Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.1% → 83.0%

Time: 28.5s

Alternatives: 23

Speedup: 2.0×

• 28.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 83.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7 \cdot 10^{-93}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot {k}^{2}}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right)}\\ \end{array} \end{array} \]

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 7e-93)
    (/
     2.0
     (* (/ (* t_m (pow k 2.0)) (pow l 2.0)) (/ (pow (sin k) 2.0) (cos k))))
    (/
     2.0
     (*
      (pow (* (cbrt (sin k)) (/ t_m (pow (cbrt l) 2.0))) 3.0)
      (* (tan k) (+ 2.0 (pow (/ k t_m) 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 tmp;
	if (t_m <= 7e-93) {
		tmp = 2.0 / (((t_m * pow(k, 2.0)) / pow(l, 2.0)) * (pow(sin(k), 2.0) / cos(k)));
	} else {
		tmp = 2.0 / (pow((cbrt(sin(k)) * (t_m / pow(cbrt(l), 2.0))), 3.0) * (tan(k) * (2.0 + pow((k / t_m), 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 tmp;
	if (t_m <= 7e-93) {
		tmp = 2.0 / (((t_m * Math.pow(k, 2.0)) / Math.pow(l, 2.0)) * (Math.pow(Math.sin(k), 2.0) / Math.cos(k)));
	} else {
		tmp = 2.0 / (Math.pow((Math.cbrt(Math.sin(k)) * (t_m / Math.pow(Math.cbrt(l), 2.0))), 3.0) * (Math.tan(k) * (2.0 + Math.pow((k / t_m), 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)
	tmp = 0.0
	if (t_m <= 7e-93)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k ^ 2.0)) / (l ^ 2.0)) * Float64((sin(k) ^ 2.0) / cos(k))));
	else
		tmp = Float64(2.0 / Float64((Float64(cbrt(sin(k)) * Float64(t_m / (cbrt(l) ^ 2.0))) ^ 3.0) * Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 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_] := N[(t$95$s * If[LessEqual[t$95$m, 7e-93], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $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[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 7e-93

   1. Initial program 53.1%

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

      1. associate-*l* 53.6%

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

      2. sqr-neg 53.6%

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

      3. sqr-neg 53.6%

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

      4. associate-/r* 57.6%

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

      5. distribute-rgt-in 57.6%

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

      6. unpow2 57.6%

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

      7. times-frac 41.6%

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

      8. sqr-neg 41.6%

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

      9. times-frac 57.6%

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

      10. unpow2 57.6%

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

      11. distribute-rgt-in 57.6%

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

   3. Simplified 57.6%

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

   5. Taylor expanded in t around 0 71.3%

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

      1. associate-*r* 71.3%

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

      2. times-frac 71.4%

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

   7. Simplified 71.4%

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

   if 7e-93 < t

   1. Initial program 65.6%

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

      1. associate-*l* 65.6%

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

      2. sqr-neg 65.6%

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

      3. sqr-neg 65.6%

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

      4. associate-/r* 72.7%

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

      5. distribute-rgt-in 72.7%

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

      6. unpow2 72.7%

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

      7. times-frac 62.8%

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

      8. sqr-neg 62.8%

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

      9. times-frac 72.7%

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

      10. unpow2 72.7%

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

      11. distribute-rgt-in 72.7%

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

   3. Simplified 72.7%

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

      1. associate-/r* 65.6%

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

      2. add-cube-cbrt 65.5%

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

      3. pow3 65.5%

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

      4. associate-/r* 72.5%

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

      5. *-commutative 72.5%

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

      6. cbrt-prod 72.5%

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

      7. associate-/r* 65.4%

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

      8. cbrt-div 68.9%

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

      9. rem-cbrt-cube 76.5%

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

      10. cbrt-prod 88.8%

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

      11. pow2 88.8%

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

   6. Applied egg-rr 88.8%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7 \cdot 10^{-93}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{2}}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 72.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\frac{k}{t\_m}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\left(\tan k \cdot \left(\sin k \cdot \frac{{t\_m}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(1 + \left(t\_2 + 1\right)\right) \leq 5 \cdot 10^{+260}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + t\_2\right)\right) \cdot \frac{\sin k \cdot \frac{{t\_m}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{\sqrt[3]{k}}{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t\_m}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (/ k t_m) 2.0)))
   (*
    t_s
    (if (<=
         (*
          (* (tan k) (* (sin k) (/ (pow t_m 3.0) (* l l))))
          (+ 1.0 (+ t_2 1.0)))
         5e+260)
      (/ 2.0 (* (* (tan k) (+ 2.0 t_2)) (/ (* (sin k) (/ (pow t_m 3.0) l)) l)))
      (/
       2.0
       (* (pow (/ (cbrt k) (/ (pow (cbrt l) 2.0) t_m)) 3.0) (* 2.0 k)))))))

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 = pow((k / t_m), 2.0);
	double tmp;
	if (((tan(k) * (sin(k) * (pow(t_m, 3.0) / (l * l)))) * (1.0 + (t_2 + 1.0))) <= 5e+260) {
		tmp = 2.0 / ((tan(k) * (2.0 + t_2)) * ((sin(k) * (pow(t_m, 3.0) / l)) / l));
	} else {
		tmp = 2.0 / (pow((cbrt(k) / (pow(cbrt(l), 2.0) / t_m)), 3.0) * (2.0 * k));
	}
	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.pow((k / t_m), 2.0);
	double tmp;
	if (((Math.tan(k) * (Math.sin(k) * (Math.pow(t_m, 3.0) / (l * l)))) * (1.0 + (t_2 + 1.0))) <= 5e+260) {
		tmp = 2.0 / ((Math.tan(k) * (2.0 + t_2)) * ((Math.sin(k) * (Math.pow(t_m, 3.0) / l)) / l));
	} else {
		tmp = 2.0 / (Math.pow((Math.cbrt(k) / (Math.pow(Math.cbrt(l), 2.0) / t_m)), 3.0) * (2.0 * k));
	}
	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(k / t_m) ^ 2.0
	tmp = 0.0
	if (Float64(Float64(tan(k) * Float64(sin(k) * Float64((t_m ^ 3.0) / Float64(l * l)))) * Float64(1.0 + Float64(t_2 + 1.0))) <= 5e+260)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(2.0 + t_2)) * Float64(Float64(sin(k) * Float64((t_m ^ 3.0) / l)) / l)));
	else
		tmp = Float64(2.0 / Float64((Float64(cbrt(k) / Float64((cbrt(l) ^ 2.0) / t_m)) ^ 3.0) * Float64(2.0 * k)));
	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[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+260], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[Power[k, 1/3], $MachinePrecision] / N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)) < 4.9999999999999996e260

   1. Initial program 87.6%

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

      1. associate-*l* 88.4%

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

      2. sqr-neg 88.4%

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

      3. sqr-neg 88.4%

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

      4. associate-/r* 90.4%

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

      5. distribute-rgt-in 90.4%

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

      6. unpow2 90.4%

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

      7. times-frac 71.8%

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

      8. sqr-neg 71.8%

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

      9. times-frac 90.4%

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

      10. unpow2 90.4%

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

      11. distribute-rgt-in 90.4%

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

   3. Simplified 90.4%

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

      1. associate-*l/ 91.2%

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

   6. Applied egg-rr 91.2%

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

   if 4.9999999999999996e260 < (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))

   1. Initial program 28.8%

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

      1. *-commutative 28.8%

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

      2. sqr-neg 28.8%

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

      3. *-commutative 28.8%

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

      4. associate-*l* 28.8%

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

      5. *-commutative 28.8%

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

      6. sqr-neg 28.8%

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

   3. Simplified 28.8%

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

   5. Taylor expanded in k around 0 37.8%

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

      1. *-commutative 37.8%

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

   7. Simplified 37.8%

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

   8. Taylor expanded in k around 0 39.8%

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

      1. associate-/l* 37.6%

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

   10. Simplified 37.6%

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

       1. add-cube-cbrt 37.6%

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

       2. pow2 37.6%

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

       3. cbrt-div 37.6%

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

       4. unpow2 37.6%

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

       5. cbrt-prod 37.6%

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

       6. pow2 37.6%

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

       7. unpow3 37.6%

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

       8. add-cbrt-cube 37.6%

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

       9. cbrt-div 37.6%

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

       10. unpow2 37.6%

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

       11. cbrt-prod 45.5%

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

       12. pow2 45.5%

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

       13. unpow3 45.5%

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

       14. add-cbrt-cube 57.2%

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

   12. Applied egg-rr 57.2%

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

       1. pow-plus 57.2%

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

       2. metadata-eval 57.2%

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

   14. Simplified 57.2%

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

       1. add-cube-cbrt 57.2%

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

       2. pow3 57.2%

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

       3. cbrt-div 57.2%

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

       4. rem-cbrt-cube 60.2%

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

   16. Applied egg-rr 60.2%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \leq 5 \cdot 10^{+260}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \frac{\sin k \cdot \frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{\sqrt[3]{k}}{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.6 \cdot 10^{-93}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot {k}^{2}}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right) \cdot \frac{{\left(\sqrt[3]{\sin k} \cdot \frac{t\_m}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}}\\ \end{array} \end{array} \]

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 2.6e-93)
    (/
     2.0
     (* (/ (* t_m (pow k 2.0)) (pow l 2.0)) (/ (pow (sin k) 2.0) (cos k))))
    (/
     2.0
     (*
      (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))
      (/ (pow (* (cbrt (sin k)) (/ 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 <= 2.6e-93) {
		tmp = 2.0 / (((t_m * pow(k, 2.0)) / pow(l, 2.0)) * (pow(sin(k), 2.0) / cos(k)));
	} else {
		tmp = 2.0 / ((tan(k) * (2.0 + pow((k / t_m), 2.0))) * (pow((cbrt(sin(k)) * (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 <= 2.6e-93) {
		tmp = 2.0 / (((t_m * Math.pow(k, 2.0)) / Math.pow(l, 2.0)) * (Math.pow(Math.sin(k), 2.0) / Math.cos(k)));
	} 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.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 <= 2.6e-93)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k ^ 2.0)) / (l ^ 2.0)) * Float64((sin(k) ^ 2.0) / cos(k))));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * Float64((Float64(cbrt(sin(k)) * 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, 2.6e-93], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $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[(N[Power[N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.5999999999999998e-93

   1. Initial program 53.1%

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

      1. associate-*l* 53.6%

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

      2. sqr-neg 53.6%

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

      3. sqr-neg 53.6%

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

      4. associate-/r* 57.6%

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

      5. distribute-rgt-in 57.6%

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

      6. unpow2 57.6%

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

      7. times-frac 41.6%

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

      8. sqr-neg 41.6%

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

      9. times-frac 57.6%

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

      10. unpow2 57.6%

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

      11. distribute-rgt-in 57.6%

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

   3. Simplified 57.6%

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

   5. Taylor expanded in t around 0 71.3%

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

      1. associate-*r* 71.3%

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

      2. times-frac 71.4%

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

   7. Simplified 71.4%

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

   if 2.5999999999999998e-93 < t

   1. Initial program 65.6%

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

      1. associate-*l* 65.6%

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

      2. sqr-neg 65.6%

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

      3. sqr-neg 65.6%

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

      4. associate-/r* 72.7%

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

      5. distribute-rgt-in 72.7%

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

      6. unpow2 72.7%

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

      7. times-frac 62.8%

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

      8. sqr-neg 62.8%

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

      9. times-frac 72.7%

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

      10. unpow2 72.7%

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

      11. distribute-rgt-in 72.7%

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

   3. Simplified 72.7%

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

      1. associate-*l/ 75.1%

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

   6. Applied egg-rr 75.1%

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

      1. add-cube-cbrt 74.8%

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

      2. pow3 74.8%

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

      3. *-commutative 74.8%

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

      4. cbrt-prod 74.8%

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

      5. cbrt-div 75.8%

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

      6. rem-cbrt-cube 85.4%

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

   8. Applied egg-rr 85.4%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.6 \cdot 10^{-93}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{2}}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \frac{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.6 \cdot 10^{-93}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot {k}^{2}}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right) \cdot {\left(\frac{t\_m}{\sqrt[3]{\ell} \cdot \sqrt[3]{\frac{\ell}{\sin k}}}\right)}^{3}}\\ \end{array} \end{array} \]

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 3.6e-93)
    (/
     2.0
     (* (/ (* t_m (pow k 2.0)) (pow l 2.0)) (/ (pow (sin k) 2.0) (cos k))))
    (/
     2.0
     (*
      (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))
      (pow (/ t_m (* (cbrt l) (cbrt (/ l (sin k))))) 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 <= 3.6e-93) {
		tmp = 2.0 / (((t_m * pow(k, 2.0)) / pow(l, 2.0)) * (pow(sin(k), 2.0) / cos(k)));
	} else {
		tmp = 2.0 / ((tan(k) * (2.0 + pow((k / t_m), 2.0))) * pow((t_m / (cbrt(l) * cbrt((l / sin(k))))), 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 <= 3.6e-93) {
		tmp = 2.0 / (((t_m * Math.pow(k, 2.0)) / Math.pow(l, 2.0)) * (Math.pow(Math.sin(k), 2.0) / Math.cos(k)));
	} else {
		tmp = 2.0 / ((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0))) * Math.pow((t_m / (Math.cbrt(l) * Math.cbrt((l / Math.sin(k))))), 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 <= 3.6e-93)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k ^ 2.0)) / (l ^ 2.0)) * Float64((sin(k) ^ 2.0) / cos(k))));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * (Float64(t_m / Float64(cbrt(l) * cbrt(Float64(l / sin(k))))) ^ 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, 3.6e-93], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $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[(t$95$m / N[(N[Power[l, 1/3], $MachinePrecision] * N[Power[N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.6000000000000002e-93

   1. Initial program 53.1%

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

      1. associate-*l* 53.6%

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

      2. sqr-neg 53.6%

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

      3. sqr-neg 53.6%

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

      4. associate-/r* 57.6%

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

      5. distribute-rgt-in 57.6%

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

      6. unpow2 57.6%

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

      7. times-frac 41.6%

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

      8. sqr-neg 41.6%

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

      9. times-frac 57.6%

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

      10. unpow2 57.6%

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

      11. distribute-rgt-in 57.6%

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

   3. Simplified 57.6%

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

   5. Taylor expanded in t around 0 71.3%

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

      1. associate-*r* 71.3%

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

      2. times-frac 71.4%

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

   7. Simplified 71.4%

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

   if 3.6000000000000002e-93 < t

   1. Initial program 65.6%

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

      1. associate-*l* 65.6%

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

      2. sqr-neg 65.6%

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

      3. sqr-neg 65.6%

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

      4. associate-/r* 72.7%

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

      5. distribute-rgt-in 72.7%

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

      6. unpow2 72.7%

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

      7. times-frac 62.8%

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

      8. sqr-neg 62.8%

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

      9. times-frac 72.7%

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

      10. unpow2 72.7%

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

      11. distribute-rgt-in 72.7%

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

   3. Simplified 72.7%

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

      1. associate-/r* 65.6%

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

      2. add-cube-cbrt 65.5%

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

      3. pow3 65.5%

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

      4. associate-/r* 72.5%

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

      5. *-commutative 72.5%

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

      6. cbrt-prod 72.5%

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

      7. associate-/r* 65.4%

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

      8. cbrt-div 68.9%

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

      9. rem-cbrt-cube 76.5%

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

      10. cbrt-prod 88.8%

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

      11. pow2 88.8%

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

   6. Applied egg-rr 88.8%

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

      1. rem-cbrt-cube 75.8%

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

      2. unpow2 75.8%

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

      3. cbrt-prod 68.9%

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

      4. cbrt-div 65.4%

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

      5. cbrt-prod 65.5%

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

      6. associate-/r* 72.5%

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

      7. cube-mult 72.6%

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

      8. unpow2 72.6%

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

      9. associate-*r/ 77.4%

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

      10. /-rgt-identity 77.4%

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

      11. *-commutative 77.4%

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

      12. associate-*l/ 79.7%

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

      13. /-rgt-identity 79.7%

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

      14. associate-*r/ 74.9%

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

      15. unpow2 74.9%

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

      16. cube-mult 74.9%

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

   8. Applied egg-rr 84.3%

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

      1. associate-/l/ 84.3%

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

   10. Simplified 84.3%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.6 \cdot 10^{-93}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{2}}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(\frac{t}{\sqrt[3]{\ell} \cdot \sqrt[3]{\frac{\ell}{\sin k}}}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.7 \cdot 10^{-93}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t\_m \cdot {\sin k}^{2}\right)}\\ \mathbf{elif}\;t\_m \leq 6 \cdot 10^{+81}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right) \cdot \frac{\sin k \cdot \frac{{t\_m}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{\sqrt[3]{k}}{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t\_m}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \end{array} \]

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 2.7e-93)
    (*
     2.0
     (/ (* (pow l 2.0) (cos k)) (* (pow k 2.0) (* t_m (pow (sin k) 2.0)))))
    (if (<= t_m 6e+81)
      (/
       2.0
       (*
        (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))
        (/ (* (sin k) (/ (pow t_m 3.0) l)) l)))
      (/
       2.0
       (* (pow (/ (cbrt k) (/ (pow (cbrt l) 2.0) t_m)) 3.0) (* 2.0 k)))))))

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 <= 2.7e-93) {
		tmp = 2.0 * ((pow(l, 2.0) * cos(k)) / (pow(k, 2.0) * (t_m * pow(sin(k), 2.0))));
	} else if (t_m <= 6e+81) {
		tmp = 2.0 / ((tan(k) * (2.0 + pow((k / t_m), 2.0))) * ((sin(k) * (pow(t_m, 3.0) / l)) / l));
	} else {
		tmp = 2.0 / (pow((cbrt(k) / (pow(cbrt(l), 2.0) / t_m)), 3.0) * (2.0 * k));
	}
	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 <= 2.7e-93) {
		tmp = 2.0 * ((Math.pow(l, 2.0) * Math.cos(k)) / (Math.pow(k, 2.0) * (t_m * Math.pow(Math.sin(k), 2.0))));
	} else if (t_m <= 6e+81) {
		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 {
		tmp = 2.0 / (Math.pow((Math.cbrt(k) / (Math.pow(Math.cbrt(l), 2.0) / t_m)), 3.0) * (2.0 * k));
	}
	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 <= 2.7e-93)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k)) / Float64((k ^ 2.0) * Float64(t_m * (sin(k) ^ 2.0)))));
	elseif (t_m <= 6e+81)
		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)));
	else
		tmp = Float64(2.0 / Float64((Float64(cbrt(k) / Float64((cbrt(l) ^ 2.0) / t_m)) ^ 3.0) * Float64(2.0 * k)));
	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, 2.7e-93], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6e+81], 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], N[(2.0 / N[(N[Power[N[(N[Power[k, 1/3], $MachinePrecision] / N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.7000000000000001e-93

   1. Initial program 53.1%

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

      1. associate-/r* 52.6%

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

      2. sqr-neg 52.6%

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

      3. associate-*l* 49.4%

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

      4. sqr-neg 49.4%

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

      5. associate-/r* 53.4%

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

      6. associate-+l+ 53.4%

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

      7. unpow2 53.4%

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

      8. times-frac 37.4%

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

      9. sqr-neg 37.4%

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

      10. times-frac 53.4%

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

      11. unpow2 53.4%

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

   3. Simplified 53.4%

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

   5. Taylor expanded in t around 0 71.3%

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

   if 2.7000000000000001e-93 < t < 5.99999999999999995e81

   1. Initial program 67.7%

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

      1. associate-*l* 67.7%

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

      2. sqr-neg 67.7%

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

      3. sqr-neg 67.7%

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

      4. associate-/r* 77.6%

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

      5. distribute-rgt-in 77.6%

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

      6. unpow2 77.6%

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

      7. times-frac 74.9%

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

      8. sqr-neg 74.9%

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

      9. times-frac 77.6%

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

      10. unpow2 77.6%

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

      11. distribute-rgt-in 77.6%

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

   3. Simplified 77.6%

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

      1. associate-*l/ 83.1%

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

   6. Applied egg-rr 83.1%

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

   if 5.99999999999999995e81 < t

   1. Initial program 64.1%

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

      1. *-commutative 64.1%

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

      2. sqr-neg 64.1%

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

      3. *-commutative 64.1%

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

      4. associate-*l* 64.1%

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

      5. *-commutative 64.1%

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

      6. sqr-neg 64.1%

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

   3. Simplified 64.1%

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

   5. Taylor expanded in k around 0 64.1%

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

      1. *-commutative 64.1%

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

   7. Simplified 64.1%

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

   8. Taylor expanded in k around 0 66.1%

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

      1. associate-/l* 64.1%

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

   10. Simplified 64.1%

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

       1. add-cube-cbrt 64.1%

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

       2. pow2 64.1%

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

       3. cbrt-div 64.1%

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

       4. unpow2 64.1%

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

       5. cbrt-prod 64.1%

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

       6. pow2 64.1%

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

       7. unpow3 64.1%

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

       8. add-cbrt-cube 64.1%

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

       9. cbrt-div 64.1%

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

       10. unpow2 64.1%

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

       11. cbrt-prod 69.0%

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

       12. pow2 69.0%

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

       13. unpow3 69.0%

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

       14. add-cbrt-cube 87.7%

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

   12. Applied egg-rr 87.7%

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

       1. pow-plus 87.7%

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

       2. metadata-eval 87.7%

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

   14. Simplified 87.7%

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

       1. add-cube-cbrt 87.7%

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

       2. pow3 87.6%

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

       3. cbrt-div 87.8%

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

       4. rem-cbrt-cube 92.0%

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

   16. Applied egg-rr 92.0%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.7 \cdot 10^{-93}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+81}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \frac{\sin k \cdot \frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{\sqrt[3]{k}}{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.8 \cdot 10^{-93}:\\ \;\;\;\;\frac{2}{t\_m \cdot {k}^{2}} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}\\ \mathbf{elif}\;t\_m \leq 2.35 \cdot 10^{+82}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right) \cdot \frac{\sin k \cdot \frac{{t\_m}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{\sqrt[3]{k}}{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t\_m}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \end{array} \]

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 6.8e-93)
    (*
     (/ 2.0 (* t_m (pow k 2.0)))
     (/ (* (pow l 2.0) (cos k)) (pow (sin k) 2.0)))
    (if (<= t_m 2.35e+82)
      (/
       2.0
       (*
        (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))
        (/ (* (sin k) (/ (pow t_m 3.0) l)) l)))
      (/
       2.0
       (* (pow (/ (cbrt k) (/ (pow (cbrt l) 2.0) t_m)) 3.0) (* 2.0 k)))))))

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 <= 6.8e-93) {
		tmp = (2.0 / (t_m * pow(k, 2.0))) * ((pow(l, 2.0) * cos(k)) / pow(sin(k), 2.0));
	} else if (t_m <= 2.35e+82) {
		tmp = 2.0 / ((tan(k) * (2.0 + pow((k / t_m), 2.0))) * ((sin(k) * (pow(t_m, 3.0) / l)) / l));
	} else {
		tmp = 2.0 / (pow((cbrt(k) / (pow(cbrt(l), 2.0) / t_m)), 3.0) * (2.0 * k));
	}
	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 <= 6.8e-93) {
		tmp = (2.0 / (t_m * Math.pow(k, 2.0))) * ((Math.pow(l, 2.0) * Math.cos(k)) / Math.pow(Math.sin(k), 2.0));
	} else if (t_m <= 2.35e+82) {
		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 {
		tmp = 2.0 / (Math.pow((Math.cbrt(k) / (Math.pow(Math.cbrt(l), 2.0) / t_m)), 3.0) * (2.0 * k));
	}
	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 <= 6.8e-93)
		tmp = Float64(Float64(2.0 / Float64(t_m * (k ^ 2.0))) * Float64(Float64((l ^ 2.0) * cos(k)) / (sin(k) ^ 2.0)));
	elseif (t_m <= 2.35e+82)
		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)));
	else
		tmp = Float64(2.0 / Float64((Float64(cbrt(k) / Float64((cbrt(l) ^ 2.0) / t_m)) ^ 3.0) * Float64(2.0 * k)));
	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, 6.8e-93], N[(N[(2.0 / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.35e+82], 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], N[(2.0 / N[(N[Power[N[(N[Power[k, 1/3], $MachinePrecision] / N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 6.80000000000000002e-93

   1. Initial program 53.1%

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

      1. associate-*l* 53.6%

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

      2. sqr-neg 53.6%

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

      3. sqr-neg 53.6%

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

      4. associate-/r* 57.6%

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

      5. distribute-rgt-in 57.6%

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

      6. unpow2 57.6%

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

      7. times-frac 41.6%

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

      8. sqr-neg 41.6%

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

      9. times-frac 57.6%

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

      10. unpow2 57.6%

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

      11. distribute-rgt-in 57.6%

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

   3. Simplified 57.6%

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

      1. associate-/r* 53.6%

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

      2. add-cube-cbrt 53.5%

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

      3. pow3 53.5%

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

      4. associate-/r* 57.5%

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

      5. *-commutative 57.5%

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

      6. cbrt-prod 57.5%

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

      7. associate-/r* 53.5%

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

      8. cbrt-div 53.5%

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

      9. rem-cbrt-cube 63.2%

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

      10. cbrt-prod 69.8%

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

      11. pow2 69.8%

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

   6. Applied egg-rr 69.8%

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

   8. Applied egg-rr 66.6%

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

      1. cube-prod 63.6%

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

      2. rem-cube-cbrt 63.7%

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

   10. Simplified 63.7%

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

   11. Taylor expanded in t around 0 71.3%

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

       1. associate-*r/ 71.3%

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

       2. associate-*r* 71.3%

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

       3. times-frac 71.3%

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

   13. Simplified 71.3%

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

   if 6.80000000000000002e-93 < t < 2.35e82

   1. Initial program 67.7%

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

      1. associate-*l* 67.7%

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

      2. sqr-neg 67.7%

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

      3. sqr-neg 67.7%

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

      4. associate-/r* 77.6%

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

      5. distribute-rgt-in 77.6%

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

      6. unpow2 77.6%

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

      7. times-frac 74.9%

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

      8. sqr-neg 74.9%

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

      9. times-frac 77.6%

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

      10. unpow2 77.6%

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

      11. distribute-rgt-in 77.6%

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

   3. Simplified 77.6%

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

      1. associate-*l/ 83.1%

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

   6. Applied egg-rr 83.1%

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

   if 2.35e82 < t

   1. Initial program 64.1%

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

      1. *-commutative 64.1%

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

      2. sqr-neg 64.1%

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

      3. *-commutative 64.1%

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

      4. associate-*l* 64.1%

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

      5. *-commutative 64.1%

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

      6. sqr-neg 64.1%

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

   3. Simplified 64.1%

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

   5. Taylor expanded in k around 0 64.1%

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

      1. *-commutative 64.1%

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

   7. Simplified 64.1%

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

   8. Taylor expanded in k around 0 66.1%

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

      1. associate-/l* 64.1%

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

   10. Simplified 64.1%

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

       1. add-cube-cbrt 64.1%

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

       2. pow2 64.1%

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

       3. cbrt-div 64.1%

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

       4. unpow2 64.1%

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

       5. cbrt-prod 64.1%

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

       6. pow2 64.1%

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

       7. unpow3 64.1%

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

       8. add-cbrt-cube 64.1%

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

       9. cbrt-div 64.1%

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

       10. unpow2 64.1%

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

       11. cbrt-prod 69.0%

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

       12. pow2 69.0%

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

       13. unpow3 69.0%

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

       14. add-cbrt-cube 87.7%

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

   12. Applied egg-rr 87.7%

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

       1. pow-plus 87.7%

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

       2. metadata-eval 87.7%

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

   14. Simplified 87.7%

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

       1. add-cube-cbrt 87.7%

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

       2. pow3 87.6%

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

       3. cbrt-div 87.8%

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

       4. rem-cbrt-cube 92.0%

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

   16. Applied egg-rr 92.0%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.8 \cdot 10^{-93}:\\ \;\;\;\;\frac{2}{t \cdot {k}^{2}} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}\\ \mathbf{elif}\;t \leq 2.35 \cdot 10^{+82}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \frac{\sin k \cdot \frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{\sqrt[3]{k}}{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.7 \cdot 10^{-93}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot {k}^{2}}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}}\\ \mathbf{elif}\;t\_m \leq 2.55 \cdot 10^{+82}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right) \cdot \frac{\sin k \cdot \frac{{t\_m}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{\sqrt[3]{k}}{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t\_m}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \end{array} \]

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 3.7e-93)
    (/
     2.0
     (* (/ (* t_m (pow k 2.0)) (pow l 2.0)) (/ (pow (sin k) 2.0) (cos k))))
    (if (<= t_m 2.55e+82)
      (/
       2.0
       (*
        (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))
        (/ (* (sin k) (/ (pow t_m 3.0) l)) l)))
      (/
       2.0
       (* (pow (/ (cbrt k) (/ (pow (cbrt l) 2.0) t_m)) 3.0) (* 2.0 k)))))))

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-93) {
		tmp = 2.0 / (((t_m * pow(k, 2.0)) / pow(l, 2.0)) * (pow(sin(k), 2.0) / cos(k)));
	} else if (t_m <= 2.55e+82) {
		tmp = 2.0 / ((tan(k) * (2.0 + pow((k / t_m), 2.0))) * ((sin(k) * (pow(t_m, 3.0) / l)) / l));
	} else {
		tmp = 2.0 / (pow((cbrt(k) / (pow(cbrt(l), 2.0) / t_m)), 3.0) * (2.0 * k));
	}
	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 <= 3.7e-93) {
		tmp = 2.0 / (((t_m * Math.pow(k, 2.0)) / Math.pow(l, 2.0)) * (Math.pow(Math.sin(k), 2.0) / Math.cos(k)));
	} else if (t_m <= 2.55e+82) {
		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 {
		tmp = 2.0 / (Math.pow((Math.cbrt(k) / (Math.pow(Math.cbrt(l), 2.0) / t_m)), 3.0) * (2.0 * k));
	}
	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-93)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k ^ 2.0)) / (l ^ 2.0)) * Float64((sin(k) ^ 2.0) / cos(k))));
	elseif (t_m <= 2.55e+82)
		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)));
	else
		tmp = Float64(2.0 / Float64((Float64(cbrt(k) / Float64((cbrt(l) ^ 2.0) / t_m)) ^ 3.0) * Float64(2.0 * k)));
	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, 3.7e-93], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.55e+82], 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], N[(2.0 / N[(N[Power[N[(N[Power[k, 1/3], $MachinePrecision] / N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 3.70000000000000002e-93

   1. Initial program 53.1%

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

      1. associate-*l* 53.6%

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

      2. sqr-neg 53.6%

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

      3. sqr-neg 53.6%

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

      4. associate-/r* 57.6%

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

      5. distribute-rgt-in 57.6%

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

      6. unpow2 57.6%

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

      7. times-frac 41.6%

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

      8. sqr-neg 41.6%

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

      9. times-frac 57.6%

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

      10. unpow2 57.6%

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

      11. distribute-rgt-in 57.6%

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

   3. Simplified 57.6%

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

   5. Taylor expanded in t around 0 71.3%

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

      1. associate-*r* 71.3%

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

      2. times-frac 71.4%

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

   7. Simplified 71.4%

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

   if 3.70000000000000002e-93 < t < 2.5500000000000001e82

   1. Initial program 67.7%

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

      1. associate-*l* 67.7%

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

      2. sqr-neg 67.7%

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

      3. sqr-neg 67.7%

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

      4. associate-/r* 77.6%

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

      5. distribute-rgt-in 77.6%

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

      6. unpow2 77.6%

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

      7. times-frac 74.9%

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

      8. sqr-neg 74.9%

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

      9. times-frac 77.6%

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

      10. unpow2 77.6%

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

      11. distribute-rgt-in 77.6%

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

   3. Simplified 77.6%

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

      1. associate-*l/ 83.1%

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

   6. Applied egg-rr 83.1%

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

   if 2.5500000000000001e82 < t

   1. Initial program 64.1%

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

      1. *-commutative 64.1%

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

      2. sqr-neg 64.1%

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

      3. *-commutative 64.1%

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

      4. associate-*l* 64.1%

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

      5. *-commutative 64.1%

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

      6. sqr-neg 64.1%

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

   3. Simplified 64.1%

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

   5. Taylor expanded in k around 0 64.1%

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

      1. *-commutative 64.1%

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

   7. Simplified 64.1%

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

   8. Taylor expanded in k around 0 66.1%

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

      1. associate-/l* 64.1%

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

   10. Simplified 64.1%

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

       1. add-cube-cbrt 64.1%

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

       2. pow2 64.1%

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

       3. cbrt-div 64.1%

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

       4. unpow2 64.1%

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

       5. cbrt-prod 64.1%

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

       6. pow2 64.1%

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

       7. unpow3 64.1%

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

       8. add-cbrt-cube 64.1%

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

       9. cbrt-div 64.1%

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

       10. unpow2 64.1%

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

       11. cbrt-prod 69.0%

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

       12. pow2 69.0%

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

       13. unpow3 69.0%

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

       14. add-cbrt-cube 87.7%

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

   12. Applied egg-rr 87.7%

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

       1. pow-plus 87.7%

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

       2. metadata-eval 87.7%

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

   14. Simplified 87.7%

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

       1. add-cube-cbrt 87.7%

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

       2. pow3 87.6%

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

       3. cbrt-div 87.8%

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

       4. rem-cbrt-cube 92.0%

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

   16. Applied egg-rr 92.0%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.7 \cdot 10^{-93}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{2}}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}}\\ \mathbf{elif}\;t \leq 2.55 \cdot 10^{+82}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \frac{\sin k \cdot \frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{\sqrt[3]{k}}{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sin k \cdot \frac{\frac{{t\_m}^{3}}{\ell}}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.2 \cdot 10^{-240}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot t\_2}{\frac{1}{\tan k}}}\\ \mathbf{elif}\;t\_m \leq 1.35 \cdot 10^{-108}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left(t\_m \cdot \sqrt[3]{\frac{k}{{\ell}^{2}}}\right)}^{3}}\\ \mathbf{elif}\;t\_m \leq 2.3 \cdot 10^{+82}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right) \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{\sqrt[3]{k}}{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t\_m}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (sin k) (/ (/ (pow t_m 3.0) l) l))))
   (*
    t_s
    (if (<= t_m 4.2e-240)
      (/ 2.0 (/ (* 2.0 t_2) (/ 1.0 (tan k))))
      (if (<= t_m 1.35e-108)
        (/ 2.0 (* (* 2.0 k) (pow (* t_m (cbrt (/ k (pow l 2.0)))) 3.0)))
        (if (<= t_m 2.3e+82)
          (/ 2.0 (* (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0))) t_2))
          (/
           2.0
           (*
            (pow (/ (cbrt k) (/ (pow (cbrt l) 2.0) t_m)) 3.0)
            (* 2.0 k)))))))))

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 = sin(k) * ((pow(t_m, 3.0) / l) / l);
	double tmp;
	if (t_m <= 4.2e-240) {
		tmp = 2.0 / ((2.0 * t_2) / (1.0 / tan(k)));
	} else if (t_m <= 1.35e-108) {
		tmp = 2.0 / ((2.0 * k) * pow((t_m * cbrt((k / pow(l, 2.0)))), 3.0));
	} else if (t_m <= 2.3e+82) {
		tmp = 2.0 / ((tan(k) * (2.0 + pow((k / t_m), 2.0))) * t_2);
	} else {
		tmp = 2.0 / (pow((cbrt(k) / (pow(cbrt(l), 2.0) / t_m)), 3.0) * (2.0 * k));
	}
	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.sin(k) * ((Math.pow(t_m, 3.0) / l) / l);
	double tmp;
	if (t_m <= 4.2e-240) {
		tmp = 2.0 / ((2.0 * t_2) / (1.0 / Math.tan(k)));
	} else if (t_m <= 1.35e-108) {
		tmp = 2.0 / ((2.0 * k) * Math.pow((t_m * Math.cbrt((k / Math.pow(l, 2.0)))), 3.0));
	} else if (t_m <= 2.3e+82) {
		tmp = 2.0 / ((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0))) * t_2);
	} else {
		tmp = 2.0 / (Math.pow((Math.cbrt(k) / (Math.pow(Math.cbrt(l), 2.0) / t_m)), 3.0) * (2.0 * k));
	}
	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(sin(k) * Float64(Float64((t_m ^ 3.0) / l) / l))
	tmp = 0.0
	if (t_m <= 4.2e-240)
		tmp = Float64(2.0 / Float64(Float64(2.0 * t_2) / Float64(1.0 / tan(k))));
	elseif (t_m <= 1.35e-108)
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * (Float64(t_m * cbrt(Float64(k / (l ^ 2.0)))) ^ 3.0)));
	elseif (t_m <= 2.3e+82)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * t_2));
	else
		tmp = Float64(2.0 / Float64((Float64(cbrt(k) / Float64((cbrt(l) ^ 2.0) / t_m)) ^ 3.0) * Float64(2.0 * k)));
	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[Sin[k], $MachinePrecision] * N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4.2e-240], N[(2.0 / N[(N[(2.0 * t$95$2), $MachinePrecision] / N[(1.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.35e-108], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[Power[N[(t$95$m * N[Power[N[(k / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.3e+82], 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] * t$95$2), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[Power[k, 1/3], $MachinePrecision] / N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 4.19999999999999987e-240

   1. Initial program 55.3%

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

      1. associate-*l* 56.0%

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

      2. sqr-neg 56.0%

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

      3. sqr-neg 56.0%

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

      4. associate-/r* 61.0%

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

      5. distribute-rgt-in 61.0%

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

      6. unpow2 61.0%

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

      7. times-frac 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 61.0%

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

      10. unpow2 61.0%

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

      11. distribute-rgt-in 61.0%

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

   3. Simplified 61.0%

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

      1. cube-mult 61.0%

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

      2. *-un-lft-identity 61.0%

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

      3. times-frac 68.8%

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

      4. pow2 68.8%

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

   6. Applied egg-rr 68.8%

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

   7. Taylor expanded in t around inf 63.0%

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

      1. associate-*r/ 63.0%

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

      2. associate-/l* 63.0%

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

   9. Simplified 63.0%

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

       1. associate-*r/ 63.0%

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

       2. *-commutative 63.0%

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

       3. frac-times 60.2%

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

       4. unpow2 60.2%

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

       5. cube-mult 60.2%

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

       6. *-un-lft-identity 60.2%

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

       7. clear-num 60.2%

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

       8. tan-quot 60.2%

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

   11. Applied egg-rr 60.2%

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

   if 4.19999999999999987e-240 < t < 1.35000000000000002e-108

   1. Initial program 35.9%

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

      1. *-commutative 35.9%

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

      2. sqr-neg 35.9%

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

      3. *-commutative 35.9%

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

      4. associate-*l* 35.9%

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

      5. *-commutative 35.9%

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

      6. sqr-neg 35.9%

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

   3. Simplified 35.9%

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

   5. Taylor expanded in k around 0 51.2%

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

      1. *-commutative 51.2%

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

   7. Simplified 51.2%

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

   8. Taylor expanded in k around 0 47.6%

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

      1. associate-/l* 47.6%

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

   10. Simplified 47.6%

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

       1. add-cube-cbrt 47.6%

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

       2. pow3 47.6%

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

       3. associate-/r/ 47.5%

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

       4. cbrt-prod 47.5%

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

       5. unpow3 47.5%

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

       6. add-cbrt-cube 62.0%

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

   12. Applied egg-rr 62.0%

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

   if 1.35000000000000002e-108 < t < 2.29999999999999988e82

   1. Initial program 69.0%

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

      1. associate-*l* 69.0%

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

      2. sqr-neg 69.0%

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

      3. sqr-neg 69.0%

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

      4. associate-/r* 77.0%

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

      5. distribute-rgt-in 77.0%

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

      6. unpow2 77.0%

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

      7. times-frac 74.8%

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

      8. sqr-neg 74.8%

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

      9. times-frac 77.0%

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

      10. unpow2 77.0%

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

      11. distribute-rgt-in 77.0%

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

   3. Simplified 77.0%

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

   if 2.29999999999999988e82 < t

   1. Initial program 64.1%

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

      1. *-commutative 64.1%

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

      2. sqr-neg 64.1%

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

      3. *-commutative 64.1%

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

      4. associate-*l* 64.1%

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

      5. *-commutative 64.1%

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

      6. sqr-neg 64.1%

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

   3. Simplified 64.1%

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

   5. Taylor expanded in k around 0 64.1%

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

      1. *-commutative 64.1%

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

   7. Simplified 64.1%

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

   8. Taylor expanded in k around 0 66.1%

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

      1. associate-/l* 64.1%

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

   10. Simplified 64.1%

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

       1. add-cube-cbrt 64.1%

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

       2. pow2 64.1%

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

       3. cbrt-div 64.1%

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

       4. unpow2 64.1%

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

       5. cbrt-prod 64.1%

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

       6. pow2 64.1%

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

       7. unpow3 64.1%

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

       8. add-cbrt-cube 64.1%

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

       9. cbrt-div 64.1%

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

       10. unpow2 64.1%

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

       11. cbrt-prod 69.0%

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

       12. pow2 69.0%

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

       13. unpow3 69.0%

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

       14. add-cbrt-cube 87.7%

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

   12. Applied egg-rr 87.7%

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

       1. pow-plus 87.7%

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

       2. metadata-eval 87.7%

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

   14. Simplified 87.7%

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

       1. add-cube-cbrt 87.7%

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

       2. pow3 87.6%

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

       3. cbrt-div 87.8%

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

       4. rem-cbrt-cube 92.0%

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

   16. Applied egg-rr 92.0%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{-240}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot \left(\sin k \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}\right)}{\frac{1}{\tan k}}}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{-108}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left(t \cdot \sqrt[3]{\frac{k}{{\ell}^{2}}}\right)}^{3}}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{+82}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\sin k \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{\sqrt[3]{k}}{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{2}{{\left(\frac{\sqrt[3]{k}}{t\_2}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \mathbf{elif}\;k \leq 2.5 \cdot 10^{+164}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot \left(\sin k \cdot \frac{\frac{{t\_m}^{3}}{\ell}}{\ell}\right)}{\frac{1}{\tan k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{\sqrt[3]{k \cdot \left(2 \cdot k\right)}}{t\_2}\right)}^{3}}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ (pow (cbrt l) 2.0) t_m)))
   (*
    t_s
    (if (<= k 3.5e-12)
      (/ 2.0 (* (pow (/ (cbrt k) t_2) 3.0) (* 2.0 k)))
      (if (<= k 2.5e+164)
        (/
         2.0
         (/ (* 2.0 (* (sin k) (/ (/ (pow t_m 3.0) l) l))) (/ 1.0 (tan k))))
        (/ 2.0 (pow (/ (cbrt (* k (* 2.0 k))) t_2) 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 = pow(cbrt(l), 2.0) / t_m;
	double tmp;
	if (k <= 3.5e-12) {
		tmp = 2.0 / (pow((cbrt(k) / t_2), 3.0) * (2.0 * k));
	} else if (k <= 2.5e+164) {
		tmp = 2.0 / ((2.0 * (sin(k) * ((pow(t_m, 3.0) / l) / l))) / (1.0 / tan(k)));
	} else {
		tmp = 2.0 / pow((cbrt((k * (2.0 * k))) / t_2), 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.pow(Math.cbrt(l), 2.0) / t_m;
	double tmp;
	if (k <= 3.5e-12) {
		tmp = 2.0 / (Math.pow((Math.cbrt(k) / t_2), 3.0) * (2.0 * k));
	} else if (k <= 2.5e+164) {
		tmp = 2.0 / ((2.0 * (Math.sin(k) * ((Math.pow(t_m, 3.0) / l) / l))) / (1.0 / Math.tan(k)));
	} else {
		tmp = 2.0 / Math.pow((Math.cbrt((k * (2.0 * k))) / t_2), 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((cbrt(l) ^ 2.0) / t_m)
	tmp = 0.0
	if (k <= 3.5e-12)
		tmp = Float64(2.0 / Float64((Float64(cbrt(k) / t_2) ^ 3.0) * Float64(2.0 * k)));
	elseif (k <= 2.5e+164)
		tmp = Float64(2.0 / Float64(Float64(2.0 * Float64(sin(k) * Float64(Float64((t_m ^ 3.0) / l) / l))) / Float64(1.0 / tan(k))));
	else
		tmp = Float64(2.0 / (Float64(cbrt(Float64(k * Float64(2.0 * k))) / t_2) ^ 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[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 3.5e-12], N[(2.0 / N[(N[Power[N[(N[Power[k, 1/3], $MachinePrecision] / t$95$2), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.5e+164], N[(2.0 / N[(N[(2.0 * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[Power[N[(k * N[(2.0 * k), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / t$95$2), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 3.5e-12

   1. Initial program 60.1%

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

      1. *-commutative 60.1%

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

      2. sqr-neg 60.1%

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

      3. *-commutative 60.1%

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

      4. associate-*l* 60.6%

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

      5. *-commutative 60.6%

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

      6. sqr-neg 60.6%

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

   3. Simplified 60.6%

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

   5. Taylor expanded in k around 0 59.6%

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

      1. *-commutative 59.6%

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

   7. Simplified 59.6%

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

   8. Taylor expanded in k around 0 61.1%

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

      1. associate-/l* 60.0%

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

   10. Simplified 60.0%

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

       1. add-cube-cbrt 59.9%

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

       2. pow2 59.9%

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

       3. cbrt-div 59.9%

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

       4. unpow2 59.9%

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

       5. cbrt-prod 59.9%

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

       6. pow2 59.9%

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

       7. unpow3 59.9%

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

       8. add-cbrt-cube 59.9%

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

       9. cbrt-div 59.9%

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

       10. unpow2 59.9%

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

       11. cbrt-prod 64.6%

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

       12. pow2 64.6%

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

       13. unpow3 64.6%

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

       14. add-cbrt-cube 74.5%

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

   12. Applied egg-rr 74.5%

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

       1. pow-plus 74.5%

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

       2. metadata-eval 74.5%

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

   14. Simplified 74.5%

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

       1. add-cube-cbrt 74.5%

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

       2. pow3 74.5%

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

       3. cbrt-div 74.4%

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

       4. rem-cbrt-cube 77.7%

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

   16. Applied egg-rr 77.7%

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

   if 3.5e-12 < k < 2.49999999999999975e164

   1. Initial program 47.2%

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

      1. associate-*l* 47.2%

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

      2. sqr-neg 47.2%

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

      3. sqr-neg 47.2%

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

      4. associate-/r* 49.2%

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

      5. distribute-rgt-in 49.2%

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

      6. unpow2 49.2%

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

      7. times-frac 44.5%

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

      8. sqr-neg 44.5%

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

      9. times-frac 49.2%

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

      10. unpow2 49.2%

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

      11. distribute-rgt-in 49.2%

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

   3. Simplified 49.2%

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

      1. cube-mult 49.2%

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

      2. *-un-lft-identity 49.2%

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

      3. times-frac 49.2%

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

      4. pow2 49.2%

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

   6. Applied egg-rr 49.2%

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

   7. Taylor expanded in t around inf 63.5%

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

      1. associate-*r/ 63.5%

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

      2. associate-/l* 63.5%

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

   9. Simplified 63.5%

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

       1. associate-*r/ 63.5%

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

       2. *-commutative 63.5%

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

       3. frac-times 63.5%

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

       4. unpow2 63.5%

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

       5. cube-mult 63.5%

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

       6. *-un-lft-identity 63.5%

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

       7. clear-num 63.5%

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

       8. tan-quot 63.5%

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

   11. Applied egg-rr 63.5%

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

   if 2.49999999999999975e164 < k

   1. Initial program 51.7%

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

      1. *-commutative 51.7%

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

      2. sqr-neg 51.7%

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

      3. *-commutative 51.7%

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

      4. associate-*l* 51.7%

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

      5. *-commutative 51.7%

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

      6. sqr-neg 51.7%

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

   3. Simplified 51.7%

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

   5. Taylor expanded in k around 0 42.8%

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

      1. *-commutative 42.8%

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

   7. Simplified 42.8%

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

   8. Taylor expanded in k around 0 52.3%

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

      1. associate-/l* 52.2%

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

   10. Simplified 52.2%

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

       1. add-cube-cbrt 52.2%

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

       2. pow3 52.2%

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

       3. associate-*l/ 51.7%

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

       4. cbrt-div 51.7%

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

       5. cbrt-div 52.3%

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

       6. unpow2 52.3%

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

       7. cbrt-prod 59.3%

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

       8. pow2 59.3%

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

       9. unpow3 59.3%

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

       10. add-cbrt-cube 74.0%

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

   12. Applied egg-rr 74.0%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{2}{{\left(\frac{\sqrt[3]{k}}{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \mathbf{elif}\;k \leq 2.5 \cdot 10^{+164}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot \left(\sin k \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}\right)}{\frac{1}{\tan k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{\sqrt[3]{k \cdot \left(2 \cdot k\right)}}{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{2}{{\left(\frac{\sqrt[3]{k}}{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t\_m}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \mathbf{elif}\;k \leq 4.7 \cdot 10^{+112}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot \left(\sin k \cdot \frac{\frac{{t\_m}^{3}}{\ell}}{\ell}\right)}{\frac{1}{\tan k}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{1}{\sqrt[3]{k}}}{t\_m \cdot \sqrt[3]{k \cdot {\ell}^{-2}}}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 2.5e-12)
    (/ 2.0 (* (pow (/ (cbrt k) (/ (pow (cbrt l) 2.0) t_m)) 3.0) (* 2.0 k)))
    (if (<= k 4.7e+112)
      (/ 2.0 (/ (* 2.0 (* (sin k) (/ (/ (pow t_m 3.0) l) l))) (/ 1.0 (tan k))))
      (pow (/ (/ 1.0 (cbrt k)) (* t_m (cbrt (* k (pow 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 (k <= 2.5e-12) {
		tmp = 2.0 / (pow((cbrt(k) / (pow(cbrt(l), 2.0) / t_m)), 3.0) * (2.0 * k));
	} else if (k <= 4.7e+112) {
		tmp = 2.0 / ((2.0 * (sin(k) * ((pow(t_m, 3.0) / l) / l))) / (1.0 / tan(k)));
	} else {
		tmp = pow(((1.0 / cbrt(k)) / (t_m * cbrt((k * pow(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 (k <= 2.5e-12) {
		tmp = 2.0 / (Math.pow((Math.cbrt(k) / (Math.pow(Math.cbrt(l), 2.0) / t_m)), 3.0) * (2.0 * k));
	} else if (k <= 4.7e+112) {
		tmp = 2.0 / ((2.0 * (Math.sin(k) * ((Math.pow(t_m, 3.0) / l) / l))) / (1.0 / Math.tan(k)));
	} else {
		tmp = Math.pow(((1.0 / Math.cbrt(k)) / (t_m * Math.cbrt((k * Math.pow(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 (k <= 2.5e-12)
		tmp = Float64(2.0 / Float64((Float64(cbrt(k) / Float64((cbrt(l) ^ 2.0) / t_m)) ^ 3.0) * Float64(2.0 * k)));
	elseif (k <= 4.7e+112)
		tmp = Float64(2.0 / Float64(Float64(2.0 * Float64(sin(k) * Float64(Float64((t_m ^ 3.0) / l) / l))) / Float64(1.0 / tan(k))));
	else
		tmp = Float64(Float64(1.0 / cbrt(k)) / Float64(t_m * cbrt(Float64(k * (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[k, 2.5e-12], N[(2.0 / N[(N[Power[N[(N[Power[k, 1/3], $MachinePrecision] / N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 4.7e+112], N[(2.0 / N[(N[(2.0 * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(1.0 / N[Power[k, 1/3], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[N[(k * N[Power[l, -2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 2.49999999999999985e-12

   1. Initial program 60.1%

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

      1. *-commutative 60.1%

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

      2. sqr-neg 60.1%

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

      3. *-commutative 60.1%

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

      4. associate-*l* 60.6%

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

      5. *-commutative 60.6%

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

      6. sqr-neg 60.6%

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

   3. Simplified 60.6%

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

   5. Taylor expanded in k around 0 59.6%

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

      1. *-commutative 59.6%

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

   7. Simplified 59.6%

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

   8. Taylor expanded in k around 0 61.1%

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

      1. associate-/l* 60.0%

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

   10. Simplified 60.0%

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

       1. add-cube-cbrt 59.9%

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

       2. pow2 59.9%

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

       3. cbrt-div 59.9%

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

       4. unpow2 59.9%

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

       5. cbrt-prod 59.9%

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

       6. pow2 59.9%

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

       7. unpow3 59.9%

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

       8. add-cbrt-cube 59.9%

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

       9. cbrt-div 59.9%

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

       10. unpow2 59.9%

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

       11. cbrt-prod 64.6%

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

       12. pow2 64.6%

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

       13. unpow3 64.6%

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

       14. add-cbrt-cube 74.5%

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

   12. Applied egg-rr 74.5%

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

       1. pow-plus 74.5%

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

       2. metadata-eval 74.5%

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

   14. Simplified 74.5%

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

       1. add-cube-cbrt 74.5%

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

       2. pow3 74.5%

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

       3. cbrt-div 74.4%

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

       4. rem-cbrt-cube 77.7%

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

   16. Applied egg-rr 77.7%

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

   if 2.49999999999999985e-12 < k < 4.69999999999999997e112

   1. Initial program 43.6%

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

      1. associate-*l* 43.6%

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

      2. sqr-neg 43.6%

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

      3. sqr-neg 43.6%

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

      4. associate-/r* 46.3%

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

      5. distribute-rgt-in 46.3%

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

      6. unpow2 46.3%

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

      7. times-frac 46.3%

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

      8. sqr-neg 46.3%

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

      9. times-frac 46.3%

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

      10. unpow2 46.3%

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

      11. distribute-rgt-in 46.3%

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

   3. Simplified 46.3%

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

      1. cube-mult 46.3%

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

      2. *-un-lft-identity 46.3%

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

      3. times-frac 46.3%

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

      4. pow2 46.3%

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

   6. Applied egg-rr 46.3%

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

   7. Taylor expanded in t around inf 64.1%

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

      1. associate-*r/ 64.1%

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

      2. associate-/l* 64.1%

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

   9. Simplified 64.1%

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

       1. associate-*r/ 64.1%

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

       2. *-commutative 64.1%

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

       3. frac-times 64.1%

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

       4. unpow2 64.1%

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

       5. cube-mult 64.1%

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

       6. *-un-lft-identity 64.1%

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

       7. clear-num 64.1%

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

       8. tan-quot 64.1%

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

   11. Applied egg-rr 64.1%

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

   if 4.69999999999999997e112 < k

   1. Initial program 53.6%

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

      1. *-commutative 53.6%

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

      2. sqr-neg 53.6%

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

      3. *-commutative 53.6%

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

      4. associate-*l* 53.6%

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

      5. *-commutative 53.6%

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

      6. sqr-neg 53.6%

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

   3. Simplified 53.6%

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

   5. Taylor expanded in k around 0 44.8%

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

      1. *-commutative 44.8%

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

   7. Simplified 44.8%

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

   8. Taylor expanded in k around 0 51.8%

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

      1. associate-/l* 51.7%

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

   10. Simplified 51.7%

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

       1. div-inv 51.7%

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

       2. associate-/r/ 51.6%

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

   12. Applied egg-rr 51.6%

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

       1. associate-*r/ 51.6%

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

       2. metadata-eval 51.6%

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

       3. *-commutative 51.6%

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

       4. associate-/r* 51.6%

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

       5. *-commutative 51.6%

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

       6. associate-/r* 51.6%

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

       7. metadata-eval 51.6%

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

       8. *-commutative 51.6%

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

   14. Simplified 51.6%

       \[\leadsto \color{blue}{\frac{\frac{1}{k}}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \]
   15. Step-by-step derivation

       1. add-cube-cbrt 51.6%

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

       2. pow3 51.6%

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

       3. cbrt-div 51.6%

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

       4. cbrt-div 51.6%

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

       5. metadata-eval 51.6%

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

       6. cbrt-prod 51.7%

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

       7. rem-cbrt-cube 65.5%

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

       8. div-inv 65.5%

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

       9. pow-flip 65.5%

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

       10. metadata-eval 65.5%

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

   16. Applied egg-rr 65.5%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{2}{{\left(\frac{\sqrt[3]{k}}{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \mathbf{elif}\;k \leq 4.7 \cdot 10^{+112}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot \left(\sin k \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}\right)}{\frac{1}{\tan k}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{1}{\sqrt[3]{k}}}{t \cdot \sqrt[3]{k \cdot {\ell}^{-2}}}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 35.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 1.6 \cdot 10^{+246}:\\ \;\;\;\;{\left(\frac{{k}^{-0.5}}{{t\_m}^{1.5} \cdot \frac{\sqrt{k}}{\ell}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot {\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}\right)}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= l 1.6e+246)
    (pow (/ (pow k -0.5) (* (pow t_m 1.5) (/ (sqrt k) l))) 2.0)
    (/ 2.0 (* (* 2.0 k) (* (sin k) (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 tmp;
	if (l <= 1.6e+246) {
		tmp = pow((pow(k, -0.5) / (pow(t_m, 1.5) * (sqrt(k) / l))), 2.0);
	} else {
		tmp = 2.0 / ((2.0 * k) * (sin(k) * 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 tmp;
	if (l <= 1.6e+246) {
		tmp = Math.pow((Math.pow(k, -0.5) / (Math.pow(t_m, 1.5) * (Math.sqrt(k) / l))), 2.0);
	} else {
		tmp = 2.0 / ((2.0 * k) * (Math.sin(k) * 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)
	tmp = 0.0
	if (l <= 1.6e+246)
		tmp = Float64((k ^ -0.5) / Float64((t_m ^ 1.5) * Float64(sqrt(k) / l))) ^ 2.0;
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * Float64(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[l, 1.6e+246], N[Power[N[(N[Power[k, -0.5], $MachinePrecision] / N[(N[Power[t$95$m, 1.5], $MachinePrecision] * N[(N[Sqrt[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * 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 2 regimes

2. if l < 1.60000000000000007e246

   1. Initial program 58.4%

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

      1. *-commutative 58.4%

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

      2. sqr-neg 58.4%

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

      3. *-commutative 58.4%

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

      4. associate-*l* 58.8%

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

      5. *-commutative 58.8%

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

      6. sqr-neg 58.8%

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

   3. Simplified 58.8%

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

   5. Taylor expanded in k around 0 56.3%

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

      1. *-commutative 56.3%

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

   7. Simplified 56.3%

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

   8. Taylor expanded in k around 0 59.2%

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

      1. associate-/l* 58.2%

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

   10. Simplified 58.2%

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

       1. div-inv 58.2%

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

       2. associate-/r/ 58.3%

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

   12. Applied egg-rr 58.3%

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

       1. associate-*r/ 58.3%

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

       2. metadata-eval 58.3%

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

       3. *-commutative 58.3%

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

       4. associate-/r* 58.3%

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

       5. *-commutative 58.3%

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

       6. associate-/r* 58.3%

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

       7. metadata-eval 58.3%

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

       8. *-commutative 58.3%

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

   14. Simplified 58.3%

       \[\leadsto \color{blue}{\frac{\frac{1}{k}}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \]
   15. Step-by-step derivation

       1. add-sqr-sqrt 44.1%

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

       2. sqrt-div 16.3%

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

       3. inv-pow 16.3%

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

       4. sqrt-pow1 16.3%

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

       5. metadata-eval 16.3%

          \[\leadsto \frac{{k}^{\color{blue}{-0.5}}}{\sqrt{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \cdot \sqrt{\frac{\frac{1}{k}}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \]

       6. sqrt-prod 16.3%

          \[\leadsto \frac{{k}^{-0.5}}{\color{blue}{\sqrt{{t}^{3}} \cdot \sqrt{\frac{k}{{\ell}^{2}}}}} \cdot \sqrt{\frac{\frac{1}{k}}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \]

       7. sqrt-pow1 16.3%

          \[\leadsto \frac{{k}^{-0.5}}{\color{blue}{{t}^{\left(\frac{3}{2}\right)}} \cdot \sqrt{\frac{k}{{\ell}^{2}}}} \cdot \sqrt{\frac{\frac{1}{k}}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \]

       8. metadata-eval 16.3%

          \[\leadsto \frac{{k}^{-0.5}}{{t}^{\color{blue}{1.5}} \cdot \sqrt{\frac{k}{{\ell}^{2}}}} \cdot \sqrt{\frac{\frac{1}{k}}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \]

       9. sqrt-div 16.3%

          \[\leadsto \frac{{k}^{-0.5}}{{t}^{1.5} \cdot \color{blue}{\frac{\sqrt{k}}{\sqrt{{\ell}^{2}}}}} \cdot \sqrt{\frac{\frac{1}{k}}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \]

       10. unpow2 16.3%

           \[\leadsto \frac{{k}^{-0.5}}{{t}^{1.5} \cdot \frac{\sqrt{k}}{\sqrt{\color{blue}{\ell \cdot \ell}}}} \cdot \sqrt{\frac{\frac{1}{k}}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \]

       11. sqrt-prod 8.5%

           \[\leadsto \frac{{k}^{-0.5}}{{t}^{1.5} \cdot \frac{\sqrt{k}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}} \cdot \sqrt{\frac{\frac{1}{k}}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \]

       12. add-sqr-sqrt 13.4%

           \[\leadsto \frac{{k}^{-0.5}}{{t}^{1.5} \cdot \frac{\sqrt{k}}{\color{blue}{\ell}}} \cdot \sqrt{\frac{\frac{1}{k}}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \]

       13. sqrt-div 13.4%

           \[\leadsto \frac{{k}^{-0.5}}{{t}^{1.5} \cdot \frac{\sqrt{k}}{\ell}} \cdot \color{blue}{\frac{\sqrt{\frac{1}{k}}}{\sqrt{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}}} \]

       14. inv-pow 13.4%

           \[\leadsto \frac{{k}^{-0.5}}{{t}^{1.5} \cdot \frac{\sqrt{k}}{\ell}} \cdot \frac{\sqrt{\color{blue}{{k}^{-1}}}}{\sqrt{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \]

       15. sqrt-pow1 13.4%

           \[\leadsto \frac{{k}^{-0.5}}{{t}^{1.5} \cdot \frac{\sqrt{k}}{\ell}} \cdot \frac{\color{blue}{{k}^{\left(\frac{-1}{2}\right)}}}{\sqrt{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \]

       16. metadata-eval 13.4%

           \[\leadsto \frac{{k}^{-0.5}}{{t}^{1.5} \cdot \frac{\sqrt{k}}{\ell}} \cdot \frac{{k}^{\color{blue}{-0.5}}}{\sqrt{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \]

       17. sqrt-prod 13.4%

           \[\leadsto \frac{{k}^{-0.5}}{{t}^{1.5} \cdot \frac{\sqrt{k}}{\ell}} \cdot \frac{{k}^{-0.5}}{\color{blue}{\sqrt{{t}^{3}} \cdot \sqrt{\frac{k}{{\ell}^{2}}}}} \]

   16. Applied egg-rr 19.9%

       \[\leadsto \color{blue}{\frac{{k}^{-0.5}}{{t}^{1.5} \cdot \frac{\sqrt{k}}{\ell}} \cdot \frac{{k}^{-0.5}}{{t}^{1.5} \cdot \frac{\sqrt{k}}{\ell}}} \]
   17. Step-by-step derivation

       1. unpow2 19.9%

          \[\leadsto \color{blue}{{\left(\frac{{k}^{-0.5}}{{t}^{1.5} \cdot \frac{\sqrt{k}}{\ell}}\right)}^{2}} \]

   18. Simplified 19.9%

       \[\leadsto \color{blue}{{\left(\frac{{k}^{-0.5}}{{t}^{1.5} \cdot \frac{\sqrt{k}}{\ell}}\right)}^{2}} \]

   if 1.60000000000000007e246 < l

   1. Initial program 39.3%

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

      1. *-commutative 39.3%

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

      2. sqr-neg 39.3%

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

      3. *-commutative 39.3%

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

      4. associate-*l* 39.3%

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

      5. *-commutative 39.3%

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

      6. sqr-neg 39.3%

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

   3. Simplified 39.3%

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

   5. Taylor expanded in k around 0 55.6%

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

      1. *-commutative 55.6%

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

   7. Simplified 55.6%

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

      1. rem-cube-cbrt 55.6%

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

      2. cbrt-div 55.6%

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

      3. rem-cbrt-cube 67.0%

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

      4. cbrt-prod 82.5%

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

      5. unpow2 82.5%

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

      6. div-inv 82.5%

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

      7. unpow-prod-down 55.6%

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

      8. pow-flip 55.6%

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

      9. metadata-eval 55.6%

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

   9. Applied egg-rr 55.6%

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

       1. cube-prod 82.5%

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

   11. Simplified 82.5%

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

4. Final simplification 24.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.6 \cdot 10^{+246}:\\ \;\;\;\;{\left(\frac{{k}^{-0.5}}{{t}^{1.5} \cdot \frac{\sqrt{k}}{\ell}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot {\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 35.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 2.8 \cdot 10^{+246}:\\ \;\;\;\;{\left(\frac{{k}^{-0.5}}{{t\_m}^{1.5} \cdot \frac{\sqrt{k}}{\ell}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot {\left(t\_m \cdot {\ell}^{-0.6666666666666666}\right)}^{3}\right)}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= l 2.8e+246)
    (pow (/ (pow k -0.5) (* (pow t_m 1.5) (/ (sqrt k) l))) 2.0)
    (/
     2.0
     (*
      (* 2.0 k)
      (* (sin k) (pow (* t_m (pow l -0.6666666666666666)) 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 (l <= 2.8e+246) {
		tmp = pow((pow(k, -0.5) / (pow(t_m, 1.5) * (sqrt(k) / l))), 2.0);
	} else {
		tmp = 2.0 / ((2.0 * k) * (sin(k) * pow((t_m * pow(l, -0.6666666666666666)), 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 (l <= 2.8d+246) then
        tmp = ((k ** (-0.5d0)) / ((t_m ** 1.5d0) * (sqrt(k) / l))) ** 2.0d0
    else
        tmp = 2.0d0 / ((2.0d0 * k) * (sin(k) * ((t_m * (l ** (-0.6666666666666666d0))) ** 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 (l <= 2.8e+246) {
		tmp = Math.pow((Math.pow(k, -0.5) / (Math.pow(t_m, 1.5) * (Math.sqrt(k) / l))), 2.0);
	} else {
		tmp = 2.0 / ((2.0 * k) * (Math.sin(k) * Math.pow((t_m * Math.pow(l, -0.6666666666666666)), 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 l <= 2.8e+246:
		tmp = math.pow((math.pow(k, -0.5) / (math.pow(t_m, 1.5) * (math.sqrt(k) / l))), 2.0)
	else:
		tmp = 2.0 / ((2.0 * k) * (math.sin(k) * math.pow((t_m * math.pow(l, -0.6666666666666666)), 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 (l <= 2.8e+246)
		tmp = Float64((k ^ -0.5) / Float64((t_m ^ 1.5) * Float64(sqrt(k) / l))) ^ 2.0;
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * Float64(sin(k) * (Float64(t_m * (l ^ -0.6666666666666666)) ^ 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 (l <= 2.8e+246)
		tmp = ((k ^ -0.5) / ((t_m ^ 1.5) * (sqrt(k) / l))) ^ 2.0;
	else
		tmp = 2.0 / ((2.0 * k) * (sin(k) * ((t_m * (l ^ -0.6666666666666666)) ^ 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[l, 2.8e+246], N[Power[N[(N[Power[k, -0.5], $MachinePrecision] / N[(N[Power[t$95$m, 1.5], $MachinePrecision] * N[(N[Sqrt[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(t$95$m * N[Power[l, -0.6666666666666666], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 2.79999999999999988e246

   1. Initial program 58.4%

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

      1. *-commutative 58.4%

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

      2. sqr-neg 58.4%

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

      3. *-commutative 58.4%

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

      4. associate-*l* 58.8%

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

      5. *-commutative 58.8%

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

      6. sqr-neg 58.8%

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

   3. Simplified 58.8%

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

   5. Taylor expanded in k around 0 56.3%

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

      1. *-commutative 56.3%

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

   7. Simplified 56.3%

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

   8. Taylor expanded in k around 0 59.2%

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

      1. associate-/l* 58.2%

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

   10. Simplified 58.2%

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

       1. div-inv 58.2%

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

       2. associate-/r/ 58.3%

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

   12. Applied egg-rr 58.3%

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

       1. associate-*r/ 58.3%

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

       2. metadata-eval 58.3%

          \[\leadsto \frac{\color{blue}{2}}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right) \cdot \left(k \cdot 2\right)} \]

       3. *-commutative 58.3%

          \[\leadsto \frac{2}{\color{blue}{\left(k \cdot 2\right) \cdot \left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right)}} \]

       4. associate-/r* 58.3%

          \[\leadsto \color{blue}{\frac{\frac{2}{k \cdot 2}}{\frac{k}{{\ell}^{2}} \cdot {t}^{3}}} \]

       5. *-commutative 58.3%

          \[\leadsto \frac{\frac{2}{\color{blue}{2 \cdot k}}}{\frac{k}{{\ell}^{2}} \cdot {t}^{3}} \]

       6. associate-/r* 58.3%

          \[\leadsto \frac{\color{blue}{\frac{\frac{2}{2}}{k}}}{\frac{k}{{\ell}^{2}} \cdot {t}^{3}} \]

       7. metadata-eval 58.3%

          \[\leadsto \frac{\frac{\color{blue}{1}}{k}}{\frac{k}{{\ell}^{2}} \cdot {t}^{3}} \]

       8. *-commutative 58.3%

          \[\leadsto \frac{\frac{1}{k}}{\color{blue}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \]

   14. Simplified 58.3%

       \[\leadsto \color{blue}{\frac{\frac{1}{k}}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \]
   15. Step-by-step derivation

       1. add-sqr-sqrt 44.1%

          \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{k}}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \cdot \sqrt{\frac{\frac{1}{k}}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}}} \]

       2. sqrt-div 16.3%

          \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{k}}}{\sqrt{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}}} \cdot \sqrt{\frac{\frac{1}{k}}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \]

       3. inv-pow 16.3%

          \[\leadsto \frac{\sqrt{\color{blue}{{k}^{-1}}}}{\sqrt{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \cdot \sqrt{\frac{\frac{1}{k}}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \]

       4. sqrt-pow1 16.3%

          \[\leadsto \frac{\color{blue}{{k}^{\left(\frac{-1}{2}\right)}}}{\sqrt{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \cdot \sqrt{\frac{\frac{1}{k}}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \]

       5. metadata-eval 16.3%

          \[\leadsto \frac{{k}^{\color{blue}{-0.5}}}{\sqrt{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \cdot \sqrt{\frac{\frac{1}{k}}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \]

       6. sqrt-prod 16.3%

          \[\leadsto \frac{{k}^{-0.5}}{\color{blue}{\sqrt{{t}^{3}} \cdot \sqrt{\frac{k}{{\ell}^{2}}}}} \cdot \sqrt{\frac{\frac{1}{k}}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \]

       7. sqrt-pow1 16.3%

          \[\leadsto \frac{{k}^{-0.5}}{\color{blue}{{t}^{\left(\frac{3}{2}\right)}} \cdot \sqrt{\frac{k}{{\ell}^{2}}}} \cdot \sqrt{\frac{\frac{1}{k}}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \]

       8. metadata-eval 16.3%

          \[\leadsto \frac{{k}^{-0.5}}{{t}^{\color{blue}{1.5}} \cdot \sqrt{\frac{k}{{\ell}^{2}}}} \cdot \sqrt{\frac{\frac{1}{k}}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \]

       9. sqrt-div 16.3%

          \[\leadsto \frac{{k}^{-0.5}}{{t}^{1.5} \cdot \color{blue}{\frac{\sqrt{k}}{\sqrt{{\ell}^{2}}}}} \cdot \sqrt{\frac{\frac{1}{k}}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \]

       10. unpow2 16.3%

           \[\leadsto \frac{{k}^{-0.5}}{{t}^{1.5} \cdot \frac{\sqrt{k}}{\sqrt{\color{blue}{\ell \cdot \ell}}}} \cdot \sqrt{\frac{\frac{1}{k}}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \]

       11. sqrt-prod 8.5%

           \[\leadsto \frac{{k}^{-0.5}}{{t}^{1.5} \cdot \frac{\sqrt{k}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}} \cdot \sqrt{\frac{\frac{1}{k}}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \]

       12. add-sqr-sqrt 13.4%

           \[\leadsto \frac{{k}^{-0.5}}{{t}^{1.5} \cdot \frac{\sqrt{k}}{\color{blue}{\ell}}} \cdot \sqrt{\frac{\frac{1}{k}}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \]

       13. sqrt-div 13.4%

           \[\leadsto \frac{{k}^{-0.5}}{{t}^{1.5} \cdot \frac{\sqrt{k}}{\ell}} \cdot \color{blue}{\frac{\sqrt{\frac{1}{k}}}{\sqrt{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}}} \]

       14. inv-pow 13.4%

           \[\leadsto \frac{{k}^{-0.5}}{{t}^{1.5} \cdot \frac{\sqrt{k}}{\ell}} \cdot \frac{\sqrt{\color{blue}{{k}^{-1}}}}{\sqrt{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \]

       15. sqrt-pow1 13.4%

           \[\leadsto \frac{{k}^{-0.5}}{{t}^{1.5} \cdot \frac{\sqrt{k}}{\ell}} \cdot \frac{\color{blue}{{k}^{\left(\frac{-1}{2}\right)}}}{\sqrt{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \]

       16. metadata-eval 13.4%

           \[\leadsto \frac{{k}^{-0.5}}{{t}^{1.5} \cdot \frac{\sqrt{k}}{\ell}} \cdot \frac{{k}^{\color{blue}{-0.5}}}{\sqrt{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \]

       17. sqrt-prod 13.4%

           \[\leadsto \frac{{k}^{-0.5}}{{t}^{1.5} \cdot \frac{\sqrt{k}}{\ell}} \cdot \frac{{k}^{-0.5}}{\color{blue}{\sqrt{{t}^{3}} \cdot \sqrt{\frac{k}{{\ell}^{2}}}}} \]

   16. Applied egg-rr 19.9%

       \[\leadsto \color{blue}{\frac{{k}^{-0.5}}{{t}^{1.5} \cdot \frac{\sqrt{k}}{\ell}} \cdot \frac{{k}^{-0.5}}{{t}^{1.5} \cdot \frac{\sqrt{k}}{\ell}}} \]
   17. Step-by-step derivation

       1. unpow2 19.9%

          \[\leadsto \color{blue}{{\left(\frac{{k}^{-0.5}}{{t}^{1.5} \cdot \frac{\sqrt{k}}{\ell}}\right)}^{2}} \]

   18. Simplified 19.9%

       \[\leadsto \color{blue}{{\left(\frac{{k}^{-0.5}}{{t}^{1.5} \cdot \frac{\sqrt{k}}{\ell}}\right)}^{2}} \]

   if 2.79999999999999988e246 < l

   1. Initial program 39.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 39.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. sqr-neg 39.3%

         \[\leadsto \frac{2}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. *-commutative 39.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. associate-*l* 39.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      5. *-commutative 39.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      6. sqr-neg 39.3%

         \[\leadsto \frac{2}{\left(\sin k \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

   3. Simplified 39.3%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 55.6%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]
   6. Step-by-step derivation

      1. *-commutative 55.6%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   7. Simplified 55.6%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]
   8. Step-by-step derivation

      1. rem-cube-cbrt 55.6%

         \[\leadsto \frac{2}{\left(\color{blue}{{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right)}^{3}} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      2. cbrt-div 55.6%

         \[\leadsto \frac{2}{\left({\color{blue}{\left(\frac{\sqrt[3]{{t}^{3}}}{\sqrt[3]{\ell \cdot \ell}}\right)}}^{3} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      3. rem-cbrt-cube 67.0%

         \[\leadsto \frac{2}{\left({\left(\frac{\color{blue}{t}}{\sqrt[3]{\ell \cdot \ell}}\right)}^{3} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      4. cbrt-prod 82.5%

         \[\leadsto \frac{2}{\left({\left(\frac{t}{\color{blue}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\right)}^{3} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      5. unpow2 82.5%

         \[\leadsto \frac{2}{\left({\left(\frac{t}{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\right)}^{3} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      6. div-inv 82.5%

         \[\leadsto \frac{2}{\left({\color{blue}{\left(t \cdot \frac{1}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}}^{3} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      7. unpow-prod-down 55.6%

         \[\leadsto \frac{2}{\left(\color{blue}{\left({t}^{3} \cdot {\left(\frac{1}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      8. pow-flip 55.6%

         \[\leadsto \frac{2}{\left(\left({t}^{3} \cdot {\color{blue}{\left({\left(\sqrt[3]{\ell}\right)}^{\left(-2\right)}\right)}}^{3}\right) \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      9. metadata-eval 55.6%

         \[\leadsto \frac{2}{\left(\left({t}^{3} \cdot {\left({\left(\sqrt[3]{\ell}\right)}^{\color{blue}{-2}}\right)}^{3}\right) \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

   9. Applied egg-rr 55.6%

      \[\leadsto \frac{2}{\left(\color{blue}{\left({t}^{3} \cdot {\left({\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}\right)} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]
   10. Step-by-step derivation

       1. cube-prod 82.5%

          \[\leadsto \frac{2}{\left(\color{blue}{{\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

   11. Simplified 82.5%

       \[\leadsto \frac{2}{\left(\color{blue}{{\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]
   12. Step-by-step derivation

       1. sqr-pow 82.5%

          \[\leadsto \frac{2}{\left({\left(t \cdot \color{blue}{\left({\left(\sqrt[3]{\ell}\right)}^{\left(\frac{-2}{2}\right)} \cdot {\left(\sqrt[3]{\ell}\right)}^{\left(\frac{-2}{2}\right)}\right)}\right)}^{3} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

       2. metadata-eval 82.5%

          \[\leadsto \frac{2}{\left({\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{\color{blue}{-1}} \cdot {\left(\sqrt[3]{\ell}\right)}^{\left(\frac{-2}{2}\right)}\right)\right)}^{3} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

       3. unpow-1 82.5%

          \[\leadsto \frac{2}{\left({\left(t \cdot \left(\color{blue}{\frac{1}{\sqrt[3]{\ell}}} \cdot {\left(\sqrt[3]{\ell}\right)}^{\left(\frac{-2}{2}\right)}\right)\right)}^{3} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

       4. metadata-eval 82.5%

          \[\leadsto \frac{2}{\left({\left(t \cdot \left(\frac{1}{\sqrt[3]{\ell}} \cdot {\left(\sqrt[3]{\ell}\right)}^{\color{blue}{-1}}\right)\right)}^{3} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

       5. unpow-1 82.5%

          \[\leadsto \frac{2}{\left({\left(t \cdot \left(\frac{1}{\sqrt[3]{\ell}} \cdot \color{blue}{\frac{1}{\sqrt[3]{\ell}}}\right)\right)}^{3} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

   13. Applied egg-rr 82.5%

       \[\leadsto \frac{2}{\left({\left(t \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{\ell}} \cdot \frac{1}{\sqrt[3]{\ell}}\right)}\right)}^{3} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]
   14. Step-by-step derivation

       1. inv-pow 82.5%

          \[\leadsto \frac{2}{\left({\left(t \cdot \left(\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{-1}} \cdot \frac{1}{\sqrt[3]{\ell}}\right)\right)}^{3} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

       2. inv-pow 82.5%

          \[\leadsto \frac{2}{\left({\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-1} \cdot \color{blue}{{\left(\sqrt[3]{\ell}\right)}^{-1}}\right)\right)}^{3} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

       3. pow-prod-up 82.5%

          \[\leadsto \frac{2}{\left({\left(t \cdot \color{blue}{{\left(\sqrt[3]{\ell}\right)}^{\left(-1 + -1\right)}}\right)}^{3} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

       4. metadata-eval 82.5%

          \[\leadsto \frac{2}{\left({\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{\color{blue}{-2}}\right)}^{3} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

       5. pow1/3 81.8%

          \[\leadsto \frac{2}{\left({\left(t \cdot {\color{blue}{\left({\ell}^{0.3333333333333333}\right)}}^{-2}\right)}^{3} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

       6. pow-pow 81.8%

          \[\leadsto \frac{2}{\left({\left(t \cdot \color{blue}{{\ell}^{\left(0.3333333333333333 \cdot -2\right)}}\right)}^{3} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

       7. metadata-eval 81.8%

          \[\leadsto \frac{2}{\left({\left(t \cdot {\ell}^{\color{blue}{-0.6666666666666666}}\right)}^{3} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

   15. Applied egg-rr 81.8%

       \[\leadsto \frac{2}{\left({\left(t \cdot \color{blue}{{\ell}^{-0.6666666666666666}}\right)}^{3} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 24.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.8 \cdot 10^{+246}:\\ \;\;\;\;{\left(\frac{{k}^{-0.5}}{{t}^{1.5} \cdot \frac{\sqrt{k}}{\ell}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot {\left(t \cdot {\ell}^{-0.6666666666666666}\right)}^{3}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 34.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 3 \cdot 10^{+246}:\\ \;\;\;\;\frac{\frac{1}{k}}{{\left({t\_m}^{1.5} \cdot \frac{\sqrt{k}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot {\left(t\_m \cdot {\ell}^{-0.6666666666666666}\right)}^{3}\right)}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= l 3e+246)
    (/ (/ 1.0 k) (pow (* (pow t_m 1.5) (/ (sqrt k) l)) 2.0))
    (/
     2.0
     (*
      (* 2.0 k)
      (* (sin k) (pow (* t_m (pow l -0.6666666666666666)) 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 (l <= 3e+246) {
		tmp = (1.0 / k) / pow((pow(t_m, 1.5) * (sqrt(k) / l)), 2.0);
	} else {
		tmp = 2.0 / ((2.0 * k) * (sin(k) * pow((t_m * pow(l, -0.6666666666666666)), 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 (l <= 3d+246) then
        tmp = (1.0d0 / k) / (((t_m ** 1.5d0) * (sqrt(k) / l)) ** 2.0d0)
    else
        tmp = 2.0d0 / ((2.0d0 * k) * (sin(k) * ((t_m * (l ** (-0.6666666666666666d0))) ** 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 (l <= 3e+246) {
		tmp = (1.0 / k) / Math.pow((Math.pow(t_m, 1.5) * (Math.sqrt(k) / l)), 2.0);
	} else {
		tmp = 2.0 / ((2.0 * k) * (Math.sin(k) * Math.pow((t_m * Math.pow(l, -0.6666666666666666)), 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 l <= 3e+246:
		tmp = (1.0 / k) / math.pow((math.pow(t_m, 1.5) * (math.sqrt(k) / l)), 2.0)
	else:
		tmp = 2.0 / ((2.0 * k) * (math.sin(k) * math.pow((t_m * math.pow(l, -0.6666666666666666)), 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 (l <= 3e+246)
		tmp = Float64(Float64(1.0 / k) / (Float64((t_m ^ 1.5) * Float64(sqrt(k) / l)) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * Float64(sin(k) * (Float64(t_m * (l ^ -0.6666666666666666)) ^ 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 (l <= 3e+246)
		tmp = (1.0 / k) / (((t_m ^ 1.5) * (sqrt(k) / l)) ^ 2.0);
	else
		tmp = 2.0 / ((2.0 * k) * (sin(k) * ((t_m * (l ^ -0.6666666666666666)) ^ 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[l, 3e+246], N[(N[(1.0 / k), $MachinePrecision] / N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] * N[(N[Sqrt[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(t$95$m * N[Power[l, -0.6666666666666666], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 3e246

   1. Initial program 58.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 58.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. sqr-neg 58.4%

         \[\leadsto \frac{2}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. *-commutative 58.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. associate-*l* 58.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      5. *-commutative 58.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      6. sqr-neg 58.8%

         \[\leadsto \frac{2}{\left(\sin k \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

   3. Simplified 58.8%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 56.3%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]
   6. Step-by-step derivation

      1. *-commutative 56.3%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   7. Simplified 56.3%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   8. Taylor expanded in k around 0 59.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot {t}^{3}}{{\ell}^{2}}} \cdot \left(k \cdot 2\right)} \]
   9. Step-by-step derivation

      1. associate-/l* 58.2%

         \[\leadsto \frac{2}{\color{blue}{\frac{k}{\frac{{\ell}^{2}}{{t}^{3}}}} \cdot \left(k \cdot 2\right)} \]

   10. Simplified 58.2%

       \[\leadsto \frac{2}{\color{blue}{\frac{k}{\frac{{\ell}^{2}}{{t}^{3}}}} \cdot \left(k \cdot 2\right)} \]
   11. Step-by-step derivation

       1. div-inv 58.2%

          \[\leadsto \color{blue}{2 \cdot \frac{1}{\frac{k}{\frac{{\ell}^{2}}{{t}^{3}}} \cdot \left(k \cdot 2\right)}} \]

       2. associate-/r/ 58.3%

          \[\leadsto 2 \cdot \frac{1}{\color{blue}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right)} \cdot \left(k \cdot 2\right)} \]

   12. Applied egg-rr 58.3%

       \[\leadsto \color{blue}{2 \cdot \frac{1}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right) \cdot \left(k \cdot 2\right)}} \]
   13. Step-by-step derivation

       1. associate-*r/ 58.3%

          \[\leadsto \color{blue}{\frac{2 \cdot 1}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right) \cdot \left(k \cdot 2\right)}} \]

       2. metadata-eval 58.3%

          \[\leadsto \frac{\color{blue}{2}}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right) \cdot \left(k \cdot 2\right)} \]

       3. *-commutative 58.3%

          \[\leadsto \frac{2}{\color{blue}{\left(k \cdot 2\right) \cdot \left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right)}} \]

       4. associate-/r* 58.3%

          \[\leadsto \color{blue}{\frac{\frac{2}{k \cdot 2}}{\frac{k}{{\ell}^{2}} \cdot {t}^{3}}} \]

       5. *-commutative 58.3%

          \[\leadsto \frac{\frac{2}{\color{blue}{2 \cdot k}}}{\frac{k}{{\ell}^{2}} \cdot {t}^{3}} \]

       6. associate-/r* 58.3%

          \[\leadsto \frac{\color{blue}{\frac{\frac{2}{2}}{k}}}{\frac{k}{{\ell}^{2}} \cdot {t}^{3}} \]

       7. metadata-eval 58.3%

          \[\leadsto \frac{\frac{\color{blue}{1}}{k}}{\frac{k}{{\ell}^{2}} \cdot {t}^{3}} \]

       8. *-commutative 58.3%

          \[\leadsto \frac{\frac{1}{k}}{\color{blue}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \]

   14. Simplified 58.3%

       \[\leadsto \color{blue}{\frac{\frac{1}{k}}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \]
   15. Step-by-step derivation

       1. add-sqr-sqrt 32.6%

          \[\leadsto \frac{\frac{1}{k}}{\color{blue}{\sqrt{{t}^{3} \cdot \frac{k}{{\ell}^{2}}} \cdot \sqrt{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}}} \]

       2. pow2 32.6%

          \[\leadsto \frac{\frac{1}{k}}{\color{blue}{{\left(\sqrt{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}\right)}^{2}}} \]

       3. sqrt-prod 18.0%

          \[\leadsto \frac{\frac{1}{k}}{{\color{blue}{\left(\sqrt{{t}^{3}} \cdot \sqrt{\frac{k}{{\ell}^{2}}}\right)}}^{2}} \]

       4. sqrt-pow1 18.4%

          \[\leadsto \frac{\frac{1}{k}}{{\left(\color{blue}{{t}^{\left(\frac{3}{2}\right)}} \cdot \sqrt{\frac{k}{{\ell}^{2}}}\right)}^{2}} \]

       5. metadata-eval 18.4%

          \[\leadsto \frac{\frac{1}{k}}{{\left({t}^{\color{blue}{1.5}} \cdot \sqrt{\frac{k}{{\ell}^{2}}}\right)}^{2}} \]

       6. sqrt-div 18.0%

          \[\leadsto \frac{\frac{1}{k}}{{\left({t}^{1.5} \cdot \color{blue}{\frac{\sqrt{k}}{\sqrt{{\ell}^{2}}}}\right)}^{2}} \]

       7. unpow2 18.0%

          \[\leadsto \frac{\frac{1}{k}}{{\left({t}^{1.5} \cdot \frac{\sqrt{k}}{\sqrt{\color{blue}{\ell \cdot \ell}}}\right)}^{2}} \]

       8. sqrt-prod 8.9%

          \[\leadsto \frac{\frac{1}{k}}{{\left({t}^{1.5} \cdot \frac{\sqrt{k}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2}} \]

       9. add-sqr-sqrt 19.9%

          \[\leadsto \frac{\frac{1}{k}}{{\left({t}^{1.5} \cdot \frac{\sqrt{k}}{\color{blue}{\ell}}\right)}^{2}} \]

   16. Applied egg-rr 19.9%

       \[\leadsto \frac{\frac{1}{k}}{\color{blue}{{\left({t}^{1.5} \cdot \frac{\sqrt{k}}{\ell}\right)}^{2}}} \]

   if 3e246 < l

   1. Initial program 39.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 39.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. sqr-neg 39.3%

         \[\leadsto \frac{2}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. *-commutative 39.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. associate-*l* 39.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      5. *-commutative 39.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      6. sqr-neg 39.3%

         \[\leadsto \frac{2}{\left(\sin k \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

   3. Simplified 39.3%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 55.6%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]
   6. Step-by-step derivation

      1. *-commutative 55.6%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   7. Simplified 55.6%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]
   8. Step-by-step derivation

      1. rem-cube-cbrt 55.6%

         \[\leadsto \frac{2}{\left(\color{blue}{{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right)}^{3}} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      2. cbrt-div 55.6%

         \[\leadsto \frac{2}{\left({\color{blue}{\left(\frac{\sqrt[3]{{t}^{3}}}{\sqrt[3]{\ell \cdot \ell}}\right)}}^{3} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      3. rem-cbrt-cube 67.0%

         \[\leadsto \frac{2}{\left({\left(\frac{\color{blue}{t}}{\sqrt[3]{\ell \cdot \ell}}\right)}^{3} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      4. cbrt-prod 82.5%

         \[\leadsto \frac{2}{\left({\left(\frac{t}{\color{blue}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\right)}^{3} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      5. unpow2 82.5%

         \[\leadsto \frac{2}{\left({\left(\frac{t}{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\right)}^{3} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      6. div-inv 82.5%

         \[\leadsto \frac{2}{\left({\color{blue}{\left(t \cdot \frac{1}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}}^{3} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      7. unpow-prod-down 55.6%

         \[\leadsto \frac{2}{\left(\color{blue}{\left({t}^{3} \cdot {\left(\frac{1}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      8. pow-flip 55.6%

         \[\leadsto \frac{2}{\left(\left({t}^{3} \cdot {\color{blue}{\left({\left(\sqrt[3]{\ell}\right)}^{\left(-2\right)}\right)}}^{3}\right) \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      9. metadata-eval 55.6%

         \[\leadsto \frac{2}{\left(\left({t}^{3} \cdot {\left({\left(\sqrt[3]{\ell}\right)}^{\color{blue}{-2}}\right)}^{3}\right) \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

   9. Applied egg-rr 55.6%

      \[\leadsto \frac{2}{\left(\color{blue}{\left({t}^{3} \cdot {\left({\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}\right)} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]
   10. Step-by-step derivation

       1. cube-prod 82.5%

          \[\leadsto \frac{2}{\left(\color{blue}{{\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

   11. Simplified 82.5%

       \[\leadsto \frac{2}{\left(\color{blue}{{\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]
   12. Step-by-step derivation

       1. sqr-pow 82.5%

          \[\leadsto \frac{2}{\left({\left(t \cdot \color{blue}{\left({\left(\sqrt[3]{\ell}\right)}^{\left(\frac{-2}{2}\right)} \cdot {\left(\sqrt[3]{\ell}\right)}^{\left(\frac{-2}{2}\right)}\right)}\right)}^{3} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

       2. metadata-eval 82.5%

          \[\leadsto \frac{2}{\left({\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{\color{blue}{-1}} \cdot {\left(\sqrt[3]{\ell}\right)}^{\left(\frac{-2}{2}\right)}\right)\right)}^{3} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

       3. unpow-1 82.5%

          \[\leadsto \frac{2}{\left({\left(t \cdot \left(\color{blue}{\frac{1}{\sqrt[3]{\ell}}} \cdot {\left(\sqrt[3]{\ell}\right)}^{\left(\frac{-2}{2}\right)}\right)\right)}^{3} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

       4. metadata-eval 82.5%

          \[\leadsto \frac{2}{\left({\left(t \cdot \left(\frac{1}{\sqrt[3]{\ell}} \cdot {\left(\sqrt[3]{\ell}\right)}^{\color{blue}{-1}}\right)\right)}^{3} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

       5. unpow-1 82.5%

          \[\leadsto \frac{2}{\left({\left(t \cdot \left(\frac{1}{\sqrt[3]{\ell}} \cdot \color{blue}{\frac{1}{\sqrt[3]{\ell}}}\right)\right)}^{3} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

   13. Applied egg-rr 82.5%

       \[\leadsto \frac{2}{\left({\left(t \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{\ell}} \cdot \frac{1}{\sqrt[3]{\ell}}\right)}\right)}^{3} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]
   14. Step-by-step derivation

       1. inv-pow 82.5%

          \[\leadsto \frac{2}{\left({\left(t \cdot \left(\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{-1}} \cdot \frac{1}{\sqrt[3]{\ell}}\right)\right)}^{3} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

       2. inv-pow 82.5%

          \[\leadsto \frac{2}{\left({\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-1} \cdot \color{blue}{{\left(\sqrt[3]{\ell}\right)}^{-1}}\right)\right)}^{3} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

       3. pow-prod-up 82.5%

          \[\leadsto \frac{2}{\left({\left(t \cdot \color{blue}{{\left(\sqrt[3]{\ell}\right)}^{\left(-1 + -1\right)}}\right)}^{3} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

       4. metadata-eval 82.5%

          \[\leadsto \frac{2}{\left({\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{\color{blue}{-2}}\right)}^{3} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

       5. pow1/3 81.8%

          \[\leadsto \frac{2}{\left({\left(t \cdot {\color{blue}{\left({\ell}^{0.3333333333333333}\right)}}^{-2}\right)}^{3} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

       6. pow-pow 81.8%

          \[\leadsto \frac{2}{\left({\left(t \cdot \color{blue}{{\ell}^{\left(0.3333333333333333 \cdot -2\right)}}\right)}^{3} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

       7. metadata-eval 81.8%

          \[\leadsto \frac{2}{\left({\left(t \cdot {\ell}^{\color{blue}{-0.6666666666666666}}\right)}^{3} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

   15. Applied egg-rr 81.8%

       \[\leadsto \frac{2}{\left({\left(t \cdot \color{blue}{{\ell}^{-0.6666666666666666}}\right)}^{3} \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 24.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3 \cdot 10^{+246}:\\ \;\;\;\;\frac{\frac{1}{k}}{{\left({t}^{1.5} \cdot \frac{\sqrt{k}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot {\left(t \cdot {\ell}^{-0.6666666666666666}\right)}^{3}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 62.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6 \cdot 10^{+93}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot \frac{t\_m \cdot \frac{{t\_m}^{2}}{\ell}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{k}}{{\left(t\_m \cdot \sqrt[3]{k \cdot {\ell}^{-2}}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 6e+93)
    (/ 2.0 (* (* 2.0 k) (* (sin k) (/ (* t_m (/ (pow t_m 2.0) l)) l))))
    (/ (/ 1.0 k) (pow (* t_m (cbrt (* k (pow 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 (k <= 6e+93) {
		tmp = 2.0 / ((2.0 * k) * (sin(k) * ((t_m * (pow(t_m, 2.0) / l)) / l)));
	} else {
		tmp = (1.0 / k) / pow((t_m * cbrt((k * pow(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 (k <= 6e+93) {
		tmp = 2.0 / ((2.0 * k) * (Math.sin(k) * ((t_m * (Math.pow(t_m, 2.0) / l)) / l)));
	} else {
		tmp = (1.0 / k) / Math.pow((t_m * Math.cbrt((k * Math.pow(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 (k <= 6e+93)
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * Float64(sin(k) * Float64(Float64(t_m * Float64((t_m ^ 2.0) / l)) / l))));
	else
		tmp = Float64(Float64(1.0 / k) / (Float64(t_m * cbrt(Float64(k * (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[k, 6e+93], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(t$95$m * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / k), $MachinePrecision] / N[Power[N[(t$95$m * N[Power[N[(k * N[Power[l, -2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 5.99999999999999957e93

   1. Initial program 59.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 59.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. sqr-neg 59.8%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. sqr-neg 59.8%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-/r* 64.5%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. distribute-rgt-in 64.5%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

      6. unpow2 64.5%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      7. times-frac 50.7%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      8. sqr-neg 50.7%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      9. times-frac 64.5%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      10. unpow2 64.5%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      11. distribute-rgt-in 64.5%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 64.5%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. cube-mult 64.5%

         \[\leadsto \frac{2}{\left(\frac{\frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      2. *-un-lft-identity 64.5%

         \[\leadsto \frac{2}{\left(\frac{\frac{t \cdot \left(t \cdot t\right)}{\color{blue}{1 \cdot \ell}}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      3. times-frac 70.6%

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\frac{t}{1} \cdot \frac{t \cdot t}{\ell}}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      4. pow2 70.6%

         \[\leadsto \frac{2}{\left(\frac{\frac{t}{1} \cdot \frac{\color{blue}{{t}^{2}}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   6. Applied egg-rr 70.6%

      \[\leadsto \frac{2}{\left(\frac{\color{blue}{\frac{t}{1} \cdot \frac{{t}^{2}}{\ell}}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   7. Taylor expanded in k around 0 67.2%

      \[\leadsto \frac{2}{\left(\frac{\frac{t}{1} \cdot \frac{{t}^{2}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]

   if 5.99999999999999957e93 < k

   1. Initial program 46.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 46.5%

         \[\leadsto \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. sqr-neg 46.5%

         \[\leadsto \frac{2}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. *-commutative 46.5%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. associate-*l* 46.5%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      5. *-commutative 46.5%

         \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      6. sqr-neg 46.5%

         \[\leadsto \frac{2}{\left(\sin k \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

   3. Simplified 46.5%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 43.5%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]
   6. Step-by-step derivation

      1. *-commutative 43.5%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   7. Simplified 43.5%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   8. Taylor expanded in k around 0 51.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot {t}^{3}}{{\ell}^{2}}} \cdot \left(k \cdot 2\right)} \]
   9. Step-by-step derivation

      1. associate-/l* 51.4%

         \[\leadsto \frac{2}{\color{blue}{\frac{k}{\frac{{\ell}^{2}}{{t}^{3}}}} \cdot \left(k \cdot 2\right)} \]

   10. Simplified 51.4%

       \[\leadsto \frac{2}{\color{blue}{\frac{k}{\frac{{\ell}^{2}}{{t}^{3}}}} \cdot \left(k \cdot 2\right)} \]
   11. Step-by-step derivation

       1. div-inv 51.4%

          \[\leadsto \color{blue}{2 \cdot \frac{1}{\frac{k}{\frac{{\ell}^{2}}{{t}^{3}}} \cdot \left(k \cdot 2\right)}} \]

       2. associate-/r/ 51.4%

          \[\leadsto 2 \cdot \frac{1}{\color{blue}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right)} \cdot \left(k \cdot 2\right)} \]

   12. Applied egg-rr 51.4%

       \[\leadsto \color{blue}{2 \cdot \frac{1}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right) \cdot \left(k \cdot 2\right)}} \]
   13. Step-by-step derivation

       1. associate-*r/ 51.4%

          \[\leadsto \color{blue}{\frac{2 \cdot 1}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right) \cdot \left(k \cdot 2\right)}} \]

       2. metadata-eval 51.4%

          \[\leadsto \frac{\color{blue}{2}}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right) \cdot \left(k \cdot 2\right)} \]

       3. *-commutative 51.4%

          \[\leadsto \frac{2}{\color{blue}{\left(k \cdot 2\right) \cdot \left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right)}} \]

       4. associate-/r* 51.4%

          \[\leadsto \color{blue}{\frac{\frac{2}{k \cdot 2}}{\frac{k}{{\ell}^{2}} \cdot {t}^{3}}} \]

       5. *-commutative 51.4%

          \[\leadsto \frac{\frac{2}{\color{blue}{2 \cdot k}}}{\frac{k}{{\ell}^{2}} \cdot {t}^{3}} \]

       6. associate-/r* 51.4%

          \[\leadsto \frac{\color{blue}{\frac{\frac{2}{2}}{k}}}{\frac{k}{{\ell}^{2}} \cdot {t}^{3}} \]

       7. metadata-eval 51.4%

          \[\leadsto \frac{\frac{\color{blue}{1}}{k}}{\frac{k}{{\ell}^{2}} \cdot {t}^{3}} \]

       8. *-commutative 51.4%

          \[\leadsto \frac{\frac{1}{k}}{\color{blue}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \]

   14. Simplified 51.4%

       \[\leadsto \color{blue}{\frac{\frac{1}{k}}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \]
   15. Step-by-step derivation

       1. add-cube-cbrt 51.4%

          \[\leadsto \frac{\frac{1}{k}}{\color{blue}{\left(\sqrt[3]{{t}^{3} \cdot \frac{k}{{\ell}^{2}}} \cdot \sqrt[3]{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}\right) \cdot \sqrt[3]{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}}} \]

       2. pow3 51.4%

          \[\leadsto \frac{\frac{1}{k}}{\color{blue}{{\left(\sqrt[3]{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}\right)}^{3}}} \]

       3. cbrt-prod 51.4%

          \[\leadsto \frac{\frac{1}{k}}{{\color{blue}{\left(\sqrt[3]{{t}^{3}} \cdot \sqrt[3]{\frac{k}{{\ell}^{2}}}\right)}}^{3}} \]

       4. rem-cbrt-cube 63.0%

          \[\leadsto \frac{\frac{1}{k}}{{\left(\color{blue}{t} \cdot \sqrt[3]{\frac{k}{{\ell}^{2}}}\right)}^{3}} \]

       5. div-inv 63.0%

          \[\leadsto \frac{\frac{1}{k}}{{\left(t \cdot \sqrt[3]{\color{blue}{k \cdot \frac{1}{{\ell}^{2}}}}\right)}^{3}} \]

       6. pow-flip 63.0%

          \[\leadsto \frac{\frac{1}{k}}{{\left(t \cdot \sqrt[3]{k \cdot \color{blue}{{\ell}^{\left(-2\right)}}}\right)}^{3}} \]

       7. metadata-eval 63.0%

          \[\leadsto \frac{\frac{1}{k}}{{\left(t \cdot \sqrt[3]{k \cdot {\ell}^{\color{blue}{-2}}}\right)}^{3}} \]

   16. Applied egg-rr 63.0%

       \[\leadsto \frac{\frac{1}{k}}{\color{blue}{{\left(t \cdot \sqrt[3]{k \cdot {\ell}^{-2}}\right)}^{3}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6 \cdot 10^{+93}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot \frac{t \cdot \frac{{t}^{2}}{\ell}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{k}}{{\left(t \cdot \sqrt[3]{k \cdot {\ell}^{-2}}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 34.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.2 \cdot 10^{+25}:\\ \;\;\;\;\frac{\frac{1}{k}}{{\left({t\_m}^{1.5} \cdot \frac{\sqrt{k}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{k}}{{\left(t\_m \cdot \sqrt[3]{k \cdot {\ell}^{-2}}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 3.2e+25)
    (/ (/ 1.0 k) (pow (* (pow t_m 1.5) (/ (sqrt k) l)) 2.0))
    (/ (/ 1.0 k) (pow (* t_m (cbrt (* k (pow 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 (k <= 3.2e+25) {
		tmp = (1.0 / k) / pow((pow(t_m, 1.5) * (sqrt(k) / l)), 2.0);
	} else {
		tmp = (1.0 / k) / pow((t_m * cbrt((k * pow(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 (k <= 3.2e+25) {
		tmp = (1.0 / k) / Math.pow((Math.pow(t_m, 1.5) * (Math.sqrt(k) / l)), 2.0);
	} else {
		tmp = (1.0 / k) / Math.pow((t_m * Math.cbrt((k * Math.pow(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 (k <= 3.2e+25)
		tmp = Float64(Float64(1.0 / k) / (Float64((t_m ^ 1.5) * Float64(sqrt(k) / l)) ^ 2.0));
	else
		tmp = Float64(Float64(1.0 / k) / (Float64(t_m * cbrt(Float64(k * (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[k, 3.2e+25], N[(N[(1.0 / k), $MachinePrecision] / N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] * N[(N[Sqrt[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / k), $MachinePrecision] / N[Power[N[(t$95$m * N[Power[N[(k * N[Power[l, -2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.1999999999999999e25

   1. Initial program 60.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 60.2%

         \[\leadsto \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. sqr-neg 60.2%

         \[\leadsto \frac{2}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. *-commutative 60.2%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. associate-*l* 60.7%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      5. *-commutative 60.7%

         \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      6. sqr-neg 60.7%

         \[\leadsto \frac{2}{\left(\sin k \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

   3. Simplified 60.7%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 59.3%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]
   6. Step-by-step derivation

      1. *-commutative 59.3%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   7. Simplified 59.3%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   8. Taylor expanded in k around 0 60.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot {t}^{3}}{{\ell}^{2}}} \cdot \left(k \cdot 2\right)} \]
   9. Step-by-step derivation

      1. associate-/l* 59.7%

         \[\leadsto \frac{2}{\color{blue}{\frac{k}{\frac{{\ell}^{2}}{{t}^{3}}}} \cdot \left(k \cdot 2\right)} \]

   10. Simplified 59.7%

       \[\leadsto \frac{2}{\color{blue}{\frac{k}{\frac{{\ell}^{2}}{{t}^{3}}}} \cdot \left(k \cdot 2\right)} \]
   11. Step-by-step derivation

       1. div-inv 59.7%

          \[\leadsto \color{blue}{2 \cdot \frac{1}{\frac{k}{\frac{{\ell}^{2}}{{t}^{3}}} \cdot \left(k \cdot 2\right)}} \]

       2. associate-/r/ 59.7%

          \[\leadsto 2 \cdot \frac{1}{\color{blue}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right)} \cdot \left(k \cdot 2\right)} \]

   12. Applied egg-rr 59.7%

       \[\leadsto \color{blue}{2 \cdot \frac{1}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right) \cdot \left(k \cdot 2\right)}} \]
   13. Step-by-step derivation

       1. associate-*r/ 59.7%

          \[\leadsto \color{blue}{\frac{2 \cdot 1}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right) \cdot \left(k \cdot 2\right)}} \]

       2. metadata-eval 59.7%

          \[\leadsto \frac{\color{blue}{2}}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right) \cdot \left(k \cdot 2\right)} \]

       3. *-commutative 59.7%

          \[\leadsto \frac{2}{\color{blue}{\left(k \cdot 2\right) \cdot \left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right)}} \]

       4. associate-/r* 59.8%

          \[\leadsto \color{blue}{\frac{\frac{2}{k \cdot 2}}{\frac{k}{{\ell}^{2}} \cdot {t}^{3}}} \]

       5. *-commutative 59.8%

          \[\leadsto \frac{\frac{2}{\color{blue}{2 \cdot k}}}{\frac{k}{{\ell}^{2}} \cdot {t}^{3}} \]

       6. associate-/r* 59.8%

          \[\leadsto \frac{\color{blue}{\frac{\frac{2}{2}}{k}}}{\frac{k}{{\ell}^{2}} \cdot {t}^{3}} \]

       7. metadata-eval 59.8%

          \[\leadsto \frac{\frac{\color{blue}{1}}{k}}{\frac{k}{{\ell}^{2}} \cdot {t}^{3}} \]

       8. *-commutative 59.8%

          \[\leadsto \frac{\frac{1}{k}}{\color{blue}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \]

   14. Simplified 59.8%

       \[\leadsto \color{blue}{\frac{\frac{1}{k}}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \]
   15. Step-by-step derivation

       1. add-sqr-sqrt 32.1%

          \[\leadsto \frac{\frac{1}{k}}{\color{blue}{\sqrt{{t}^{3} \cdot \frac{k}{{\ell}^{2}}} \cdot \sqrt{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}}} \]

       2. pow2 32.1%

          \[\leadsto \frac{\frac{1}{k}}{\color{blue}{{\left(\sqrt{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}\right)}^{2}}} \]

       3. sqrt-prod 14.1%

          \[\leadsto \frac{\frac{1}{k}}{{\color{blue}{\left(\sqrt{{t}^{3}} \cdot \sqrt{\frac{k}{{\ell}^{2}}}\right)}}^{2}} \]

       4. sqrt-pow1 15.2%

          \[\leadsto \frac{\frac{1}{k}}{{\left(\color{blue}{{t}^{\left(\frac{3}{2}\right)}} \cdot \sqrt{\frac{k}{{\ell}^{2}}}\right)}^{2}} \]

       5. metadata-eval 15.2%

          \[\leadsto \frac{\frac{1}{k}}{{\left({t}^{\color{blue}{1.5}} \cdot \sqrt{\frac{k}{{\ell}^{2}}}\right)}^{2}} \]

       6. sqrt-div 15.2%

          \[\leadsto \frac{\frac{1}{k}}{{\left({t}^{1.5} \cdot \color{blue}{\frac{\sqrt{k}}{\sqrt{{\ell}^{2}}}}\right)}^{2}} \]

       7. unpow2 15.2%

          \[\leadsto \frac{\frac{1}{k}}{{\left({t}^{1.5} \cdot \frac{\sqrt{k}}{\sqrt{\color{blue}{\ell \cdot \ell}}}\right)}^{2}} \]

       8. sqrt-prod 8.3%

          \[\leadsto \frac{\frac{1}{k}}{{\left({t}^{1.5} \cdot \frac{\sqrt{k}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2}} \]

       9. add-sqr-sqrt 17.0%

          \[\leadsto \frac{\frac{1}{k}}{{\left({t}^{1.5} \cdot \frac{\sqrt{k}}{\color{blue}{\ell}}\right)}^{2}} \]

   16. Applied egg-rr 17.0%

       \[\leadsto \frac{\frac{1}{k}}{\color{blue}{{\left({t}^{1.5} \cdot \frac{\sqrt{k}}{\ell}\right)}^{2}}} \]

   if 3.1999999999999999e25 < k

   1. Initial program 47.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 47.6%

         \[\leadsto \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. sqr-neg 47.6%

         \[\leadsto \frac{2}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. *-commutative 47.6%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. associate-*l* 47.6%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      5. *-commutative 47.6%

         \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      6. sqr-neg 47.6%

         \[\leadsto \frac{2}{\left(\sin k \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

   3. Simplified 47.6%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 47.2%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]
   6. Step-by-step derivation

      1. *-commutative 47.2%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   7. Simplified 47.2%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   8. Taylor expanded in k around 0 50.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot {t}^{3}}{{\ell}^{2}}} \cdot \left(k \cdot 2\right)} \]
   9. Step-by-step derivation

      1. associate-/l* 50.1%

         \[\leadsto \frac{2}{\color{blue}{\frac{k}{\frac{{\ell}^{2}}{{t}^{3}}}} \cdot \left(k \cdot 2\right)} \]

   10. Simplified 50.1%

       \[\leadsto \frac{2}{\color{blue}{\frac{k}{\frac{{\ell}^{2}}{{t}^{3}}}} \cdot \left(k \cdot 2\right)} \]
   11. Step-by-step derivation

       1. div-inv 50.1%

          \[\leadsto \color{blue}{2 \cdot \frac{1}{\frac{k}{\frac{{\ell}^{2}}{{t}^{3}}} \cdot \left(k \cdot 2\right)}} \]

       2. associate-/r/ 50.0%

          \[\leadsto 2 \cdot \frac{1}{\color{blue}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right)} \cdot \left(k \cdot 2\right)} \]

   12. Applied egg-rr 50.0%

       \[\leadsto \color{blue}{2 \cdot \frac{1}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right) \cdot \left(k \cdot 2\right)}} \]
   13. Step-by-step derivation

       1. associate-*r/ 50.0%

          \[\leadsto \color{blue}{\frac{2 \cdot 1}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right) \cdot \left(k \cdot 2\right)}} \]

       2. metadata-eval 50.0%

          \[\leadsto \frac{\color{blue}{2}}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right) \cdot \left(k \cdot 2\right)} \]

       3. *-commutative 50.0%

          \[\leadsto \frac{2}{\color{blue}{\left(k \cdot 2\right) \cdot \left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right)}} \]

       4. associate-/r* 50.0%

          \[\leadsto \color{blue}{\frac{\frac{2}{k \cdot 2}}{\frac{k}{{\ell}^{2}} \cdot {t}^{3}}} \]

       5. *-commutative 50.0%

          \[\leadsto \frac{\frac{2}{\color{blue}{2 \cdot k}}}{\frac{k}{{\ell}^{2}} \cdot {t}^{3}} \]

       6. associate-/r* 50.0%

          \[\leadsto \frac{\color{blue}{\frac{\frac{2}{2}}{k}}}{\frac{k}{{\ell}^{2}} \cdot {t}^{3}} \]

       7. metadata-eval 50.0%

          \[\leadsto \frac{\frac{\color{blue}{1}}{k}}{\frac{k}{{\ell}^{2}} \cdot {t}^{3}} \]

       8. *-commutative 50.0%

          \[\leadsto \frac{\frac{1}{k}}{\color{blue}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \]

   14. Simplified 50.0%

       \[\leadsto \color{blue}{\frac{\frac{1}{k}}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \]
   15. Step-by-step derivation

       1. add-cube-cbrt 50.0%

          \[\leadsto \frac{\frac{1}{k}}{\color{blue}{\left(\sqrt[3]{{t}^{3} \cdot \frac{k}{{\ell}^{2}}} \cdot \sqrt[3]{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}\right) \cdot \sqrt[3]{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}}} \]

       2. pow3 50.0%

          \[\leadsto \frac{\frac{1}{k}}{\color{blue}{{\left(\sqrt[3]{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}\right)}^{3}}} \]

       3. cbrt-prod 50.0%

          \[\leadsto \frac{\frac{1}{k}}{{\color{blue}{\left(\sqrt[3]{{t}^{3}} \cdot \sqrt[3]{\frac{k}{{\ell}^{2}}}\right)}}^{3}} \]

       4. rem-cbrt-cube 58.4%

          \[\leadsto \frac{\frac{1}{k}}{{\left(\color{blue}{t} \cdot \sqrt[3]{\frac{k}{{\ell}^{2}}}\right)}^{3}} \]

       5. div-inv 58.4%

          \[\leadsto \frac{\frac{1}{k}}{{\left(t \cdot \sqrt[3]{\color{blue}{k \cdot \frac{1}{{\ell}^{2}}}}\right)}^{3}} \]

       6. pow-flip 58.4%

          \[\leadsto \frac{\frac{1}{k}}{{\left(t \cdot \sqrt[3]{k \cdot \color{blue}{{\ell}^{\left(-2\right)}}}\right)}^{3}} \]

       7. metadata-eval 58.4%

          \[\leadsto \frac{\frac{1}{k}}{{\left(t \cdot \sqrt[3]{k \cdot {\ell}^{\color{blue}{-2}}}\right)}^{3}} \]

   16. Applied egg-rr 58.4%

       \[\leadsto \frac{\frac{1}{k}}{\color{blue}{{\left(t \cdot \sqrt[3]{k \cdot {\ell}^{-2}}\right)}^{3}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 27.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.2 \cdot 10^{+25}:\\ \;\;\;\;\frac{\frac{1}{k}}{{\left({t}^{1.5} \cdot \frac{\sqrt{k}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{k}}{{\left(t \cdot \sqrt[3]{k \cdot {\ell}^{-2}}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 59.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.35 \cdot 10^{+166}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot \frac{t\_m \cdot \frac{{t\_m}^{2}}{\ell}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{2} \cdot {t\_m}^{3}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.35e+166)
    (/ 2.0 (* (* 2.0 k) (* (sin k) (/ (* t_m (/ (pow t_m 2.0) l)) l))))
    (/ (pow l 2.0) (* (pow k 2.0) (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 (k <= 1.35e+166) {
		tmp = 2.0 / ((2.0 * k) * (sin(k) * ((t_m * (pow(t_m, 2.0) / l)) / l)));
	} else {
		tmp = pow(l, 2.0) / (pow(k, 2.0) * 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 (k <= 1.35d+166) then
        tmp = 2.0d0 / ((2.0d0 * k) * (sin(k) * ((t_m * ((t_m ** 2.0d0) / l)) / l)))
    else
        tmp = (l ** 2.0d0) / ((k ** 2.0d0) * (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 (k <= 1.35e+166) {
		tmp = 2.0 / ((2.0 * k) * (Math.sin(k) * ((t_m * (Math.pow(t_m, 2.0) / l)) / l)));
	} else {
		tmp = Math.pow(l, 2.0) / (Math.pow(k, 2.0) * 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 k <= 1.35e+166:
		tmp = 2.0 / ((2.0 * k) * (math.sin(k) * ((t_m * (math.pow(t_m, 2.0) / l)) / l)))
	else:
		tmp = math.pow(l, 2.0) / (math.pow(k, 2.0) * 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 (k <= 1.35e+166)
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * Float64(sin(k) * Float64(Float64(t_m * Float64((t_m ^ 2.0) / l)) / l))));
	else
		tmp = Float64((l ^ 2.0) / Float64((k ^ 2.0) * (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 (k <= 1.35e+166)
		tmp = 2.0 / ((2.0 * k) * (sin(k) * ((t_m * ((t_m ^ 2.0) / l)) / l)));
	else
		tmp = (l ^ 2.0) / ((k ^ 2.0) * (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[k, 1.35e+166], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(t$95$m * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[l, 2.0], $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.35000000000000006e166

   1. Initial program 57.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 57.9%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. sqr-neg 57.9%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. sqr-neg 57.9%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-/r* 62.6%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. distribute-rgt-in 62.6%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

      6. unpow2 62.6%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      7. times-frac 49.0%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      8. sqr-neg 49.0%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      9. times-frac 62.6%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      10. unpow2 62.6%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      11. distribute-rgt-in 62.6%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 62.6%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. cube-mult 62.6%

         \[\leadsto \frac{2}{\left(\frac{\frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      2. *-un-lft-identity 62.6%

         \[\leadsto \frac{2}{\left(\frac{\frac{t \cdot \left(t \cdot t\right)}{\color{blue}{1 \cdot \ell}}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      3. times-frac 68.6%

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\frac{t}{1} \cdot \frac{t \cdot t}{\ell}}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      4. pow2 68.6%

         \[\leadsto \frac{2}{\left(\frac{\frac{t}{1} \cdot \frac{\color{blue}{{t}^{2}}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   6. Applied egg-rr 68.6%

      \[\leadsto \frac{2}{\left(\frac{\color{blue}{\frac{t}{1} \cdot \frac{{t}^{2}}{\ell}}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   7. Taylor expanded in k around 0 65.6%

      \[\leadsto \frac{2}{\left(\frac{\frac{t}{1} \cdot \frac{{t}^{2}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]

   if 1.35000000000000006e166 < k

   1. Initial program 53.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 53.6%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. sqr-neg 53.6%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. associate-*l* 53.6%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. sqr-neg 53.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. associate-/r* 60.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. associate-+l+ 60.7%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{\color{blue}{1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)}} \]

      7. unpow2 60.7%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right)} \]

      8. times-frac 42.9%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k \cdot k}{t \cdot t}} + 1\right)} \]

      9. sqr-neg 42.9%

         \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}} + 1\right)} \]

      10. times-frac 60.7%

          \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}} + 1\right)} \]

      11. unpow2 60.7%

          \[\leadsto \frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{1 + \left(\color{blue}{{\left(\frac{k}{-t}\right)}^{2}} + 1\right)} \]

   3. Simplified 60.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 54.1%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.35 \cdot 10^{+166}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot \frac{t \cdot \frac{{t}^{2}}{\ell}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 60.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.6 \cdot 10^{-21} \lor \neg \left(t\_m \leq 4.2 \cdot 10^{+106}\right):\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot \frac{t\_m \cdot \frac{{t\_m}^{2}}{\ell}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \frac{1}{\frac{{\ell}^{2}}{\frac{k}{{t\_m}^{-3}}}}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (or (<= t_m 7.6e-21) (not (<= t_m 4.2e+106)))
    (/ 2.0 (* (* 2.0 k) (* (sin k) (/ (* t_m (/ (pow t_m 2.0) l)) l))))
    (/ 2.0 (* (* 2.0 k) (/ 1.0 (/ (pow l 2.0) (/ k (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 <= 7.6e-21) || !(t_m <= 4.2e+106)) {
		tmp = 2.0 / ((2.0 * k) * (sin(k) * ((t_m * (pow(t_m, 2.0) / l)) / l)));
	} else {
		tmp = 2.0 / ((2.0 * k) * (1.0 / (pow(l, 2.0) / (k / 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 <= 7.6d-21) .or. (.not. (t_m <= 4.2d+106))) then
        tmp = 2.0d0 / ((2.0d0 * k) * (sin(k) * ((t_m * ((t_m ** 2.0d0) / l)) / l)))
    else
        tmp = 2.0d0 / ((2.0d0 * k) * (1.0d0 / ((l ** 2.0d0) / (k / (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 <= 7.6e-21) || !(t_m <= 4.2e+106)) {
		tmp = 2.0 / ((2.0 * k) * (Math.sin(k) * ((t_m * (Math.pow(t_m, 2.0) / l)) / l)));
	} else {
		tmp = 2.0 / ((2.0 * k) * (1.0 / (Math.pow(l, 2.0) / (k / 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 <= 7.6e-21) or not (t_m <= 4.2e+106):
		tmp = 2.0 / ((2.0 * k) * (math.sin(k) * ((t_m * (math.pow(t_m, 2.0) / l)) / l)))
	else:
		tmp = 2.0 / ((2.0 * k) * (1.0 / (math.pow(l, 2.0) / (k / 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 <= 7.6e-21) || !(t_m <= 4.2e+106))
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * Float64(sin(k) * Float64(Float64(t_m * Float64((t_m ^ 2.0) / l)) / l))));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * Float64(1.0 / Float64((l ^ 2.0) / Float64(k / (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 <= 7.6e-21) || ~((t_m <= 4.2e+106)))
		tmp = 2.0 / ((2.0 * k) * (sin(k) * ((t_m * ((t_m ^ 2.0) / l)) / l)));
	else
		tmp = 2.0 / ((2.0 * k) * (1.0 / ((l ^ 2.0) / (k / (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[Or[LessEqual[t$95$m, 7.6e-21], N[Not[LessEqual[t$95$m, 4.2e+106]], $MachinePrecision]], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(t$95$m * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[(1.0 / N[(N[Power[l, 2.0], $MachinePrecision] / N[(k / N[Power[t$95$m, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 7.5999999999999995e-21 or 4.2000000000000001e106 < t

   1. Initial program 55.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 55.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. sqr-neg 55.4%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. sqr-neg 55.4%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-/r* 60.5%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. distribute-rgt-in 60.5%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

      6. unpow2 60.5%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      7. times-frac 45.3%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      8. sqr-neg 45.3%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      9. times-frac 60.5%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      10. unpow2 60.5%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      11. distribute-rgt-in 60.5%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 60.5%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. cube-mult 60.5%

         \[\leadsto \frac{2}{\left(\frac{\frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      2. *-un-lft-identity 60.5%

         \[\leadsto \frac{2}{\left(\frac{\frac{t \cdot \left(t \cdot t\right)}{\color{blue}{1 \cdot \ell}}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      3. times-frac 66.9%

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\frac{t}{1} \cdot \frac{t \cdot t}{\ell}}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

      4. pow2 66.9%

         \[\leadsto \frac{2}{\left(\frac{\frac{t}{1} \cdot \frac{\color{blue}{{t}^{2}}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   6. Applied egg-rr 66.9%

      \[\leadsto \frac{2}{\left(\frac{\color{blue}{\frac{t}{1} \cdot \frac{{t}^{2}}{\ell}}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   7. Taylor expanded in k around 0 63.9%

      \[\leadsto \frac{2}{\left(\frac{\frac{t}{1} \cdot \frac{{t}^{2}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]

   if 7.5999999999999995e-21 < t < 4.2000000000000001e106

   1. Initial program 76.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 76.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. sqr-neg 76.3%

         \[\leadsto \frac{2}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. *-commutative 76.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. associate-*l* 76.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      5. *-commutative 76.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      6. sqr-neg 76.3%

         \[\leadsto \frac{2}{\left(\sin k \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

   3. Simplified 76.3%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 65.5%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]
   6. Step-by-step derivation

      1. *-commutative 65.5%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   7. Simplified 65.5%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   8. Taylor expanded in k around 0 80.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot {t}^{3}}{{\ell}^{2}}} \cdot \left(k \cdot 2\right)} \]
   9. Step-by-step derivation

      1. associate-/l* 68.9%

         \[\leadsto \frac{2}{\color{blue}{\frac{k}{\frac{{\ell}^{2}}{{t}^{3}}}} \cdot \left(k \cdot 2\right)} \]

   10. Simplified 68.9%

       \[\leadsto \frac{2}{\color{blue}{\frac{k}{\frac{{\ell}^{2}}{{t}^{3}}}} \cdot \left(k \cdot 2\right)} \]
   11. Step-by-step derivation

       1. clear-num 68.9%

          \[\leadsto \frac{2}{\color{blue}{\frac{1}{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{k}}} \cdot \left(k \cdot 2\right)} \]

       2. inv-pow 68.9%

          \[\leadsto \frac{2}{\color{blue}{{\left(\frac{\frac{{\ell}^{2}}{{t}^{3}}}{k}\right)}^{-1}} \cdot \left(k \cdot 2\right)} \]

       3. div-inv 68.9%

          \[\leadsto \frac{2}{{\left(\frac{\color{blue}{{\ell}^{2} \cdot \frac{1}{{t}^{3}}}}{k}\right)}^{-1} \cdot \left(k \cdot 2\right)} \]

       4. pow-flip 69.0%

          \[\leadsto \frac{2}{{\left(\frac{{\ell}^{2} \cdot \color{blue}{{t}^{\left(-3\right)}}}{k}\right)}^{-1} \cdot \left(k \cdot 2\right)} \]

       5. metadata-eval 69.0%

          \[\leadsto \frac{2}{{\left(\frac{{\ell}^{2} \cdot {t}^{\color{blue}{-3}}}{k}\right)}^{-1} \cdot \left(k \cdot 2\right)} \]

   12. Applied egg-rr 69.0%

       \[\leadsto \frac{2}{\color{blue}{{\left(\frac{{\ell}^{2} \cdot {t}^{-3}}{k}\right)}^{-1}} \cdot \left(k \cdot 2\right)} \]
   13. Step-by-step derivation

       1. unpow-1 69.0%

          \[\leadsto \frac{2}{\color{blue}{\frac{1}{\frac{{\ell}^{2} \cdot {t}^{-3}}{k}}} \cdot \left(k \cdot 2\right)} \]

       2. associate-/l* 80.2%

          \[\leadsto \frac{2}{\frac{1}{\color{blue}{\frac{{\ell}^{2}}{\frac{k}{{t}^{-3}}}}} \cdot \left(k \cdot 2\right)} \]

   14. Simplified 80.2%

       \[\leadsto \frac{2}{\color{blue}{\frac{1}{\frac{{\ell}^{2}}{\frac{k}{{t}^{-3}}}}} \cdot \left(k \cdot 2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.6 \cdot 10^{-21} \lor \neg \left(t \leq 4.2 \cdot 10^{+106}\right):\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot \frac{t \cdot \frac{{t}^{2}}{\ell}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \frac{1}{\frac{{\ell}^{2}}{\frac{k}{{t}^{-3}}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 55.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 6.5 \cdot 10^{+245}:\\ \;\;\;\;\frac{\frac{1}{k}}{\frac{k \cdot {t\_m}^{3}}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot \frac{{t\_m}^{3}}{\ell \cdot \ell}\right)}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= l 6.5e+245)
    (/ (/ 1.0 k) (/ (* k (pow t_m 3.0)) (pow l 2.0)))
    (/ 2.0 (* (* 2.0 k) (* (sin k) (/ (pow t_m 3.0) (* l 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 (l <= 6.5e+245) {
		tmp = (1.0 / k) / ((k * pow(t_m, 3.0)) / pow(l, 2.0));
	} else {
		tmp = 2.0 / ((2.0 * k) * (sin(k) * (pow(t_m, 3.0) / (l * l))));
	}
	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 (l <= 6.5d+245) then
        tmp = (1.0d0 / k) / ((k * (t_m ** 3.0d0)) / (l ** 2.0d0))
    else
        tmp = 2.0d0 / ((2.0d0 * k) * (sin(k) * ((t_m ** 3.0d0) / (l * l))))
    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 (l <= 6.5e+245) {
		tmp = (1.0 / k) / ((k * Math.pow(t_m, 3.0)) / Math.pow(l, 2.0));
	} else {
		tmp = 2.0 / ((2.0 * k) * (Math.sin(k) * (Math.pow(t_m, 3.0) / (l * l))));
	}
	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 l <= 6.5e+245:
		tmp = (1.0 / k) / ((k * math.pow(t_m, 3.0)) / math.pow(l, 2.0))
	else:
		tmp = 2.0 / ((2.0 * k) * (math.sin(k) * (math.pow(t_m, 3.0) / (l * 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 (l <= 6.5e+245)
		tmp = Float64(Float64(1.0 / k) / Float64(Float64(k * (t_m ^ 3.0)) / (l ^ 2.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * Float64(sin(k) * Float64((t_m ^ 3.0) / Float64(l * l)))));
	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 (l <= 6.5e+245)
		tmp = (1.0 / k) / ((k * (t_m ^ 3.0)) / (l ^ 2.0));
	else
		tmp = 2.0 / ((2.0 * k) * (sin(k) * ((t_m ^ 3.0) / (l * l))));
	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[l, 6.5e+245], N[(N[(1.0 / k), $MachinePrecision] / N[(N[(k * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 6.50000000000000035e245

   1. Initial program 58.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 58.6%

         \[\leadsto \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. sqr-neg 58.6%

         \[\leadsto \frac{2}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. *-commutative 58.6%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. associate-*l* 59.0%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      5. *-commutative 59.0%

         \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      6. sqr-neg 59.0%

         \[\leadsto \frac{2}{\left(\sin k \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

   3. Simplified 59.0%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 56.6%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]
   6. Step-by-step derivation

      1. *-commutative 56.6%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   7. Simplified 56.6%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   8. Taylor expanded in k around 0 59.4%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot {t}^{3}}{{\ell}^{2}}} \cdot \left(k \cdot 2\right)} \]
   9. Step-by-step derivation

      1. associate-/l* 58.5%

         \[\leadsto \frac{2}{\color{blue}{\frac{k}{\frac{{\ell}^{2}}{{t}^{3}}}} \cdot \left(k \cdot 2\right)} \]

   10. Simplified 58.5%

       \[\leadsto \frac{2}{\color{blue}{\frac{k}{\frac{{\ell}^{2}}{{t}^{3}}}} \cdot \left(k \cdot 2\right)} \]
   11. Step-by-step derivation

       1. div-inv 58.5%

          \[\leadsto \color{blue}{2 \cdot \frac{1}{\frac{k}{\frac{{\ell}^{2}}{{t}^{3}}} \cdot \left(k \cdot 2\right)}} \]

       2. associate-/r/ 58.5%

          \[\leadsto 2 \cdot \frac{1}{\color{blue}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right)} \cdot \left(k \cdot 2\right)} \]

   12. Applied egg-rr 58.5%

       \[\leadsto \color{blue}{2 \cdot \frac{1}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right) \cdot \left(k \cdot 2\right)}} \]
   13. Step-by-step derivation

       1. associate-*r/ 58.5%

          \[\leadsto \color{blue}{\frac{2 \cdot 1}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right) \cdot \left(k \cdot 2\right)}} \]

       2. metadata-eval 58.5%

          \[\leadsto \frac{\color{blue}{2}}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right) \cdot \left(k \cdot 2\right)} \]

       3. *-commutative 58.5%

          \[\leadsto \frac{2}{\color{blue}{\left(k \cdot 2\right) \cdot \left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right)}} \]

       4. associate-/r* 58.5%

          \[\leadsto \color{blue}{\frac{\frac{2}{k \cdot 2}}{\frac{k}{{\ell}^{2}} \cdot {t}^{3}}} \]

       5. *-commutative 58.5%

          \[\leadsto \frac{\frac{2}{\color{blue}{2 \cdot k}}}{\frac{k}{{\ell}^{2}} \cdot {t}^{3}} \]

       6. associate-/r* 58.5%

          \[\leadsto \frac{\color{blue}{\frac{\frac{2}{2}}{k}}}{\frac{k}{{\ell}^{2}} \cdot {t}^{3}} \]

       7. metadata-eval 58.5%

          \[\leadsto \frac{\frac{\color{blue}{1}}{k}}{\frac{k}{{\ell}^{2}} \cdot {t}^{3}} \]

       8. *-commutative 58.5%

          \[\leadsto \frac{\frac{1}{k}}{\color{blue}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \]

   14. Simplified 58.5%

       \[\leadsto \color{blue}{\frac{\frac{1}{k}}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \]
   15. Step-by-step derivation

       1. associate-*r/ 59.4%

          \[\leadsto \frac{\frac{1}{k}}{\color{blue}{\frac{{t}^{3} \cdot k}{{\ell}^{2}}}} \]

   16. Applied egg-rr 59.4%

       \[\leadsto \frac{\frac{1}{k}}{\color{blue}{\frac{{t}^{3} \cdot k}{{\ell}^{2}}}} \]

   if 6.50000000000000035e245 < l

   1. Initial program 37.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 37.2%

         \[\leadsto \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. sqr-neg 37.2%

         \[\leadsto \frac{2}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. *-commutative 37.2%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. associate-*l* 37.2%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      5. *-commutative 37.2%

         \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      6. sqr-neg 37.2%

         \[\leadsto \frac{2}{\left(\sin k \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

   3. Simplified 37.2%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 52.6%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]
   6. Step-by-step derivation

      1. *-commutative 52.6%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   7. Simplified 52.6%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 6.5 \cdot 10^{+245}:\\ \;\;\;\;\frac{\frac{1}{k}}{\frac{k \cdot {t}^{3}}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 59.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.1 \cdot 10^{+166}:\\ \;\;\;\;\frac{2}{\frac{\sin k \cdot \frac{{t\_m}^{3}}{\ell}}{\ell} \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(2 \cdot k\right)}{{\ell}^{2} \cdot {t\_m}^{-3}}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 3.1e+166)
    (/ 2.0 (* (/ (* (sin k) (/ (pow t_m 3.0) l)) l) (* 2.0 k)))
    (/ 2.0 (/ (* k (* 2.0 k)) (* (pow l 2.0) (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 (k <= 3.1e+166) {
		tmp = 2.0 / (((sin(k) * (pow(t_m, 3.0) / l)) / l) * (2.0 * k));
	} else {
		tmp = 2.0 / ((k * (2.0 * k)) / (pow(l, 2.0) * 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 (k <= 3.1d+166) then
        tmp = 2.0d0 / (((sin(k) * ((t_m ** 3.0d0) / l)) / l) * (2.0d0 * k))
    else
        tmp = 2.0d0 / ((k * (2.0d0 * k)) / ((l ** 2.0d0) * (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 (k <= 3.1e+166) {
		tmp = 2.0 / (((Math.sin(k) * (Math.pow(t_m, 3.0) / l)) / l) * (2.0 * k));
	} else {
		tmp = 2.0 / ((k * (2.0 * k)) / (Math.pow(l, 2.0) * 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 k <= 3.1e+166:
		tmp = 2.0 / (((math.sin(k) * (math.pow(t_m, 3.0) / l)) / l) * (2.0 * k))
	else:
		tmp = 2.0 / ((k * (2.0 * k)) / (math.pow(l, 2.0) * 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 (k <= 3.1e+166)
		tmp = Float64(2.0 / Float64(Float64(Float64(sin(k) * Float64((t_m ^ 3.0) / l)) / l) * Float64(2.0 * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(k * Float64(2.0 * k)) / Float64((l ^ 2.0) * (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 (k <= 3.1e+166)
		tmp = 2.0 / (((sin(k) * ((t_m ^ 3.0) / l)) / l) * (2.0 * k));
	else
		tmp = 2.0 / ((k * (2.0 * k)) / ((l ^ 2.0) * (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[k, 3.1e+166], N[(2.0 / N[(N[(N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * N[(2.0 * k), $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Power[t$95$m, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.09999999999999983e166

   1. Initial program 57.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 57.9%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. sqr-neg 57.9%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. sqr-neg 57.9%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-/r* 62.6%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. distribute-rgt-in 62.6%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

      6. unpow2 62.6%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      7. times-frac 49.0%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      8. sqr-neg 49.0%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      9. times-frac 62.6%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      10. unpow2 62.6%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      11. distribute-rgt-in 62.6%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 62.6%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l/ 63.9%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell}} \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   6. Applied egg-rr 63.9%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell}} \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   7. Taylor expanded in k around 0 63.0%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k}{\ell} \cdot \color{blue}{\left(2 \cdot k\right)}} \]

   if 3.09999999999999983e166 < k

   1. Initial program 53.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 53.6%

         \[\leadsto \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. sqr-neg 53.6%

         \[\leadsto \frac{2}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. *-commutative 53.6%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. associate-*l* 53.6%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      5. *-commutative 53.6%

         \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      6. sqr-neg 53.6%

         \[\leadsto \frac{2}{\left(\sin k \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

   3. Simplified 53.6%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 44.4%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]
   6. Step-by-step derivation

      1. *-commutative 44.4%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   7. Simplified 44.4%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   8. Taylor expanded in k around 0 54.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot {t}^{3}}{{\ell}^{2}}} \cdot \left(k \cdot 2\right)} \]
   9. Step-by-step derivation

      1. associate-/l* 54.1%

         \[\leadsto \frac{2}{\color{blue}{\frac{k}{\frac{{\ell}^{2}}{{t}^{3}}}} \cdot \left(k \cdot 2\right)} \]

   10. Simplified 54.1%

       \[\leadsto \frac{2}{\color{blue}{\frac{k}{\frac{{\ell}^{2}}{{t}^{3}}}} \cdot \left(k \cdot 2\right)} \]
   11. Step-by-step derivation

       1. associate-*l/ 53.6%

          \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot \left(k \cdot 2\right)}{\frac{{\ell}^{2}}{{t}^{3}}}}} \]

       2. div-inv 53.6%

          \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot 2\right)}{\color{blue}{{\ell}^{2} \cdot \frac{1}{{t}^{3}}}}} \]

       3. pow-flip 53.6%

          \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot 2\right)}{{\ell}^{2} \cdot \color{blue}{{t}^{\left(-3\right)}}}} \]

       4. metadata-eval 53.6%

          \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot 2\right)}{{\ell}^{2} \cdot {t}^{\color{blue}{-3}}}} \]

   12. Applied egg-rr 53.6%

       \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot \left(k \cdot 2\right)}{{\ell}^{2} \cdot {t}^{-3}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.1 \cdot 10^{+166}:\\ \;\;\;\;\frac{2}{\frac{\sin k \cdot \frac{{t}^{3}}{\ell}}{\ell} \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(2 \cdot k\right)}{{\ell}^{2} \cdot {t}^{-3}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 54.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{\frac{1}{k}}{k \cdot \left({t\_m}^{3} \cdot {\ell}^{-2}\right)} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ (/ 1.0 k) (* k (* (pow t_m 3.0) (pow l -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) {
	return t_s * ((1.0 / k) / (k * (pow(t_m, 3.0) * pow(l, -2.0))));
}

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 * ((1.0d0 / k) / (k * ((t_m ** 3.0d0) * (l ** (-2.0d0)))))
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 * ((1.0 / k) / (k * (Math.pow(t_m, 3.0) * Math.pow(l, -2.0))));
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * ((1.0 / k) / (k * (math.pow(t_m, 3.0) * math.pow(l, -2.0))))

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(1.0 / k) / Float64(k * Float64((t_m ^ 3.0) * (l ^ -2.0)))))
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 * ((1.0 / k) / (k * ((t_m ^ 3.0) * (l ^ -2.0))));
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[(1.0 / k), $MachinePrecision] / N[(k * N[(N[Power[t$95$m, 3.0], $MachinePrecision] * N[Power[l, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.0%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. *-commutative 57.0%

      \[\leadsto \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. sqr-neg 57.0%

      \[\leadsto \frac{2}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. *-commutative 57.0%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. associate-*l* 57.4%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   5. *-commutative 57.4%

      \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

   6. sqr-neg 57.4%

      \[\leadsto \frac{2}{\left(\sin k \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

3. Simplified 57.4%

   \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in k around 0 56.3%

   \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]
6. Step-by-step derivation

   1. *-commutative 56.3%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]

7. Simplified 56.3%

   \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]

8. Taylor expanded in k around 0 58.1%

   \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot {t}^{3}}{{\ell}^{2}}} \cdot \left(k \cdot 2\right)} \]
9. Step-by-step derivation

   1. associate-/l* 57.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{k}{\frac{{\ell}^{2}}{{t}^{3}}}} \cdot \left(k \cdot 2\right)} \]

10. Simplified 57.3%

    \[\leadsto \frac{2}{\color{blue}{\frac{k}{\frac{{\ell}^{2}}{{t}^{3}}}} \cdot \left(k \cdot 2\right)} \]
11. Step-by-step derivation

    1. div-inv 57.3%

       \[\leadsto \color{blue}{2 \cdot \frac{1}{\frac{k}{\frac{{\ell}^{2}}{{t}^{3}}} \cdot \left(k \cdot 2\right)}} \]

    2. associate-/r/ 57.3%

       \[\leadsto 2 \cdot \frac{1}{\color{blue}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right)} \cdot \left(k \cdot 2\right)} \]

12. Applied egg-rr 57.3%

    \[\leadsto \color{blue}{2 \cdot \frac{1}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right) \cdot \left(k \cdot 2\right)}} \]
13. Step-by-step derivation

    1. associate-*r/ 57.3%

       \[\leadsto \color{blue}{\frac{2 \cdot 1}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right) \cdot \left(k \cdot 2\right)}} \]

    2. metadata-eval 57.3%

       \[\leadsto \frac{\color{blue}{2}}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right) \cdot \left(k \cdot 2\right)} \]

    3. *-commutative 57.3%

       \[\leadsto \frac{2}{\color{blue}{\left(k \cdot 2\right) \cdot \left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right)}} \]

    4. associate-/r* 57.3%

       \[\leadsto \color{blue}{\frac{\frac{2}{k \cdot 2}}{\frac{k}{{\ell}^{2}} \cdot {t}^{3}}} \]

    5. *-commutative 57.3%

       \[\leadsto \frac{\frac{2}{\color{blue}{2 \cdot k}}}{\frac{k}{{\ell}^{2}} \cdot {t}^{3}} \]

    6. associate-/r* 57.3%

       \[\leadsto \frac{\color{blue}{\frac{\frac{2}{2}}{k}}}{\frac{k}{{\ell}^{2}} \cdot {t}^{3}} \]

    7. metadata-eval 57.3%

       \[\leadsto \frac{\frac{\color{blue}{1}}{k}}{\frac{k}{{\ell}^{2}} \cdot {t}^{3}} \]

    8. *-commutative 57.3%

       \[\leadsto \frac{\frac{1}{k}}{\color{blue}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \]

14. Simplified 57.3%

    \[\leadsto \color{blue}{\frac{\frac{1}{k}}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \]
15. Step-by-step derivation

    1. pow1 57.3%

       \[\leadsto \frac{\frac{1}{k}}{\color{blue}{{\left({t}^{3} \cdot \frac{k}{{\ell}^{2}}\right)}^{1}}} \]

    2. div-inv 56.8%

       \[\leadsto \frac{\frac{1}{k}}{{\left({t}^{3} \cdot \color{blue}{\left(k \cdot \frac{1}{{\ell}^{2}}\right)}\right)}^{1}} \]

    3. pow-flip 56.8%

       \[\leadsto \frac{\frac{1}{k}}{{\left({t}^{3} \cdot \left(k \cdot \color{blue}{{\ell}^{\left(-2\right)}}\right)\right)}^{1}} \]

    4. metadata-eval 56.8%

       \[\leadsto \frac{\frac{1}{k}}{{\left({t}^{3} \cdot \left(k \cdot {\ell}^{\color{blue}{-2}}\right)\right)}^{1}} \]

16. Applied egg-rr 56.8%

    \[\leadsto \frac{\frac{1}{k}}{\color{blue}{{\left({t}^{3} \cdot \left(k \cdot {\ell}^{-2}\right)\right)}^{1}}} \]
17. Step-by-step derivation

    1. unpow1 56.8%

       \[\leadsto \frac{\frac{1}{k}}{\color{blue}{{t}^{3} \cdot \left(k \cdot {\ell}^{-2}\right)}} \]

    2. *-commutative 56.8%

       \[\leadsto \frac{\frac{1}{k}}{\color{blue}{\left(k \cdot {\ell}^{-2}\right) \cdot {t}^{3}}} \]

    3. associate-*l* 56.8%

       \[\leadsto \frac{\frac{1}{k}}{\color{blue}{k \cdot \left({\ell}^{-2} \cdot {t}^{3}\right)}} \]

    4. *-commutative 56.8%

       \[\leadsto \frac{\frac{1}{k}}{k \cdot \color{blue}{\left({t}^{3} \cdot {\ell}^{-2}\right)}} \]

18. Simplified 56.8%

    \[\leadsto \frac{\frac{1}{k}}{\color{blue}{k \cdot \left({t}^{3} \cdot {\ell}^{-2}\right)}} \]

19. Final simplification 56.8%

    \[\leadsto \frac{\frac{1}{k}}{k \cdot \left({t}^{3} \cdot {\ell}^{-2}\right)} \]
20. Add Preprocessing

Alternative 21: 55.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{\frac{1}{k}}{{t\_m}^{3} \cdot \left(k \cdot {\ell}^{-2}\right)} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ (/ 1.0 k) (* (pow t_m 3.0) (* k (pow l -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) {
	return t_s * ((1.0 / k) / (pow(t_m, 3.0) * (k * pow(l, -2.0))));
}

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 * ((1.0d0 / k) / ((t_m ** 3.0d0) * (k * (l ** (-2.0d0)))))
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 * ((1.0 / k) / (Math.pow(t_m, 3.0) * (k * Math.pow(l, -2.0))));
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * ((1.0 / k) / (math.pow(t_m, 3.0) * (k * math.pow(l, -2.0))))

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(1.0 / k) / Float64((t_m ^ 3.0) * Float64(k * (l ^ -2.0)))))
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 * ((1.0 / k) / ((t_m ^ 3.0) * (k * (l ^ -2.0))));
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[(1.0 / k), $MachinePrecision] / N[(N[Power[t$95$m, 3.0], $MachinePrecision] * N[(k * N[Power[l, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.0%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. *-commutative 57.0%

      \[\leadsto \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. sqr-neg 57.0%

      \[\leadsto \frac{2}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. *-commutative 57.0%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. associate-*l* 57.4%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   5. *-commutative 57.4%

      \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

   6. sqr-neg 57.4%

      \[\leadsto \frac{2}{\left(\sin k \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

3. Simplified 57.4%

   \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in k around 0 56.3%

   \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]
6. Step-by-step derivation

   1. *-commutative 56.3%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]

7. Simplified 56.3%

   \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]

8. Taylor expanded in k around 0 58.1%

   \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot {t}^{3}}{{\ell}^{2}}} \cdot \left(k \cdot 2\right)} \]
9. Step-by-step derivation

   1. associate-/l* 57.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{k}{\frac{{\ell}^{2}}{{t}^{3}}}} \cdot \left(k \cdot 2\right)} \]

10. Simplified 57.3%

    \[\leadsto \frac{2}{\color{blue}{\frac{k}{\frac{{\ell}^{2}}{{t}^{3}}}} \cdot \left(k \cdot 2\right)} \]
11. Step-by-step derivation

    1. div-inv 57.3%

       \[\leadsto \color{blue}{2 \cdot \frac{1}{\frac{k}{\frac{{\ell}^{2}}{{t}^{3}}} \cdot \left(k \cdot 2\right)}} \]

    2. associate-/r/ 57.3%

       \[\leadsto 2 \cdot \frac{1}{\color{blue}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right)} \cdot \left(k \cdot 2\right)} \]

12. Applied egg-rr 57.3%

    \[\leadsto \color{blue}{2 \cdot \frac{1}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right) \cdot \left(k \cdot 2\right)}} \]
13. Step-by-step derivation

    1. associate-*r/ 57.3%

       \[\leadsto \color{blue}{\frac{2 \cdot 1}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right) \cdot \left(k \cdot 2\right)}} \]

    2. metadata-eval 57.3%

       \[\leadsto \frac{\color{blue}{2}}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right) \cdot \left(k \cdot 2\right)} \]

    3. *-commutative 57.3%

       \[\leadsto \frac{2}{\color{blue}{\left(k \cdot 2\right) \cdot \left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right)}} \]

    4. associate-/r* 57.3%

       \[\leadsto \color{blue}{\frac{\frac{2}{k \cdot 2}}{\frac{k}{{\ell}^{2}} \cdot {t}^{3}}} \]

    5. *-commutative 57.3%

       \[\leadsto \frac{\frac{2}{\color{blue}{2 \cdot k}}}{\frac{k}{{\ell}^{2}} \cdot {t}^{3}} \]

    6. associate-/r* 57.3%

       \[\leadsto \frac{\color{blue}{\frac{\frac{2}{2}}{k}}}{\frac{k}{{\ell}^{2}} \cdot {t}^{3}} \]

    7. metadata-eval 57.3%

       \[\leadsto \frac{\frac{\color{blue}{1}}{k}}{\frac{k}{{\ell}^{2}} \cdot {t}^{3}} \]

    8. *-commutative 57.3%

       \[\leadsto \frac{\frac{1}{k}}{\color{blue}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \]

14. Simplified 57.3%

    \[\leadsto \color{blue}{\frac{\frac{1}{k}}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \]
15. Step-by-step derivation

    1. div-inv 56.8%

       \[\leadsto \frac{\frac{1}{k}}{{t}^{3} \cdot \color{blue}{\left(k \cdot \frac{1}{{\ell}^{2}}\right)}} \]

    2. pow-flip 56.8%

       \[\leadsto \frac{\frac{1}{k}}{{t}^{3} \cdot \left(k \cdot \color{blue}{{\ell}^{\left(-2\right)}}\right)} \]

    3. metadata-eval 56.8%

       \[\leadsto \frac{\frac{1}{k}}{{t}^{3} \cdot \left(k \cdot {\ell}^{\color{blue}{-2}}\right)} \]

16. Applied egg-rr 56.8%

    \[\leadsto \frac{\frac{1}{k}}{{t}^{3} \cdot \color{blue}{\left(k \cdot {\ell}^{-2}\right)}} \]

17. Final simplification 56.8%

    \[\leadsto \frac{\frac{1}{k}}{{t}^{3} \cdot \left(k \cdot {\ell}^{-2}\right)} \]
18. Add Preprocessing

Alternative 22: 55.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{\frac{1}{k}}{{t\_m}^{3} \cdot \frac{k}{{\ell}^{2}}} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ (/ 1.0 k) (* (pow t_m 3.0) (/ k (pow l 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) {
	return t_s * ((1.0 / k) / (pow(t_m, 3.0) * (k / pow(l, 2.0))));
}

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 * ((1.0d0 / k) / ((t_m ** 3.0d0) * (k / (l ** 2.0d0))))
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 * ((1.0 / k) / (Math.pow(t_m, 3.0) * (k / Math.pow(l, 2.0))));
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * ((1.0 / k) / (math.pow(t_m, 3.0) * (k / math.pow(l, 2.0))))

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(1.0 / k) / Float64((t_m ^ 3.0) * Float64(k / (l ^ 2.0)))))
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 * ((1.0 / k) / ((t_m ^ 3.0) * (k / (l ^ 2.0))));
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[(1.0 / k), $MachinePrecision] / N[(N[Power[t$95$m, 3.0], $MachinePrecision] * N[(k / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.0%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. *-commutative 57.0%

      \[\leadsto \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. sqr-neg 57.0%

      \[\leadsto \frac{2}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. *-commutative 57.0%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. associate-*l* 57.4%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   5. *-commutative 57.4%

      \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

   6. sqr-neg 57.4%

      \[\leadsto \frac{2}{\left(\sin k \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

3. Simplified 57.4%

   \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in k around 0 56.3%

   \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]
6. Step-by-step derivation

   1. *-commutative 56.3%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]

7. Simplified 56.3%

   \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]

8. Taylor expanded in k around 0 58.1%

   \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot {t}^{3}}{{\ell}^{2}}} \cdot \left(k \cdot 2\right)} \]
9. Step-by-step derivation

   1. associate-/l* 57.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{k}{\frac{{\ell}^{2}}{{t}^{3}}}} \cdot \left(k \cdot 2\right)} \]

10. Simplified 57.3%

    \[\leadsto \frac{2}{\color{blue}{\frac{k}{\frac{{\ell}^{2}}{{t}^{3}}}} \cdot \left(k \cdot 2\right)} \]
11. Step-by-step derivation

    1. div-inv 57.3%

       \[\leadsto \color{blue}{2 \cdot \frac{1}{\frac{k}{\frac{{\ell}^{2}}{{t}^{3}}} \cdot \left(k \cdot 2\right)}} \]

    2. associate-/r/ 57.3%

       \[\leadsto 2 \cdot \frac{1}{\color{blue}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right)} \cdot \left(k \cdot 2\right)} \]

12. Applied egg-rr 57.3%

    \[\leadsto \color{blue}{2 \cdot \frac{1}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right) \cdot \left(k \cdot 2\right)}} \]
13. Step-by-step derivation

    1. associate-*r/ 57.3%

       \[\leadsto \color{blue}{\frac{2 \cdot 1}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right) \cdot \left(k \cdot 2\right)}} \]

    2. metadata-eval 57.3%

       \[\leadsto \frac{\color{blue}{2}}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right) \cdot \left(k \cdot 2\right)} \]

    3. *-commutative 57.3%

       \[\leadsto \frac{2}{\color{blue}{\left(k \cdot 2\right) \cdot \left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right)}} \]

    4. associate-/r* 57.3%

       \[\leadsto \color{blue}{\frac{\frac{2}{k \cdot 2}}{\frac{k}{{\ell}^{2}} \cdot {t}^{3}}} \]

    5. *-commutative 57.3%

       \[\leadsto \frac{\frac{2}{\color{blue}{2 \cdot k}}}{\frac{k}{{\ell}^{2}} \cdot {t}^{3}} \]

    6. associate-/r* 57.3%

       \[\leadsto \frac{\color{blue}{\frac{\frac{2}{2}}{k}}}{\frac{k}{{\ell}^{2}} \cdot {t}^{3}} \]

    7. metadata-eval 57.3%

       \[\leadsto \frac{\frac{\color{blue}{1}}{k}}{\frac{k}{{\ell}^{2}} \cdot {t}^{3}} \]

    8. *-commutative 57.3%

       \[\leadsto \frac{\frac{1}{k}}{\color{blue}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \]

14. Simplified 57.3%

    \[\leadsto \color{blue}{\frac{\frac{1}{k}}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \]

15. Final simplification 57.3%

    \[\leadsto \frac{\frac{1}{k}}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}} \]
16. Add Preprocessing

Alternative 23: 55.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{\frac{1}{k}}{\frac{k \cdot {t\_m}^{3}}{{\ell}^{2}}} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ (/ 1.0 k) (/ (* k (pow t_m 3.0)) (pow l 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) {
	return t_s * ((1.0 / k) / ((k * pow(t_m, 3.0)) / pow(l, 2.0)));
}

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 * ((1.0d0 / k) / ((k * (t_m ** 3.0d0)) / (l ** 2.0d0)))
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 * ((1.0 / k) / ((k * Math.pow(t_m, 3.0)) / Math.pow(l, 2.0)));
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * ((1.0 / k) / ((k * math.pow(t_m, 3.0)) / math.pow(l, 2.0)))

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(1.0 / k) / Float64(Float64(k * (t_m ^ 3.0)) / (l ^ 2.0))))
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 * ((1.0 / k) / ((k * (t_m ^ 3.0)) / (l ^ 2.0)));
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[(1.0 / k), $MachinePrecision] / N[(N[(k * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.0%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. *-commutative 57.0%

      \[\leadsto \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. sqr-neg 57.0%

      \[\leadsto \frac{2}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. *-commutative 57.0%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. associate-*l* 57.4%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   5. *-commutative 57.4%

      \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

   6. sqr-neg 57.4%

      \[\leadsto \frac{2}{\left(\sin k \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

3. Simplified 57.4%

   \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in k around 0 56.3%

   \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]
6. Step-by-step derivation

   1. *-commutative 56.3%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]

7. Simplified 56.3%

   \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]

8. Taylor expanded in k around 0 58.1%

   \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot {t}^{3}}{{\ell}^{2}}} \cdot \left(k \cdot 2\right)} \]
9. Step-by-step derivation

   1. associate-/l* 57.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{k}{\frac{{\ell}^{2}}{{t}^{3}}}} \cdot \left(k \cdot 2\right)} \]

10. Simplified 57.3%

    \[\leadsto \frac{2}{\color{blue}{\frac{k}{\frac{{\ell}^{2}}{{t}^{3}}}} \cdot \left(k \cdot 2\right)} \]
11. Step-by-step derivation

    1. div-inv 57.3%

       \[\leadsto \color{blue}{2 \cdot \frac{1}{\frac{k}{\frac{{\ell}^{2}}{{t}^{3}}} \cdot \left(k \cdot 2\right)}} \]

    2. associate-/r/ 57.3%

       \[\leadsto 2 \cdot \frac{1}{\color{blue}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right)} \cdot \left(k \cdot 2\right)} \]

12. Applied egg-rr 57.3%

    \[\leadsto \color{blue}{2 \cdot \frac{1}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right) \cdot \left(k \cdot 2\right)}} \]
13. Step-by-step derivation

    1. associate-*r/ 57.3%

       \[\leadsto \color{blue}{\frac{2 \cdot 1}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right) \cdot \left(k \cdot 2\right)}} \]

    2. metadata-eval 57.3%

       \[\leadsto \frac{\color{blue}{2}}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right) \cdot \left(k \cdot 2\right)} \]

    3. *-commutative 57.3%

       \[\leadsto \frac{2}{\color{blue}{\left(k \cdot 2\right) \cdot \left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right)}} \]

    4. associate-/r* 57.3%

       \[\leadsto \color{blue}{\frac{\frac{2}{k \cdot 2}}{\frac{k}{{\ell}^{2}} \cdot {t}^{3}}} \]

    5. *-commutative 57.3%

       \[\leadsto \frac{\frac{2}{\color{blue}{2 \cdot k}}}{\frac{k}{{\ell}^{2}} \cdot {t}^{3}} \]

    6. associate-/r* 57.3%

       \[\leadsto \frac{\color{blue}{\frac{\frac{2}{2}}{k}}}{\frac{k}{{\ell}^{2}} \cdot {t}^{3}} \]

    7. metadata-eval 57.3%

       \[\leadsto \frac{\frac{\color{blue}{1}}{k}}{\frac{k}{{\ell}^{2}} \cdot {t}^{3}} \]

    8. *-commutative 57.3%

       \[\leadsto \frac{\frac{1}{k}}{\color{blue}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \]

14. Simplified 57.3%

    \[\leadsto \color{blue}{\frac{\frac{1}{k}}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \]
15. Step-by-step derivation

    1. associate-*r/ 58.1%

       \[\leadsto \frac{\frac{1}{k}}{\color{blue}{\frac{{t}^{3} \cdot k}{{\ell}^{2}}}} \]

16. Applied egg-rr 58.1%

    \[\leadsto \frac{\frac{1}{k}}{\color{blue}{\frac{{t}^{3} \cdot k}{{\ell}^{2}}}} \]

17. Final simplification 58.1%

    \[\leadsto \frac{\frac{1}{k}}{\frac{k \cdot {t}^{3}}{{\ell}^{2}}} \]
18. Add Preprocessing

Reproduce

herbie shell --seed 2024034 
(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))))