2}}{k}}}{\sin k}\right)}^{2} \]

       5. associate-/r* 75.2%

          \[\leadsto \frac{\cos k}{t} \cdot {\color{blue}{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k}\right)}}^{2} \]

       6. *-rgt-identity 75.2%

          \[\leadsto \frac{\cos k}{t} \cdot {\left(\frac{\color{blue}{\left(\ell \cdot \sqrt{2}\right) \cdot 1}}{k \cdot \sin k}\right)}^{2} \]

       7. associate-*r/ 75.2%

          \[\leadsto \frac{\cos k}{t} \cdot {\color{blue}{\left(\left(\ell \cdot \sqrt{2}\right) \cdot \frac{1}{k \cdot \sin k}\right)}}^{2} \]

       8. associate-*l* 75.2%

          \[\leadsto \frac{\cos k}{t} \cdot {\color{blue}{\left(\ell \cdot \left(\sqrt{2} \cdot \frac{1}{k \cdot \sin k}\right)\right)}}^{2} \]

       9. associate-*r/ 75.2%

          \[\leadsto \frac{\cos k}{t} \cdot {\left(\ell \cdot \color{blue}{\frac{\sqrt{2} \cdot 1}{k \cdot \sin k}}\right)}^{2} \]

       10. *-commutative 75.2%

           \[\leadsto \frac{\cos k}{t} \cdot {\left(\ell \cdot \frac{\color{blue}{1 \cdot \sqrt{2}}}{k \cdot \sin k}\right)}^{2} \]

       11. *-lft-identity 75.2%

           \[\leadsto \frac{\cos k}{t} \cdot {\left(\ell \cdot \frac{\color{blue}{\sqrt{2}}}{k \cdot \sin k}\right)}^{2} \]

   12. Simplified 75.2%

       \[\leadsto \color{blue}{\frac{\cos k}{t} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2}} \]

   if 4.5e-42 < t < 4.20000000000000003e102

   1. Initial program 76.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 81.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 71.5%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \cdot \sqrt{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}}} \]

   5. Applied egg-rr 71.6%

      \[\leadsto \color{blue}{\frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}} \]
   6. Step-by-step derivation

      1. unpow2 71.6%

         \[\leadsto \color{blue}{{\left(\frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right)}^{2}} \]

      2. associate-/l* 71.8%

         \[\leadsto {\color{blue}{\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}}\right)}}^{2} \]

      3. associate-*r* 71.8%

         \[\leadsto {\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\color{blue}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}}}}\right)}^{2} \]

      4. *-commutative 71.8%

         \[\leadsto {\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\color{blue}{\left({t}^{3} \cdot \tan k\right)} \cdot \sin k}}}}\right)}^{2} \]

   7. Simplified 71.8%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\left({t}^{3} \cdot \tan k\right) \cdot \sin k}}}}\right)}^{2}} \]

   8. Taylor expanded in k around 0 81.6%

      \[\leadsto {\left(\frac{\ell}{\color{blue}{k \cdot \sqrt{{t}^{3}}}}\right)}^{2} \]

   if 4.20000000000000003e102 < t

   1. Initial program 56.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 56.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. *-commutative 56.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. *-commutative 56.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-/r* 63.3%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. distribute-rgt-in 63.3%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

      6. unpow2 63.3%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      7. times-frac 42.9%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      8. sqr-neg 42.9%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      9. times-frac 63.3%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      10. unpow2 63.3%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      11. distribute-rgt-in 63.3%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

      12. +-commutative 63.3%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)\right)}\right)} \]

   3. Simplified 63.3%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 63.3%

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      2. *-un-lft-identity 63.3%

         \[\leadsto \frac{2}{\left(\frac{\left(\sqrt[3]{\frac{{t}^{3}}{\ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}}{\color{blue}{1 \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      3. times-frac 63.3%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{\sqrt[3]{\frac{{t}^{3}}{\ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}}{1} \cdot \frac{\sqrt[3]{\frac{{t}^{3}}{\ell}}}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      4. pow2 63.3%

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{\left(\sqrt[3]{\frac{{t}^{3}}{\ell}}\right)}^{2}}}{1} \cdot \frac{\sqrt[3]{\frac{{t}^{3}}{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      5. cbrt-div 63.3%

         \[\leadsto \frac{2}{\left(\left(\frac{{\color{blue}{\left(\frac{\sqrt[3]{{t}^{3}}}{\sqrt[3]{\ell}}\right)}}^{2}}{1} \cdot \frac{\sqrt[3]{\frac{{t}^{3}}{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      6. rem-cbrt-cube 63.3%

         \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{\color{blue}{t}}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{\frac{{t}^{3}}{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      7. cbrt-div 63.3%

         \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\color{blue}{\frac{\sqrt[3]{{t}^{3}}}{\sqrt[3]{\ell}}}}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      8. rem-cbrt-cube 85.1%

         \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{\color{blue}{t}}{\sqrt[3]{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   5. Applied egg-rr 85.1%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   6. Taylor expanded in k around 0 80.8%

      \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]
   7. Step-by-step derivation

      1. *-commutative 80.8%

         \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   8. Simplified 80.8%

      \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 80.3%

         \[\leadsto \frac{2}{\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right)\right)} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      2. expm1-udef 80.3%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right)} - 1\right)} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      3. frac-times 73.9%

         \[\leadsto \frac{2}{\left(\left(e^{\mathsf{log1p}\left(\color{blue}{\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2} \cdot \frac{t}{\sqrt[3]{\ell}}}{1 \cdot \ell}}\right)} - 1\right) \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      4. pow-plus 73.9%

         \[\leadsto \frac{2}{\left(\left(e^{\mathsf{log1p}\left(\frac{\color{blue}{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{\left(2 + 1\right)}}}{1 \cdot \ell}\right)} - 1\right) \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      5. metadata-eval 73.9%

         \[\leadsto \frac{2}{\left(\left(e^{\mathsf{log1p}\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{\color{blue}{3}}}{1 \cdot \ell}\right)} - 1\right) \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      6. *-un-lft-identity 73.9%

         \[\leadsto \frac{2}{\left(\left(e^{\mathsf{log1p}\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\color{blue}{\ell}}\right)} - 1\right) \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

   10. Applied egg-rr 73.9%

       \[\leadsto \frac{2}{\left(\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}\right)} - 1\right)} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]
   11. Step-by-step derivation

       1. expm1-def 73.9%

          \[\leadsto \frac{2}{\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}\right)\right)} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

       2. expm1-log1p 74.2%

          \[\leadsto \frac{2}{\left(\color{blue}{\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

   12. Simplified 74.2%

       \[\leadsto \frac{2}{\left(\color{blue}{\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.5 \cdot 10^{-42}:\\ \;\;\;\;\frac{\cos k}{t} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{+102}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot \sqrt{{t}^{3}}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot \frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}\right)}\\ \end{array} \]

Alternative 12: 70.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 3.5 \cdot 10^{-51}:\\ \;\;\;\;{\left(\sqrt{\frac{1}{t_m}} \cdot \frac{\ell}{\frac{{k}^{2}}{\sqrt{2}}}\right)}^{2}\\ \mathbf{elif}\;t_m \leq 1.15 \cdot 10^{+102}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot \sqrt{{t_m}^{3}}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot \frac{{\left(\frac{t_m}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.5e-51)
    (pow (* (sqrt (/ 1.0 t_m)) (/ l (/ (pow k 2.0) (sqrt 2.0)))) 2.0)
    (if (<= t_m 1.15e+102)
      (pow (/ l (* k (sqrt (pow t_m 3.0)))) 2.0)
      (/ 2.0 (* (* k 2.0) (* (sin k) (/ (pow (/ t_m (cbrt l)) 3.0) l))))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.5e-51) {
		tmp = pow((sqrt((1.0 / t_m)) * (l / (pow(k, 2.0) / sqrt(2.0)))), 2.0);
	} else if (t_m <= 1.15e+102) {
		tmp = pow((l / (k * sqrt(pow(t_m, 3.0)))), 2.0);
	} else {
		tmp = 2.0 / ((k * 2.0) * (sin(k) * (pow((t_m / cbrt(l)), 3.0) / l)));
	}
	return t_s * tmp;
}

Java

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) {
	double tmp;
	if (t_m <= 3.5e-51) {
		tmp = Math.pow((Math.sqrt((1.0 / t_m)) * (l / (Math.pow(k, 2.0) / Math.sqrt(2.0)))), 2.0);
	} else if (t_m <= 1.15e+102) {
		tmp = Math.pow((l / (k * Math.sqrt(Math.pow(t_m, 3.0)))), 2.0);
	} else {
		tmp = 2.0 / ((k * 2.0) * (Math.sin(k) * (Math.pow((t_m / Math.cbrt(l)), 3.0) / l)));
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 3.5e-51)
		tmp = Float64(sqrt(Float64(1.0 / t_m)) * Float64(l / Float64((k ^ 2.0) / sqrt(2.0)))) ^ 2.0;
	elseif (t_m <= 1.15e+102)
		tmp = Float64(l / Float64(k * sqrt((t_m ^ 3.0)))) ^ 2.0;
	else
		tmp = Float64(2.0 / Float64(Float64(k * 2.0) * Float64(sin(k) * Float64((Float64(t_m / cbrt(l)) ^ 3.0) / l))));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[t$95$m, 3.5e-51], N[Power[N[(N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision] * N[(l / N[(N[Power[k, 2.0], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[t$95$m, 1.15e+102], N[Power[N[(l / N[(k * N[Sqrt[N[Power[t$95$m, 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 / N[(N[(k * 2.0), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 3.4999999999999997e-51

   1. Initial program 48.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 48.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 37.7%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \cdot \sqrt{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}}} \]

   5. Applied egg-rr 43.2%

      \[\leadsto \color{blue}{\frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}} \]
   6. Step-by-step derivation

      1. unpow2 43.2%

         \[\leadsto \color{blue}{{\left(\frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right)}^{2}} \]

      2. associate-/l* 43.2%

         \[\leadsto {\color{blue}{\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}}\right)}}^{2} \]

      3. associate-*r* 43.2%

         \[\leadsto {\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\color{blue}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}}}}\right)}^{2} \]

      4. *-commutative 43.2%

         \[\leadsto {\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\color{blue}{\left({t}^{3} \cdot \tan k\right)} \cdot \sin k}}}}\right)}^{2} \]

   7. Simplified 43.2%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\left({t}^{3} \cdot \tan k\right) \cdot \sin k}}}}\right)}^{2}} \]

   8. Taylor expanded in k around inf 39.1%

      \[\leadsto {\color{blue}{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t}}\right)}}^{2} \]

   9. Taylor expanded in k around 0 22.7%

      \[\leadsto {\color{blue}{\left(\frac{\ell \cdot \sqrt{2}}{{k}^{2}} \cdot \sqrt{\frac{1}{t}}\right)}}^{2} \]
   10. Step-by-step derivation

       1. associate-/l* 22.7%

          \[\leadsto {\left(\color{blue}{\frac{\ell}{\frac{{k}^{2}}{\sqrt{2}}}} \cdot \sqrt{\frac{1}{t}}\right)}^{2} \]

   11. Simplified 22.7%

       \[\leadsto {\color{blue}{\left(\frac{\ell}{\frac{{k}^{2}}{\sqrt{2}}} \cdot \sqrt{\frac{1}{t}}\right)}}^{2} \]

   if 3.4999999999999997e-51 < t < 1.1499999999999999e102

   1. Initial program 78.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 82.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 74.0%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \cdot \sqrt{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}}} \]

   5. Applied egg-rr 74.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}} \]
   6. Step-by-step derivation

      1. unpow2 74.1%

         \[\leadsto \color{blue}{{\left(\frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right)}^{2}} \]

      2. associate-/l* 74.2%

         \[\leadsto {\color{blue}{\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}}\right)}}^{2} \]

      3. associate-*r* 74.2%

         \[\leadsto {\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\color{blue}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}}}}\right)}^{2} \]

      4. *-commutative 74.2%

         \[\leadsto {\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\color{blue}{\left({t}^{3} \cdot \tan k\right)} \cdot \sin k}}}}\right)}^{2} \]

   7. Simplified 74.2%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\left({t}^{3} \cdot \tan k\right) \cdot \sin k}}}}\right)}^{2}} \]

   8. Taylor expanded in k around 0 83.2%

      \[\leadsto {\left(\frac{\ell}{\color{blue}{k \cdot \sqrt{{t}^{3}}}}\right)}^{2} \]

   if 1.1499999999999999e102 < t

   1. Initial program 56.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 56.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. *-commutative 56.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. *-commutative 56.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-/r* 63.3%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. distribute-rgt-in 63.3%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

      6. unpow2 63.3%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      7. times-frac 42.9%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      8. sqr-neg 42.9%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      9. times-frac 63.3%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      10. unpow2 63.3%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      11. distribute-rgt-in 63.3%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

      12. +-commutative 63.3%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)\right)}\right)} \]

   3. Simplified 63.3%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 63.3%

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      2. *-un-lft-identity 63.3%

         \[\leadsto \frac{2}{\left(\frac{\left(\sqrt[3]{\frac{{t}^{3}}{\ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}}{\color{blue}{1 \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      3. times-frac 63.3%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{\sqrt[3]{\frac{{t}^{3}}{\ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}}{1} \cdot \frac{\sqrt[3]{\frac{{t}^{3}}{\ell}}}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      4. pow2 63.3%

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{\left(\sqrt[3]{\frac{{t}^{3}}{\ell}}\right)}^{2}}}{1} \cdot \frac{\sqrt[3]{\frac{{t}^{3}}{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      5. cbrt-div 63.3%

         \[\leadsto \frac{2}{\left(\left(\frac{{\color{blue}{\left(\frac{\sqrt[3]{{t}^{3}}}{\sqrt[3]{\ell}}\right)}}^{2}}{1} \cdot \frac{\sqrt[3]{\frac{{t}^{3}}{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      6. rem-cbrt-cube 63.3%

         \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{\color{blue}{t}}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{\frac{{t}^{3}}{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      7. cbrt-div 63.3%

         \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\color{blue}{\frac{\sqrt[3]{{t}^{3}}}{\sqrt[3]{\ell}}}}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      8. rem-cbrt-cube 85.1%

         \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{\color{blue}{t}}{\sqrt[3]{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   5. Applied egg-rr 85.1%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   6. Taylor expanded in k around 0 80.8%

      \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]
   7. Step-by-step derivation

      1. *-commutative 80.8%

         \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   8. Simplified 80.8%

      \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 80.3%

         \[\leadsto \frac{2}{\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right)\right)} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      2. expm1-udef 80.3%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right)} - 1\right)} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      3. frac-times 73.9%

         \[\leadsto \frac{2}{\left(\left(e^{\mathsf{log1p}\left(\color{blue}{\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2} \cdot \frac{t}{\sqrt[3]{\ell}}}{1 \cdot \ell}}\right)} - 1\right) \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      4. pow-plus 73.9%

         \[\leadsto \frac{2}{\left(\left(e^{\mathsf{log1p}\left(\frac{\color{blue}{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{\left(2 + 1\right)}}}{1 \cdot \ell}\right)} - 1\right) \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      5. metadata-eval 73.9%

         \[\leadsto \frac{2}{\left(\left(e^{\mathsf{log1p}\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{\color{blue}{3}}}{1 \cdot \ell}\right)} - 1\right) \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      6. *-un-lft-identity 73.9%

         \[\leadsto \frac{2}{\left(\left(e^{\mathsf{log1p}\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\color{blue}{\ell}}\right)} - 1\right) \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

   10. Applied egg-rr 73.9%

       \[\leadsto \frac{2}{\left(\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}\right)} - 1\right)} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]
   11. Step-by-step derivation

       1. expm1-def 73.9%

          \[\leadsto \frac{2}{\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}\right)\right)} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

       2. expm1-log1p 74.2%

          \[\leadsto \frac{2}{\left(\color{blue}{\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

   12. Simplified 74.2%

       \[\leadsto \frac{2}{\left(\color{blue}{\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 37.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.5 \cdot 10^{-51}:\\ \;\;\;\;{\left(\sqrt{\frac{1}{t}} \cdot \frac{\ell}{\frac{{k}^{2}}{\sqrt{2}}}\right)}^{2}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+102}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot \sqrt{{t}^{3}}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot \frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}\right)}\\ \end{array} \]

Alternative 13: 70.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 2.2 \cdot 10^{-51}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{{k}^{2}} \cdot \sqrt{\frac{1}{t_m}}\right)}^{2}\\ \mathbf{elif}\;t_m \leq 4.2 \cdot 10^{+102}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot \sqrt{{t_m}^{3}}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot \frac{{\left(\frac{t_m}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.2e-51)
    (pow (* (/ (* l (sqrt 2.0)) (pow k 2.0)) (sqrt (/ 1.0 t_m))) 2.0)
    (if (<= t_m 4.2e+102)
      (pow (/ l (* k (sqrt (pow t_m 3.0)))) 2.0)
      (/ 2.0 (* (* k 2.0) (* (sin k) (/ (pow (/ t_m (cbrt l)) 3.0) l))))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.2e-51) {
		tmp = pow((((l * sqrt(2.0)) / pow(k, 2.0)) * sqrt((1.0 / t_m))), 2.0);
	} else if (t_m <= 4.2e+102) {
		tmp = pow((l / (k * sqrt(pow(t_m, 3.0)))), 2.0);
	} else {
		tmp = 2.0 / ((k * 2.0) * (sin(k) * (pow((t_m / cbrt(l)), 3.0) / l)));
	}
	return t_s * tmp;
}

Java

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) {
	double tmp;
	if (t_m <= 2.2e-51) {
		tmp = Math.pow((((l * Math.sqrt(2.0)) / Math.pow(k, 2.0)) * Math.sqrt((1.0 / t_m))), 2.0);
	} else if (t_m <= 4.2e+102) {
		tmp = Math.pow((l / (k * Math.sqrt(Math.pow(t_m, 3.0)))), 2.0);
	} else {
		tmp = 2.0 / ((k * 2.0) * (Math.sin(k) * (Math.pow((t_m / Math.cbrt(l)), 3.0) / l)));
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.2e-51)
		tmp = Float64(Float64(Float64(l * sqrt(2.0)) / (k ^ 2.0)) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	elseif (t_m <= 4.2e+102)
		tmp = Float64(l / Float64(k * sqrt((t_m ^ 3.0)))) ^ 2.0;
	else
		tmp = Float64(2.0 / Float64(Float64(k * 2.0) * Float64(sin(k) * Float64((Float64(t_m / cbrt(l)) ^ 3.0) / l))));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[t$95$m, 2.2e-51], N[Power[N[(N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[t$95$m, 4.2e+102], N[Power[N[(l / N[(k * N[Sqrt[N[Power[t$95$m, 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 / N[(N[(k * 2.0), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.2e-51

   1. Initial program 48.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 48.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 37.7%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \cdot \sqrt{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}}} \]

   5. Applied egg-rr 43.2%

      \[\leadsto \color{blue}{\frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}} \]
   6. Step-by-step derivation

      1. unpow2 43.2%

         \[\leadsto \color{blue}{{\left(\frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right)}^{2}} \]

      2. associate-/l* 43.2%

         \[\leadsto {\color{blue}{\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}}\right)}}^{2} \]

      3. associate-*r* 43.2%

         \[\leadsto {\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\color{blue}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}}}}\right)}^{2} \]

      4. *-commutative 43.2%

         \[\leadsto {\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\color{blue}{\left({t}^{3} \cdot \tan k\right)} \cdot \sin k}}}}\right)}^{2} \]

   7. Simplified 43.2%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\left({t}^{3} \cdot \tan k\right) \cdot \sin k}}}}\right)}^{2}} \]

   8. Taylor expanded in k around inf 39.1%

      \[\leadsto {\color{blue}{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t}}\right)}}^{2} \]

   9. Taylor expanded in k around 0 22.7%

      \[\leadsto {\color{blue}{\left(\frac{\ell \cdot \sqrt{2}}{{k}^{2}} \cdot \sqrt{\frac{1}{t}}\right)}}^{2} \]

   if 2.2e-51 < t < 4.20000000000000003e102

   1. Initial program 78.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 82.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 74.0%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \cdot \sqrt{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}}} \]

   5. Applied egg-rr 74.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}} \]
   6. Step-by-step derivation

      1. unpow2 74.1%

         \[\leadsto \color{blue}{{\left(\frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right)}^{2}} \]

      2. associate-/l* 74.2%

         \[\leadsto {\color{blue}{\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}}\right)}}^{2} \]

      3. associate-*r* 74.2%

         \[\leadsto {\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\color{blue}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}}}}\right)}^{2} \]

      4. *-commutative 74.2%

         \[\leadsto {\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\color{blue}{\left({t}^{3} \cdot \tan k\right)} \cdot \sin k}}}}\right)}^{2} \]

   7. Simplified 74.2%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\left({t}^{3} \cdot \tan k\right) \cdot \sin k}}}}\right)}^{2}} \]

   8. Taylor expanded in k around 0 83.2%

      \[\leadsto {\left(\frac{\ell}{\color{blue}{k \cdot \sqrt{{t}^{3}}}}\right)}^{2} \]

   if 4.20000000000000003e102 < t

   1. Initial program 56.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 56.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. *-commutative 56.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. *-commutative 56.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-/r* 63.3%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. distribute-rgt-in 63.3%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

      6. unpow2 63.3%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      7. times-frac 42.9%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      8. sqr-neg 42.9%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      9. times-frac 63.3%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      10. unpow2 63.3%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      11. distribute-rgt-in 63.3%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

      12. +-commutative 63.3%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)\right)}\right)} \]

   3. Simplified 63.3%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 63.3%

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      2. *-un-lft-identity 63.3%

         \[\leadsto \frac{2}{\left(\frac{\left(\sqrt[3]{\frac{{t}^{3}}{\ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}}{\color{blue}{1 \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      3. times-frac 63.3%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{\sqrt[3]{\frac{{t}^{3}}{\ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}}{1} \cdot \frac{\sqrt[3]{\frac{{t}^{3}}{\ell}}}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      4. pow2 63.3%

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{\left(\sqrt[3]{\frac{{t}^{3}}{\ell}}\right)}^{2}}}{1} \cdot \frac{\sqrt[3]{\frac{{t}^{3}}{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      5. cbrt-div 63.3%

         \[\leadsto \frac{2}{\left(\left(\frac{{\color{blue}{\left(\frac{\sqrt[3]{{t}^{3}}}{\sqrt[3]{\ell}}\right)}}^{2}}{1} \cdot \frac{\sqrt[3]{\frac{{t}^{3}}{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      6. rem-cbrt-cube 63.3%

         \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{\color{blue}{t}}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{\frac{{t}^{3}}{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      7. cbrt-div 63.3%

         \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\color{blue}{\frac{\sqrt[3]{{t}^{3}}}{\sqrt[3]{\ell}}}}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      8. rem-cbrt-cube 85.1%

         \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{\color{blue}{t}}{\sqrt[3]{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   5. Applied egg-rr 85.1%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   6. Taylor expanded in k around 0 80.8%

      \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]
   7. Step-by-step derivation

      1. *-commutative 80.8%

         \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   8. Simplified 80.8%

      \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 80.3%

         \[\leadsto \frac{2}{\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right)\right)} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      2. expm1-udef 80.3%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right)} - 1\right)} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      3. frac-times 73.9%

         \[\leadsto \frac{2}{\left(\left(e^{\mathsf{log1p}\left(\color{blue}{\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2} \cdot \frac{t}{\sqrt[3]{\ell}}}{1 \cdot \ell}}\right)} - 1\right) \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      4. pow-plus 73.9%

         \[\leadsto \frac{2}{\left(\left(e^{\mathsf{log1p}\left(\frac{\color{blue}{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{\left(2 + 1\right)}}}{1 \cdot \ell}\right)} - 1\right) \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      5. metadata-eval 73.9%

         \[\leadsto \frac{2}{\left(\left(e^{\mathsf{log1p}\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{\color{blue}{3}}}{1 \cdot \ell}\right)} - 1\right) \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      6. *-un-lft-identity 73.9%

         \[\leadsto \frac{2}{\left(\left(e^{\mathsf{log1p}\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\color{blue}{\ell}}\right)} - 1\right) \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

   10. Applied egg-rr 73.9%

       \[\leadsto \frac{2}{\left(\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}\right)} - 1\right)} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]
   11. Step-by-step derivation

       1. expm1-def 73.9%

          \[\leadsto \frac{2}{\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}\right)\right)} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

       2. expm1-log1p 74.2%

          \[\leadsto \frac{2}{\left(\color{blue}{\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

   12. Simplified 74.2%

       \[\leadsto \frac{2}{\left(\color{blue}{\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 37.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.2 \cdot 10^{-51}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{{k}^{2}} \cdot \sqrt{\frac{1}{t}}\right)}^{2}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{+102}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot \sqrt{{t}^{3}}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot \frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}\right)}\\ \end{array} \]

Alternative 14: 70.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 2.25 \cdot 10^{-51}:\\ \;\;\;\;{\left(\frac{\ell}{{k}^{2}} \cdot \frac{\sqrt{2}}{\sqrt{t_m}}\right)}^{2}\\ \mathbf{elif}\;t_m \leq 4.2 \cdot 10^{+102}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot \sqrt{{t_m}^{3}}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot \frac{{\left(\frac{t_m}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.25e-51)
    (pow (* (/ l (pow k 2.0)) (/ (sqrt 2.0) (sqrt t_m))) 2.0)
    (if (<= t_m 4.2e+102)
      (pow (/ l (* k (sqrt (pow t_m 3.0)))) 2.0)
      (/ 2.0 (* (* k 2.0) (* (sin k) (/ (pow (/ t_m (cbrt l)) 3.0) l))))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.25e-51) {
		tmp = pow(((l / pow(k, 2.0)) * (sqrt(2.0) / sqrt(t_m))), 2.0);
	} else if (t_m <= 4.2e+102) {
		tmp = pow((l / (k * sqrt(pow(t_m, 3.0)))), 2.0);
	} else {
		tmp = 2.0 / ((k * 2.0) * (sin(k) * (pow((t_m / cbrt(l)), 3.0) / l)));
	}
	return t_s * tmp;
}

Java

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) {
	double tmp;
	if (t_m <= 2.25e-51) {
		tmp = Math.pow(((l / Math.pow(k, 2.0)) * (Math.sqrt(2.0) / Math.sqrt(t_m))), 2.0);
	} else if (t_m <= 4.2e+102) {
		tmp = Math.pow((l / (k * Math.sqrt(Math.pow(t_m, 3.0)))), 2.0);
	} else {
		tmp = 2.0 / ((k * 2.0) * (Math.sin(k) * (Math.pow((t_m / Math.cbrt(l)), 3.0) / l)));
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.25e-51)
		tmp = Float64(Float64(l / (k ^ 2.0)) * Float64(sqrt(2.0) / sqrt(t_m))) ^ 2.0;
	elseif (t_m <= 4.2e+102)
		tmp = Float64(l / Float64(k * sqrt((t_m ^ 3.0)))) ^ 2.0;
	else
		tmp = Float64(2.0 / Float64(Float64(k * 2.0) * Float64(sin(k) * Float64((Float64(t_m / cbrt(l)) ^ 3.0) / l))));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[t$95$m, 2.25e-51], N[Power[N[(N[(l / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[t$95$m, 4.2e+102], N[Power[N[(l / N[(k * N[Sqrt[N[Power[t$95$m, 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 / N[(N[(k * 2.0), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.24999999999999987e-51

   1. Initial program 48.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 48.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 37.7%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \cdot \sqrt{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}}} \]

   5. Applied egg-rr 43.2%

      \[\leadsto \color{blue}{\frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}} \]
   6. Step-by-step derivation

      1. unpow2 43.2%

         \[\leadsto \color{blue}{{\left(\frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right)}^{2}} \]

      2. associate-/l* 43.2%

         \[\leadsto {\color{blue}{\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}}\right)}}^{2} \]

      3. associate-*r* 43.2%

         \[\leadsto {\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\color{blue}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}}}}\right)}^{2} \]

      4. *-commutative 43.2%

         \[\leadsto {\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\color{blue}{\left({t}^{3} \cdot \tan k\right)} \cdot \sin k}}}}\right)}^{2} \]

   7. Simplified 43.2%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\left({t}^{3} \cdot \tan k\right) \cdot \sin k}}}}\right)}^{2}} \]

   8. Taylor expanded in k around 0 39.8%

      \[\leadsto {\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\color{blue}{{k}^{2} \cdot {t}^{3}}}}}}\right)}^{2} \]

   9. Taylor expanded in k around inf 22.3%

      \[\leadsto {\left(\frac{\ell}{\color{blue}{\frac{{k}^{2}}{\sqrt{2}} \cdot \sqrt{t}}}\right)}^{2} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 14.4%

          \[\leadsto {\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\ell}{\frac{{k}^{2}}{\sqrt{2}} \cdot \sqrt{t}}\right)\right)\right)}}^{2} \]

       2. expm1-udef 14.2%

          \[\leadsto {\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\ell}{\frac{{k}^{2}}{\sqrt{2}} \cdot \sqrt{t}}\right)} - 1\right)}}^{2} \]

   11. Applied egg-rr 14.2%

       \[\leadsto {\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\ell}{\frac{{k}^{2}}{\sqrt{2}} \cdot \sqrt{t}}\right)} - 1\right)}}^{2} \]
   12. Step-by-step derivation

       1. expm1-def 14.4%

          \[\leadsto {\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\ell}{\frac{{k}^{2}}{\sqrt{2}} \cdot \sqrt{t}}\right)\right)\right)}}^{2} \]

       2. expm1-log1p 22.3%

          \[\leadsto {\color{blue}{\left(\frac{\ell}{\frac{{k}^{2}}{\sqrt{2}} \cdot \sqrt{t}}\right)}}^{2} \]

       3. associate-/r/ 22.3%

          \[\leadsto {\left(\frac{\ell}{\color{blue}{\frac{{k}^{2}}{\frac{\sqrt{2}}{\sqrt{t}}}}}\right)}^{2} \]

       4. associate-/r/ 22.6%

          \[\leadsto {\color{blue}{\left(\frac{\ell}{{k}^{2}} \cdot \frac{\sqrt{2}}{\sqrt{t}}\right)}}^{2} \]

   13. Simplified 22.6%

       \[\leadsto {\color{blue}{\left(\frac{\ell}{{k}^{2}} \cdot \frac{\sqrt{2}}{\sqrt{t}}\right)}}^{2} \]

   if 2.24999999999999987e-51 < t < 4.20000000000000003e102

   1. Initial program 78.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 82.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 74.0%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \cdot \sqrt{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}}} \]

   5. Applied egg-rr 74.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}} \]
   6. Step-by-step derivation

      1. unpow2 74.1%

         \[\leadsto \color{blue}{{\left(\frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right)}^{2}} \]

      2. associate-/l* 74.2%

         \[\leadsto {\color{blue}{\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}}\right)}}^{2} \]

      3. associate-*r* 74.2%

         \[\leadsto {\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\color{blue}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}}}}\right)}^{2} \]

      4. *-commutative 74.2%

         \[\leadsto {\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\color{blue}{\left({t}^{3} \cdot \tan k\right)} \cdot \sin k}}}}\right)}^{2} \]

   7. Simplified 74.2%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\left({t}^{3} \cdot \tan k\right) \cdot \sin k}}}}\right)}^{2}} \]

   8. Taylor expanded in k around 0 83.2%

      \[\leadsto {\left(\frac{\ell}{\color{blue}{k \cdot \sqrt{{t}^{3}}}}\right)}^{2} \]

   if 4.20000000000000003e102 < t

   1. Initial program 56.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 56.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. *-commutative 56.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. *-commutative 56.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-/r* 63.3%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. distribute-rgt-in 63.3%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

      6. unpow2 63.3%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      7. times-frac 42.9%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      8. sqr-neg 42.9%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      9. times-frac 63.3%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      10. unpow2 63.3%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      11. distribute-rgt-in 63.3%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

      12. +-commutative 63.3%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)\right)}\right)} \]

   3. Simplified 63.3%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 63.3%

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      2. *-un-lft-identity 63.3%

         \[\leadsto \frac{2}{\left(\frac{\left(\sqrt[3]{\frac{{t}^{3}}{\ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}}{\color{blue}{1 \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      3. times-frac 63.3%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{\sqrt[3]{\frac{{t}^{3}}{\ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}}{1} \cdot \frac{\sqrt[3]{\frac{{t}^{3}}{\ell}}}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      4. pow2 63.3%

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{\left(\sqrt[3]{\frac{{t}^{3}}{\ell}}\right)}^{2}}}{1} \cdot \frac{\sqrt[3]{\frac{{t}^{3}}{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      5. cbrt-div 63.3%

         \[\leadsto \frac{2}{\left(\left(\frac{{\color{blue}{\left(\frac{\sqrt[3]{{t}^{3}}}{\sqrt[3]{\ell}}\right)}}^{2}}{1} \cdot \frac{\sqrt[3]{\frac{{t}^{3}}{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      6. rem-cbrt-cube 63.3%

         \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{\color{blue}{t}}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{\frac{{t}^{3}}{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      7. cbrt-div 63.3%

         \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\color{blue}{\frac{\sqrt[3]{{t}^{3}}}{\sqrt[3]{\ell}}}}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      8. rem-cbrt-cube 85.1%

         \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{\color{blue}{t}}{\sqrt[3]{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   5. Applied egg-rr 85.1%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   6. Taylor expanded in k around 0 80.8%

      \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]
   7. Step-by-step derivation

      1. *-commutative 80.8%

         \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   8. Simplified 80.8%

      \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 80.3%

         \[\leadsto \frac{2}{\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right)\right)} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      2. expm1-udef 80.3%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right)} - 1\right)} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      3. frac-times 73.9%

         \[\leadsto \frac{2}{\left(\left(e^{\mathsf{log1p}\left(\color{blue}{\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2} \cdot \frac{t}{\sqrt[3]{\ell}}}{1 \cdot \ell}}\right)} - 1\right) \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      4. pow-plus 73.9%

         \[\leadsto \frac{2}{\left(\left(e^{\mathsf{log1p}\left(\frac{\color{blue}{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{\left(2 + 1\right)}}}{1 \cdot \ell}\right)} - 1\right) \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      5. metadata-eval 73.9%

         \[\leadsto \frac{2}{\left(\left(e^{\mathsf{log1p}\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{\color{blue}{3}}}{1 \cdot \ell}\right)} - 1\right) \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      6. *-un-lft-identity 73.9%

         \[\leadsto \frac{2}{\left(\left(e^{\mathsf{log1p}\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\color{blue}{\ell}}\right)} - 1\right) \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

   10. Applied egg-rr 73.9%

       \[\leadsto \frac{2}{\left(\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}\right)} - 1\right)} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]
   11. Step-by-step derivation

       1. expm1-def 73.9%

          \[\leadsto \frac{2}{\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}\right)\right)} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

       2. expm1-log1p 74.2%

          \[\leadsto \frac{2}{\left(\color{blue}{\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

   12. Simplified 74.2%

       \[\leadsto \frac{2}{\left(\color{blue}{\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 36.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.25 \cdot 10^{-51}:\\ \;\;\;\;{\left(\frac{\ell}{{k}^{2}} \cdot \frac{\sqrt{2}}{\sqrt{t}}\right)}^{2}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{+102}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot \sqrt{{t}^{3}}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot \frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}\right)}\\ \end{array} \]

Alternative 15: 63.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.7 \cdot 10^{-247}:\\ \;\;\;\;{\left(\frac{\ell}{k} \cdot \sqrt{\frac{1}{{t_m}^{3}}}\right)}^{2}\\ \mathbf{elif}\;k \leq 2.2 \cdot 10^{+97}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot \frac{{\left(\frac{t_m}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{\ell \cdot \left(\ell \cdot {k}^{-2}\right)}}{t_m}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 2.7e-247)
    (pow (* (/ l k) (sqrt (/ 1.0 (pow t_m 3.0)))) 2.0)
    (if (<= k 2.2e+97)
      (/ 2.0 (* (* k 2.0) (* (sin k) (/ (pow (/ t_m (cbrt l)) 3.0) l))))
      (pow (/ (cbrt (* l (* l (pow k -2.0)))) t_m) 3.0)))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.7e-247) {
		tmp = pow(((l / k) * sqrt((1.0 / pow(t_m, 3.0)))), 2.0);
	} else if (k <= 2.2e+97) {
		tmp = 2.0 / ((k * 2.0) * (sin(k) * (pow((t_m / cbrt(l)), 3.0) / l)));
	} else {
		tmp = pow((cbrt((l * (l * pow(k, -2.0)))) / t_m), 3.0);
	}
	return t_s * tmp;
}

Java

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) {
	double tmp;
	if (k <= 2.7e-247) {
		tmp = Math.pow(((l / k) * Math.sqrt((1.0 / Math.pow(t_m, 3.0)))), 2.0);
	} else if (k <= 2.2e+97) {
		tmp = 2.0 / ((k * 2.0) * (Math.sin(k) * (Math.pow((t_m / Math.cbrt(l)), 3.0) / l)));
	} else {
		tmp = Math.pow((Math.cbrt((l * (l * Math.pow(k, -2.0)))) / t_m), 3.0);
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 2.7e-247)
		tmp = Float64(Float64(l / k) * sqrt(Float64(1.0 / (t_m ^ 3.0)))) ^ 2.0;
	elseif (k <= 2.2e+97)
		tmp = Float64(2.0 / Float64(Float64(k * 2.0) * Float64(sin(k) * Float64((Float64(t_m / cbrt(l)) ^ 3.0) / l))));
	else
		tmp = Float64(cbrt(Float64(l * Float64(l * (k ^ -2.0)))) / t_m) ^ 3.0;
	end
	return Float64(t_s * tmp)
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[k, 2.7e-247], N[Power[N[(N[(l / k), $MachinePrecision] * N[Sqrt[N[(1.0 / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[k, 2.2e+97], N[(2.0 / N[(N[(k * 2.0), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Power[N[(l * N[(l * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / t$95$m), $MachinePrecision], 3.0], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 2.70000000000000008e-247

   1. Initial program 53.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 53.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 46.5%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \cdot \sqrt{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}}} \]

   5. Applied egg-rr 52.9%

      \[\leadsto \color{blue}{\frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}} \]
   6. Step-by-step derivation

      1. unpow2 52.9%

         \[\leadsto \color{blue}{{\left(\frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right)}^{2}} \]

      2. associate-/l* 53.0%

         \[\leadsto {\color{blue}{\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}}\right)}}^{2} \]

      3. associate-*r* 53.0%

         \[\leadsto {\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\color{blue}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}}}}\right)}^{2} \]

      4. *-commutative 53.0%

         \[\leadsto {\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\color{blue}{\left({t}^{3} \cdot \tan k\right)} \cdot \sin k}}}}\right)}^{2} \]

   7. Simplified 53.0%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\left({t}^{3} \cdot \tan k\right) \cdot \sin k}}}}\right)}^{2}} \]

   8. Taylor expanded in k around 0 48.9%

      \[\leadsto {\color{blue}{\left(\frac{\ell}{k} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}}^{2} \]

   if 2.70000000000000008e-247 < k < 2.2000000000000001e97

   1. Initial program 50.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 50.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. *-commutative 50.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. *-commutative 50.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-/r* 57.8%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. distribute-rgt-in 57.8%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

      6. unpow2 57.8%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      7. times-frac 51.9%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      8. sqr-neg 51.9%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      9. times-frac 57.8%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      10. unpow2 57.8%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      11. distribute-rgt-in 57.8%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

      12. +-commutative 57.8%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)\right)}\right)} \]

   3. Simplified 57.8%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 57.6%

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      2. *-un-lft-identity 57.6%

         \[\leadsto \frac{2}{\left(\frac{\left(\sqrt[3]{\frac{{t}^{3}}{\ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}}{\color{blue}{1 \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      3. times-frac 57.6%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{\sqrt[3]{\frac{{t}^{3}}{\ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}}{1} \cdot \frac{\sqrt[3]{\frac{{t}^{3}}{\ell}}}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      4. pow2 57.6%

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{\left(\sqrt[3]{\frac{{t}^{3}}{\ell}}\right)}^{2}}}{1} \cdot \frac{\sqrt[3]{\frac{{t}^{3}}{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      5. cbrt-div 57.5%

         \[\leadsto \frac{2}{\left(\left(\frac{{\color{blue}{\left(\frac{\sqrt[3]{{t}^{3}}}{\sqrt[3]{\ell}}\right)}}^{2}}{1} \cdot \frac{\sqrt[3]{\frac{{t}^{3}}{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      6. rem-cbrt-cube 57.6%

         \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{\color{blue}{t}}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{\frac{{t}^{3}}{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      7. cbrt-div 57.5%

         \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\color{blue}{\frac{\sqrt[3]{{t}^{3}}}{\sqrt[3]{\ell}}}}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      8. rem-cbrt-cube 68.6%

         \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{\color{blue}{t}}{\sqrt[3]{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   5. Applied egg-rr 68.6%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   6. Taylor expanded in k around 0 71.2%

      \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]
   7. Step-by-step derivation

      1. *-commutative 71.2%

         \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   8. Simplified 71.2%

      \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 50.7%

         \[\leadsto \frac{2}{\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right)\right)} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      2. expm1-udef 34.4%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right)} - 1\right)} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      3. frac-times 34.4%

         \[\leadsto \frac{2}{\left(\left(e^{\mathsf{log1p}\left(\color{blue}{\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2} \cdot \frac{t}{\sqrt[3]{\ell}}}{1 \cdot \ell}}\right)} - 1\right) \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      4. pow-plus 34.4%

         \[\leadsto \frac{2}{\left(\left(e^{\mathsf{log1p}\left(\frac{\color{blue}{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{\left(2 + 1\right)}}}{1 \cdot \ell}\right)} - 1\right) \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      5. metadata-eval 34.4%

         \[\leadsto \frac{2}{\left(\left(e^{\mathsf{log1p}\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{\color{blue}{3}}}{1 \cdot \ell}\right)} - 1\right) \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      6. *-un-lft-identity 34.4%

         \[\leadsto \frac{2}{\left(\left(e^{\mathsf{log1p}\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\color{blue}{\ell}}\right)} - 1\right) \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

   10. Applied egg-rr 34.4%

       \[\leadsto \frac{2}{\left(\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}\right)} - 1\right)} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]
   11. Step-by-step derivation

       1. expm1-def 50.7%

          \[\leadsto \frac{2}{\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}\right)\right)} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

       2. expm1-log1p 71.2%

          \[\leadsto \frac{2}{\left(\color{blue}{\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

   12. Simplified 71.2%

       \[\leadsto \frac{2}{\left(\color{blue}{\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

   if 2.2000000000000001e97 < k

   1. Initial program 52.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 52.5%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

   4. Taylor expanded in k around 0 50.8%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 50.8%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   6. Applied egg-rr 50.8%

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]
   7. Step-by-step derivation

      1. times-frac 55.4%

         \[\leadsto \color{blue}{\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{{t}^{3}}} \]

   8. Applied egg-rr 55.4%

      \[\leadsto \color{blue}{\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{{t}^{3}}} \]
   9. Step-by-step derivation

      1. add-cube-cbrt 55.4%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{{t}^{3}}} \cdot \sqrt[3]{\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{{t}^{3}}}\right) \cdot \sqrt[3]{\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{{t}^{3}}}} \]

      2. pow3 55.4%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{{t}^{3}}}\right)}^{3}} \]

      3. associate-*r/ 55.6%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\frac{\frac{\ell}{{k}^{2}} \cdot \ell}{{t}^{3}}}}\right)}^{3} \]

      4. cbrt-div 55.6%

         \[\leadsto {\color{blue}{\left(\frac{\sqrt[3]{\frac{\ell}{{k}^{2}} \cdot \ell}}{\sqrt[3]{{t}^{3}}}\right)}}^{3} \]

      5. div-inv 55.6%

         \[\leadsto {\left(\frac{\sqrt[3]{\color{blue}{\left(\ell \cdot \frac{1}{{k}^{2}}\right)} \cdot \ell}}{\sqrt[3]{{t}^{3}}}\right)}^{3} \]

      6. pow-flip 55.6%

         \[\leadsto {\left(\frac{\sqrt[3]{\left(\ell \cdot \color{blue}{{k}^{\left(-2\right)}}\right) \cdot \ell}}{\sqrt[3]{{t}^{3}}}\right)}^{3} \]

      7. metadata-eval 55.6%

         \[\leadsto {\left(\frac{\sqrt[3]{\left(\ell \cdot {k}^{\color{blue}{-2}}\right) \cdot \ell}}{\sqrt[3]{{t}^{3}}}\right)}^{3} \]

      8. unpow3 55.6%

         \[\leadsto {\left(\frac{\sqrt[3]{\left(\ell \cdot {k}^{-2}\right) \cdot \ell}}{\sqrt[3]{\color{blue}{\left(t \cdot t\right) \cdot t}}}\right)}^{3} \]

      9. add-cbrt-cube 70.2%

         \[\leadsto {\left(\frac{\sqrt[3]{\left(\ell \cdot {k}^{-2}\right) \cdot \ell}}{\color{blue}{t}}\right)}^{3} \]

   10. Applied egg-rr 70.2%

       \[\leadsto \color{blue}{{\left(\frac{\sqrt[3]{\left(\ell \cdot {k}^{-2}\right) \cdot \ell}}{t}\right)}^{3}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.7 \cdot 10^{-247}:\\ \;\;\;\;{\left(\frac{\ell}{k} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2}\\ \mathbf{elif}\;k \leq 2.2 \cdot 10^{+97}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot \frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{\ell \cdot \left(\ell \cdot {k}^{-2}\right)}}{t}\right)}^{3}\\ \end{array} \]

Alternative 16: 63.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 7.5 \cdot 10^{-247}:\\ \;\;\;\;{\left(\frac{\ell}{k} \cdot \sqrt{\frac{1}{{t_m}^{3}}}\right)}^{2}\\ \mathbf{elif}\;k \leq 8.2 \cdot 10^{+96}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot \left(\frac{{t_m}^{2}}{\ell} \cdot \frac{t_m}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{\ell \cdot \left(\ell \cdot {k}^{-2}\right)}}{t_m}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 7.5e-247)
    (pow (* (/ l k) (sqrt (/ 1.0 (pow t_m 3.0)))) 2.0)
    (if (<= k 8.2e+96)
      (/ 2.0 (* (* k 2.0) (* (sin k) (* (/ (pow t_m 2.0) l) (/ t_m l)))))
      (pow (/ (cbrt (* l (* l (pow k -2.0)))) t_m) 3.0)))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 7.5e-247) {
		tmp = pow(((l / k) * sqrt((1.0 / pow(t_m, 3.0)))), 2.0);
	} else if (k <= 8.2e+96) {
		tmp = 2.0 / ((k * 2.0) * (sin(k) * ((pow(t_m, 2.0) / l) * (t_m / l))));
	} else {
		tmp = pow((cbrt((l * (l * pow(k, -2.0)))) / t_m), 3.0);
	}
	return t_s * tmp;
}

Java

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) {
	double tmp;
	if (k <= 7.5e-247) {
		tmp = Math.pow(((l / k) * Math.sqrt((1.0 / Math.pow(t_m, 3.0)))), 2.0);
	} else if (k <= 8.2e+96) {
		tmp = 2.0 / ((k * 2.0) * (Math.sin(k) * ((Math.pow(t_m, 2.0) / l) * (t_m / l))));
	} else {
		tmp = Math.pow((Math.cbrt((l * (l * Math.pow(k, -2.0)))) / t_m), 3.0);
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 7.5e-247)
		tmp = Float64(Float64(l / k) * sqrt(Float64(1.0 / (t_m ^ 3.0)))) ^ 2.0;
	elseif (k <= 8.2e+96)
		tmp = Float64(2.0 / Float64(Float64(k * 2.0) * Float64(sin(k) * Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l)))));
	else
		tmp = Float64(cbrt(Float64(l * Float64(l * (k ^ -2.0)))) / t_m) ^ 3.0;
	end
	return Float64(t_s * tmp)
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[k, 7.5e-247], N[Power[N[(N[(l / k), $MachinePrecision] * N[Sqrt[N[(1.0 / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[k, 8.2e+96], N[(2.0 / N[(N[(k * 2.0), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Power[N[(l * N[(l * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / t$95$m), $MachinePrecision], 3.0], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 7.5e-247

   1. Initial program 53.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 53.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 46.5%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \cdot \sqrt{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}}} \]

   5. Applied egg-rr 52.9%

      \[\leadsto \color{blue}{\frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}} \]
   6. Step-by-step derivation

      1. unpow2 52.9%

         \[\leadsto \color{blue}{{\left(\frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right)}^{2}} \]

      2. associate-/l* 53.0%

         \[\leadsto {\color{blue}{\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}}\right)}}^{2} \]

      3. associate-*r* 53.0%

         \[\leadsto {\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\color{blue}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}}}}\right)}^{2} \]

      4. *-commutative 53.0%

         \[\leadsto {\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\color{blue}{\left({t}^{3} \cdot \tan k\right)} \cdot \sin k}}}}\right)}^{2} \]

   7. Simplified 53.0%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\left({t}^{3} \cdot \tan k\right) \cdot \sin k}}}}\right)}^{2}} \]

   8. Taylor expanded in k around 0 48.9%

      \[\leadsto {\color{blue}{\left(\frac{\ell}{k} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}}^{2} \]

   if 7.5e-247 < k < 8.19999999999999996e96

   1. Initial program 50.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 50.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. *-commutative 50.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. *-commutative 50.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-/r* 57.8%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. distribute-rgt-in 57.8%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

      6. unpow2 57.8%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      7. times-frac 51.9%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      8. sqr-neg 51.9%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      9. times-frac 57.8%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      10. unpow2 57.8%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      11. distribute-rgt-in 57.8%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

      12. +-commutative 57.8%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)\right)}\right)} \]

   3. Simplified 57.8%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-/r* 50.3%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      2. unpow3 50.3%

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      3. times-frac 67.0%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      4. pow2 67.0%

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{2}}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   5. Applied egg-rr 67.0%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   6. Taylor expanded in k around 0 71.4%

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]
   7. Step-by-step derivation

      1. *-commutative 71.2%

         \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   8. Simplified 71.4%

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   if 8.19999999999999996e96 < k

   1. Initial program 52.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 52.5%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

   4. Taylor expanded in k around 0 50.8%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 50.8%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   6. Applied egg-rr 50.8%

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]
   7. Step-by-step derivation

      1. times-frac 55.4%

         \[\leadsto \color{blue}{\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{{t}^{3}}} \]

   8. Applied egg-rr 55.4%

      \[\leadsto \color{blue}{\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{{t}^{3}}} \]
   9. Step-by-step derivation

      1. add-cube-cbrt 55.4%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{{t}^{3}}} \cdot \sqrt[3]{\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{{t}^{3}}}\right) \cdot \sqrt[3]{\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{{t}^{3}}}} \]

      2. pow3 55.4%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{{t}^{3}}}\right)}^{3}} \]

      3. associate-*r/ 55.6%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\frac{\frac{\ell}{{k}^{2}} \cdot \ell}{{t}^{3}}}}\right)}^{3} \]

      4. cbrt-div 55.6%

         \[\leadsto {\color{blue}{\left(\frac{\sqrt[3]{\frac{\ell}{{k}^{2}} \cdot \ell}}{\sqrt[3]{{t}^{3}}}\right)}}^{3} \]

      5. div-inv 55.6%

         \[\leadsto {\left(\frac{\sqrt[3]{\color{blue}{\left(\ell \cdot \frac{1}{{k}^{2}}\right)} \cdot \ell}}{\sqrt[3]{{t}^{3}}}\right)}^{3} \]

      6. pow-flip 55.6%

         \[\leadsto {\left(\frac{\sqrt[3]{\left(\ell \cdot \color{blue}{{k}^{\left(-2\right)}}\right) \cdot \ell}}{\sqrt[3]{{t}^{3}}}\right)}^{3} \]

      7. metadata-eval 55.6%

         \[\leadsto {\left(\frac{\sqrt[3]{\left(\ell \cdot {k}^{\color{blue}{-2}}\right) \cdot \ell}}{\sqrt[3]{{t}^{3}}}\right)}^{3} \]

      8. unpow3 55.6%

         \[\leadsto {\left(\frac{\sqrt[3]{\left(\ell \cdot {k}^{-2}\right) \cdot \ell}}{\sqrt[3]{\color{blue}{\left(t \cdot t\right) \cdot t}}}\right)}^{3} \]

      9. add-cbrt-cube 70.2%

         \[\leadsto {\left(\frac{\sqrt[3]{\left(\ell \cdot {k}^{-2}\right) \cdot \ell}}{\color{blue}{t}}\right)}^{3} \]

   10. Applied egg-rr 70.2%

       \[\leadsto \color{blue}{{\left(\frac{\sqrt[3]{\left(\ell \cdot {k}^{-2}\right) \cdot \ell}}{t}\right)}^{3}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.5 \cdot 10^{-247}:\\ \;\;\;\;{\left(\frac{\ell}{k} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2}\\ \mathbf{elif}\;k \leq 8.2 \cdot 10^{+96}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{\ell \cdot \left(\ell \cdot {k}^{-2}\right)}}{t}\right)}^{3}\\ \end{array} \]

Alternative 17: 62.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.7 \cdot 10^{-230}:\\ \;\;\;\;\frac{\ell \cdot \ell}{{\left(t_m \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \mathbf{elif}\;k \leq 1.7 \cdot 10^{+128}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot \left(\frac{{t_m}^{2}}{\ell} \cdot \frac{t_m}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{2}{t_m}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 2.7e-230)
    (/ (* l l) (pow (* t_m (pow (cbrt k) 2.0)) 3.0))
    (if (<= k 1.7e+128)
      (/ 2.0 (* (* k 2.0) (* (sin k) (* (/ (pow t_m 2.0) l) (/ t_m l)))))
      (* (/ (pow l 2.0) (pow k 4.0)) (/ 2.0 t_m))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.7e-230) {
		tmp = (l * l) / pow((t_m * pow(cbrt(k), 2.0)), 3.0);
	} else if (k <= 1.7e+128) {
		tmp = 2.0 / ((k * 2.0) * (sin(k) * ((pow(t_m, 2.0) / l) * (t_m / l))));
	} else {
		tmp = (pow(l, 2.0) / pow(k, 4.0)) * (2.0 / t_m);
	}
	return t_s * tmp;
}

Java

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) {
	double tmp;
	if (k <= 2.7e-230) {
		tmp = (l * l) / Math.pow((t_m * Math.pow(Math.cbrt(k), 2.0)), 3.0);
	} else if (k <= 1.7e+128) {
		tmp = 2.0 / ((k * 2.0) * (Math.sin(k) * ((Math.pow(t_m, 2.0) / l) * (t_m / l))));
	} else {
		tmp = (Math.pow(l, 2.0) / Math.pow(k, 4.0)) * (2.0 / t_m);
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 2.7e-230)
		tmp = Float64(Float64(l * l) / (Float64(t_m * (cbrt(k) ^ 2.0)) ^ 3.0));
	elseif (k <= 1.7e+128)
		tmp = Float64(2.0 / Float64(Float64(k * 2.0) * Float64(sin(k) * Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l)))));
	else
		tmp = Float64(Float64((l ^ 2.0) / (k ^ 4.0)) * Float64(2.0 / t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[k, 2.7e-230], N[(N[(l * l), $MachinePrecision] / N[Power[N[(t$95$m * N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.7e+128], N[(2.0 / N[(N[(k * 2.0), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 / t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 2.70000000000000011e-230

   1. Initial program 54.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 46.6%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

   4. Taylor expanded in k around 0 46.8%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 46.8%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   6. Applied egg-rr 46.8%

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]
   7. Step-by-step derivation

      1. add-cube-cbrt 46.8%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\sqrt[3]{{k}^{2} \cdot {t}^{3}} \cdot \sqrt[3]{{k}^{2} \cdot {t}^{3}}\right) \cdot \sqrt[3]{{k}^{2} \cdot {t}^{3}}}} \]

      2. pow3 46.8%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{\left(\sqrt[3]{{k}^{2} \cdot {t}^{3}}\right)}^{3}}} \]

      3. *-commutative 46.8%

         \[\leadsto \frac{\ell \cdot \ell}{{\left(\sqrt[3]{\color{blue}{{t}^{3} \cdot {k}^{2}}}\right)}^{3}} \]

      4. cbrt-prod 46.8%

         \[\leadsto \frac{\ell \cdot \ell}{{\color{blue}{\left(\sqrt[3]{{t}^{3}} \cdot \sqrt[3]{{k}^{2}}\right)}}^{3}} \]

      5. unpow3 46.8%

         \[\leadsto \frac{\ell \cdot \ell}{{\left(\sqrt[3]{\color{blue}{\left(t \cdot t\right) \cdot t}} \cdot \sqrt[3]{{k}^{2}}\right)}^{3}} \]

      6. add-cbrt-cube 55.3%

         \[\leadsto \frac{\ell \cdot \ell}{{\left(\color{blue}{t} \cdot \sqrt[3]{{k}^{2}}\right)}^{3}} \]

      7. unpow2 55.3%

         \[\leadsto \frac{\ell \cdot \ell}{{\left(t \cdot \sqrt[3]{\color{blue}{k \cdot k}}\right)}^{3}} \]

      8. cbrt-prod 62.2%

         \[\leadsto \frac{\ell \cdot \ell}{{\left(t \cdot \color{blue}{\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)}\right)}^{3}} \]

      9. pow2 62.2%

         \[\leadsto \frac{\ell \cdot \ell}{{\left(t \cdot \color{blue}{{\left(\sqrt[3]{k}\right)}^{2}}\right)}^{3}} \]

   8. Applied egg-rr 62.2%

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}} \]

   if 2.70000000000000011e-230 < k < 1.6999999999999999e128

   1. Initial program 48.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 48.9%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. *-commutative 48.9%

         \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. *-commutative 48.9%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-/r* 56.2%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. distribute-rgt-in 56.2%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

      6. unpow2 56.2%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      7. times-frac 50.5%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      8. sqr-neg 50.5%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      9. times-frac 56.2%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      10. unpow2 56.2%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      11. distribute-rgt-in 56.2%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

      12. +-commutative 56.2%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)\right)}\right)} \]

   3. Simplified 56.2%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-/r* 48.9%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      2. unpow3 49.0%

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      3. times-frac 66.5%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      4. pow2 66.5%

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{2}}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   5. Applied egg-rr 66.5%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   6. Taylor expanded in k around 0 69.5%

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]
   7. Step-by-step derivation

      1. *-commutative 69.3%

         \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   8. Simplified 69.5%

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   if 1.6999999999999999e128 < k

   1. Initial program 52.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 52.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 52.6%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \cdot \sqrt{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}}} \]

   5. Applied egg-rr 38.0%

      \[\leadsto \color{blue}{\frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}} \]
   6. Step-by-step derivation

      1. unpow2 38.0%

         \[\leadsto \color{blue}{{\left(\frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right)}^{2}} \]

      2. associate-/l* 38.0%

         \[\leadsto {\color{blue}{\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}}\right)}}^{2} \]

      3. associate-*r* 38.0%

         \[\leadsto {\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\color{blue}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}}}}\right)}^{2} \]

      4. *-commutative 38.0%

         \[\leadsto {\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\color{blue}{\left({t}^{3} \cdot \tan k\right)} \cdot \sin k}}}}\right)}^{2} \]

   7. Simplified 38.0%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\left({t}^{3} \cdot \tan k\right) \cdot \sin k}}}}\right)}^{2}} \]

   8. Taylor expanded in k around inf 47.3%

      \[\leadsto {\color{blue}{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t}}\right)}}^{2} \]

   9. Taylor expanded in k around 0 69.0%

      \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}{{k}^{4} \cdot t}} \]
   10. Step-by-step derivation

       1. times-frac 69.0%

          \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{t}} \]

       2. unpow2 69.0%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}{t} \]

       3. rem-square-sqrt 69.0%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{\color{blue}{2}}{t} \]

   11. Simplified 69.0%

       \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{2}{t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.7 \cdot 10^{-230}:\\ \;\;\;\;\frac{\ell \cdot \ell}{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \mathbf{elif}\;k \leq 1.7 \cdot 10^{+128}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{2}{t}\\ \end{array} \]

Alternative 18: 62.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.05 \cdot 10^{-247}:\\ \;\;\;\;{\left(\frac{\ell}{k} \cdot \sqrt{\frac{1}{{t_m}^{3}}}\right)}^{2}\\ \mathbf{elif}\;k \leq 6.5 \cdot 10^{+125}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot \left(\frac{{t_m}^{2}}{\ell} \cdot \frac{t_m}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{2}{t_m}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 2.05e-247)
    (pow (* (/ l k) (sqrt (/ 1.0 (pow t_m 3.0)))) 2.0)
    (if (<= k 6.5e+125)
      (/ 2.0 (* (* k 2.0) (* (sin k) (* (/ (pow t_m 2.0) l) (/ t_m l)))))
      (* (/ (pow l 2.0) (pow k 4.0)) (/ 2.0 t_m))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.05e-247) {
		tmp = pow(((l / k) * sqrt((1.0 / pow(t_m, 3.0)))), 2.0);
	} else if (k <= 6.5e+125) {
		tmp = 2.0 / ((k * 2.0) * (sin(k) * ((pow(t_m, 2.0) / l) * (t_m / l))));
	} else {
		tmp = (pow(l, 2.0) / pow(k, 4.0)) * (2.0 / t_m);
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.05d-247) then
        tmp = ((l / k) * sqrt((1.0d0 / (t_m ** 3.0d0)))) ** 2.0d0
    else if (k <= 6.5d+125) then
        tmp = 2.0d0 / ((k * 2.0d0) * (sin(k) * (((t_m ** 2.0d0) / l) * (t_m / l))))
    else
        tmp = ((l ** 2.0d0) / (k ** 4.0d0)) * (2.0d0 / t_m)
    end if
    code = t_s * tmp
end function

Java

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) {
	double tmp;
	if (k <= 2.05e-247) {
		tmp = Math.pow(((l / k) * Math.sqrt((1.0 / Math.pow(t_m, 3.0)))), 2.0);
	} else if (k <= 6.5e+125) {
		tmp = 2.0 / ((k * 2.0) * (Math.sin(k) * ((Math.pow(t_m, 2.0) / l) * (t_m / l))));
	} else {
		tmp = (Math.pow(l, 2.0) / Math.pow(k, 4.0)) * (2.0 / t_m);
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 2.05e-247:
		tmp = math.pow(((l / k) * math.sqrt((1.0 / math.pow(t_m, 3.0)))), 2.0)
	elif k <= 6.5e+125:
		tmp = 2.0 / ((k * 2.0) * (math.sin(k) * ((math.pow(t_m, 2.0) / l) * (t_m / l))))
	else:
		tmp = (math.pow(l, 2.0) / math.pow(k, 4.0)) * (2.0 / t_m)
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 2.05e-247)
		tmp = Float64(Float64(l / k) * sqrt(Float64(1.0 / (t_m ^ 3.0)))) ^ 2.0;
	elseif (k <= 6.5e+125)
		tmp = Float64(2.0 / Float64(Float64(k * 2.0) * Float64(sin(k) * Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l)))));
	else
		tmp = Float64(Float64((l ^ 2.0) / (k ^ 4.0)) * Float64(2.0 / t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 2.05e-247)
		tmp = ((l / k) * sqrt((1.0 / (t_m ^ 3.0)))) ^ 2.0;
	elseif (k <= 6.5e+125)
		tmp = 2.0 / ((k * 2.0) * (sin(k) * (((t_m ^ 2.0) / l) * (t_m / l))));
	else
		tmp = ((l ^ 2.0) / (k ^ 4.0)) * (2.0 / t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[k, 2.05e-247], N[Power[N[(N[(l / k), $MachinePrecision] * N[Sqrt[N[(1.0 / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[k, 6.5e+125], N[(2.0 / N[(N[(k * 2.0), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 / t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 2.0499999999999999e-247

   1. Initial program 53.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 53.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 46.5%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \cdot \sqrt{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}}} \]

   5. Applied egg-rr 52.9%

      \[\leadsto \color{blue}{\frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}} \]
   6. Step-by-step derivation

      1. unpow2 52.9%

         \[\leadsto \color{blue}{{\left(\frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right)}^{2}} \]

      2. associate-/l* 53.0%

         \[\leadsto {\color{blue}{\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}}\right)}}^{2} \]

      3. associate-*r* 53.0%

         \[\leadsto {\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\color{blue}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}}}}\right)}^{2} \]

      4. *-commutative 53.0%

         \[\leadsto {\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\color{blue}{\left({t}^{3} \cdot \tan k\right)} \cdot \sin k}}}}\right)}^{2} \]

   7. Simplified 53.0%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\left({t}^{3} \cdot \tan k\right) \cdot \sin k}}}}\right)}^{2}} \]

   8. Taylor expanded in k around 0 48.9%

      \[\leadsto {\color{blue}{\left(\frac{\ell}{k} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}}^{2} \]

   if 2.0499999999999999e-247 < k < 6.4999999999999999e125

   1. Initial program 50.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 50.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. *-commutative 50.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. *-commutative 50.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-/r* 57.4%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. distribute-rgt-in 57.4%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

      6. unpow2 57.4%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      7. times-frac 51.9%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      8. sqr-neg 51.9%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      9. times-frac 57.4%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      10. unpow2 57.4%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      11. distribute-rgt-in 57.4%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

      12. +-commutative 57.4%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)\right)}\right)} \]

   3. Simplified 57.4%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-/r* 50.3%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      2. unpow3 50.4%

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      3. times-frac 67.4%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      4. pow2 67.4%

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{2}}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   5. Applied egg-rr 67.4%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   6. Taylor expanded in k around 0 70.3%

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]
   7. Step-by-step derivation

      1. *-commutative 70.1%

         \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   8. Simplified 70.3%

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   if 6.4999999999999999e125 < k

   1. Initial program 52.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 52.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 52.6%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \cdot \sqrt{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}}} \]

   5. Applied egg-rr 38.0%

      \[\leadsto \color{blue}{\frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}} \]
   6. Step-by-step derivation

      1. unpow2 38.0%

         \[\leadsto \color{blue}{{\left(\frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right)}^{2}} \]

      2. associate-/l* 38.0%

         \[\leadsto {\color{blue}{\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}}\right)}}^{2} \]

      3. associate-*r* 38.0%

         \[\leadsto {\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\color{blue}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}}}}\right)}^{2} \]

      4. *-commutative 38.0%

         \[\leadsto {\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\color{blue}{\left({t}^{3} \cdot \tan k\right)} \cdot \sin k}}}}\right)}^{2} \]

   7. Simplified 38.0%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\left({t}^{3} \cdot \tan k\right) \cdot \sin k}}}}\right)}^{2}} \]

   8. Taylor expanded in k around inf 47.3%

      \[\leadsto {\color{blue}{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t}}\right)}}^{2} \]

   9. Taylor expanded in k around 0 69.0%

      \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}{{k}^{4} \cdot t}} \]
   10. Step-by-step derivation

       1. times-frac 69.0%

          \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{t}} \]

       2. unpow2 69.0%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}{t} \]

       3. rem-square-sqrt 69.0%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{\color{blue}{2}}{t} \]

   11. Simplified 69.0%

       \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{2}{t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.05 \cdot 10^{-247}:\\ \;\;\;\;{\left(\frac{\ell}{k} \cdot \sqrt{\frac{1}{{t}^{3}}}\right)}^{2}\\ \mathbf{elif}\;k \leq 6.5 \cdot 10^{+125}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{2}{t}\\ \end{array} \]

Alternative 19: 62.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-247}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot \sqrt{{t_m}^{3}}}\right)}^{2}\\ \mathbf{elif}\;k \leq 1.15 \cdot 10^{+128}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot \left(\frac{{t_m}^{2}}{\ell} \cdot \frac{t_m}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{2}{t_m}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 2e-247)
    (pow (/ l (* k (sqrt (pow t_m 3.0)))) 2.0)
    (if (<= k 1.15e+128)
      (/ 2.0 (* (* k 2.0) (* (sin k) (* (/ (pow t_m 2.0) l) (/ t_m l)))))
      (* (/ (pow l 2.0) (pow k 4.0)) (/ 2.0 t_m))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2e-247) {
		tmp = pow((l / (k * sqrt(pow(t_m, 3.0)))), 2.0);
	} else if (k <= 1.15e+128) {
		tmp = 2.0 / ((k * 2.0) * (sin(k) * ((pow(t_m, 2.0) / l) * (t_m / l))));
	} else {
		tmp = (pow(l, 2.0) / pow(k, 4.0)) * (2.0 / t_m);
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2d-247) then
        tmp = (l / (k * sqrt((t_m ** 3.0d0)))) ** 2.0d0
    else if (k <= 1.15d+128) then
        tmp = 2.0d0 / ((k * 2.0d0) * (sin(k) * (((t_m ** 2.0d0) / l) * (t_m / l))))
    else
        tmp = ((l ** 2.0d0) / (k ** 4.0d0)) * (2.0d0 / t_m)
    end if
    code = t_s * tmp
end function

Java

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) {
	double tmp;
	if (k <= 2e-247) {
		tmp = Math.pow((l / (k * Math.sqrt(Math.pow(t_m, 3.0)))), 2.0);
	} else if (k <= 1.15e+128) {
		tmp = 2.0 / ((k * 2.0) * (Math.sin(k) * ((Math.pow(t_m, 2.0) / l) * (t_m / l))));
	} else {
		tmp = (Math.pow(l, 2.0) / Math.pow(k, 4.0)) * (2.0 / t_m);
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 2e-247:
		tmp = math.pow((l / (k * math.sqrt(math.pow(t_m, 3.0)))), 2.0)
	elif k <= 1.15e+128:
		tmp = 2.0 / ((k * 2.0) * (math.sin(k) * ((math.pow(t_m, 2.0) / l) * (t_m / l))))
	else:
		tmp = (math.pow(l, 2.0) / math.pow(k, 4.0)) * (2.0 / t_m)
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 2e-247)
		tmp = Float64(l / Float64(k * sqrt((t_m ^ 3.0)))) ^ 2.0;
	elseif (k <= 1.15e+128)
		tmp = Float64(2.0 / Float64(Float64(k * 2.0) * Float64(sin(k) * Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l)))));
	else
		tmp = Float64(Float64((l ^ 2.0) / (k ^ 4.0)) * Float64(2.0 / t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 2e-247)
		tmp = (l / (k * sqrt((t_m ^ 3.0)))) ^ 2.0;
	elseif (k <= 1.15e+128)
		tmp = 2.0 / ((k * 2.0) * (sin(k) * (((t_m ^ 2.0) / l) * (t_m / l))));
	else
		tmp = ((l ^ 2.0) / (k ^ 4.0)) * (2.0 / t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[k, 2e-247], N[Power[N[(l / N[(k * N[Sqrt[N[Power[t$95$m, 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[k, 1.15e+128], N[(2.0 / N[(N[(k * 2.0), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 / t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 2e-247

   1. Initial program 53.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 53.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 46.5%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \cdot \sqrt{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}}} \]

   5. Applied egg-rr 52.9%

      \[\leadsto \color{blue}{\frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}} \]
   6. Step-by-step derivation

      1. unpow2 52.9%

         \[\leadsto \color{blue}{{\left(\frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right)}^{2}} \]

      2. associate-/l* 53.0%

         \[\leadsto {\color{blue}{\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}}\right)}}^{2} \]

      3. associate-*r* 53.0%

         \[\leadsto {\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\color{blue}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}}}}\right)}^{2} \]

      4. *-commutative 53.0%

         \[\leadsto {\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\color{blue}{\left({t}^{3} \cdot \tan k\right)} \cdot \sin k}}}}\right)}^{2} \]

   7. Simplified 53.0%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\left({t}^{3} \cdot \tan k\right) \cdot \sin k}}}}\right)}^{2}} \]

   8. Taylor expanded in k around 0 34.3%

      \[\leadsto {\left(\frac{\ell}{\color{blue}{k \cdot \sqrt{{t}^{3}}}}\right)}^{2} \]

   if 2e-247 < k < 1.14999999999999999e128

   1. Initial program 50.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 50.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. *-commutative 50.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. *-commutative 50.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-/r* 57.4%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. distribute-rgt-in 57.4%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

      6. unpow2 57.4%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      7. times-frac 51.9%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      8. sqr-neg 51.9%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      9. times-frac 57.4%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      10. unpow2 57.4%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      11. distribute-rgt-in 57.4%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

      12. +-commutative 57.4%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)\right)}\right)} \]

   3. Simplified 57.4%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-/r* 50.3%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      2. unpow3 50.4%

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      3. times-frac 67.4%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      4. pow2 67.4%

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{2}}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   5. Applied egg-rr 67.4%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   6. Taylor expanded in k around 0 70.3%

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]
   7. Step-by-step derivation

      1. *-commutative 70.1%

         \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   8. Simplified 70.3%

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   if 1.14999999999999999e128 < k

   1. Initial program 52.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 52.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 52.6%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \cdot \sqrt{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}}} \]

   5. Applied egg-rr 38.0%

      \[\leadsto \color{blue}{\frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}} \]
   6. Step-by-step derivation

      1. unpow2 38.0%

         \[\leadsto \color{blue}{{\left(\frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right)}^{2}} \]

      2. associate-/l* 38.0%

         \[\leadsto {\color{blue}{\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}}\right)}}^{2} \]

      3. associate-*r* 38.0%

         \[\leadsto {\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\color{blue}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}}}}\right)}^{2} \]

      4. *-commutative 38.0%

         \[\leadsto {\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\color{blue}{\left({t}^{3} \cdot \tan k\right)} \cdot \sin k}}}}\right)}^{2} \]

   7. Simplified 38.0%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\left({t}^{3} \cdot \tan k\right) \cdot \sin k}}}}\right)}^{2}} \]

   8. Taylor expanded in k around inf 47.3%

      \[\leadsto {\color{blue}{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t}}\right)}}^{2} \]

   9. Taylor expanded in k around 0 69.0%

      \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}{{k}^{4} \cdot t}} \]
   10. Step-by-step derivation

       1. times-frac 69.0%

          \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{t}} \]

       2. unpow2 69.0%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}{t} \]

       3. rem-square-sqrt 69.0%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{\color{blue}{2}}{t} \]

   11. Simplified 69.0%

       \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{2}{t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-247}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot \sqrt{{t}^{3}}}\right)}^{2}\\ \mathbf{elif}\;k \leq 1.15 \cdot 10^{+128}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{2}{t}\\ \end{array} \]

Alternative 20: 61.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{+128}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot \left(\frac{{t_m}^{2}}{\ell} \cdot \frac{t_m}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{2}{t_m}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.5e+128)
    (/ 2.0 (* (* k 2.0) (* (sin k) (* (/ (pow t_m 2.0) l) (/ t_m l)))))
    (* (/ (pow l 2.0) (pow k 4.0)) (/ 2.0 t_m)))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.5e+128) {
		tmp = 2.0 / ((k * 2.0) * (sin(k) * ((pow(t_m, 2.0) / l) * (t_m / l))));
	} else {
		tmp = (pow(l, 2.0) / pow(k, 4.0)) * (2.0 / t_m);
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.5d+128) then
        tmp = 2.0d0 / ((k * 2.0d0) * (sin(k) * (((t_m ** 2.0d0) / l) * (t_m / l))))
    else
        tmp = ((l ** 2.0d0) / (k ** 4.0d0)) * (2.0d0 / t_m)
    end if
    code = t_s * tmp
end function

Java

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) {
	double tmp;
	if (k <= 1.5e+128) {
		tmp = 2.0 / ((k * 2.0) * (Math.sin(k) * ((Math.pow(t_m, 2.0) / l) * (t_m / l))));
	} else {
		tmp = (Math.pow(l, 2.0) / Math.pow(k, 4.0)) * (2.0 / t_m);
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 1.5e+128:
		tmp = 2.0 / ((k * 2.0) * (math.sin(k) * ((math.pow(t_m, 2.0) / l) * (t_m / l))))
	else:
		tmp = (math.pow(l, 2.0) / math.pow(k, 4.0)) * (2.0 / t_m)
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.5e+128)
		tmp = Float64(2.0 / Float64(Float64(k * 2.0) * Float64(sin(k) * Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l)))));
	else
		tmp = Float64(Float64((l ^ 2.0) / (k ^ 4.0)) * Float64(2.0 / t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 1.5e+128)
		tmp = 2.0 / ((k * 2.0) * (sin(k) * (((t_m ^ 2.0) / l) * (t_m / l))));
	else
		tmp = ((l ^ 2.0) / (k ^ 4.0)) * (2.0 / t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[k, 1.5e+128], N[(2.0 / N[(N[(k * 2.0), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 / t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.4999999999999999e128

   1. Initial program 52.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 52.6%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. *-commutative 52.6%

         \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. *-commutative 52.6%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-/r* 60.0%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. distribute-rgt-in 60.0%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

      6. unpow2 60.0%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      7. times-frac 45.7%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      8. sqr-neg 45.7%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      9. times-frac 60.0%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      10. unpow2 60.0%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      11. distribute-rgt-in 60.0%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

      12. +-commutative 60.0%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)\right)}\right)} \]

   3. Simplified 60.0%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-/r* 52.6%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      2. unpow3 52.6%

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      3. times-frac 67.0%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      4. pow2 67.0%

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{2}}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   5. Applied egg-rr 67.0%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   6. Taylor expanded in k around 0 63.5%

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]
   7. Step-by-step derivation

      1. *-commutative 65.7%

         \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   8. Simplified 63.5%

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   if 1.4999999999999999e128 < k

   1. Initial program 52.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 52.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 52.6%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \cdot \sqrt{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}}} \]

   5. Applied egg-rr 38.0%

      \[\leadsto \color{blue}{\frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}} \]
   6. Step-by-step derivation

      1. unpow2 38.0%

         \[\leadsto \color{blue}{{\left(\frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right)}^{2}} \]

      2. associate-/l* 38.0%

         \[\leadsto {\color{blue}{\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}}\right)}}^{2} \]

      3. associate-*r* 38.0%

         \[\leadsto {\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\color{blue}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}}}}\right)}^{2} \]

      4. *-commutative 38.0%

         \[\leadsto {\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\color{blue}{\left({t}^{3} \cdot \tan k\right)} \cdot \sin k}}}}\right)}^{2} \]

   7. Simplified 38.0%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\left({t}^{3} \cdot \tan k\right) \cdot \sin k}}}}\right)}^{2}} \]

   8. Taylor expanded in k around inf 47.3%

      \[\leadsto {\color{blue}{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t}}\right)}}^{2} \]

   9. Taylor expanded in k around 0 69.0%

      \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}{{k}^{4} \cdot t}} \]
   10. Step-by-step derivation

       1. times-frac 69.0%

          \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{t}} \]

       2. unpow2 69.0%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}{t} \]

       3. rem-square-sqrt 69.0%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{\color{blue}{2}}{t} \]

   11. Simplified 69.0%

       \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{2}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{+128}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{2}{t}\\ \end{array} \]

Alternative 21: 59.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 5.8 \cdot 10^{+87}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot \frac{\frac{{t_m}^{3}}{\ell}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{2}{t_m}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 5.8e+87)
    (/ 2.0 (* (* k 2.0) (* (sin k) (/ (/ (pow t_m 3.0) l) l))))
    (* (/ (pow l 2.0) (pow k 4.0)) (/ 2.0 t_m)))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 5.8e+87) {
		tmp = 2.0 / ((k * 2.0) * (sin(k) * ((pow(t_m, 3.0) / l) / l)));
	} else {
		tmp = (pow(l, 2.0) / pow(k, 4.0)) * (2.0 / t_m);
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 5.8d+87) then
        tmp = 2.0d0 / ((k * 2.0d0) * (sin(k) * (((t_m ** 3.0d0) / l) / l)))
    else
        tmp = ((l ** 2.0d0) / (k ** 4.0d0)) * (2.0d0 / t_m)
    end if
    code = t_s * tmp
end function

Java

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) {
	double tmp;
	if (k <= 5.8e+87) {
		tmp = 2.0 / ((k * 2.0) * (Math.sin(k) * ((Math.pow(t_m, 3.0) / l) / l)));
	} else {
		tmp = (Math.pow(l, 2.0) / Math.pow(k, 4.0)) * (2.0 / t_m);
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 5.8e+87:
		tmp = 2.0 / ((k * 2.0) * (math.sin(k) * ((math.pow(t_m, 3.0) / l) / l)))
	else:
		tmp = (math.pow(l, 2.0) / math.pow(k, 4.0)) * (2.0 / t_m)
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 5.8e+87)
		tmp = Float64(2.0 / Float64(Float64(k * 2.0) * Float64(sin(k) * Float64(Float64((t_m ^ 3.0) / l) / l))));
	else
		tmp = Float64(Float64((l ^ 2.0) / (k ^ 4.0)) * Float64(2.0 / t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 5.8e+87)
		tmp = 2.0 / ((k * 2.0) * (sin(k) * (((t_m ^ 3.0) / l) / l)));
	else
		tmp = ((l ^ 2.0) / (k ^ 4.0)) * (2.0 / t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[k, 5.8e+87], N[(2.0 / N[(N[(k * 2.0), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 / t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 5.7999999999999996e87

   1. Initial program 53.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 53.9%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. *-commutative 53.9%

         \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. *-commutative 53.9%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-/r* 61.6%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. distribute-rgt-in 61.6%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

      6. unpow2 61.6%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      7. times-frac 46.7%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      8. sqr-neg 46.7%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      9. times-frac 61.6%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      10. unpow2 61.6%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      11. distribute-rgt-in 61.6%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

      12. +-commutative 61.6%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)\right)}\right)} \]

   3. Simplified 61.6%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 61.5%

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      2. *-un-lft-identity 61.5%

         \[\leadsto \frac{2}{\left(\frac{\left(\sqrt[3]{\frac{{t}^{3}}{\ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}}{\color{blue}{1 \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      3. times-frac 61.5%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{\sqrt[3]{\frac{{t}^{3}}{\ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}}{1} \cdot \frac{\sqrt[3]{\frac{{t}^{3}}{\ell}}}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      4. pow2 61.5%

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{\left(\sqrt[3]{\frac{{t}^{3}}{\ell}}\right)}^{2}}}{1} \cdot \frac{\sqrt[3]{\frac{{t}^{3}}{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      5. cbrt-div 61.5%

         \[\leadsto \frac{2}{\left(\left(\frac{{\color{blue}{\left(\frac{\sqrt[3]{{t}^{3}}}{\sqrt[3]{\ell}}\right)}}^{2}}{1} \cdot \frac{\sqrt[3]{\frac{{t}^{3}}{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      6. rem-cbrt-cube 61.5%

         \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{\color{blue}{t}}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{\frac{{t}^{3}}{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      7. cbrt-div 61.5%

         \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\color{blue}{\frac{\sqrt[3]{{t}^{3}}}{\sqrt[3]{\ell}}}}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      8. rem-cbrt-cube 72.7%

         \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{\color{blue}{t}}{\sqrt[3]{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   5. Applied egg-rr 72.7%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   6. Taylor expanded in k around 0 67.0%

      \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]
   7. Step-by-step derivation

      1. *-commutative 67.0%

         \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   8. Simplified 67.0%

      \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 45.7%

         \[\leadsto \frac{2}{\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right)\right)} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      2. expm1-udef 35.4%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right)} - 1\right)} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      3. frac-times 34.0%

         \[\leadsto \frac{2}{\left(\left(e^{\mathsf{log1p}\left(\color{blue}{\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2} \cdot \frac{t}{\sqrt[3]{\ell}}}{1 \cdot \ell}}\right)} - 1\right) \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      4. pow-plus 34.0%

         \[\leadsto \frac{2}{\left(\left(e^{\mathsf{log1p}\left(\frac{\color{blue}{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{\left(2 + 1\right)}}}{1 \cdot \ell}\right)} - 1\right) \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      5. metadata-eval 34.0%

         \[\leadsto \frac{2}{\left(\left(e^{\mathsf{log1p}\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{\color{blue}{3}}}{1 \cdot \ell}\right)} - 1\right) \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      6. *-un-lft-identity 34.0%

         \[\leadsto \frac{2}{\left(\left(e^{\mathsf{log1p}\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\color{blue}{\ell}}\right)} - 1\right) \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

   10. Applied egg-rr 34.0%

       \[\leadsto \frac{2}{\left(\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}\right)} - 1\right)} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]
   11. Step-by-step derivation

       1. expm1-def 44.4%

          \[\leadsto \frac{2}{\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}\right)\right)} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

       2. expm1-log1p 65.1%

          \[\leadsto \frac{2}{\left(\color{blue}{\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

       3. cube-div 62.2%

          \[\leadsto \frac{2}{\left(\frac{\color{blue}{\frac{{t}^{3}}{{\left(\sqrt[3]{\ell}\right)}^{3}}}}{\ell} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

       4. rem-cube-cbrt 62.2%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\color{blue}{\ell}}}{\ell} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

   12. Simplified 62.2%

       \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

   if 5.7999999999999996e87 < k

   1. Initial program 47.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 47.2%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 47.2%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \cdot \sqrt{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}}} \]

   5. Applied egg-rr 37.2%

      \[\leadsto \color{blue}{\frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}} \]
   6. Step-by-step derivation

      1. unpow2 37.2%

         \[\leadsto \color{blue}{{\left(\frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right)}^{2}} \]

      2. associate-/l* 37.3%

         \[\leadsto {\color{blue}{\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}}\right)}}^{2} \]

      3. associate-*r* 37.3%

         \[\leadsto {\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\color{blue}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}}}}\right)}^{2} \]

      4. *-commutative 37.3%

         \[\leadsto {\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\color{blue}{\left({t}^{3} \cdot \tan k\right)} \cdot \sin k}}}}\right)}^{2} \]

   7. Simplified 37.3%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\left({t}^{3} \cdot \tan k\right) \cdot \sin k}}}}\right)}^{2}} \]

   8. Taylor expanded in k around inf 46.7%

      \[\leadsto {\color{blue}{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t}}\right)}}^{2} \]

   9. Taylor expanded in k around 0 62.8%

      \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}{{k}^{4} \cdot t}} \]
   10. Step-by-step derivation

       1. times-frac 62.8%

          \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{t}} \]

       2. unpow2 62.8%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}{t} \]

       3. rem-square-sqrt 62.8%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{\color{blue}{2}}{t} \]

   11. Simplified 62.8%

       \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{2}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.8 \cdot 10^{+87}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{2}{t}\\ \end{array} \]

Alternative 22: 56.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 9.5 \cdot 10^{-160}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \frac{k}{\frac{{\ell}^{2}}{{t_m}^{3}}}}\\ \mathbf{elif}\;k \leq 1.1 \cdot 10^{+89}:\\ \;\;\;\;\left(\ell \cdot {k}^{-2}\right) \cdot \frac{\ell}{{t_m}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{2}{t_m}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 9.5e-160)
    (/ 2.0 (* (* k 2.0) (/ k (/ (pow l 2.0) (pow t_m 3.0)))))
    (if (<= k 1.1e+89)
      (* (* l (pow k -2.0)) (/ l (pow t_m 3.0)))
      (* (/ (pow l 2.0) (pow k 4.0)) (/ 2.0 t_m))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 9.5e-160) {
		tmp = 2.0 / ((k * 2.0) * (k / (pow(l, 2.0) / pow(t_m, 3.0))));
	} else if (k <= 1.1e+89) {
		tmp = (l * pow(k, -2.0)) * (l / pow(t_m, 3.0));
	} else {
		tmp = (pow(l, 2.0) / pow(k, 4.0)) * (2.0 / t_m);
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 9.5d-160) then
        tmp = 2.0d0 / ((k * 2.0d0) * (k / ((l ** 2.0d0) / (t_m ** 3.0d0))))
    else if (k <= 1.1d+89) then
        tmp = (l * (k ** (-2.0d0))) * (l / (t_m ** 3.0d0))
    else
        tmp = ((l ** 2.0d0) / (k ** 4.0d0)) * (2.0d0 / t_m)
    end if
    code = t_s * tmp
end function

Java

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) {
	double tmp;
	if (k <= 9.5e-160) {
		tmp = 2.0 / ((k * 2.0) * (k / (Math.pow(l, 2.0) / Math.pow(t_m, 3.0))));
	} else if (k <= 1.1e+89) {
		tmp = (l * Math.pow(k, -2.0)) * (l / Math.pow(t_m, 3.0));
	} else {
		tmp = (Math.pow(l, 2.0) / Math.pow(k, 4.0)) * (2.0 / t_m);
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 9.5e-160:
		tmp = 2.0 / ((k * 2.0) * (k / (math.pow(l, 2.0) / math.pow(t_m, 3.0))))
	elif k <= 1.1e+89:
		tmp = (l * math.pow(k, -2.0)) * (l / math.pow(t_m, 3.0))
	else:
		tmp = (math.pow(l, 2.0) / math.pow(k, 4.0)) * (2.0 / t_m)
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 9.5e-160)
		tmp = Float64(2.0 / Float64(Float64(k * 2.0) * Float64(k / Float64((l ^ 2.0) / (t_m ^ 3.0)))));
	elseif (k <= 1.1e+89)
		tmp = Float64(Float64(l * (k ^ -2.0)) * Float64(l / (t_m ^ 3.0)));
	else
		tmp = Float64(Float64((l ^ 2.0) / (k ^ 4.0)) * Float64(2.0 / t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 9.5e-160)
		tmp = 2.0 / ((k * 2.0) * (k / ((l ^ 2.0) / (t_m ^ 3.0))));
	elseif (k <= 1.1e+89)
		tmp = (l * (k ^ -2.0)) * (l / (t_m ^ 3.0));
	else
		tmp = ((l ^ 2.0) / (k ^ 4.0)) * (2.0 / t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[k, 9.5e-160], N[(2.0 / N[(N[(k * 2.0), $MachinePrecision] * N[(k / N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.1e+89], N[(N[(l * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision] * N[(l / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 / t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 9.5000000000000002e-160

   1. Initial program 54.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 54.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. *-commutative 54.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. *-commutative 54.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-/r* 62.1%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. distribute-rgt-in 62.1%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

      6. unpow2 62.1%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      7. times-frac 43.7%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      8. sqr-neg 43.7%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      9. times-frac 62.1%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      10. unpow2 62.1%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      11. distribute-rgt-in 62.1%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

      12. +-commutative 62.1%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)\right)}\right)} \]

   3. Simplified 62.1%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 62.0%

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      2. *-un-lft-identity 62.0%

         \[\leadsto \frac{2}{\left(\frac{\left(\sqrt[3]{\frac{{t}^{3}}{\ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}}{\color{blue}{1 \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      3. times-frac 62.0%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{\sqrt[3]{\frac{{t}^{3}}{\ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}}{1} \cdot \frac{\sqrt[3]{\frac{{t}^{3}}{\ell}}}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      4. pow2 62.0%

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{\left(\sqrt[3]{\frac{{t}^{3}}{\ell}}\right)}^{2}}}{1} \cdot \frac{\sqrt[3]{\frac{{t}^{3}}{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      5. cbrt-div 62.0%

         \[\leadsto \frac{2}{\left(\left(\frac{{\color{blue}{\left(\frac{\sqrt[3]{{t}^{3}}}{\sqrt[3]{\ell}}\right)}}^{2}}{1} \cdot \frac{\sqrt[3]{\frac{{t}^{3}}{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      6. rem-cbrt-cube 62.0%

         \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{\color{blue}{t}}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{\frac{{t}^{3}}{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      7. cbrt-div 62.0%

         \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\color{blue}{\frac{\sqrt[3]{{t}^{3}}}{\sqrt[3]{\ell}}}}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      8. rem-cbrt-cube 73.7%

         \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{\color{blue}{t}}{\sqrt[3]{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   5. Applied egg-rr 73.7%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   6. Taylor expanded in k around 0 64.7%

      \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]
   7. Step-by-step derivation

      1. *-commutative 64.7%

         \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   8. Simplified 64.7%

      \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   9. Taylor expanded in k around 0 54.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot {t}^{3}}{{\ell}^{2}}} \cdot \left(k \cdot 2\right)} \]
   10. Step-by-step derivation

       1. associate-/l* 52.8%

          \[\leadsto \frac{2}{\color{blue}{\frac{k}{\frac{{\ell}^{2}}{{t}^{3}}}} \cdot \left(k \cdot 2\right)} \]

   11. Simplified 52.8%

       \[\leadsto \frac{2}{\color{blue}{\frac{k}{\frac{{\ell}^{2}}{{t}^{3}}}} \cdot \left(k \cdot 2\right)} \]

   if 9.5000000000000002e-160 < k < 1.1e89

   1. Initial program 53.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 53.0%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

   4. Taylor expanded in k around 0 60.4%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 60.4%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   6. Applied egg-rr 60.4%

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]
   7. Step-by-step derivation

      1. times-frac 66.2%

         \[\leadsto \color{blue}{\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{{t}^{3}}} \]

   8. Applied egg-rr 66.2%

      \[\leadsto \color{blue}{\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{{t}^{3}}} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 36.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\ell}{{k}^{2}}\right)\right)} \cdot \frac{\ell}{{t}^{3}} \]

      2. expm1-udef 34.1%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\ell}{{k}^{2}}\right)} - 1\right)} \cdot \frac{\ell}{{t}^{3}} \]

      3. div-inv 34.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\ell \cdot \frac{1}{{k}^{2}}}\right)} - 1\right) \cdot \frac{\ell}{{t}^{3}} \]

      4. pow-flip 34.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\ell \cdot \color{blue}{{k}^{\left(-2\right)}}\right)} - 1\right) \cdot \frac{\ell}{{t}^{3}} \]

      5. metadata-eval 34.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\ell \cdot {k}^{\color{blue}{-2}}\right)} - 1\right) \cdot \frac{\ell}{{t}^{3}} \]

   10. Applied egg-rr 34.1%

       \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\ell \cdot {k}^{-2}\right)} - 1\right)} \cdot \frac{\ell}{{t}^{3}} \]
   11. Step-by-step derivation

       1. expm1-def 36.1%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\ell \cdot {k}^{-2}\right)\right)} \cdot \frac{\ell}{{t}^{3}} \]

       2. expm1-log1p 66.1%

          \[\leadsto \color{blue}{\left(\ell \cdot {k}^{-2}\right)} \cdot \frac{\ell}{{t}^{3}} \]

   12. Simplified 66.1%

       \[\leadsto \color{blue}{\left(\ell \cdot {k}^{-2}\right)} \cdot \frac{\ell}{{t}^{3}} \]

   if 1.1e89 < k

   1. Initial program 47.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 47.2%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 47.2%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \cdot \sqrt{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}}} \]

   5. Applied egg-rr 37.2%

      \[\leadsto \color{blue}{\frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}} \]
   6. Step-by-step derivation

      1. unpow2 37.2%

         \[\leadsto \color{blue}{{\left(\frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right)}^{2}} \]

      2. associate-/l* 37.3%

         \[\leadsto {\color{blue}{\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}}\right)}}^{2} \]

      3. associate-*r* 37.3%

         \[\leadsto {\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\color{blue}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}}}}\right)}^{2} \]

      4. *-commutative 37.3%

         \[\leadsto {\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\color{blue}{\left({t}^{3} \cdot \tan k\right)} \cdot \sin k}}}}\right)}^{2} \]

   7. Simplified 37.3%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\left({t}^{3} \cdot \tan k\right) \cdot \sin k}}}}\right)}^{2}} \]

   8. Taylor expanded in k around inf 46.7%

      \[\leadsto {\color{blue}{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t}}\right)}}^{2} \]

   9. Taylor expanded in k around 0 62.8%

      \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}{{k}^{4} \cdot t}} \]
   10. Step-by-step derivation

       1. times-frac 62.8%

          \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{t}} \]

       2. unpow2 62.8%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}{t} \]

       3. rem-square-sqrt 62.8%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{\color{blue}{2}}{t} \]

   11. Simplified 62.8%

       \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{2}{t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.5 \cdot 10^{-160}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \frac{k}{\frac{{\ell}^{2}}{{t}^{3}}}}\\ \mathbf{elif}\;k \leq 1.1 \cdot 10^{+89}:\\ \;\;\;\;\left(\ell \cdot {k}^{-2}\right) \cdot \frac{\ell}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{2}{t}\\ \end{array} \]

Alternative 23: 57.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 9.5 \cdot 10^{-160}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \frac{k \cdot {t_m}^{3}}{{\ell}^{2}}}\\ \mathbf{elif}\;k \leq 1.25 \cdot 10^{+89}:\\ \;\;\;\;\left(\ell \cdot {k}^{-2}\right) \cdot \frac{\ell}{{t_m}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{2}{t_m}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 9.5e-160)
    (/ 2.0 (* (* k 2.0) (/ (* k (pow t_m 3.0)) (pow l 2.0))))
    (if (<= k 1.25e+89)
      (* (* l (pow k -2.0)) (/ l (pow t_m 3.0)))
      (* (/ (pow l 2.0) (pow k 4.0)) (/ 2.0 t_m))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 9.5e-160) {
		tmp = 2.0 / ((k * 2.0) * ((k * pow(t_m, 3.0)) / pow(l, 2.0)));
	} else if (k <= 1.25e+89) {
		tmp = (l * pow(k, -2.0)) * (l / pow(t_m, 3.0));
	} else {
		tmp = (pow(l, 2.0) / pow(k, 4.0)) * (2.0 / t_m);
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 9.5d-160) then
        tmp = 2.0d0 / ((k * 2.0d0) * ((k * (t_m ** 3.0d0)) / (l ** 2.0d0)))
    else if (k <= 1.25d+89) then
        tmp = (l * (k ** (-2.0d0))) * (l / (t_m ** 3.0d0))
    else
        tmp = ((l ** 2.0d0) / (k ** 4.0d0)) * (2.0d0 / t_m)
    end if
    code = t_s * tmp
end function

Java

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) {
	double tmp;
	if (k <= 9.5e-160) {
		tmp = 2.0 / ((k * 2.0) * ((k * Math.pow(t_m, 3.0)) / Math.pow(l, 2.0)));
	} else if (k <= 1.25e+89) {
		tmp = (l * Math.pow(k, -2.0)) * (l / Math.pow(t_m, 3.0));
	} else {
		tmp = (Math.pow(l, 2.0) / Math.pow(k, 4.0)) * (2.0 / t_m);
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 9.5e-160:
		tmp = 2.0 / ((k * 2.0) * ((k * math.pow(t_m, 3.0)) / math.pow(l, 2.0)))
	elif k <= 1.25e+89:
		tmp = (l * math.pow(k, -2.0)) * (l / math.pow(t_m, 3.0))
	else:
		tmp = (math.pow(l, 2.0) / math.pow(k, 4.0)) * (2.0 / t_m)
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 9.5e-160)
		tmp = Float64(2.0 / Float64(Float64(k * 2.0) * Float64(Float64(k * (t_m ^ 3.0)) / (l ^ 2.0))));
	elseif (k <= 1.25e+89)
		tmp = Float64(Float64(l * (k ^ -2.0)) * Float64(l / (t_m ^ 3.0)));
	else
		tmp = Float64(Float64((l ^ 2.0) / (k ^ 4.0)) * Float64(2.0 / t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 9.5e-160)
		tmp = 2.0 / ((k * 2.0) * ((k * (t_m ^ 3.0)) / (l ^ 2.0)));
	elseif (k <= 1.25e+89)
		tmp = (l * (k ^ -2.0)) * (l / (t_m ^ 3.0));
	else
		tmp = ((l ^ 2.0) / (k ^ 4.0)) * (2.0 / t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[k, 9.5e-160], N[(2.0 / N[(N[(k * 2.0), $MachinePrecision] * N[(N[(k * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.25e+89], N[(N[(l * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision] * N[(l / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 / t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 9.5000000000000002e-160

   1. Initial program 54.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 54.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. *-commutative 54.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. *-commutative 54.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-/r* 62.1%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. distribute-rgt-in 62.1%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

      6. unpow2 62.1%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      7. times-frac 43.7%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      8. sqr-neg 43.7%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      9. times-frac 62.1%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      10. unpow2 62.1%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      11. distribute-rgt-in 62.1%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

      12. +-commutative 62.1%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)\right)}\right)} \]

   3. Simplified 62.1%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 62.0%

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      2. *-un-lft-identity 62.0%

         \[\leadsto \frac{2}{\left(\frac{\left(\sqrt[3]{\frac{{t}^{3}}{\ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}}{\color{blue}{1 \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      3. times-frac 62.0%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{\sqrt[3]{\frac{{t}^{3}}{\ell}} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell}}}{1} \cdot \frac{\sqrt[3]{\frac{{t}^{3}}{\ell}}}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      4. pow2 62.0%

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{\left(\sqrt[3]{\frac{{t}^{3}}{\ell}}\right)}^{2}}}{1} \cdot \frac{\sqrt[3]{\frac{{t}^{3}}{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      5. cbrt-div 62.0%

         \[\leadsto \frac{2}{\left(\left(\frac{{\color{blue}{\left(\frac{\sqrt[3]{{t}^{3}}}{\sqrt[3]{\ell}}\right)}}^{2}}{1} \cdot \frac{\sqrt[3]{\frac{{t}^{3}}{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      6. rem-cbrt-cube 62.0%

         \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{\color{blue}{t}}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{\frac{{t}^{3}}{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      7. cbrt-div 62.0%

         \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\color{blue}{\frac{\sqrt[3]{{t}^{3}}}{\sqrt[3]{\ell}}}}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      8. rem-cbrt-cube 73.7%

         \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{\color{blue}{t}}{\sqrt[3]{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   5. Applied egg-rr 73.7%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   6. Taylor expanded in k around 0 64.7%

      \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]
   7. Step-by-step derivation

      1. *-commutative 64.7%

         \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   8. Simplified 64.7%

      \[\leadsto \frac{2}{\left(\left(\frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2}}{1} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right) \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   9. Taylor expanded in k around 0 54.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot {t}^{3}}{{\ell}^{2}}} \cdot \left(k \cdot 2\right)} \]

   if 9.5000000000000002e-160 < k < 1.24999999999999996e89

   1. Initial program 53.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 53.0%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

   4. Taylor expanded in k around 0 60.4%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 60.4%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   6. Applied egg-rr 60.4%

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]
   7. Step-by-step derivation

      1. times-frac 66.2%

         \[\leadsto \color{blue}{\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{{t}^{3}}} \]

   8. Applied egg-rr 66.2%

      \[\leadsto \color{blue}{\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{{t}^{3}}} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 36.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\ell}{{k}^{2}}\right)\right)} \cdot \frac{\ell}{{t}^{3}} \]

      2. expm1-udef 34.1%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\ell}{{k}^{2}}\right)} - 1\right)} \cdot \frac{\ell}{{t}^{3}} \]

      3. div-inv 34.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\ell \cdot \frac{1}{{k}^{2}}}\right)} - 1\right) \cdot \frac{\ell}{{t}^{3}} \]

      4. pow-flip 34.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\ell \cdot \color{blue}{{k}^{\left(-2\right)}}\right)} - 1\right) \cdot \frac{\ell}{{t}^{3}} \]

      5. metadata-eval 34.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\ell \cdot {k}^{\color{blue}{-2}}\right)} - 1\right) \cdot \frac{\ell}{{t}^{3}} \]

   10. Applied egg-rr 34.1%

       \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\ell \cdot {k}^{-2}\right)} - 1\right)} \cdot \frac{\ell}{{t}^{3}} \]
   11. Step-by-step derivation

       1. expm1-def 36.1%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\ell \cdot {k}^{-2}\right)\right)} \cdot \frac{\ell}{{t}^{3}} \]

       2. expm1-log1p 66.1%

          \[\leadsto \color{blue}{\left(\ell \cdot {k}^{-2}\right)} \cdot \frac{\ell}{{t}^{3}} \]

   12. Simplified 66.1%

       \[\leadsto \color{blue}{\left(\ell \cdot {k}^{-2}\right)} \cdot \frac{\ell}{{t}^{3}} \]

   if 1.24999999999999996e89 < k

   1. Initial program 47.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 47.2%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 47.2%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \cdot \sqrt{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}}} \]

   5. Applied egg-rr 37.2%

      \[\leadsto \color{blue}{\frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}} \]
   6. Step-by-step derivation

      1. unpow2 37.2%

         \[\leadsto \color{blue}{{\left(\frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right)}^{2}} \]

      2. associate-/l* 37.3%

         \[\leadsto {\color{blue}{\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}}\right)}}^{2} \]

      3. associate-*r* 37.3%

         \[\leadsto {\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\color{blue}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}}}}\right)}^{2} \]

      4. *-commutative 37.3%

         \[\leadsto {\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\color{blue}{\left({t}^{3} \cdot \tan k\right)} \cdot \sin k}}}}\right)}^{2} \]

   7. Simplified 37.3%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\left({t}^{3} \cdot \tan k\right) \cdot \sin k}}}}\right)}^{2}} \]

   8. Taylor expanded in k around inf 46.7%

      \[\leadsto {\color{blue}{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t}}\right)}}^{2} \]

   9. Taylor expanded in k around 0 62.8%

      \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}{{k}^{4} \cdot t}} \]
   10. Step-by-step derivation

       1. times-frac 62.8%

          \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{t}} \]

       2. unpow2 62.8%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}{t} \]

       3. rem-square-sqrt 62.8%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{\color{blue}{2}}{t} \]

   11. Simplified 62.8%

       \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{2}{t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.5 \cdot 10^{-160}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \frac{k \cdot {t}^{3}}{{\ell}^{2}}}\\ \mathbf{elif}\;k \leq 1.25 \cdot 10^{+89}:\\ \;\;\;\;\left(\ell \cdot {k}^{-2}\right) \cdot \frac{\ell}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{2}{t}\\ \end{array} \]

Alternative 24: 58.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 3.7 \cdot 10^{-88}:\\ \;\;\;\;\frac{{\ell}^{2}}{t_m} \cdot \frac{2}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{{t_m}^{3}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.7e-88)
    (* (/ (pow l 2.0) t_m) (/ 2.0 (pow k 4.0)))
    (* (/ l (pow k 2.0)) (/ l (pow t_m 3.0))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.7e-88) {
		tmp = (pow(l, 2.0) / t_m) * (2.0 / pow(k, 4.0));
	} else {
		tmp = (l / pow(k, 2.0)) * (l / pow(t_m, 3.0));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 3.7d-88) then
        tmp = ((l ** 2.0d0) / t_m) * (2.0d0 / (k ** 4.0d0))
    else
        tmp = (l / (k ** 2.0d0)) * (l / (t_m ** 3.0d0))
    end if
    code = t_s * tmp
end function

Java

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) {
	double tmp;
	if (t_m <= 3.7e-88) {
		tmp = (Math.pow(l, 2.0) / t_m) * (2.0 / Math.pow(k, 4.0));
	} else {
		tmp = (l / Math.pow(k, 2.0)) * (l / Math.pow(t_m, 3.0));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 3.7e-88:
		tmp = (math.pow(l, 2.0) / t_m) * (2.0 / math.pow(k, 4.0))
	else:
		tmp = (l / math.pow(k, 2.0)) * (l / math.pow(t_m, 3.0))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 3.7e-88)
		tmp = Float64(Float64((l ^ 2.0) / t_m) * Float64(2.0 / (k ^ 4.0)));
	else
		tmp = Float64(Float64(l / (k ^ 2.0)) * Float64(l / (t_m ^ 3.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 3.7e-88)
		tmp = ((l ^ 2.0) / t_m) * (2.0 / (k ^ 4.0));
	else
		tmp = (l / (k ^ 2.0)) * (l / (t_m ^ 3.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[t$95$m, 3.7e-88], N[(N[(N[Power[l, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] * N[(2.0 / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(l / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.6999999999999997e-88

   1. Initial program 47.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 47.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 37.4%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \cdot \sqrt{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}}} \]

   5. Applied egg-rr 42.5%

      \[\leadsto \color{blue}{\frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}} \]
   6. Step-by-step derivation

      1. unpow2 42.5%

         \[\leadsto \color{blue}{{\left(\frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right)}^{2}} \]

      2. associate-/l* 42.6%

         \[\leadsto {\color{blue}{\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}}\right)}}^{2} \]

      3. associate-*r* 42.6%

         \[\leadsto {\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\color{blue}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}}}}\right)}^{2} \]

      4. *-commutative 42.6%

         \[\leadsto {\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\color{blue}{\left({t}^{3} \cdot \tan k\right)} \cdot \sin k}}}}\right)}^{2} \]

   7. Simplified 42.6%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\left({t}^{3} \cdot \tan k\right) \cdot \sin k}}}}\right)}^{2}} \]

   8. Taylor expanded in k around 0 39.5%

      \[\leadsto {\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\color{blue}{{k}^{2} \cdot {t}^{3}}}}}}\right)}^{2} \]

   9. Taylor expanded in k around inf 21.7%

      \[\leadsto {\left(\frac{\ell}{\color{blue}{\frac{{k}^{2}}{\sqrt{2}} \cdot \sqrt{t}}}\right)}^{2} \]

   10. Taylor expanded in l around 0 52.1%

       \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}{{k}^{4} \cdot t}} \]
   11. Step-by-step derivation

       1. unpow2 52.1%

          \[\leadsto \frac{{\ell}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)}}{{k}^{4} \cdot t} \]

       2. rem-square-sqrt 52.1%

          \[\leadsto \frac{{\ell}^{2} \cdot \color{blue}{2}}{{k}^{4} \cdot t} \]

       3. *-commutative 52.1%

          \[\leadsto \frac{{\ell}^{2} \cdot 2}{\color{blue}{t \cdot {k}^{4}}} \]

       4. times-frac 52.5%

          \[\leadsto \color{blue}{\frac{{\ell}^{2}}{t} \cdot \frac{2}{{k}^{4}}} \]

   12. Simplified 52.5%

       \[\leadsto \color{blue}{\frac{{\ell}^{2}}{t} \cdot \frac{2}{{k}^{4}}} \]

   if 3.6999999999999997e-88 < t

   1. Initial program 64.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 54.8%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

   4. Taylor expanded in k around 0 49.3%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 49.3%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   6. Applied egg-rr 49.3%

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]
   7. Step-by-step derivation

      1. times-frac 54.7%

         \[\leadsto \color{blue}{\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{{t}^{3}}} \]

   8. Applied egg-rr 54.7%

      \[\leadsto \color{blue}{\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{{t}^{3}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.7 \cdot 10^{-88}:\\ \;\;\;\;\frac{{\ell}^{2}}{t} \cdot \frac{2}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{{t}^{3}}\\ \end{array} \]

Alternative 25: 58.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 2.45 \cdot 10^{-92}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{2}{t_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{{t_m}^{3}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.45e-92)
    (* (/ (pow l 2.0) (pow k 4.0)) (/ 2.0 t_m))
    (* (/ l (pow k 2.0)) (/ l (pow t_m 3.0))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.45e-92) {
		tmp = (pow(l, 2.0) / pow(k, 4.0)) * (2.0 / t_m);
	} else {
		tmp = (l / pow(k, 2.0)) * (l / pow(t_m, 3.0));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2.45d-92) then
        tmp = ((l ** 2.0d0) / (k ** 4.0d0)) * (2.0d0 / t_m)
    else
        tmp = (l / (k ** 2.0d0)) * (l / (t_m ** 3.0d0))
    end if
    code = t_s * tmp
end function

Java

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) {
	double tmp;
	if (t_m <= 2.45e-92) {
		tmp = (Math.pow(l, 2.0) / Math.pow(k, 4.0)) * (2.0 / t_m);
	} else {
		tmp = (l / Math.pow(k, 2.0)) * (l / Math.pow(t_m, 3.0));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2.45e-92:
		tmp = (math.pow(l, 2.0) / math.pow(k, 4.0)) * (2.0 / t_m)
	else:
		tmp = (l / math.pow(k, 2.0)) * (l / math.pow(t_m, 3.0))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.45e-92)
		tmp = Float64(Float64((l ^ 2.0) / (k ^ 4.0)) * Float64(2.0 / t_m));
	else
		tmp = Float64(Float64(l / (k ^ 2.0)) * Float64(l / (t_m ^ 3.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2.45e-92)
		tmp = ((l ^ 2.0) / (k ^ 4.0)) * (2.0 / t_m);
	else
		tmp = (l / (k ^ 2.0)) * (l / (t_m ^ 3.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[t$95$m, 2.45e-92], N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 / t$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(l / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.45e-92

   1. Initial program 47.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 47.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 36.4%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \cdot \sqrt{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}}} \]

   5. Applied egg-rr 42.7%

      \[\leadsto \color{blue}{\frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}} \]
   6. Step-by-step derivation

      1. unpow2 42.7%

         \[\leadsto \color{blue}{{\left(\frac{\ell \cdot \sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right)}^{2}} \]

      2. associate-/l* 42.7%

         \[\leadsto {\color{blue}{\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}}}\right)}}^{2} \]

      3. associate-*r* 42.7%

         \[\leadsto {\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\color{blue}{\left(\tan k \cdot {t}^{3}\right) \cdot \sin k}}}}}\right)}^{2} \]

      4. *-commutative 42.7%

         \[\leadsto {\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\color{blue}{\left({t}^{3} \cdot \tan k\right)} \cdot \sin k}}}}\right)}^{2} \]

   7. Simplified 42.7%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{\frac{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\sqrt{\frac{2}{\left({t}^{3} \cdot \tan k\right) \cdot \sin k}}}}\right)}^{2}} \]

   8. Taylor expanded in k around inf 38.4%

      \[\leadsto {\color{blue}{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t}}\right)}}^{2} \]

   9. Taylor expanded in k around 0 51.3%

      \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}{{k}^{4} \cdot t}} \]
   10. Step-by-step derivation

       1. times-frac 51.4%

          \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{t}} \]

       2. unpow2 51.4%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}{t} \]

       3. rem-square-sqrt 51.4%

          \[\leadsto \frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{\color{blue}{2}}{t} \]

   11. Simplified 51.4%

       \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{2}{t}} \]

   if 2.45e-92 < t

   1. Initial program 66.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 56.6%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

   4. Taylor expanded in k around 0 51.4%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 51.4%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   6. Applied egg-rr 51.4%

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]
   7. Step-by-step derivation

      1. times-frac 56.5%

         \[\leadsto \color{blue}{\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{{t}^{3}}} \]

   8. Applied egg-rr 56.5%

      \[\leadsto \color{blue}{\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{{t}^{3}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.45 \cdot 10^{-92}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{{t}^{3}}\\ \end{array} \]

Alternative 26: 53.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \left(\left(\ell \cdot {k}^{-2}\right) \cdot \frac{\ell}{{t_m}^{3}}\right) \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* (* l (pow k -2.0)) (/ l (pow t_m 3.0)))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l * pow(k, -2.0)) * (l / pow(t_m, 3.0)));
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((l * (k ** (-2.0d0))) * (l / (t_m ** 3.0d0)))
end function

Java

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) {
	return t_s * ((l * Math.pow(k, -2.0)) * (l / Math.pow(t_m, 3.0)));
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * ((l * math.pow(k, -2.0)) * (l / math.pow(t_m, 3.0)))

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(l * (k ^ -2.0)) * Float64(l / (t_m ^ 3.0))))
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * ((l * (k ^ -2.0)) * (l / (t_m ^ 3.0)));
end

Wolfram

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_] := N[(t$95$s * N[(N[(l * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision] * N[(l / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.6%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

3. Simplified 47.8%

   \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

4. Taylor expanded in k around 0 49.3%

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
5. Step-by-step derivation

   1. unpow2 49.3%

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

6. Applied egg-rr 49.3%

   \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]
7. Step-by-step derivation

   1. times-frac 55.2%

      \[\leadsto \color{blue}{\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{{t}^{3}}} \]

8. Applied egg-rr 55.2%

   \[\leadsto \color{blue}{\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{{t}^{3}}} \]
9. Step-by-step derivation

   1. expm1-log1p-u 41.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\ell}{{k}^{2}}\right)\right)} \cdot \frac{\ell}{{t}^{3}} \]

   2. expm1-udef 40.2%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\ell}{{k}^{2}}\right)} - 1\right)} \cdot \frac{\ell}{{t}^{3}} \]

   3. div-inv 40.2%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\ell \cdot \frac{1}{{k}^{2}}}\right)} - 1\right) \cdot \frac{\ell}{{t}^{3}} \]

   4. pow-flip 40.2%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\ell \cdot \color{blue}{{k}^{\left(-2\right)}}\right)} - 1\right) \cdot \frac{\ell}{{t}^{3}} \]

   5. metadata-eval 40.2%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\ell \cdot {k}^{\color{blue}{-2}}\right)} - 1\right) \cdot \frac{\ell}{{t}^{3}} \]

10. Applied egg-rr 40.2%

    \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\ell \cdot {k}^{-2}\right)} - 1\right)} \cdot \frac{\ell}{{t}^{3}} \]
11. Step-by-step derivation

    1. expm1-def 41.1%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\ell \cdot {k}^{-2}\right)\right)} \cdot \frac{\ell}{{t}^{3}} \]

    2. expm1-log1p 55.1%

       \[\leadsto \color{blue}{\left(\ell \cdot {k}^{-2}\right)} \cdot \frac{\ell}{{t}^{3}} \]

12. Simplified 55.1%

    \[\leadsto \color{blue}{\left(\ell \cdot {k}^{-2}\right)} \cdot \frac{\ell}{{t}^{3}} \]

13. Final simplification 55.1%

    \[\leadsto \left(\ell \cdot {k}^{-2}\right) \cdot \frac{\ell}{{t}^{3}} \]

Alternative 27: 54.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \left(\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{{t_m}^{3}}\right) \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* (/ l (pow k 2.0)) (/ l (pow t_m 3.0)))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l / pow(k, 2.0)) * (l / pow(t_m, 3.0)));
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((l / (k ** 2.0d0)) * (l / (t_m ** 3.0d0)))
end function

Java

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) {
	return t_s * ((l / Math.pow(k, 2.0)) * (l / Math.pow(t_m, 3.0)));
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * ((l / math.pow(k, 2.0)) * (l / math.pow(t_m, 3.0)))

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(l / (k ^ 2.0)) * Float64(l / (t_m ^ 3.0))))
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * ((l / (k ^ 2.0)) * (l / (t_m ^ 3.0)));
end

Wolfram

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_] := N[(t$95$s * N[(N[(l / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(l / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.6%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

3. Simplified 47.8%

   \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

4. Taylor expanded in k around 0 49.3%

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
5. Step-by-step derivation

   1. unpow2 49.3%

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

6. Applied egg-rr 49.3%

   \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]
7. Step-by-step derivation

   1. times-frac 55.2%

      \[\leadsto \color{blue}{\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{{t}^{3}}} \]

8. Applied egg-rr 55.2%

   \[\leadsto \color{blue}{\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{{t}^{3}}} \]

9. Final simplification 55.2%

   \[\leadsto \frac{\ell}{{k}^{2}} \cdot \frac{\ell}{{t}^{3}} \]

Reproduce

herbie shell --seed 2023334 
(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))))