Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.5% → 87.8%

Time: 20.6s

Alternatives: 19

Speedup: 3.7×

• Could not converge on a ground truth

  1. reg0 = (bf "2.178316400001935280896770928875684521663e175")
  2. reg1 = (bf "1.302995543437300232266073515068774213263e-282")
  3. reg2 = (bf "2.320220386807941338700921247281073169353e-100")

• 27.8% 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: 55.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 87.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.2 \cdot 10^{-116}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(\frac{t\_m}{\sqrt[3]{\ell}} \cdot \sqrt[3]{\frac{1}{\ell}}\right) \cdot \left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)}\right)\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4.2e-116)
    (pow (* (/ (* l (sqrt 2.0)) (* k (sin k))) (sqrt (/ (cos k) t_m))) 2.0)
    (/
     2.0
     (pow
      (*
       (* (/ t_m (cbrt l)) (cbrt (/ 1.0 l)))
       (* (cbrt (sin k)) (cbrt (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0))))))
      3.0)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.2e-116) {
		tmp = pow((((l * sqrt(2.0)) / (k * sin(k))) * sqrt((cos(k) / t_m))), 2.0);
	} else {
		tmp = 2.0 / pow((((t_m / cbrt(l)) * cbrt((1.0 / l))) * (cbrt(sin(k)) * cbrt((tan(k) * (2.0 + pow((k / t_m), 2.0)))))), 3.0);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.2e-116) {
		tmp = Math.pow((((l * Math.sqrt(2.0)) / (k * Math.sin(k))) * Math.sqrt((Math.cos(k) / t_m))), 2.0);
	} else {
		tmp = 2.0 / Math.pow((((t_m / Math.cbrt(l)) * Math.cbrt((1.0 / l))) * (Math.cbrt(Math.sin(k)) * Math.cbrt((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0)))))), 3.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 4.2e-116)
		tmp = Float64(Float64(Float64(l * sqrt(2.0)) / Float64(k * sin(k))) * sqrt(Float64(cos(k) / t_m))) ^ 2.0;
	else
		tmp = Float64(2.0 / (Float64(Float64(Float64(t_m / cbrt(l)) * cbrt(Float64(1.0 / l))) * Float64(cbrt(sin(k)) * cbrt(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0)))))) ^ 3.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 4.2e-116], N[Power[N[(N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 / N[Power[N[(N[(N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[(1.0 / l), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.1999999999999998e-116

   1. Initial program 53.3%

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

   3. Simplified 50.4%

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

      1. add-sqr-sqrt 28.3%

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

      2. pow2 28.3%

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

   6. Applied egg-rr 31.7%

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

   7. Taylor expanded in t around 0 31.3%

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

   if 4.1999999999999998e-116 < t

   1. Initial program 62.1%

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

   3. Simplified 64.1%

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

      1. unpow2 64.1%

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

      2. frac-2neg 64.1%

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

      3. frac-times 53.1%

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

   6. Applied egg-rr 53.1%

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

      1. add-cube-cbrt 53.0%

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

      2. pow3 53.0%

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

   8. Applied egg-rr 82.5%

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

      1. associate-*r* 82.5%

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

   10. Simplified 82.5%

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

       1. pow1/3 61.4%

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

       2. associate-*l* 61.4%

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

       3. unpow-prod-down 36.9%

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

       4. pow1/3 50.4%

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

   12. Applied egg-rr 50.4%

       \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \color{blue}{\left(\sqrt[3]{\sin k} \cdot {\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}^{0.3333333333333333}\right)}\right)}^{3}} \]
   13. Step-by-step derivation

       1. unpow1/3 96.4%

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

   14. Simplified 96.4%

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

       1. add-cbrt-cube 73.8%

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

       2. unpow3 73.8%

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

       3. unpow2 73.8%

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

       4. cbrt-unprod 64.1%

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

       5. cbrt-div 61.9%

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

       6. associate-/l/ 70.5%

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

       7. div-inv 70.5%

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

       8. cbrt-prod 72.5%

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

       9. cbrt-div 73.8%

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

       10. unpow3 73.8%

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

       11. add-cbrt-cube 96.5%

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

   16. Applied egg-rr 96.5%

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

Alternative 2: 77.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \left({\left(\frac{k}{t\_m}\right)}^{2} + 1\right) + 1\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \frac{{t\_m}^{3}}{\ell \cdot \ell}\right)\right) \cdot t\_2} \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\left(\frac{2}{\sin k \cdot {t\_m}^{3}} \cdot \frac{\ell}{\tan k}\right) \cdot \frac{\ell}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t\_m}}\right)}^{2}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (+ (+ (pow (/ k t_m) 2.0) 1.0) 1.0)))
   (*
    t_s
    (if (<=
         (/ 2.0 (* (* (tan k) (* (sin k) (/ (pow t_m 3.0) (* l l)))) t_2))
         2e+307)
      (* (* (/ 2.0 (* (sin k) (pow t_m 3.0))) (/ l (tan k))) (/ l t_2))
      (pow
       (* (/ (* l (sqrt 2.0)) (* k (sin k))) (sqrt (/ (cos 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 t_2 = (pow((k / t_m), 2.0) + 1.0) + 1.0;
	double tmp;
	if ((2.0 / ((tan(k) * (sin(k) * (pow(t_m, 3.0) / (l * l)))) * t_2)) <= 2e+307) {
		tmp = ((2.0 / (sin(k) * pow(t_m, 3.0))) * (l / tan(k))) * (l / t_2);
	} else {
		tmp = pow((((l * sqrt(2.0)) / (k * sin(k))) * sqrt((cos(k) / t_m))), 2.0);
	}
	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) :: t_2
    real(8) :: tmp
    t_2 = (((k / t_m) ** 2.0d0) + 1.0d0) + 1.0d0
    if ((2.0d0 / ((tan(k) * (sin(k) * ((t_m ** 3.0d0) / (l * l)))) * t_2)) <= 2d+307) then
        tmp = ((2.0d0 / (sin(k) * (t_m ** 3.0d0))) * (l / tan(k))) * (l / t_2)
    else
        tmp = (((l * sqrt(2.0d0)) / (k * sin(k))) * sqrt((cos(k) / t_m))) ** 2.0d0
    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 t_2 = (Math.pow((k / t_m), 2.0) + 1.0) + 1.0;
	double tmp;
	if ((2.0 / ((Math.tan(k) * (Math.sin(k) * (Math.pow(t_m, 3.0) / (l * l)))) * t_2)) <= 2e+307) {
		tmp = ((2.0 / (Math.sin(k) * Math.pow(t_m, 3.0))) * (l / Math.tan(k))) * (l / t_2);
	} else {
		tmp = Math.pow((((l * Math.sqrt(2.0)) / (k * Math.sin(k))) * Math.sqrt((Math.cos(k) / t_m))), 2.0);
	}
	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):
	t_2 = (math.pow((k / t_m), 2.0) + 1.0) + 1.0
	tmp = 0
	if (2.0 / ((math.tan(k) * (math.sin(k) * (math.pow(t_m, 3.0) / (l * l)))) * t_2)) <= 2e+307:
		tmp = ((2.0 / (math.sin(k) * math.pow(t_m, 3.0))) * (l / math.tan(k))) * (l / t_2)
	else:
		tmp = math.pow((((l * math.sqrt(2.0)) / (k * math.sin(k))) * math.sqrt((math.cos(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)
	t_2 = Float64(Float64((Float64(k / t_m) ^ 2.0) + 1.0) + 1.0)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(tan(k) * Float64(sin(k) * Float64((t_m ^ 3.0) / Float64(l * l)))) * t_2)) <= 2e+307)
		tmp = Float64(Float64(Float64(2.0 / Float64(sin(k) * (t_m ^ 3.0))) * Float64(l / tan(k))) * Float64(l / t_2));
	else
		tmp = Float64(Float64(Float64(l * sqrt(2.0)) / Float64(k * sin(k))) * sqrt(Float64(cos(k) / t_m))) ^ 2.0;
	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)
	t_2 = (((k / t_m) ^ 2.0) + 1.0) + 1.0;
	tmp = 0.0;
	if ((2.0 / ((tan(k) * (sin(k) * ((t_m ^ 3.0) / (l * l)))) * t_2)) <= 2e+307)
		tmp = ((2.0 / (sin(k) * (t_m ^ 3.0))) * (l / tan(k))) * (l / t_2);
	else
		tmp = (((l * sqrt(2.0)) / (k * sin(k))) * sqrt((cos(k) / t_m))) ^ 2.0;
	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_] := Block[{t$95$2 = N[(N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(2.0 / 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] * t$95$2), $MachinePrecision]), $MachinePrecision], 2e+307], N[(N[(N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / t$95$2), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 1.99999999999999997e307

   1. Initial program 80.6%

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

   3. Simplified 74.5%

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

      1. associate-*r* 77.7%

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

      2. add-sqr-sqrt 77.6%

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

      3. times-frac 77.6%

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

   6. Applied egg-rr 86.2%

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

      1. frac-times 85.7%

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

      2. associate-*l/ 85.6%

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

      3. pow2 85.6%

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

   8. Applied egg-rr 85.6%

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

      1. associate-/l* 84.6%

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

      2. times-frac 87.1%

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

      3. *-commutative 87.1%

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

      4. unpow2 87.1%

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

      5. hypot-undefine 87.1%

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

      6. hypot-undefine 87.1%

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

      7. rem-square-sqrt 87.2%

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

      8. metadata-eval 87.2%

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

      9. hypot-undefine 87.2%

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

      10. metadata-eval 87.2%

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

      11. unpow2 87.2%

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

      12. hypot-undefine 87.2%

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

      13. metadata-eval 87.2%

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

      14. unpow2 87.2%

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

   10. Simplified 87.2%

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

   if 1.99999999999999997e307 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 23.1%

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

   3. Simplified 23.1%

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

      1. add-sqr-sqrt 23.1%

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

      2. pow2 23.1%

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

   6. Applied egg-rr 34.9%

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

   7. Taylor expanded in t around 0 54.1%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)} \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\left(\frac{2}{\sin k \cdot {t}^{3}} \cdot \frac{\ell}{\tan k}\right) \cdot \frac{\ell}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t}}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.9% accurate, 0.4× speedup

Math

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

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (+ (+ (pow (/ k t_m) 2.0) 1.0) 1.0)))
   (*
    t_s
    (if (<=
         (/ 2.0 (* (* (tan k) (* (sin k) (/ (pow t_m 3.0) (* l l)))) t_2))
         INFINITY)
      (* (* (/ 2.0 (* (sin k) (pow t_m 3.0))) (/ l (tan k))) (/ l t_2))
      (/
       2.0
       (pow
        (* t_m (* (pow (cbrt l) -2.0) (cbrt (* 2.0 (pow k 2.0)))))
        3.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = (pow((k / t_m), 2.0) + 1.0) + 1.0;
	double tmp;
	if ((2.0 / ((tan(k) * (sin(k) * (pow(t_m, 3.0) / (l * l)))) * t_2)) <= ((double) INFINITY)) {
		tmp = ((2.0 / (sin(k) * pow(t_m, 3.0))) * (l / tan(k))) * (l / t_2);
	} else {
		tmp = 2.0 / pow((t_m * (pow(cbrt(l), -2.0) * cbrt((2.0 * pow(k, 2.0))))), 3.0);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = (Math.pow((k / t_m), 2.0) + 1.0) + 1.0;
	double tmp;
	if ((2.0 / ((Math.tan(k) * (Math.sin(k) * (Math.pow(t_m, 3.0) / (l * l)))) * t_2)) <= Double.POSITIVE_INFINITY) {
		tmp = ((2.0 / (Math.sin(k) * Math.pow(t_m, 3.0))) * (l / Math.tan(k))) * (l / t_2);
	} else {
		tmp = 2.0 / Math.pow((t_m * (Math.pow(Math.cbrt(l), -2.0) * Math.cbrt((2.0 * Math.pow(k, 2.0))))), 3.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(Float64((Float64(k / t_m) ^ 2.0) + 1.0) + 1.0)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(tan(k) * Float64(sin(k) * Float64((t_m ^ 3.0) / Float64(l * l)))) * t_2)) <= Inf)
		tmp = Float64(Float64(Float64(2.0 / Float64(sin(k) * (t_m ^ 3.0))) * Float64(l / tan(k))) * Float64(l / t_2));
	else
		tmp = Float64(2.0 / (Float64(t_m * Float64((cbrt(l) ^ -2.0) * cbrt(Float64(2.0 * (k ^ 2.0))))) ^ 3.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < +inf.0

   1. Initial program 82.1%

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

   3. Simplified 76.9%

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

      1. associate-*r* 80.2%

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

      2. add-sqr-sqrt 80.1%

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

      3. times-frac 80.6%

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

   6. Applied egg-rr 87.9%

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

      1. frac-times 86.9%

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

      2. associate-*l/ 86.9%

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

      3. pow2 86.9%

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

   8. Applied egg-rr 86.9%

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

      1. associate-/l* 85.9%

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

      2. times-frac 88.0%

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

      3. *-commutative 88.0%

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

      4. unpow2 88.0%

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

      5. hypot-undefine 88.0%

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

      6. hypot-undefine 88.0%

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

      7. rem-square-sqrt 88.1%

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

      8. metadata-eval 88.1%

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

      9. hypot-undefine 88.1%

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

      10. metadata-eval 88.1%

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

      11. unpow2 88.1%

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

      12. hypot-undefine 88.1%

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

      13. metadata-eval 88.1%

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

      14. unpow2 88.1%

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

   10. Simplified 88.1%

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

   if +inf.0 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 0.0%

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

   3. Simplified 11.0%

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

   5. Taylor expanded in k around 0 22.6%

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

      1. add-exp-log 13.4%

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

      2. log-pow 11.9%

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

   7. Applied egg-rr 11.9%

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

      1. pow2 11.9%

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

      2. associate-/l/ 3.9%

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

      3. *-commutative 3.9%

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

      4. pow-to-exp 10.2%

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

      5. associate-/l/ 22.6%

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

      6. add-cube-cbrt 22.6%

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

      7. pow3 22.6%

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

   9. Applied egg-rr 41.0%

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

       1. associate-*l* 41.1%

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

   11. Simplified 41.1%

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

4. Final simplification 73.2%

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

Alternative 4: 70.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \left({\left(\frac{k}{t\_m}\right)}^{2} + 1\right) + 1\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \frac{{t\_m}^{3}}{\ell \cdot \ell}\right)\right) \cdot t\_2} \leq \infty:\\ \;\;\;\;\left(\frac{2}{\sin k \cdot {t\_m}^{3}} \cdot \frac{\ell}{\tan k}\right) \cdot \frac{\ell}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left({\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{2}\right)}^{1.5} \cdot \left(2 \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (+ (+ (pow (/ k t_m) 2.0) 1.0) 1.0)))
   (*
    t_s
    (if (<=
         (/ 2.0 (* (* (tan k) (* (sin k) (/ (pow t_m 3.0) (* l l)))) t_2))
         INFINITY)
      (* (* (/ 2.0 (* (sin k) (pow t_m 3.0))) (/ l (tan k))) (/ l t_2))
      (/
       2.0
       (* (pow (pow (* t_m (pow (cbrt l) -2.0)) 2.0) 1.5) (* 2.0 (* k 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) + 1.0) + 1.0;
	double tmp;
	if ((2.0 / ((tan(k) * (sin(k) * (pow(t_m, 3.0) / (l * l)))) * t_2)) <= ((double) INFINITY)) {
		tmp = ((2.0 / (sin(k) * pow(t_m, 3.0))) * (l / tan(k))) * (l / t_2);
	} else {
		tmp = 2.0 / (pow(pow((t_m * pow(cbrt(l), -2.0)), 2.0), 1.5) * (2.0 * (k * 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) + 1.0) + 1.0;
	double tmp;
	if ((2.0 / ((Math.tan(k) * (Math.sin(k) * (Math.pow(t_m, 3.0) / (l * l)))) * t_2)) <= Double.POSITIVE_INFINITY) {
		tmp = ((2.0 / (Math.sin(k) * Math.pow(t_m, 3.0))) * (l / Math.tan(k))) * (l / t_2);
	} else {
		tmp = 2.0 / (Math.pow(Math.pow((t_m * Math.pow(Math.cbrt(l), -2.0)), 2.0), 1.5) * (2.0 * (k * 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(Float64((Float64(k / t_m) ^ 2.0) + 1.0) + 1.0)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(tan(k) * Float64(sin(k) * Float64((t_m ^ 3.0) / Float64(l * l)))) * t_2)) <= Inf)
		tmp = Float64(Float64(Float64(2.0 / Float64(sin(k) * (t_m ^ 3.0))) * Float64(l / tan(k))) * Float64(l / t_2));
	else
		tmp = Float64(2.0 / Float64(((Float64(t_m * (cbrt(l) ^ -2.0)) ^ 2.0) ^ 1.5) * Float64(2.0 * Float64(k * 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[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(2.0 / 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] * t$95$2), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / t$95$2), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[Power[N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 1.5], $MachinePrecision] * N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < +inf.0

   1. Initial program 82.1%

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

   3. Simplified 76.9%

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

      1. associate-*r* 80.2%

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

      2. add-sqr-sqrt 80.1%

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

      3. times-frac 80.6%

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

   6. Applied egg-rr 87.9%

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

      1. frac-times 86.9%

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

      2. associate-*l/ 86.9%

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

      3. pow2 86.9%

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

   8. Applied egg-rr 86.9%

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

      1. associate-/l* 85.9%

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

      2. times-frac 88.0%

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

      3. *-commutative 88.0%

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

      4. unpow2 88.0%

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

      5. hypot-undefine 88.0%

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

      6. hypot-undefine 88.0%

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

      7. rem-square-sqrt 88.1%

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

      8. metadata-eval 88.1%

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

      9. hypot-undefine 88.1%

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

      10. metadata-eval 88.1%

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

      11. unpow2 88.1%

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

      12. hypot-undefine 88.1%

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

      13. metadata-eval 88.1%

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

      14. unpow2 88.1%

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

   10. Simplified 88.1%

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

   if +inf.0 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 0.0%

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

   3. Simplified 11.0%

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

   5. Taylor expanded in k around 0 22.6%

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

      1. unpow2 22.6%

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

   7. Applied egg-rr 22.6%

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

      1. associate-/r* 10.2%

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

      2. unpow3 10.2%

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

      3. times-frac 27.6%

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

      4. pow2 27.6%

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

   9. Applied egg-rr 27.6%

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

       1. frac-times 10.2%

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

       2. unpow2 10.2%

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

       3. unpow3 10.2%

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

       4. add-cube-cbrt 10.2%

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

       5. pow3 10.2%

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

       6. add-cube-cbrt 10.2%

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

       7. pow3 10.2%

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

       8. unpow-prod-down 10.2%

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

       9. unpow2 10.2%

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

       10. cube-div 34.4%

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

       11. sqr-pow 23.1%

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

       12. pow-prod-down 25.1%

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

       13. pow2 25.1%

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

       14. div-inv 25.1%

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

       15. pow-flip 25.1%

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

       16. metadata-eval 25.1%

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

       17. metadata-eval 25.1%

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

   11. Applied egg-rr 25.1%

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

4. Final simplification 68.2%

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

Alternative 5: 87.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5 \cdot 10^{-116}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)}\right) \cdot \frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 5e-116)
    (pow (* (/ (* l (sqrt 2.0)) (* k (sin k))) (sqrt (/ (cos k) t_m))) 2.0)
    (/
     2.0
     (pow
      (*
       (* (cbrt (sin k)) (cbrt (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))))
       (/ t_m (pow (cbrt l) 2.0)))
      3.0)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 5e-116) {
		tmp = pow((((l * sqrt(2.0)) / (k * sin(k))) * sqrt((cos(k) / t_m))), 2.0);
	} else {
		tmp = 2.0 / pow(((cbrt(sin(k)) * cbrt((tan(k) * (2.0 + pow((k / t_m), 2.0))))) * (t_m / pow(cbrt(l), 2.0))), 3.0);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 5e-116) {
		tmp = Math.pow((((l * Math.sqrt(2.0)) / (k * Math.sin(k))) * Math.sqrt((Math.cos(k) / t_m))), 2.0);
	} else {
		tmp = 2.0 / Math.pow(((Math.cbrt(Math.sin(k)) * Math.cbrt((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0))))) * (t_m / Math.pow(Math.cbrt(l), 2.0))), 3.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 5e-116)
		tmp = Float64(Float64(Float64(l * sqrt(2.0)) / Float64(k * sin(k))) * sqrt(Float64(cos(k) / t_m))) ^ 2.0;
	else
		tmp = Float64(2.0 / (Float64(Float64(cbrt(sin(k)) * cbrt(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))))) * Float64(t_m / (cbrt(l) ^ 2.0))) ^ 3.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 5e-116], N[Power[N[(N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 / N[Power[N[(N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5.0000000000000003e-116

   1. Initial program 53.3%

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

   3. Simplified 50.4%

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

      1. add-sqr-sqrt 28.3%

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

      2. pow2 28.3%

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

   6. Applied egg-rr 31.7%

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

   7. Taylor expanded in t around 0 31.3%

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

   if 5.0000000000000003e-116 < t

   1. Initial program 62.1%

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

   3. Simplified 64.1%

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

      1. unpow2 64.1%

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

      2. frac-2neg 64.1%

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

      3. frac-times 53.1%

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

   6. Applied egg-rr 53.1%

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

      1. add-cube-cbrt 53.0%

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

      2. pow3 53.0%

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

   8. Applied egg-rr 82.5%

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

      1. associate-*r* 82.5%

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

   10. Simplified 82.5%

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

       1. pow1/3 61.4%

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

       2. associate-*l* 61.4%

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

       3. unpow-prod-down 36.9%

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

       4. pow1/3 50.4%

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

   12. Applied egg-rr 50.4%

       \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \color{blue}{\left(\sqrt[3]{\sin k} \cdot {\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}^{0.3333333333333333}\right)}\right)}^{3}} \]
   13. Step-by-step derivation

       1. unpow1/3 96.4%

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

   14. Simplified 96.4%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5 \cdot 10^{-116}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right) \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.9 \cdot 10^{-118}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t\_m}}\right)}^{2}\\ \mathbf{elif}\;t\_m \leq 5.1 \cdot 10^{+29}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\ell \cdot \cos k}{k}}{{\sin k}^{2} \cdot {t\_m}^{2}}\right) \cdot \frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t\_m}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right) \cdot {\left(\sqrt[3]{\sin k} \cdot \left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 6.9e-118)
    (pow (* (/ (* l (sqrt 2.0)) (* k (sin k))) (sqrt (/ (cos k) t_m))) 2.0)
    (if (<= t_m 5.1e+29)
      (*
       (* 2.0 (/ (/ (* l (cos k)) k) (* (pow (sin k) 2.0) (pow t_m 2.0))))
       (/ l (hypot 1.0 (hypot 1.0 (/ k t_m)))))
      (/
       2.0
       (*
        (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))
        (pow (* (cbrt (sin k)) (* t_m (pow (cbrt l) -2.0))) 3.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 6.9e-118) {
		tmp = pow((((l * sqrt(2.0)) / (k * sin(k))) * sqrt((cos(k) / t_m))), 2.0);
	} else if (t_m <= 5.1e+29) {
		tmp = (2.0 * (((l * cos(k)) / k) / (pow(sin(k), 2.0) * pow(t_m, 2.0)))) * (l / hypot(1.0, hypot(1.0, (k / t_m))));
	} else {
		tmp = 2.0 / ((tan(k) * (2.0 + pow((k / t_m), 2.0))) * pow((cbrt(sin(k)) * (t_m * pow(cbrt(l), -2.0))), 3.0));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 6.9e-118) {
		tmp = Math.pow((((l * Math.sqrt(2.0)) / (k * Math.sin(k))) * Math.sqrt((Math.cos(k) / t_m))), 2.0);
	} else if (t_m <= 5.1e+29) {
		tmp = (2.0 * (((l * Math.cos(k)) / k) / (Math.pow(Math.sin(k), 2.0) * Math.pow(t_m, 2.0)))) * (l / Math.hypot(1.0, Math.hypot(1.0, (k / t_m))));
	} else {
		tmp = 2.0 / ((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0))) * Math.pow((Math.cbrt(Math.sin(k)) * (t_m * Math.pow(Math.cbrt(l), -2.0))), 3.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 6.9e-118)
		tmp = Float64(Float64(Float64(l * sqrt(2.0)) / Float64(k * sin(k))) * sqrt(Float64(cos(k) / t_m))) ^ 2.0;
	elseif (t_m <= 5.1e+29)
		tmp = Float64(Float64(2.0 * Float64(Float64(Float64(l * cos(k)) / k) / Float64((sin(k) ^ 2.0) * (t_m ^ 2.0)))) * Float64(l / hypot(1.0, hypot(1.0, Float64(k / t_m)))));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * (Float64(cbrt(sin(k)) * Float64(t_m * (cbrt(l) ^ -2.0))) ^ 3.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 6.9e-118], N[Power[N[(N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[t$95$m, 5.1e+29], N[(N[(2.0 * N[(N[(N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $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[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 6.9000000000000002e-118

   1. Initial program 53.6%

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

   3. Simplified 50.7%

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

      1. add-sqr-sqrt 28.5%

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

      2. pow2 28.5%

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

   6. Applied egg-rr 31.9%

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

   7. Taylor expanded in t around 0 31.5%

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

   if 6.9000000000000002e-118 < t < 5.1000000000000001e29

   1. Initial program 64.1%

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

   3. Simplified 63.8%

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

      1. associate-*r* 64.5%

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

      2. add-sqr-sqrt 64.4%

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

      3. times-frac 64.6%

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

   6. Applied egg-rr 69.1%

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

   7. Taylor expanded in t around 0 55.2%

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

      1. associate-/r* 59.4%

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

      2. *-commutative 59.4%

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

   9. Simplified 59.4%

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

   if 5.1000000000000001e29 < t

   1. Initial program 60.3%

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

   3. Simplified 60.9%

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

      1. unpow2 60.9%

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

      2. frac-2neg 60.9%

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

      3. frac-times 46.1%

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

   6. Applied egg-rr 46.1%

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

      1. add-cube-cbrt 46.1%

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

      2. pow3 46.1%

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

   8. Applied egg-rr 80.8%

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

      1. associate-*r* 80.8%

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

   10. Simplified 80.8%

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

       1. pow1/3 54.5%

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

       2. associate-*l* 54.5%

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

       3. unpow-prod-down 35.3%

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

       4. pow1/3 48.4%

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

   12. Applied egg-rr 48.4%

       \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \color{blue}{\left(\sqrt[3]{\sin k} \cdot {\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}^{0.3333333333333333}\right)}\right)}^{3}} \]
   13. Step-by-step derivation

       1. unpow1/3 97.1%

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

   14. Simplified 97.1%

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

       1. associate-*r* 97.1%

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

       2. unpow-prod-down 94.2%

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

       3. div-inv 94.2%

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

       4. pow-flip 94.1%

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

       5. metadata-eval 94.1%

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

       6. pow3 94.2%

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

       7. add-cube-cbrt 94.1%

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

   16. Applied egg-rr 94.1%

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

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.9 \cdot 10^{-118}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t}}\right)}^{2}\\ \mathbf{elif}\;t \leq 5.1 \cdot 10^{+29}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\ell \cdot \cos k}{k}}{{\sin k}^{2} \cdot {t}^{2}}\right) \cdot \frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(\sqrt[3]{\sin k} \cdot \left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7 \cdot 10^{-118}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t\_m}}\right)}^{2}\\ \mathbf{elif}\;t\_m \leq 39000:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\ell \cdot \cos k}{k}}{{\sin k}^{2} \cdot {t\_m}^{2}}\right) \cdot \frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t\_m}\right)\right)}\\ \mathbf{elif}\;t\_m \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\left(\frac{2}{\sin k \cdot {t\_m}^{3}} \cdot \frac{\ell}{\tan k}\right) \cdot \frac{\ell}{\left({\left(\frac{k}{t\_m}\right)}^{2} + 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\sqrt[3]{\sin k} \cdot \sqrt[3]{2 \cdot k}\right)\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 7e-118)
    (pow (* (/ (* l (sqrt 2.0)) (* k (sin k))) (sqrt (/ (cos k) t_m))) 2.0)
    (if (<= t_m 39000.0)
      (*
       (* 2.0 (/ (/ (* l (cos k)) k) (* (pow (sin k) 2.0) (pow t_m 2.0))))
       (/ l (hypot 1.0 (hypot 1.0 (/ k t_m)))))
      (if (<= t_m 5.6e+102)
        (*
         (* (/ 2.0 (* (sin k) (pow t_m 3.0))) (/ l (tan k)))
         (/ l (+ (+ (pow (/ k t_m) 2.0) 1.0) 1.0)))
        (/
         2.0
         (pow
          (* (/ t_m (pow (cbrt l) 2.0)) (* (cbrt (sin k)) (cbrt (* 2.0 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 <= 7e-118) {
		tmp = pow((((l * sqrt(2.0)) / (k * sin(k))) * sqrt((cos(k) / t_m))), 2.0);
	} else if (t_m <= 39000.0) {
		tmp = (2.0 * (((l * cos(k)) / k) / (pow(sin(k), 2.0) * pow(t_m, 2.0)))) * (l / hypot(1.0, hypot(1.0, (k / t_m))));
	} else if (t_m <= 5.6e+102) {
		tmp = ((2.0 / (sin(k) * pow(t_m, 3.0))) * (l / tan(k))) * (l / ((pow((k / t_m), 2.0) + 1.0) + 1.0));
	} else {
		tmp = 2.0 / pow(((t_m / pow(cbrt(l), 2.0)) * (cbrt(sin(k)) * cbrt((2.0 * 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 <= 7e-118) {
		tmp = Math.pow((((l * Math.sqrt(2.0)) / (k * Math.sin(k))) * Math.sqrt((Math.cos(k) / t_m))), 2.0);
	} else if (t_m <= 39000.0) {
		tmp = (2.0 * (((l * Math.cos(k)) / k) / (Math.pow(Math.sin(k), 2.0) * Math.pow(t_m, 2.0)))) * (l / Math.hypot(1.0, Math.hypot(1.0, (k / t_m))));
	} else if (t_m <= 5.6e+102) {
		tmp = ((2.0 / (Math.sin(k) * Math.pow(t_m, 3.0))) * (l / Math.tan(k))) * (l / ((Math.pow((k / t_m), 2.0) + 1.0) + 1.0));
	} else {
		tmp = 2.0 / Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * (Math.cbrt(Math.sin(k)) * Math.cbrt((2.0 * 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 <= 7e-118)
		tmp = Float64(Float64(Float64(l * sqrt(2.0)) / Float64(k * sin(k))) * sqrt(Float64(cos(k) / t_m))) ^ 2.0;
	elseif (t_m <= 39000.0)
		tmp = Float64(Float64(2.0 * Float64(Float64(Float64(l * cos(k)) / k) / Float64((sin(k) ^ 2.0) * (t_m ^ 2.0)))) * Float64(l / hypot(1.0, hypot(1.0, Float64(k / t_m)))));
	elseif (t_m <= 5.6e+102)
		tmp = Float64(Float64(Float64(2.0 / Float64(sin(k) * (t_m ^ 3.0))) * Float64(l / tan(k))) * Float64(l / Float64(Float64((Float64(k / t_m) ^ 2.0) + 1.0) + 1.0)));
	else
		tmp = Float64(2.0 / (Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * Float64(cbrt(sin(k)) * cbrt(Float64(2.0 * 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, 7e-118], N[Power[N[(N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[t$95$m, 39000.0], N[(N[(2.0 * N[(N[(N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.6e+102], N[(N[(N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(2.0 * k), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if t < 7e-118

   1. Initial program 53.6%

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

   3. Simplified 50.7%

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

      1. add-sqr-sqrt 28.5%

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

      2. pow2 28.5%

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

   6. Applied egg-rr 31.9%

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

   7. Taylor expanded in t around 0 31.5%

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

   if 7e-118 < t < 39000

   1. Initial program 59.5%

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

   3. Simplified 59.0%

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

      1. associate-*r* 59.9%

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

      2. add-sqr-sqrt 59.8%

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

      3. times-frac 59.8%

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

   6. Applied egg-rr 60.0%

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

   7. Taylor expanded in t around 0 53.5%

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

      1. associate-/r* 53.5%

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

      2. *-commutative 53.5%

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

   9. Simplified 53.5%

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

   if 39000 < t < 5.60000000000000037e102

   1. Initial program 82.5%

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

   3. Simplified 72.7%

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

      1. associate-*r* 79.6%

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

      2. add-sqr-sqrt 79.3%

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

      3. times-frac 79.6%

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

   6. Applied egg-rr 96.2%

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

      1. frac-times 92.5%

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

      2. associate-*l/ 92.3%

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

      3. pow2 92.3%

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

   8. Applied egg-rr 92.3%

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

      1. associate-/l* 92.6%

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

      2. times-frac 95.9%

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

      3. *-commutative 95.9%

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

      4. unpow2 95.9%

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

      5. hypot-undefine 95.9%

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

      6. hypot-undefine 95.9%

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

      7. rem-square-sqrt 96.4%

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

      8. metadata-eval 96.4%

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

      9. hypot-undefine 96.4%

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

      10. metadata-eval 96.4%

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

      11. unpow2 96.4%

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

      12. hypot-undefine 96.4%

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

      13. metadata-eval 96.4%

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

      14. unpow2 96.4%

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

   10. Simplified 96.4%

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

   if 5.60000000000000037e102 < t

   1. Initial program 46.9%

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

   3. Simplified 50.3%

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

      1. unpow2 50.3%

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

      2. frac-2neg 50.3%

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

      3. frac-times 26.8%

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

   6. Applied egg-rr 26.8%

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

      1. add-cube-cbrt 26.8%

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

      2. pow3 26.8%

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

   8. Applied egg-rr 81.9%

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

      1. associate-*r* 82.0%

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

   10. Simplified 82.0%

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

       1. pow1/3 53.3%

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

       2. associate-*l* 53.3%

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

       3. unpow-prod-down 34.2%

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

       4. pow1/3 47.3%

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

   12. Applied egg-rr 47.3%

       \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \color{blue}{\left(\sqrt[3]{\sin k} \cdot {\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}^{0.3333333333333333}\right)}\right)}^{3}} \]
   13. Step-by-step derivation

       1. unpow1/3 96.4%

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

   14. Simplified 96.4%

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

   15. Taylor expanded in k around 0 77.6%

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

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7 \cdot 10^{-118}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t}}\right)}^{2}\\ \mathbf{elif}\;t \leq 39000:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\ell \cdot \cos k}{k}}{{\sin k}^{2} \cdot {t}^{2}}\right) \cdot \frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\left(\frac{2}{\sin k \cdot {t}^{3}} \cdot \frac{\ell}{\tan k}\right) \cdot \frac{\ell}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\sqrt[3]{\sin k} \cdot \sqrt[3]{2 \cdot k}\right)\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.5 \cdot 10^{-118}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t\_m}}\right)}^{2}\\ \mathbf{elif}\;t\_m \leq 31500:\\ \;\;\;\;\frac{\ell \cdot \left(2 \cdot \left(\frac{\ell}{k} \cdot \frac{\cos k}{{\left(t\_m \cdot \sin k\right)}^{2}}\right)\right)}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t\_m}\right)\right)}\\ \mathbf{elif}\;t\_m \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\left(\frac{2}{\sin k \cdot {t\_m}^{3}} \cdot \frac{\ell}{\tan k}\right) \cdot \frac{\ell}{\left({\left(\frac{k}{t\_m}\right)}^{2} + 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\sqrt[3]{\sin k} \cdot \sqrt[3]{2 \cdot k}\right)\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 7.5e-118)
    (pow (* (/ (* l (sqrt 2.0)) (* k (sin k))) (sqrt (/ (cos k) t_m))) 2.0)
    (if (<= t_m 31500.0)
      (/
       (* l (* 2.0 (* (/ l k) (/ (cos k) (pow (* t_m (sin k)) 2.0)))))
       (hypot 1.0 (hypot 1.0 (/ k t_m))))
      (if (<= t_m 5.6e+102)
        (*
         (* (/ 2.0 (* (sin k) (pow t_m 3.0))) (/ l (tan k)))
         (/ l (+ (+ (pow (/ k t_m) 2.0) 1.0) 1.0)))
        (/
         2.0
         (pow
          (* (/ t_m (pow (cbrt l) 2.0)) (* (cbrt (sin k)) (cbrt (* 2.0 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 <= 7.5e-118) {
		tmp = pow((((l * sqrt(2.0)) / (k * sin(k))) * sqrt((cos(k) / t_m))), 2.0);
	} else if (t_m <= 31500.0) {
		tmp = (l * (2.0 * ((l / k) * (cos(k) / pow((t_m * sin(k)), 2.0))))) / hypot(1.0, hypot(1.0, (k / t_m)));
	} else if (t_m <= 5.6e+102) {
		tmp = ((2.0 / (sin(k) * pow(t_m, 3.0))) * (l / tan(k))) * (l / ((pow((k / t_m), 2.0) + 1.0) + 1.0));
	} else {
		tmp = 2.0 / pow(((t_m / pow(cbrt(l), 2.0)) * (cbrt(sin(k)) * cbrt((2.0 * 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 <= 7.5e-118) {
		tmp = Math.pow((((l * Math.sqrt(2.0)) / (k * Math.sin(k))) * Math.sqrt((Math.cos(k) / t_m))), 2.0);
	} else if (t_m <= 31500.0) {
		tmp = (l * (2.0 * ((l / k) * (Math.cos(k) / Math.pow((t_m * Math.sin(k)), 2.0))))) / Math.hypot(1.0, Math.hypot(1.0, (k / t_m)));
	} else if (t_m <= 5.6e+102) {
		tmp = ((2.0 / (Math.sin(k) * Math.pow(t_m, 3.0))) * (l / Math.tan(k))) * (l / ((Math.pow((k / t_m), 2.0) + 1.0) + 1.0));
	} else {
		tmp = 2.0 / Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * (Math.cbrt(Math.sin(k)) * Math.cbrt((2.0 * 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 <= 7.5e-118)
		tmp = Float64(Float64(Float64(l * sqrt(2.0)) / Float64(k * sin(k))) * sqrt(Float64(cos(k) / t_m))) ^ 2.0;
	elseif (t_m <= 31500.0)
		tmp = Float64(Float64(l * Float64(2.0 * Float64(Float64(l / k) * Float64(cos(k) / (Float64(t_m * sin(k)) ^ 2.0))))) / hypot(1.0, hypot(1.0, Float64(k / t_m))));
	elseif (t_m <= 5.6e+102)
		tmp = Float64(Float64(Float64(2.0 / Float64(sin(k) * (t_m ^ 3.0))) * Float64(l / tan(k))) * Float64(l / Float64(Float64((Float64(k / t_m) ^ 2.0) + 1.0) + 1.0)));
	else
		tmp = Float64(2.0 / (Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * Float64(cbrt(sin(k)) * cbrt(Float64(2.0 * 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, 7.5e-118], N[Power[N[(N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[t$95$m, 31500.0], N[(N[(l * N[(2.0 * N[(N[(l / k), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Power[N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.6e+102], N[(N[(N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(2.0 * k), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if t < 7.49999999999999978e-118

   1. Initial program 53.6%

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

   3. Simplified 50.7%

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

      1. add-sqr-sqrt 28.5%

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

      2. pow2 28.5%

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

   6. Applied egg-rr 31.9%

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

   7. Taylor expanded in t around 0 31.5%

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

   if 7.49999999999999978e-118 < t < 31500

   1. Initial program 59.5%

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

   3. Simplified 59.0%

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

      1. associate-*r* 59.9%

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

      2. add-sqr-sqrt 59.8%

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

      3. times-frac 59.8%

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

   6. Applied egg-rr 60.0%

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

   7. Taylor expanded in t around 0 53.5%

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

      1. *-commutative 53.5%

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

   9. Simplified 53.5%

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

       1. associate-*r/ 53.4%

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

       2. times-frac 53.5%

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

       3. pow-prod-down 53.4%

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

   11. Applied egg-rr 53.4%

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

   if 31500 < t < 5.60000000000000037e102

   1. Initial program 82.5%

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

   3. Simplified 72.7%

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

      1. associate-*r* 79.6%

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

      2. add-sqr-sqrt 79.3%

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

      3. times-frac 79.6%

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

   6. Applied egg-rr 96.2%

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

      1. frac-times 92.5%

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

      2. associate-*l/ 92.3%

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

      3. pow2 92.3%

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

   8. Applied egg-rr 92.3%

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

      1. associate-/l* 92.6%

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

      2. times-frac 95.9%

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

      3. *-commutative 95.9%

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

      4. unpow2 95.9%

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

      5. hypot-undefine 95.9%

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

      6. hypot-undefine 95.9%

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

      7. rem-square-sqrt 96.4%

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

      8. metadata-eval 96.4%

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

      9. hypot-undefine 96.4%

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

      10. metadata-eval 96.4%

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

      11. unpow2 96.4%

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

      12. hypot-undefine 96.4%

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

      13. metadata-eval 96.4%

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

      14. unpow2 96.4%

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

   10. Simplified 96.4%

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

   if 5.60000000000000037e102 < t

   1. Initial program 46.9%

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

   3. Simplified 50.3%

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

      1. unpow2 50.3%

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

      2. frac-2neg 50.3%

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

      3. frac-times 26.8%

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

   6. Applied egg-rr 26.8%

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

      1. add-cube-cbrt 26.8%

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

      2. pow3 26.8%

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

   8. Applied egg-rr 81.9%

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

      1. associate-*r* 82.0%

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

   10. Simplified 82.0%

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

       1. pow1/3 53.3%

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

       2. associate-*l* 53.3%

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

       3. unpow-prod-down 34.2%

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

       4. pow1/3 47.3%

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

   12. Applied egg-rr 47.3%

       \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \color{blue}{\left(\sqrt[3]{\sin k} \cdot {\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}^{0.3333333333333333}\right)}\right)}^{3}} \]
   13. Step-by-step derivation

       1. unpow1/3 96.4%

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

   14. Simplified 96.4%

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

   15. Taylor expanded in k around 0 77.6%

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

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.5 \cdot 10^{-118}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t}}\right)}^{2}\\ \mathbf{elif}\;t \leq 31500:\\ \;\;\;\;\frac{\ell \cdot \left(2 \cdot \left(\frac{\ell}{k} \cdot \frac{\cos k}{{\left(t \cdot \sin k\right)}^{2}}\right)\right)}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\left(\frac{2}{\sin k \cdot {t}^{3}} \cdot \frac{\ell}{\tan k}\right) \cdot \frac{\ell}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\sqrt[3]{\sin k} \cdot \sqrt[3]{2 \cdot k}\right)\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\frac{k}{t\_m}\right)}^{2}\\ t_3 := 2 + t\_2\\ t_4 := \sin k \cdot {t\_m}^{3}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7 \cdot 10^{-118}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t\_m}}\right)}^{2}\\ \mathbf{elif}\;t\_m \leq 40000:\\ \;\;\;\;\frac{\ell \cdot \left(2 \cdot \left(\frac{\ell}{k} \cdot \frac{\cos k}{{\left(t\_m \cdot \sin k\right)}^{2}}\right)\right)}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t\_m}\right)\right)}\\ \mathbf{elif}\;t\_m \leq 2.6 \cdot 10^{+107}:\\ \;\;\;\;\left(\frac{2}{t\_4} \cdot \frac{\ell}{\tan k}\right) \cdot \frac{\ell}{\left(t\_2 + 1\right) + 1}\\ \mathbf{elif}\;t\_m \leq 7.8 \cdot 10^{+194}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2} \cdot \left(\sin k \cdot \left(\tan k \cdot t\_3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \frac{2}{\tan k \cdot t\_4}\right) \cdot \frac{\ell}{t\_3}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (/ k t_m) 2.0))
        (t_3 (+ 2.0 t_2))
        (t_4 (* (sin k) (pow t_m 3.0))))
   (*
    t_s
    (if (<= t_m 7e-118)
      (pow (* (/ (* l (sqrt 2.0)) (* k (sin k))) (sqrt (/ (cos k) t_m))) 2.0)
      (if (<= t_m 40000.0)
        (/
         (* l (* 2.0 (* (/ l k) (/ (cos k) (pow (* t_m (sin k)) 2.0)))))
         (hypot 1.0 (hypot 1.0 (/ k t_m))))
        (if (<= t_m 2.6e+107)
          (* (* (/ 2.0 t_4) (/ l (tan k))) (/ l (+ (+ t_2 1.0) 1.0)))
          (if (<= t_m 7.8e+194)
            (/
             2.0
             (* (pow (/ (pow t_m 1.5) l) 2.0) (* (sin k) (* (tan k) t_3))))
            (* (* l (/ 2.0 (* (tan k) t_4))) (/ l t_3)))))))))

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 t_3 = 2.0 + t_2;
	double t_4 = sin(k) * pow(t_m, 3.0);
	double tmp;
	if (t_m <= 7e-118) {
		tmp = pow((((l * sqrt(2.0)) / (k * sin(k))) * sqrt((cos(k) / t_m))), 2.0);
	} else if (t_m <= 40000.0) {
		tmp = (l * (2.0 * ((l / k) * (cos(k) / pow((t_m * sin(k)), 2.0))))) / hypot(1.0, hypot(1.0, (k / t_m)));
	} else if (t_m <= 2.6e+107) {
		tmp = ((2.0 / t_4) * (l / tan(k))) * (l / ((t_2 + 1.0) + 1.0));
	} else if (t_m <= 7.8e+194) {
		tmp = 2.0 / (pow((pow(t_m, 1.5) / l), 2.0) * (sin(k) * (tan(k) * t_3)));
	} else {
		tmp = (l * (2.0 / (tan(k) * t_4))) * (l / t_3);
	}
	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 t_3 = 2.0 + t_2;
	double t_4 = Math.sin(k) * Math.pow(t_m, 3.0);
	double tmp;
	if (t_m <= 7e-118) {
		tmp = Math.pow((((l * Math.sqrt(2.0)) / (k * Math.sin(k))) * Math.sqrt((Math.cos(k) / t_m))), 2.0);
	} else if (t_m <= 40000.0) {
		tmp = (l * (2.0 * ((l / k) * (Math.cos(k) / Math.pow((t_m * Math.sin(k)), 2.0))))) / Math.hypot(1.0, Math.hypot(1.0, (k / t_m)));
	} else if (t_m <= 2.6e+107) {
		tmp = ((2.0 / t_4) * (l / Math.tan(k))) * (l / ((t_2 + 1.0) + 1.0));
	} else if (t_m <= 7.8e+194) {
		tmp = 2.0 / (Math.pow((Math.pow(t_m, 1.5) / l), 2.0) * (Math.sin(k) * (Math.tan(k) * t_3)));
	} else {
		tmp = (l * (2.0 / (Math.tan(k) * t_4))) * (l / t_3);
	}
	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):
	t_2 = math.pow((k / t_m), 2.0)
	t_3 = 2.0 + t_2
	t_4 = math.sin(k) * math.pow(t_m, 3.0)
	tmp = 0
	if t_m <= 7e-118:
		tmp = math.pow((((l * math.sqrt(2.0)) / (k * math.sin(k))) * math.sqrt((math.cos(k) / t_m))), 2.0)
	elif t_m <= 40000.0:
		tmp = (l * (2.0 * ((l / k) * (math.cos(k) / math.pow((t_m * math.sin(k)), 2.0))))) / math.hypot(1.0, math.hypot(1.0, (k / t_m)))
	elif t_m <= 2.6e+107:
		tmp = ((2.0 / t_4) * (l / math.tan(k))) * (l / ((t_2 + 1.0) + 1.0))
	elif t_m <= 7.8e+194:
		tmp = 2.0 / (math.pow((math.pow(t_m, 1.5) / l), 2.0) * (math.sin(k) * (math.tan(k) * t_3)))
	else:
		tmp = (l * (2.0 / (math.tan(k) * t_4))) * (l / t_3)
	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
	t_3 = Float64(2.0 + t_2)
	t_4 = Float64(sin(k) * (t_m ^ 3.0))
	tmp = 0.0
	if (t_m <= 7e-118)
		tmp = Float64(Float64(Float64(l * sqrt(2.0)) / Float64(k * sin(k))) * sqrt(Float64(cos(k) / t_m))) ^ 2.0;
	elseif (t_m <= 40000.0)
		tmp = Float64(Float64(l * Float64(2.0 * Float64(Float64(l / k) * Float64(cos(k) / (Float64(t_m * sin(k)) ^ 2.0))))) / hypot(1.0, hypot(1.0, Float64(k / t_m))));
	elseif (t_m <= 2.6e+107)
		tmp = Float64(Float64(Float64(2.0 / t_4) * Float64(l / tan(k))) * Float64(l / Float64(Float64(t_2 + 1.0) + 1.0)));
	elseif (t_m <= 7.8e+194)
		tmp = Float64(2.0 / Float64((Float64((t_m ^ 1.5) / l) ^ 2.0) * Float64(sin(k) * Float64(tan(k) * t_3))));
	else
		tmp = Float64(Float64(l * Float64(2.0 / Float64(tan(k) * t_4))) * Float64(l / t_3));
	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)
	t_2 = (k / t_m) ^ 2.0;
	t_3 = 2.0 + t_2;
	t_4 = sin(k) * (t_m ^ 3.0);
	tmp = 0.0;
	if (t_m <= 7e-118)
		tmp = (((l * sqrt(2.0)) / (k * sin(k))) * sqrt((cos(k) / t_m))) ^ 2.0;
	elseif (t_m <= 40000.0)
		tmp = (l * (2.0 * ((l / k) * (cos(k) / ((t_m * sin(k)) ^ 2.0))))) / hypot(1.0, hypot(1.0, (k / t_m)));
	elseif (t_m <= 2.6e+107)
		tmp = ((2.0 / t_4) * (l / tan(k))) * (l / ((t_2 + 1.0) + 1.0));
	elseif (t_m <= 7.8e+194)
		tmp = 2.0 / ((((t_m ^ 1.5) / l) ^ 2.0) * (sin(k) * (tan(k) * t_3)));
	else
		tmp = (l * (2.0 / (tan(k) * t_4))) * (l / t_3);
	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_] := Block[{t$95$2 = N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(2.0 + t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 7e-118], N[Power[N[(N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[t$95$m, 40000.0], N[(N[(l * N[(2.0 * N[(N[(l / k), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Power[N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.6e+107], N[(N[(N[(2.0 / t$95$4), $MachinePrecision] * N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(N[(t$95$2 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 7.8e+194], N[(2.0 / N[(N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / t$95$3), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 5 regimes

2. if t < 7e-118

   1. Initial program 53.6%

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

   3. Simplified 50.7%

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

      1. add-sqr-sqrt 28.5%

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

      2. pow2 28.5%

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

   6. Applied egg-rr 31.9%

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

   7. Taylor expanded in t around 0 31.5%

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

   if 7e-118 < t < 4e4

   1. Initial program 59.5%

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

   3. Simplified 59.0%

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

      1. associate-*r* 59.9%

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

      2. add-sqr-sqrt 59.8%

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

      3. times-frac 59.8%

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

   6. Applied egg-rr 60.0%

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

   7. Taylor expanded in t around 0 53.5%

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

      1. *-commutative 53.5%

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

   9. Simplified 53.5%

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

       1. associate-*r/ 53.4%

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

       2. times-frac 53.5%

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

       3. pow-prod-down 53.4%

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

   11. Applied egg-rr 53.4%

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

   if 4e4 < t < 2.6000000000000001e107

   1. Initial program 79.8%

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

   3. Simplified 70.1%

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

      1. associate-*r* 76.9%

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

      2. add-sqr-sqrt 76.6%

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

      3. times-frac 76.8%

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

   6. Applied egg-rr 93.0%

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

      1. frac-times 89.4%

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

      2. associate-*l/ 89.2%

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

      3. pow2 89.2%

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

   8. Applied egg-rr 89.2%

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

      1. associate-/l* 89.5%

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

      2. times-frac 92.7%

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

      3. *-commutative 92.7%

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

      4. unpow2 92.7%

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

      5. hypot-undefine 92.7%

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

      6. hypot-undefine 92.7%

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

      7. rem-square-sqrt 93.2%

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

      8. metadata-eval 93.2%

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

      9. hypot-undefine 93.2%

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

      10. metadata-eval 93.2%

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

      11. unpow2 93.2%

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

      12. hypot-undefine 93.2%

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

      13. metadata-eval 93.2%

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

      14. unpow2 93.2%

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

   10. Simplified 93.2%

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

   if 2.6000000000000001e107 < t < 7.80000000000000031e194

   1. Initial program 37.2%

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

   3. Simplified 38.2%

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

      1. add-sqr-sqrt 38.2%

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

      2. pow2 38.2%

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

      3. associate-/r* 36.9%

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

      4. sqrt-div 36.9%

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

      5. sqrt-pow1 59.9%

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

      6. metadata-eval 59.9%

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

      7. sqrt-prod 36.4%

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

      8. add-sqr-sqrt 86.1%

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

   6. Applied egg-rr 86.1%

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

      1. distribute-lft-in 86.1%

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

   8. Applied egg-rr 86.1%

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

      1. distribute-lft-out 86.1%

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

      2. associate-*r* 86.2%

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

   10. Simplified 86.2%

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

   if 7.80000000000000031e194 < t

   1. Initial program 63.1%

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

   3. Simplified 56.3%

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

      1. associate-*r* 69.9%

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

      2. *-un-lft-identity 69.9%

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

      3. times-frac 69.9%

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

      4. associate-*r* 76.7%

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

   6. Applied egg-rr 76.7%

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

      1. /-rgt-identity 76.7%

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

      2. *-commutative 76.7%

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

      3. *-commutative 76.7%

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

   8. Simplified 76.7%

      \[\leadsto \color{blue}{\left(\ell \cdot \frac{2}{\left(\sin k \cdot {t}^{3}\right) \cdot \tan k}\right) \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7 \cdot 10^{-118}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t}}\right)}^{2}\\ \mathbf{elif}\;t \leq 40000:\\ \;\;\;\;\frac{\ell \cdot \left(2 \cdot \left(\frac{\ell}{k} \cdot \frac{\cos k}{{\left(t \cdot \sin k\right)}^{2}}\right)\right)}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{+107}:\\ \;\;\;\;\left(\frac{2}{\sin k \cdot {t}^{3}} \cdot \frac{\ell}{\tan k}\right) \cdot \frac{\ell}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1}\\ \mathbf{elif}\;t \leq 7.8 \cdot 10^{+194}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot {t}^{3}\right)}\right) \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 78.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\frac{k}{t\_m}\right)}^{2}\\ t_3 := 2 + t\_2\\ t_4 := \sin k \cdot {t\_m}^{3}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 10^{-63}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t\_m}}\right)}^{2}\\ \mathbf{elif}\;t\_m \leq 2.6 \cdot 10^{+107}:\\ \;\;\;\;\left(\frac{2}{t\_4} \cdot \frac{\ell}{\tan k}\right) \cdot \frac{\ell}{\left(t\_2 + 1\right) + 1}\\ \mathbf{elif}\;t\_m \leq 1.5 \cdot 10^{+195}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2} \cdot \left(\sin k \cdot \left(\tan k \cdot t\_3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \frac{2}{\tan k \cdot t\_4}\right) \cdot \frac{\ell}{t\_3}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (/ k t_m) 2.0))
        (t_3 (+ 2.0 t_2))
        (t_4 (* (sin k) (pow t_m 3.0))))
   (*
    t_s
    (if (<= t_m 1e-63)
      (pow (* (/ (* l (sqrt 2.0)) (* k (sin k))) (sqrt (/ (cos k) t_m))) 2.0)
      (if (<= t_m 2.6e+107)
        (* (* (/ 2.0 t_4) (/ l (tan k))) (/ l (+ (+ t_2 1.0) 1.0)))
        (if (<= t_m 1.5e+195)
          (/ 2.0 (* (pow (/ (pow t_m 1.5) l) 2.0) (* (sin k) (* (tan k) t_3))))
          (* (* l (/ 2.0 (* (tan k) t_4))) (/ l t_3))))))))

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 t_3 = 2.0 + t_2;
	double t_4 = sin(k) * pow(t_m, 3.0);
	double tmp;
	if (t_m <= 1e-63) {
		tmp = pow((((l * sqrt(2.0)) / (k * sin(k))) * sqrt((cos(k) / t_m))), 2.0);
	} else if (t_m <= 2.6e+107) {
		tmp = ((2.0 / t_4) * (l / tan(k))) * (l / ((t_2 + 1.0) + 1.0));
	} else if (t_m <= 1.5e+195) {
		tmp = 2.0 / (pow((pow(t_m, 1.5) / l), 2.0) * (sin(k) * (tan(k) * t_3)));
	} else {
		tmp = (l * (2.0 / (tan(k) * t_4))) * (l / t_3);
	}
	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) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_2 = (k / t_m) ** 2.0d0
    t_3 = 2.0d0 + t_2
    t_4 = sin(k) * (t_m ** 3.0d0)
    if (t_m <= 1d-63) then
        tmp = (((l * sqrt(2.0d0)) / (k * sin(k))) * sqrt((cos(k) / t_m))) ** 2.0d0
    else if (t_m <= 2.6d+107) then
        tmp = ((2.0d0 / t_4) * (l / tan(k))) * (l / ((t_2 + 1.0d0) + 1.0d0))
    else if (t_m <= 1.5d+195) then
        tmp = 2.0d0 / ((((t_m ** 1.5d0) / l) ** 2.0d0) * (sin(k) * (tan(k) * t_3)))
    else
        tmp = (l * (2.0d0 / (tan(k) * t_4))) * (l / t_3)
    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 t_2 = Math.pow((k / t_m), 2.0);
	double t_3 = 2.0 + t_2;
	double t_4 = Math.sin(k) * Math.pow(t_m, 3.0);
	double tmp;
	if (t_m <= 1e-63) {
		tmp = Math.pow((((l * Math.sqrt(2.0)) / (k * Math.sin(k))) * Math.sqrt((Math.cos(k) / t_m))), 2.0);
	} else if (t_m <= 2.6e+107) {
		tmp = ((2.0 / t_4) * (l / Math.tan(k))) * (l / ((t_2 + 1.0) + 1.0));
	} else if (t_m <= 1.5e+195) {
		tmp = 2.0 / (Math.pow((Math.pow(t_m, 1.5) / l), 2.0) * (Math.sin(k) * (Math.tan(k) * t_3)));
	} else {
		tmp = (l * (2.0 / (Math.tan(k) * t_4))) * (l / t_3);
	}
	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):
	t_2 = math.pow((k / t_m), 2.0)
	t_3 = 2.0 + t_2
	t_4 = math.sin(k) * math.pow(t_m, 3.0)
	tmp = 0
	if t_m <= 1e-63:
		tmp = math.pow((((l * math.sqrt(2.0)) / (k * math.sin(k))) * math.sqrt((math.cos(k) / t_m))), 2.0)
	elif t_m <= 2.6e+107:
		tmp = ((2.0 / t_4) * (l / math.tan(k))) * (l / ((t_2 + 1.0) + 1.0))
	elif t_m <= 1.5e+195:
		tmp = 2.0 / (math.pow((math.pow(t_m, 1.5) / l), 2.0) * (math.sin(k) * (math.tan(k) * t_3)))
	else:
		tmp = (l * (2.0 / (math.tan(k) * t_4))) * (l / t_3)
	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
	t_3 = Float64(2.0 + t_2)
	t_4 = Float64(sin(k) * (t_m ^ 3.0))
	tmp = 0.0
	if (t_m <= 1e-63)
		tmp = Float64(Float64(Float64(l * sqrt(2.0)) / Float64(k * sin(k))) * sqrt(Float64(cos(k) / t_m))) ^ 2.0;
	elseif (t_m <= 2.6e+107)
		tmp = Float64(Float64(Float64(2.0 / t_4) * Float64(l / tan(k))) * Float64(l / Float64(Float64(t_2 + 1.0) + 1.0)));
	elseif (t_m <= 1.5e+195)
		tmp = Float64(2.0 / Float64((Float64((t_m ^ 1.5) / l) ^ 2.0) * Float64(sin(k) * Float64(tan(k) * t_3))));
	else
		tmp = Float64(Float64(l * Float64(2.0 / Float64(tan(k) * t_4))) * Float64(l / t_3));
	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)
	t_2 = (k / t_m) ^ 2.0;
	t_3 = 2.0 + t_2;
	t_4 = sin(k) * (t_m ^ 3.0);
	tmp = 0.0;
	if (t_m <= 1e-63)
		tmp = (((l * sqrt(2.0)) / (k * sin(k))) * sqrt((cos(k) / t_m))) ^ 2.0;
	elseif (t_m <= 2.6e+107)
		tmp = ((2.0 / t_4) * (l / tan(k))) * (l / ((t_2 + 1.0) + 1.0));
	elseif (t_m <= 1.5e+195)
		tmp = 2.0 / ((((t_m ^ 1.5) / l) ^ 2.0) * (sin(k) * (tan(k) * t_3)));
	else
		tmp = (l * (2.0 / (tan(k) * t_4))) * (l / t_3);
	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_] := Block[{t$95$2 = N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(2.0 + t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1e-63], N[Power[N[(N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[t$95$m, 2.6e+107], N[(N[(N[(2.0 / t$95$4), $MachinePrecision] * N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(N[(t$95$2 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.5e+195], N[(2.0 / N[(N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / t$95$3), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < 1.00000000000000007e-63

   1. Initial program 52.9%

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

   3. Simplified 50.1%

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

      1. add-sqr-sqrt 29.1%

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

      2. pow2 29.1%

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

   6. Applied egg-rr 33.0%

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

   7. Taylor expanded in t around 0 33.7%

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

   if 1.00000000000000007e-63 < t < 2.6000000000000001e107

   1. Initial program 81.0%

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

   3. Simplified 73.1%

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

      1. associate-*r* 78.6%

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

      2. add-sqr-sqrt 78.4%

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

      3. times-frac 78.6%

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

   6. Applied egg-rr 91.6%

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

      1. frac-times 88.7%

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

      2. associate-*l/ 88.6%

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

      3. pow2 88.6%

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

   8. Applied egg-rr 88.6%

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

      1. associate-/l* 88.8%

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

      2. times-frac 94.0%

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

      3. *-commutative 94.0%

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

      4. unpow2 94.0%

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

      5. hypot-undefine 94.0%

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

      6. hypot-undefine 94.0%

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

      7. rem-square-sqrt 94.5%

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

      8. metadata-eval 94.5%

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

      9. hypot-undefine 94.5%

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

      10. metadata-eval 94.5%

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

      11. unpow2 94.5%

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

      12. hypot-undefine 94.5%

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

      13. metadata-eval 94.5%

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

      14. unpow2 94.5%

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

   10. Simplified 94.5%

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

   if 2.6000000000000001e107 < t < 1.5e195

   1. Initial program 37.2%

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

   3. Simplified 38.2%

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

      1. add-sqr-sqrt 38.2%

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

      2. pow2 38.2%

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

      3. associate-/r* 36.9%

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

      4. sqrt-div 36.9%

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

      5. sqrt-pow1 59.9%

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

      6. metadata-eval 59.9%

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

      7. sqrt-prod 36.4%

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

      8. add-sqr-sqrt 86.1%

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

   6. Applied egg-rr 86.1%

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

      1. distribute-lft-in 86.1%

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

   8. Applied egg-rr 86.1%

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

      1. distribute-lft-out 86.1%

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

      2. associate-*r* 86.2%

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

   10. Simplified 86.2%

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

   if 1.5e195 < t

   1. Initial program 63.1%

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

   3. Simplified 56.3%

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

      1. associate-*r* 69.9%

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

      2. *-un-lft-identity 69.9%

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

      3. times-frac 69.9%

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

      4. associate-*r* 76.7%

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

   6. Applied egg-rr 76.7%

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

      1. /-rgt-identity 76.7%

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

      2. *-commutative 76.7%

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

      3. *-commutative 76.7%

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

   8. Simplified 76.7%

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

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 10^{-63}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t}}\right)}^{2}\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{+107}:\\ \;\;\;\;\left(\frac{2}{\sin k \cdot {t}^{3}} \cdot \frac{\ell}{\tan k}\right) \cdot \frac{\ell}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1}\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{+195}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot {t}^{3}\right)}\right) \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot {\left(\frac{\frac{t\_m}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot {t\_m}^{3}\right)}\right) \cdot \frac{\ell}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 9e-18)
    (/ 2.0 (* (* 2.0 (* k k)) (pow (/ (/ t_m (cbrt l)) (cbrt l)) 3.0)))
    (*
     (* l (/ 2.0 (* (tan k) (* (sin k) (pow t_m 3.0)))))
     (/ l (+ 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 <= 9e-18) {
		tmp = 2.0 / ((2.0 * (k * k)) * pow(((t_m / cbrt(l)) / cbrt(l)), 3.0));
	} else {
		tmp = (l * (2.0 / (tan(k) * (sin(k) * pow(t_m, 3.0))))) * (l / (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 <= 9e-18) {
		tmp = 2.0 / ((2.0 * (k * k)) * Math.pow(((t_m / Math.cbrt(l)) / Math.cbrt(l)), 3.0));
	} else {
		tmp = (l * (2.0 / (Math.tan(k) * (Math.sin(k) * Math.pow(t_m, 3.0))))) * (l / (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 <= 9e-18)
		tmp = Float64(2.0 / Float64(Float64(2.0 * Float64(k * k)) * (Float64(Float64(t_m / cbrt(l)) / cbrt(l)) ^ 3.0)));
	else
		tmp = Float64(Float64(l * Float64(2.0 / Float64(tan(k) * Float64(sin(k) * (t_m ^ 3.0))))) * Float64(l / 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, 9e-18], N[(2.0 / N[(N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 8.99999999999999987e-18

   1. Initial program 54.1%

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

   3. Simplified 56.5%

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

   5. Taylor expanded in k around 0 57.3%

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

      1. unpow2 57.3%

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

   7. Applied egg-rr 57.3%

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

      1. add-log-exp 32.1%

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

   9. Applied egg-rr 32.1%

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

       1. add-cube-cbrt 32.1%

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

       2. pow3 32.1%

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

       3. cbrt-div 32.1%

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

       4. rem-log-exp 57.2%

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

       5. cbrt-div 57.3%

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

       6. unpow3 57.2%

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

       7. add-cbrt-cube 61.2%

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

   11. Applied egg-rr 61.2%

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

   if 8.99999999999999987e-18 < t

   1. Initial program 61.8%

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

   3. Simplified 55.9%

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

      1. associate-*r* 62.5%

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

      2. *-un-lft-identity 62.5%

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

      3. times-frac 62.6%

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

      4. associate-*r* 70.0%

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

   6. Applied egg-rr 70.0%

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

      1. /-rgt-identity 70.0%

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

      2. *-commutative 70.0%

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

      3. *-commutative 70.0%

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

   8. Simplified 70.0%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot {\left(\frac{\frac{t}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot {t}^{3}\right)}\right) \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 67.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.2 \cdot 10^{-142}:\\ \;\;\;\;\frac{2}{\frac{\frac{{t\_m}^{3}}{\ell}}{\ell} \cdot \left(2 \cdot \frac{{\sin k}^{2}}{\cos k}\right)}\\ \mathbf{elif}\;t\_m \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\left(\frac{{t\_m}^{2}}{\ell} \cdot \frac{t\_m}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{k \cdot k}{t\_m \cdot t\_m}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot {\left(\frac{\frac{t\_m}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.2e-142)
    (/ 2.0 (* (/ (/ (pow t_m 3.0) l) l) (* 2.0 (/ (pow (sin k) 2.0) (cos k)))))
    (if (<= t_m 1.4e+154)
      (/
       2.0
       (*
        (* (/ (pow t_m 2.0) l) (/ t_m l))
        (* (* (sin k) (tan k)) (+ 2.0 (/ (* k k) (* t_m t_m))))))
      (/ 2.0 (* (* 2.0 (* k k)) (pow (/ (/ t_m (cbrt l)) (cbrt l)) 3.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.2e-142) {
		tmp = 2.0 / (((pow(t_m, 3.0) / l) / l) * (2.0 * (pow(sin(k), 2.0) / cos(k))));
	} else if (t_m <= 1.4e+154) {
		tmp = 2.0 / (((pow(t_m, 2.0) / l) * (t_m / l)) * ((sin(k) * tan(k)) * (2.0 + ((k * k) / (t_m * t_m)))));
	} else {
		tmp = 2.0 / ((2.0 * (k * k)) * pow(((t_m / cbrt(l)) / cbrt(l)), 3.0));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.2e-142) {
		tmp = 2.0 / (((Math.pow(t_m, 3.0) / l) / l) * (2.0 * (Math.pow(Math.sin(k), 2.0) / Math.cos(k))));
	} else if (t_m <= 1.4e+154) {
		tmp = 2.0 / (((Math.pow(t_m, 2.0) / l) * (t_m / l)) * ((Math.sin(k) * Math.tan(k)) * (2.0 + ((k * k) / (t_m * t_m)))));
	} else {
		tmp = 2.0 / ((2.0 * (k * k)) * Math.pow(((t_m / Math.cbrt(l)) / Math.cbrt(l)), 3.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.2e-142)
		tmp = Float64(2.0 / Float64(Float64(Float64((t_m ^ 3.0) / l) / l) * Float64(2.0 * Float64((sin(k) ^ 2.0) / cos(k)))));
	elseif (t_m <= 1.4e+154)
		tmp = Float64(2.0 / Float64(Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l)) * Float64(Float64(sin(k) * tan(k)) * Float64(2.0 + Float64(Float64(k * k) / Float64(t_m * t_m))))));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * Float64(k * k)) * (Float64(Float64(t_m / cbrt(l)) / cbrt(l)) ^ 3.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.2e-142], N[(2.0 / N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * N[(2.0 * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.4e+154], N[(2.0 / N[(N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(N[(k * k), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.19999999999999994e-142

   1. Initial program 53.6%

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

   3. Simplified 55.5%

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

   5. Taylor expanded in t around inf 55.0%

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

   if 1.19999999999999994e-142 < t < 1.4e154

   1. Initial program 67.1%

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

   3. Simplified 68.1%

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

      1. unpow2 68.1%

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

      2. frac-2neg 68.1%

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

      3. frac-times 68.1%

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

   6. Applied egg-rr 68.1%

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

      1. associate-/r* 54.5%

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

      2. unpow3 54.5%

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

      3. times-frac 63.0%

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

      4. pow2 63.0%

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

   8. Applied egg-rr 79.1%

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

   if 1.4e154 < t

   1. Initial program 50.9%

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

   3. Simplified 54.6%

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

   5. Taylor expanded in k around 0 54.6%

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

      1. unpow2 54.6%

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

   7. Applied egg-rr 54.6%

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

      1. add-log-exp 54.6%

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

   9. Applied egg-rr 54.6%

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

       1. add-cube-cbrt 54.6%

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

       2. pow3 54.6%

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

       3. cbrt-div 54.6%

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

       4. rem-log-exp 54.6%

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

       5. cbrt-div 54.6%

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

       6. unpow3 54.6%

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

       7. add-cbrt-cube 66.8%

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

   11. Applied egg-rr 66.8%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.2 \cdot 10^{-142}:\\ \;\;\;\;\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(2 \cdot \frac{{\sin k}^{2}}{\cos k}\right)}\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot {\left(\frac{\frac{t}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 65.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.2 \cdot 10^{-87} \lor \neg \left(t\_m \leq 1.4 \cdot 10^{+154}\right):\\ \;\;\;\;\frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot {\left(\frac{\frac{t\_m}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{t\_m}^{2}}{\ell} \cdot \frac{t\_m}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{k \cdot k}{t\_m \cdot t\_m}\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (or (<= t_m 5.2e-87) (not (<= t_m 1.4e+154)))
    (/ 2.0 (* (* 2.0 (* k k)) (pow (/ (/ t_m (cbrt l)) (cbrt l)) 3.0)))
    (/
     2.0
     (*
      (* (/ (pow t_m 2.0) l) (/ t_m l))
      (* (* (sin k) (tan k)) (+ 2.0 (/ (* k k) (* t_m t_m)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((t_m <= 5.2e-87) || !(t_m <= 1.4e+154)) {
		tmp = 2.0 / ((2.0 * (k * k)) * pow(((t_m / cbrt(l)) / cbrt(l)), 3.0));
	} else {
		tmp = 2.0 / (((pow(t_m, 2.0) / l) * (t_m / l)) * ((sin(k) * tan(k)) * (2.0 + ((k * k) / (t_m * t_m)))));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((t_m <= 5.2e-87) || !(t_m <= 1.4e+154)) {
		tmp = 2.0 / ((2.0 * (k * k)) * Math.pow(((t_m / Math.cbrt(l)) / Math.cbrt(l)), 3.0));
	} else {
		tmp = 2.0 / (((Math.pow(t_m, 2.0) / l) * (t_m / l)) * ((Math.sin(k) * Math.tan(k)) * (2.0 + ((k * k) / (t_m * t_m)))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if ((t_m <= 5.2e-87) || !(t_m <= 1.4e+154))
		tmp = Float64(2.0 / Float64(Float64(2.0 * Float64(k * k)) * (Float64(Float64(t_m / cbrt(l)) / cbrt(l)) ^ 3.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l)) * Float64(Float64(sin(k) * tan(k)) * Float64(2.0 + Float64(Float64(k * k) / Float64(t_m * t_m))))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[Or[LessEqual[t$95$m, 5.2e-87], N[Not[LessEqual[t$95$m, 1.4e+154]], $MachinePrecision]], N[(2.0 / N[(N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(N[(k * k), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5.20000000000000005e-87 or 1.4e154 < t

   1. Initial program 52.1%

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

   3. Simplified 54.4%

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

   5. Taylor expanded in k around 0 55.1%

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

      1. unpow2 55.1%

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

   7. Applied egg-rr 55.1%

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

      1. add-log-exp 34.8%

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

   9. Applied egg-rr 34.8%

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

       1. add-cube-cbrt 34.8%

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

       2. pow3 34.8%

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

       3. cbrt-div 34.8%

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

       4. rem-log-exp 55.1%

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

       5. cbrt-div 55.1%

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

       6. unpow3 55.1%

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

       7. add-cbrt-cube 60.5%

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

   11. Applied egg-rr 60.5%

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

   if 5.20000000000000005e-87 < t < 1.4e154

   1. Initial program 74.4%

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

   3. Simplified 74.6%

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

      1. unpow2 74.6%

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

      2. frac-2neg 74.6%

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

      3. frac-times 74.6%

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

   6. Applied egg-rr 74.6%

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

      1. associate-/r* 59.6%

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

      2. unpow3 59.6%

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

      3. times-frac 62.1%

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

      4. pow2 62.1%

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

   8. Applied egg-rr 80.9%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.2 \cdot 10^{-87} \lor \neg \left(t \leq 1.4 \cdot 10^{+154}\right):\\ \;\;\;\;\frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot {\left(\frac{\frac{t}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 65.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.2 \cdot 10^{-87} \lor \neg \left(t\_m \leq 3.7 \cdot 10^{+98}\right):\\ \;\;\;\;\frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot {\left(\frac{\frac{t\_m}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{{t\_m}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{k}{t\_m} \cdot \frac{k}{t\_m}\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (or (<= t_m 5.2e-87) (not (<= t_m 3.7e+98)))
    (/ 2.0 (* (* 2.0 (* k k)) (pow (/ (/ t_m (cbrt l)) (cbrt l)) 3.0)))
    (/
     2.0
     (*
      (/ (/ (pow t_m 3.0) l) l)
      (* (* (sin k) (tan k)) (+ 2.0 (* (/ k t_m) (/ k t_m)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((t_m <= 5.2e-87) || !(t_m <= 3.7e+98)) {
		tmp = 2.0 / ((2.0 * (k * k)) * pow(((t_m / cbrt(l)) / cbrt(l)), 3.0));
	} else {
		tmp = 2.0 / (((pow(t_m, 3.0) / l) / l) * ((sin(k) * tan(k)) * (2.0 + ((k / t_m) * (k / t_m)))));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((t_m <= 5.2e-87) || !(t_m <= 3.7e+98)) {
		tmp = 2.0 / ((2.0 * (k * k)) * Math.pow(((t_m / Math.cbrt(l)) / Math.cbrt(l)), 3.0));
	} else {
		tmp = 2.0 / (((Math.pow(t_m, 3.0) / l) / l) * ((Math.sin(k) * Math.tan(k)) * (2.0 + ((k / t_m) * (k / t_m)))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if ((t_m <= 5.2e-87) || !(t_m <= 3.7e+98))
		tmp = Float64(2.0 / Float64(Float64(2.0 * Float64(k * k)) * (Float64(Float64(t_m / cbrt(l)) / cbrt(l)) ^ 3.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((t_m ^ 3.0) / l) / l) * Float64(Float64(sin(k) * tan(k)) * Float64(2.0 + Float64(Float64(k / t_m) * Float64(k / t_m))))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[Or[LessEqual[t$95$m, 5.2e-87], N[Not[LessEqual[t$95$m, 3.7e+98]], $MachinePrecision]], N[(2.0 / N[(N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5.20000000000000005e-87 or 3.6999999999999999e98 < t

   1. Initial program 52.2%

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

   3. Simplified 53.7%

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

   5. Taylor expanded in k around 0 54.3%

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

      1. unpow2 54.3%

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

   7. Applied egg-rr 54.3%

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

      1. add-log-exp 34.9%

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

   9. Applied egg-rr 34.9%

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

       1. add-cube-cbrt 34.9%

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

       2. pow3 34.9%

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

       3. cbrt-div 34.9%

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

       4. rem-log-exp 54.3%

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

       5. cbrt-div 54.3%

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

       6. unpow3 54.3%

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

       7. add-cbrt-cube 59.9%

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

   11. Applied egg-rr 59.9%

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

   if 5.20000000000000005e-87 < t < 3.6999999999999999e98

   1. Initial program 80.6%

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

   3. Simplified 85.6%

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

      1. unpow2 85.6%

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

   6. Applied egg-rr 85.6%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.2 \cdot 10^{-87} \lor \neg \left(t \leq 3.7 \cdot 10^{+98}\right):\\ \;\;\;\;\frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot {\left(\frac{\frac{t}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{k}{t} \cdot \frac{k}{t}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 61.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot {\left(\frac{\frac{t\_m}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{3}} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ 2.0 (* (* 2.0 (* k k)) (pow (/ (/ t_m (cbrt l)) (cbrt l)) 3.0)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / ((2.0 * (k * k)) * pow(((t_m / cbrt(l)) / cbrt(l)), 3.0)));
}

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 * (2.0 / ((2.0 * (k * k)) * Math.pow(((t_m / Math.cbrt(l)) / Math.cbrt(l)), 3.0)));
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 / Float64(Float64(2.0 * Float64(k * k)) * (Float64(Float64(t_m / cbrt(l)) / cbrt(l)) ^ 3.0))))
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(2.0 / N[(N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.1%

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

3. Simplified 58.0%

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

5. Taylor expanded in k around 0 56.0%

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

   1. unpow2 56.0%

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

7. Applied egg-rr 56.0%

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

   1. add-log-exp 35.8%

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

9. Applied egg-rr 35.8%

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

    1. add-cube-cbrt 35.8%

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

    2. pow3 35.8%

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

    3. cbrt-div 35.8%

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

    4. rem-log-exp 55.9%

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

    5. cbrt-div 55.9%

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

    6. unpow3 55.9%

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

    7. add-cbrt-cube 60.7%

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

11. Applied egg-rr 60.7%

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

12. Final simplification 60.7%

    \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot {\left(\frac{\frac{t}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{3}} \]
13. Add Preprocessing

Alternative 16: 61.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot {\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ 2.0 (* (* 2.0 (* k 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) {
	return t_s * (2.0 / ((2.0 * (k * k)) * pow((t_m * pow(cbrt(l), -2.0)), 3.0)));
}

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 * (2.0 / ((2.0 * (k * k)) * Math.pow((t_m * Math.pow(Math.cbrt(l), -2.0)), 3.0)));
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 / Float64(Float64(2.0 * Float64(k * k)) * (Float64(t_m * (cbrt(l) ^ -2.0)) ^ 3.0))))
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(2.0 / N[(N[(2.0 * N[(k * k), $MachinePrecision]), $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]

Derivation

1. Initial program 56.1%

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

3. Simplified 58.0%

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

5. Taylor expanded in k around 0 56.0%

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

   1. unpow2 56.0%

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

7. Applied egg-rr 56.0%

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

   1. associate-/r* 52.0%

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

   2. unpow3 52.0%

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

   3. times-frac 57.9%

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

   4. pow2 57.9%

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

9. Applied egg-rr 57.9%

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

    1. frac-times 52.0%

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

    2. unpow2 52.0%

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

    3. unpow3 52.0%

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

    4. add-cube-cbrt 52.0%

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

    5. pow3 52.0%

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

    6. add-cube-cbrt 52.0%

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

    7. pow3 52.0%

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

    8. unpow-prod-down 52.0%

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

    9. unpow2 52.0%

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

    10. cube-div 60.7%

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

    11. div-inv 60.7%

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

    12. pow-flip 60.7%

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

    13. metadata-eval 60.7%

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

11. Applied egg-rr 60.7%

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

12. Final simplification 60.7%

    \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot {\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}} \]
13. Add Preprocessing

Alternative 17: 60.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{{\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ 2.0 (* (pow (/ (pow t_m 1.5) l) 2.0) (* 2.0 (* k k))))))

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 * (2.0 / (pow((pow(t_m, 1.5) / l), 2.0) * (2.0 * (k * k))));
}

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 * (2.0d0 / ((((t_m ** 1.5d0) / l) ** 2.0d0) * (2.0d0 * (k * k))))
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 * (2.0 / (Math.pow((Math.pow(t_m, 1.5) / l), 2.0) * (2.0 * (k * k))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 / (math.pow((math.pow(t_m, 1.5) / l), 2.0) * (2.0 * (k * k))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 / Float64((Float64((t_m ^ 1.5) / l) ^ 2.0) * Float64(2.0 * Float64(k * k)))))
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 * (2.0 / ((((t_m ^ 1.5) / l) ^ 2.0) * (2.0 * (k * k))));
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[(2.0 / N[(N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision] * N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.1%

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

3. Simplified 58.0%

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

5. Taylor expanded in k around 0 56.0%

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

   1. unpow2 56.0%

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

7. Applied egg-rr 56.0%

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

   1. add-sqr-sqrt 27.8%

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

   2. pow2 27.8%

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

   3. associate-/r* 24.4%

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

   4. sqrt-div 24.4%

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

   5. sqrt-pow1 28.3%

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

   6. metadata-eval 28.3%

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

   7. sqrt-prod 18.2%

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

   8. add-sqr-sqrt 33.9%

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

9. Applied egg-rr 30.3%

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

Alternative 18: 58.1% accurate, 3.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{{t\_m}^{2}}{\ell} \cdot \frac{t\_m}{\ell}\right)} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ 2.0 (* (* 2.0 (* k k)) (* (/ (pow t_m 2.0) l) (/ t_m l))))))

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 * (2.0 / ((2.0 * (k * k)) * ((pow(t_m, 2.0) / l) * (t_m / l))));
}

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 * (2.0d0 / ((2.0d0 * (k * k)) * (((t_m ** 2.0d0) / l) * (t_m / l))))
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 * (2.0 / ((2.0 * (k * k)) * ((Math.pow(t_m, 2.0) / l) * (t_m / l))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 / ((2.0 * (k * k)) * ((math.pow(t_m, 2.0) / l) * (t_m / l))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 / Float64(Float64(2.0 * Float64(k * k)) * Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l)))))
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 * (2.0 / ((2.0 * (k * k)) * (((t_m ^ 2.0) / l) * (t_m / l))));
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[(2.0 / N[(N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.1%

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

3. Simplified 58.0%

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

5. Taylor expanded in k around 0 56.0%

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

   1. unpow2 56.0%

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

7. Applied egg-rr 56.0%

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

   1. associate-/r* 52.0%

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

   2. unpow3 52.0%

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

   3. times-frac 57.9%

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

   4. pow2 57.9%

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

9. Applied egg-rr 57.9%

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

10. Final simplification 57.9%

    \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \]
11. Add Preprocessing

Alternative 19: 55.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{{t\_m}^{3}}{\ell}}{\ell}} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ 2.0 (* (* 2.0 (* k 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) {
	return t_s * (2.0 / ((2.0 * (k * k)) * ((pow(t_m, 3.0) / l) / l)));
}

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 * (2.0d0 / ((2.0d0 * (k * k)) * (((t_m ** 3.0d0) / l) / l)))
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 * (2.0 / ((2.0 * (k * k)) * ((Math.pow(t_m, 3.0) / l) / l)));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 / ((2.0 * (k * k)) * ((math.pow(t_m, 3.0) / l) / l)))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 / Float64(Float64(2.0 * Float64(k * k)) * Float64(Float64((t_m ^ 3.0) / l) / l))))
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 * (2.0 / ((2.0 * (k * k)) * (((t_m ^ 3.0) / l) / l)));
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[(2.0 / N[(N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.1%

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

3. Simplified 58.0%

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

5. Taylor expanded in k around 0 56.0%

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

   1. unpow2 56.0%

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

7. Applied egg-rr 56.0%

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

8. Final simplification 56.0%

   \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024148 
(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))))