Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.9% → 86.1%

Time: 18.8s

Alternatives: 21

Speedup: 24.8×

• 27.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.9% 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: 86.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7 \cdot 10^{-73}:\\ \;\;\;\;\frac{2}{\frac{\left(t\_m \cdot {k}^{2}\right) \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 7e-73)
    (/
     2.0
     (/ (* (* t_m (pow k 2.0)) (pow (sin k) 2.0)) (* (pow l 2.0) (cos k))))
    (/
     2.0
     (pow
      (*
       (* (* t_m (pow (cbrt l) -2.0)) (cbrt (sin k)))
       (cbrt (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))))
      3.0)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 7e-73) {
		tmp = 2.0 / (((t_m * pow(k, 2.0)) * pow(sin(k), 2.0)) / (pow(l, 2.0) * cos(k)));
	} else {
		tmp = 2.0 / pow((((t_m * pow(cbrt(l), -2.0)) * cbrt(sin(k))) * cbrt((tan(k) * (2.0 + pow((k / t_m), 2.0))))), 3.0);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 7e-73) {
		tmp = 2.0 / (((t_m * Math.pow(k, 2.0)) * Math.pow(Math.sin(k), 2.0)) / (Math.pow(l, 2.0) * Math.cos(k)));
	} else {
		tmp = 2.0 / Math.pow((((t_m * Math.pow(Math.cbrt(l), -2.0)) * Math.cbrt(Math.sin(k))) * Math.cbrt((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0))))), 3.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 7e-73)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k ^ 2.0)) * (sin(k) ^ 2.0)) / Float64((l ^ 2.0) * cos(k))));
	else
		tmp = Float64(2.0 / (Float64(Float64(Float64(t_m * (cbrt(l) ^ -2.0)) * cbrt(sin(k))) * cbrt(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))))) ^ 3.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 7e-73], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 6.9999999999999995e-73

   1. Initial program 50.0%

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

   3. Simplified 50.0%

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

   5. Taylor expanded in t around 0 69.8%

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

      1. associate-*r* 69.8%

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

   7. Simplified 69.8%

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

   if 6.9999999999999995e-73 < t

   1. Initial program 70.4%

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

   3. Simplified 70.4%

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

      1. unpow2 70.4%

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

      2. clear-num 70.4%

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

      3. un-div-inv 70.4%

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

   6. Applied egg-rr 70.4%

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

      1. add-cube-cbrt 70.2%

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

      2. pow3 70.3%

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

      3. associate-/r* 73.3%

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

      4. *-commutative 73.3%

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

      5. cbrt-prod 73.2%

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

      6. associate-/r* 70.3%

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

      7. div-inv 70.3%

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

      8. cbrt-prod 70.2%

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

      9. rem-cbrt-cube 77.1%

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

      10. pow2 77.1%

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

      11. pow-flip 77.6%

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

      12. metadata-eval 77.6%

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

   8. Applied egg-rr 77.6%

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

      1. pow1 77.6%

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

      2. associate-*r* 77.7%

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

   10. Applied egg-rr 77.7%

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

       1. unpow1 77.7%

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

       2. associate-*l* 77.6%

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

       3. *-commutative 77.6%

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

       4. unpow1/3 76.4%

          \[\leadsto \frac{2}{{\left(\left(t \cdot \color{blue}{{\left({\ell}^{-2}\right)}^{0.3333333333333333}}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)} \]

       5. exp-to-pow 36.5%

          \[\leadsto \frac{2}{{\left(\left(t \cdot {\color{blue}{\left(e^{\log \ell \cdot -2}\right)}}^{0.3333333333333333}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)} \]

       6. *-commutative 36.5%

          \[\leadsto \frac{2}{{\left(\left(t \cdot {\left(e^{\color{blue}{-2 \cdot \log \ell}}\right)}^{0.3333333333333333}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)} \]

       7. exp-prod 40.6%

          \[\leadsto \frac{2}{{\left(\left(t \cdot \color{blue}{e^{\left(-2 \cdot \log \ell\right) \cdot 0.3333333333333333}}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)} \]

       8. associate-*l* 40.6%

          \[\leadsto \frac{2}{{\left(\left(t \cdot e^{\color{blue}{-2 \cdot \left(\log \ell \cdot 0.3333333333333333\right)}}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)} \]

       9. rem-log-exp 40.6%

          \[\leadsto \frac{2}{{\left(\left(t \cdot e^{-2 \cdot \color{blue}{\log \left(e^{\log \ell \cdot 0.3333333333333333}\right)}}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)} \]

       10. exp-to-pow 40.7%

           \[\leadsto \frac{2}{{\left(\left(t \cdot e^{-2 \cdot \log \color{blue}{\left({\ell}^{0.3333333333333333}\right)}}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)} \]

       11. unpow1/3 40.8%

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

       12. *-commutative 40.8%

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

       13. exp-to-pow 88.7%

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

   12. Simplified 88.7%

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

       1. add-cube-cbrt 88.7%

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

       2. pow3 88.7%

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

   14. Applied egg-rr 95.1%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7 \cdot 10^{-73}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot {k}^{2}\right) \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t\_m}\right)\right)}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.25 \cdot 10^{-89}:\\ \;\;\;\;\frac{2}{\frac{\left(t\_m \cdot {k}^{2}\right) \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\\ \mathbf{elif}\;t\_m \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;t\_2 \cdot \left(\frac{2}{\tan k \cdot \left(\sin k \cdot {t\_m}^{3}\right)} \cdot t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t\_m \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k}\right)\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t\_m}}{\frac{t\_m}{k}}\right)\right)\right)}\\ \end{array} \end{array} \end{array} \]

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 (/ l (hypot 1.0 (hypot 1.0 (/ k t_m))))))
   (*
    t_s
    (if (<= t_m 1.25e-89)
      (/
       2.0
       (/ (* (* t_m (pow k 2.0)) (pow (sin k) 2.0)) (* (pow l 2.0) (cos k))))
      (if (<= t_m 5.6e+102)
        (* t_2 (* (/ 2.0 (* (tan k) (* (sin k) (pow t_m 3.0)))) t_2))
        (/
         2.0
         (*
          (pow (* t_m (* (pow (cbrt l) -2.0) (cbrt (sin k)))) 3.0)
          (* (tan k) (+ 1.0 (+ 1.0 (/ (/ k t_m) (/ t_m 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 = l / hypot(1.0, hypot(1.0, (k / t_m)));
	double tmp;
	if (t_m <= 1.25e-89) {
		tmp = 2.0 / (((t_m * pow(k, 2.0)) * pow(sin(k), 2.0)) / (pow(l, 2.0) * cos(k)));
	} else if (t_m <= 5.6e+102) {
		tmp = t_2 * ((2.0 / (tan(k) * (sin(k) * pow(t_m, 3.0)))) * t_2);
	} else {
		tmp = 2.0 / (pow((t_m * (pow(cbrt(l), -2.0) * cbrt(sin(k)))), 3.0) * (tan(k) * (1.0 + (1.0 + ((k / t_m) / (t_m / 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 = l / Math.hypot(1.0, Math.hypot(1.0, (k / t_m)));
	double tmp;
	if (t_m <= 1.25e-89) {
		tmp = 2.0 / (((t_m * Math.pow(k, 2.0)) * Math.pow(Math.sin(k), 2.0)) / (Math.pow(l, 2.0) * Math.cos(k)));
	} else if (t_m <= 5.6e+102) {
		tmp = t_2 * ((2.0 / (Math.tan(k) * (Math.sin(k) * Math.pow(t_m, 3.0)))) * t_2);
	} else {
		tmp = 2.0 / (Math.pow((t_m * (Math.pow(Math.cbrt(l), -2.0) * Math.cbrt(Math.sin(k)))), 3.0) * (Math.tan(k) * (1.0 + (1.0 + ((k / t_m) / (t_m / 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(l / hypot(1.0, hypot(1.0, Float64(k / t_m))))
	tmp = 0.0
	if (t_m <= 1.25e-89)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k ^ 2.0)) * (sin(k) ^ 2.0)) / Float64((l ^ 2.0) * cos(k))));
	elseif (t_m <= 5.6e+102)
		tmp = Float64(t_2 * Float64(Float64(2.0 / Float64(tan(k) * Float64(sin(k) * (t_m ^ 3.0)))) * t_2));
	else
		tmp = Float64(2.0 / Float64((Float64(t_m * Float64((cbrt(l) ^ -2.0) * cbrt(sin(k)))) ^ 3.0) * Float64(tan(k) * Float64(1.0 + Float64(1.0 + Float64(Float64(k / t_m) / Float64(t_m / 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[(l / N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.25e-89], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.6e+102], N[(t$95$2 * N[(N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(t$95$m * N[(N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(1.0 + N[(N[(k / t$95$m), $MachinePrecision] / N[(t$95$m / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.24999999999999992e-89

   1. Initial program 48.8%

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

   3. Simplified 48.8%

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

   5. Taylor expanded in t around 0 69.3%

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

      1. associate-*r* 69.3%

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

   7. Simplified 69.3%

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

   if 1.24999999999999992e-89 < t < 5.60000000000000037e102

   1. Initial program 75.1%

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

   3. Simplified 77.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing
   5. 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 80.0%

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

      3. times-frac 82.5%

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

   6. Applied egg-rr 83.6%

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

      1. associate-/l* 85.9%

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

      2. *-commutative 85.9%

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

   8. Simplified 85.9%

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

   if 5.60000000000000037e102 < t

   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 67.1%

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

      1. unpow2 67.1%

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

      2. clear-num 67.1%

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

      3. un-div-inv 67.1%

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

   6. Applied egg-rr 67.1%

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

      1. add-cube-cbrt 67.1%

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

      2. pow3 67.1%

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

      3. associate-/r* 70.4%

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

      4. *-commutative 70.4%

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

      5. cbrt-prod 70.4%

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

      6. associate-/r* 67.1%

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

      7. div-inv 67.1%

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

      8. cbrt-prod 67.1%

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

      9. rem-cbrt-cube 80.8%

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

      10. pow2 80.8%

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

      11. pow-flip 81.5%

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

      12. metadata-eval 81.5%

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

   8. Applied egg-rr 81.5%

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

      1. pow1 81.5%

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

      2. associate-*r* 81.6%

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

   10. Applied egg-rr 81.6%

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

       1. unpow1 81.6%

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

       2. associate-*l* 81.5%

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

       3. *-commutative 81.5%

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

       4. unpow1/3 80.6%

          \[\leadsto \frac{2}{{\left(\left(t \cdot \color{blue}{{\left({\ell}^{-2}\right)}^{0.3333333333333333}}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)} \]

       5. exp-to-pow 44.9%

          \[\leadsto \frac{2}{{\left(\left(t \cdot {\color{blue}{\left(e^{\log \ell \cdot -2}\right)}}^{0.3333333333333333}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)} \]

       6. *-commutative 44.9%

          \[\leadsto \frac{2}{{\left(\left(t \cdot {\left(e^{\color{blue}{-2 \cdot \log \ell}}\right)}^{0.3333333333333333}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)} \]

       7. exp-prod 51.1%

          \[\leadsto \frac{2}{{\left(\left(t \cdot \color{blue}{e^{\left(-2 \cdot \log \ell\right) \cdot 0.3333333333333333}}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)} \]

       8. associate-*l* 51.1%

          \[\leadsto \frac{2}{{\left(\left(t \cdot e^{\color{blue}{-2 \cdot \left(\log \ell \cdot 0.3333333333333333\right)}}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)} \]

       9. rem-log-exp 51.1%

          \[\leadsto \frac{2}{{\left(\left(t \cdot e^{-2 \cdot \color{blue}{\log \left(e^{\log \ell \cdot 0.3333333333333333}\right)}}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)} \]

       10. exp-to-pow 51.2%

           \[\leadsto \frac{2}{{\left(\left(t \cdot e^{-2 \cdot \log \color{blue}{\left({\ell}^{0.3333333333333333}\right)}}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)} \]

       11. unpow1/3 51.2%

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

       12. *-commutative 51.2%

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

       13. exp-to-pow 96.4%

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

   12. Simplified 96.4%

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

       1. pow1 96.4%

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

   14. Applied egg-rr 96.4%

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

       1. unpow1 96.4%

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

       2. associate-*l* 96.6%

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

   16. Simplified 96.6%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.25 \cdot 10^{-89}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot {k}^{2}\right) \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \left(\frac{2}{\tan k \cdot \left(\sin k \cdot {t}^{3}\right)} \cdot \frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k}\right)\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.1 \cdot 10^{-70}:\\ \;\;\;\;\frac{2}{\frac{\left(t\_m \cdot {k}^{2}\right) \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t\_m \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k}\right)\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t\_m}}{\frac{t\_m}{k}}\right)\right)\right)}\\ \end{array} \end{array} \]

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 (<= t_m 2.1e-70)
    (/
     2.0
     (/ (* (* t_m (pow k 2.0)) (pow (sin k) 2.0)) (* (pow l 2.0) (cos k))))
    (/
     2.0
     (*
      (pow (* t_m (* (pow (cbrt l) -2.0) (cbrt (sin k)))) 3.0)
      (* (tan k) (+ 1.0 (+ 1.0 (/ (/ k t_m) (/ t_m k))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.1e-70) {
		tmp = 2.0 / (((t_m * pow(k, 2.0)) * pow(sin(k), 2.0)) / (pow(l, 2.0) * cos(k)));
	} else {
		tmp = 2.0 / (pow((t_m * (pow(cbrt(l), -2.0) * cbrt(sin(k)))), 3.0) * (tan(k) * (1.0 + (1.0 + ((k / t_m) / (t_m / k))))));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.1e-70) {
		tmp = 2.0 / (((t_m * Math.pow(k, 2.0)) * Math.pow(Math.sin(k), 2.0)) / (Math.pow(l, 2.0) * Math.cos(k)));
	} else {
		tmp = 2.0 / (Math.pow((t_m * (Math.pow(Math.cbrt(l), -2.0) * Math.cbrt(Math.sin(k)))), 3.0) * (Math.tan(k) * (1.0 + (1.0 + ((k / t_m) / (t_m / k))))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.1e-70)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k ^ 2.0)) * (sin(k) ^ 2.0)) / Float64((l ^ 2.0) * cos(k))));
	else
		tmp = Float64(2.0 / Float64((Float64(t_m * Float64((cbrt(l) ^ -2.0) * cbrt(sin(k)))) ^ 3.0) * Float64(tan(k) * Float64(1.0 + Float64(1.0 + Float64(Float64(k / t_m) / Float64(t_m / k)))))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.1e-70], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(t$95$m * N[(N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(1.0 + N[(N[(k / t$95$m), $MachinePrecision] / N[(t$95$m / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.1000000000000001e-70

   1. Initial program 50.0%

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

   3. Simplified 50.0%

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

   5. Taylor expanded in t around 0 69.8%

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

      1. associate-*r* 69.8%

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

   7. Simplified 69.8%

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

   if 2.1000000000000001e-70 < t

   1. Initial program 70.4%

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

   3. Simplified 70.4%

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

      1. unpow2 70.4%

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

      2. clear-num 70.4%

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

      3. un-div-inv 70.4%

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

   6. Applied egg-rr 70.4%

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

      1. add-cube-cbrt 70.2%

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

      2. pow3 70.3%

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

      3. associate-/r* 73.3%

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

      4. *-commutative 73.3%

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

      5. cbrt-prod 73.2%

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

      6. associate-/r* 70.3%

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

      7. div-inv 70.3%

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

      8. cbrt-prod 70.2%

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

      9. rem-cbrt-cube 77.1%

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

      10. pow2 77.1%

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

      11. pow-flip 77.6%

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

      12. metadata-eval 77.6%

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

   8. Applied egg-rr 77.6%

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

      1. pow1 77.6%

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

      2. associate-*r* 77.7%

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

   10. Applied egg-rr 77.7%

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

       1. unpow1 77.7%

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

       2. associate-*l* 77.6%

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

       3. *-commutative 77.6%

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

       4. unpow1/3 76.4%

          \[\leadsto \frac{2}{{\left(\left(t \cdot \color{blue}{{\left({\ell}^{-2}\right)}^{0.3333333333333333}}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)} \]

       5. exp-to-pow 36.5%

          \[\leadsto \frac{2}{{\left(\left(t \cdot {\color{blue}{\left(e^{\log \ell \cdot -2}\right)}}^{0.3333333333333333}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)} \]

       6. *-commutative 36.5%

          \[\leadsto \frac{2}{{\left(\left(t \cdot {\left(e^{\color{blue}{-2 \cdot \log \ell}}\right)}^{0.3333333333333333}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)} \]

       7. exp-prod 40.6%

          \[\leadsto \frac{2}{{\left(\left(t \cdot \color{blue}{e^{\left(-2 \cdot \log \ell\right) \cdot 0.3333333333333333}}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)} \]

       8. associate-*l* 40.6%

          \[\leadsto \frac{2}{{\left(\left(t \cdot e^{\color{blue}{-2 \cdot \left(\log \ell \cdot 0.3333333333333333\right)}}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)} \]

       9. rem-log-exp 40.6%

          \[\leadsto \frac{2}{{\left(\left(t \cdot e^{-2 \cdot \color{blue}{\log \left(e^{\log \ell \cdot 0.3333333333333333}\right)}}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)} \]

       10. exp-to-pow 40.7%

           \[\leadsto \frac{2}{{\left(\left(t \cdot e^{-2 \cdot \log \color{blue}{\left({\ell}^{0.3333333333333333}\right)}}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)} \]

       11. unpow1/3 40.8%

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

       12. *-commutative 40.8%

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

       13. exp-to-pow 88.7%

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

   12. Simplified 88.7%

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

       1. pow1 88.7%

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

   14. Applied egg-rr 88.7%

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

       1. unpow1 88.7%

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

       2. associate-*l* 88.8%

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

   16. Simplified 88.8%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.1 \cdot 10^{-70}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot {k}^{2}\right) \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k}\right)\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 55.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.5 \cdot 10^{-73}:\\ \;\;\;\;\frac{2}{\frac{\left(t\_m \cdot {k}^{2}\right) \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t\_m}}{\frac{t\_m}{k}}\right)\right)\right) \cdot {\left(\sqrt[3]{\sin k} \cdot \left(t\_m \cdot {\ell}^{-0.6666666666666666}\right)\right)}^{3}}\\ \end{array} \end{array} \]

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-73)
    (/
     2.0
     (/ (* (* t_m (pow k 2.0)) (pow (sin k) 2.0)) (* (pow l 2.0) (cos k))))
    (/
     2.0
     (*
      (* (tan k) (+ 1.0 (+ 1.0 (/ (/ k t_m) (/ t_m k)))))
      (pow (* (cbrt (sin k)) (* t_m (pow l -0.6666666666666666))) 3.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 7.5e-73) {
		tmp = 2.0 / (((t_m * pow(k, 2.0)) * pow(sin(k), 2.0)) / (pow(l, 2.0) * cos(k)));
	} else {
		tmp = 2.0 / ((tan(k) * (1.0 + (1.0 + ((k / t_m) / (t_m / k))))) * pow((cbrt(sin(k)) * (t_m * pow(l, -0.6666666666666666))), 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-73) {
		tmp = 2.0 / (((t_m * Math.pow(k, 2.0)) * Math.pow(Math.sin(k), 2.0)) / (Math.pow(l, 2.0) * Math.cos(k)));
	} else {
		tmp = 2.0 / ((Math.tan(k) * (1.0 + (1.0 + ((k / t_m) / (t_m / k))))) * Math.pow((Math.cbrt(Math.sin(k)) * (t_m * Math.pow(l, -0.6666666666666666))), 3.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 7.5e-73)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k ^ 2.0)) * (sin(k) ^ 2.0)) / Float64((l ^ 2.0) * cos(k))));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(1.0 + Float64(1.0 + Float64(Float64(k / t_m) / Float64(t_m / k))))) * (Float64(cbrt(sin(k)) * Float64(t_m * (l ^ -0.6666666666666666))) ^ 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-73], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(1.0 + N[(N[(k / t$95$m), $MachinePrecision] / N[(t$95$m / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(t$95$m * N[Power[l, -0.6666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 7.5e-73

   1. Initial program 50.0%

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

   3. Simplified 50.0%

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

   5. Taylor expanded in t around 0 69.8%

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

      1. associate-*r* 69.8%

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

   7. Simplified 69.8%

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

   if 7.5e-73 < t

   1. Initial program 70.4%

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

   3. Simplified 70.4%

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

      1. unpow2 70.4%

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

      2. clear-num 70.4%

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

      3. un-div-inv 70.4%

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

   6. Applied egg-rr 70.4%

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

      1. add-cube-cbrt 70.2%

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

      2. pow3 70.3%

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

      3. associate-/r* 73.3%

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

      4. *-commutative 73.3%

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

      5. cbrt-prod 73.2%

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

      6. associate-/r* 70.3%

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

      7. div-inv 70.3%

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

      8. cbrt-prod 70.2%

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

      9. rem-cbrt-cube 77.1%

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

      10. pow2 77.1%

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

      11. pow-flip 77.6%

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

      12. metadata-eval 77.6%

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

   8. Applied egg-rr 77.6%

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

      1. pow1/3 76.4%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \left(t \cdot \color{blue}{{\left({\ell}^{-2}\right)}^{0.3333333333333333}}\right)\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)} \]

      2. pow-pow 40.7%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \left(t \cdot \color{blue}{{\ell}^{\left(-2 \cdot 0.3333333333333333\right)}}\right)\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)} \]

      3. metadata-eval 40.7%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \left(t \cdot {\ell}^{\color{blue}{-0.6666666666666666}}\right)\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)} \]

   10. Applied egg-rr 40.7%

       \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \left(t \cdot \color{blue}{{\ell}^{-0.6666666666666666}}\right)\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.5 \cdot 10^{-73}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot {k}^{2}\right) \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right) \cdot {\left(\sqrt[3]{\sin k} \cdot \left(t \cdot {\ell}^{-0.6666666666666666}\right)\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.4 \cdot 10^{-70}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \frac{\cos k \cdot \left({\ell}^{2} \cdot {\sin k}^{-2}\right)}{t\_m}\\ \mathbf{elif}\;t\_m \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{2 \cdot \ell}{\sin k \cdot \left(\tan k \cdot {t\_m}^{3}\right)} \cdot \frac{\ell}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \end{array} \]

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 (<= t_m 6.4e-70)
    (*
     (/ 2.0 (pow k 2.0))
     (/ (* (cos k) (* (pow l 2.0) (pow (sin k) -2.0))) t_m))
    (if (<= t_m 5.6e+102)
      (*
       (/ (* 2.0 l) (* (sin k) (* (tan k) (pow t_m 3.0))))
       (/ l (+ 2.0 (pow (/ k t_m) 2.0))))
      (/
       2.0
       (*
        (pow (* (* t_m (pow (cbrt l) -2.0)) (cbrt (sin k))) 3.0)
        (* 2.0 k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 6.4e-70) {
		tmp = (2.0 / pow(k, 2.0)) * ((cos(k) * (pow(l, 2.0) * pow(sin(k), -2.0))) / t_m);
	} else if (t_m <= 5.6e+102) {
		tmp = ((2.0 * l) / (sin(k) * (tan(k) * pow(t_m, 3.0)))) * (l / (2.0 + pow((k / t_m), 2.0)));
	} else {
		tmp = 2.0 / (pow(((t_m * pow(cbrt(l), -2.0)) * cbrt(sin(k))), 3.0) * (2.0 * k));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 6.4e-70) {
		tmp = (2.0 / Math.pow(k, 2.0)) * ((Math.cos(k) * (Math.pow(l, 2.0) * Math.pow(Math.sin(k), -2.0))) / t_m);
	} else if (t_m <= 5.6e+102) {
		tmp = ((2.0 * l) / (Math.sin(k) * (Math.tan(k) * Math.pow(t_m, 3.0)))) * (l / (2.0 + Math.pow((k / t_m), 2.0)));
	} else {
		tmp = 2.0 / (Math.pow(((t_m * Math.pow(Math.cbrt(l), -2.0)) * Math.cbrt(Math.sin(k))), 3.0) * (2.0 * k));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 6.4e-70)
		tmp = Float64(Float64(2.0 / (k ^ 2.0)) * Float64(Float64(cos(k) * Float64((l ^ 2.0) * (sin(k) ^ -2.0))) / t_m));
	elseif (t_m <= 5.6e+102)
		tmp = Float64(Float64(Float64(2.0 * l) / Float64(sin(k) * Float64(tan(k) * (t_m ^ 3.0)))) * Float64(l / Float64(2.0 + (Float64(k / t_m) ^ 2.0))));
	else
		tmp = Float64(2.0 / Float64((Float64(Float64(t_m * (cbrt(l) ^ -2.0)) * cbrt(sin(k))) ^ 3.0) * Float64(2.0 * k)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 6.4e-70], N[(N[(2.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.6e+102], N[(N[(N[(2.0 * l), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 6.3999999999999995e-70

   1. Initial program 50.0%

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

   3. Simplified 50.0%

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

   5. Taylor expanded in t around 0 69.8%

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

      1. associate-*r/ 69.8%

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

      2. times-frac 68.7%

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

      3. *-commutative 68.7%

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

      4. times-frac 68.8%

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

   7. Simplified 68.8%

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

      1. associate-*r/ 68.8%

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

      2. div-inv 68.7%

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

      3. pow-flip 68.8%

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

      4. metadata-eval 68.8%

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

   9. Applied egg-rr 68.8%

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

   if 6.3999999999999995e-70 < t < 5.60000000000000037e102

   1. Initial program 73.7%

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

   3. Simplified 76.7%

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

      1. associate-*r* 79.7%

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

      2. *-un-lft-identity 79.7%

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

      3. times-frac 82.4%

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

      4. associate-*r* 83.7%

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

   6. Applied egg-rr 83.7%

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

      1. /-rgt-identity 83.7%

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

      2. *-commutative 83.7%

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

      3. *-commutative 83.7%

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

   8. Simplified 83.7%

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

      1. associate-*r/ 84.8%

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

      2. associate-*l* 84.8%

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

   10. Applied egg-rr 84.8%

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

   if 5.60000000000000037e102 < t

   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 67.1%

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

      1. unpow2 67.1%

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

      2. clear-num 67.1%

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

      3. un-div-inv 67.1%

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

   6. Applied egg-rr 67.1%

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

      1. add-cube-cbrt 67.1%

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

      2. pow3 67.1%

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

      3. associate-/r* 70.4%

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

      4. *-commutative 70.4%

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

      5. cbrt-prod 70.4%

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

      6. associate-/r* 67.1%

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

      7. div-inv 67.1%

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

      8. cbrt-prod 67.1%

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

      9. rem-cbrt-cube 80.8%

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

      10. pow2 80.8%

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

      11. pow-flip 81.5%

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

      12. metadata-eval 81.5%

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

   8. Applied egg-rr 81.5%

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

      1. pow1 81.5%

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

      2. associate-*r* 81.6%

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

   10. Applied egg-rr 81.6%

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

       1. unpow1 81.6%

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

       2. associate-*l* 81.5%

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

       3. *-commutative 81.5%

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

       4. unpow1/3 80.6%

          \[\leadsto \frac{2}{{\left(\left(t \cdot \color{blue}{{\left({\ell}^{-2}\right)}^{0.3333333333333333}}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)} \]

       5. exp-to-pow 44.9%

          \[\leadsto \frac{2}{{\left(\left(t \cdot {\color{blue}{\left(e^{\log \ell \cdot -2}\right)}}^{0.3333333333333333}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)} \]

       6. *-commutative 44.9%

          \[\leadsto \frac{2}{{\left(\left(t \cdot {\left(e^{\color{blue}{-2 \cdot \log \ell}}\right)}^{0.3333333333333333}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)} \]

       7. exp-prod 51.1%

          \[\leadsto \frac{2}{{\left(\left(t \cdot \color{blue}{e^{\left(-2 \cdot \log \ell\right) \cdot 0.3333333333333333}}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)} \]

       8. associate-*l* 51.1%

          \[\leadsto \frac{2}{{\left(\left(t \cdot e^{\color{blue}{-2 \cdot \left(\log \ell \cdot 0.3333333333333333\right)}}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)} \]

       9. rem-log-exp 51.1%

          \[\leadsto \frac{2}{{\left(\left(t \cdot e^{-2 \cdot \color{blue}{\log \left(e^{\log \ell \cdot 0.3333333333333333}\right)}}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)} \]

       10. exp-to-pow 51.2%

           \[\leadsto \frac{2}{{\left(\left(t \cdot e^{-2 \cdot \log \color{blue}{\left({\ell}^{0.3333333333333333}\right)}}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)} \]

       11. unpow1/3 51.2%

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

       12. *-commutative 51.2%

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

       13. exp-to-pow 96.4%

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

   12. Simplified 96.4%

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

   13. Taylor expanded in k around 0 91.3%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.4 \cdot 10^{-70}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \frac{\cos k \cdot \left({\ell}^{2} \cdot {\sin k}^{-2}\right)}{t}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{2 \cdot \ell}{\sin k \cdot \left(\tan k \cdot {t}^{3}\right)} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.8 \cdot 10^{-71}:\\ \;\;\;\;\frac{2}{k \cdot k} \cdot \left(\frac{{\ell}^{2}}{{\sin k}^{2}} \cdot \frac{\cos k}{t\_m}\right)\\ \mathbf{elif}\;t\_m \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{2 \cdot \ell}{\sin k \cdot \left(\tan k \cdot {t\_m}^{3}\right)} \cdot \frac{\ell}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \end{array} \]

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 (<= t_m 1.8e-71)
    (* (/ 2.0 (* k k)) (* (/ (pow l 2.0) (pow (sin k) 2.0)) (/ (cos k) t_m)))
    (if (<= t_m 5.6e+102)
      (*
       (/ (* 2.0 l) (* (sin k) (* (tan k) (pow t_m 3.0))))
       (/ l (+ 2.0 (pow (/ k t_m) 2.0))))
      (/
       2.0
       (*
        (pow (* (* t_m (pow (cbrt l) -2.0)) (cbrt (sin k))) 3.0)
        (* 2.0 k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.8e-71) {
		tmp = (2.0 / (k * k)) * ((pow(l, 2.0) / pow(sin(k), 2.0)) * (cos(k) / t_m));
	} else if (t_m <= 5.6e+102) {
		tmp = ((2.0 * l) / (sin(k) * (tan(k) * pow(t_m, 3.0)))) * (l / (2.0 + pow((k / t_m), 2.0)));
	} else {
		tmp = 2.0 / (pow(((t_m * pow(cbrt(l), -2.0)) * cbrt(sin(k))), 3.0) * (2.0 * k));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.8e-71) {
		tmp = (2.0 / (k * k)) * ((Math.pow(l, 2.0) / Math.pow(Math.sin(k), 2.0)) * (Math.cos(k) / t_m));
	} else if (t_m <= 5.6e+102) {
		tmp = ((2.0 * l) / (Math.sin(k) * (Math.tan(k) * Math.pow(t_m, 3.0)))) * (l / (2.0 + Math.pow((k / t_m), 2.0)));
	} else {
		tmp = 2.0 / (Math.pow(((t_m * Math.pow(Math.cbrt(l), -2.0)) * Math.cbrt(Math.sin(k))), 3.0) * (2.0 * k));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.8e-71)
		tmp = Float64(Float64(2.0 / Float64(k * k)) * Float64(Float64((l ^ 2.0) / (sin(k) ^ 2.0)) * Float64(cos(k) / t_m)));
	elseif (t_m <= 5.6e+102)
		tmp = Float64(Float64(Float64(2.0 * l) / Float64(sin(k) * Float64(tan(k) * (t_m ^ 3.0)))) * Float64(l / Float64(2.0 + (Float64(k / t_m) ^ 2.0))));
	else
		tmp = Float64(2.0 / Float64((Float64(Float64(t_m * (cbrt(l) ^ -2.0)) * cbrt(sin(k))) ^ 3.0) * Float64(2.0 * k)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.8e-71], N[(N[(2.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.6e+102], N[(N[(N[(2.0 * l), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.8e-71

   1. Initial program 50.0%

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

   3. Simplified 50.0%

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

   5. Taylor expanded in t around 0 69.8%

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

      1. associate-*r/ 69.8%

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

      2. times-frac 68.7%

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

      3. *-commutative 68.7%

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

      4. times-frac 68.8%

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

   7. Simplified 68.8%

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

      1. unpow2 68.8%

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

   9. Applied egg-rr 68.8%

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

   if 1.8e-71 < t < 5.60000000000000037e102

   1. Initial program 73.7%

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

   3. Simplified 76.7%

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

      1. associate-*r* 79.7%

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

      2. *-un-lft-identity 79.7%

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

      3. times-frac 82.4%

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

      4. associate-*r* 83.7%

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

   6. Applied egg-rr 83.7%

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

      1. /-rgt-identity 83.7%

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

      2. *-commutative 83.7%

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

      3. *-commutative 83.7%

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

   8. Simplified 83.7%

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

      1. associate-*r/ 84.8%

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

      2. associate-*l* 84.8%

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

   10. Applied egg-rr 84.8%

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

   if 5.60000000000000037e102 < t

   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 67.1%

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

      1. unpow2 67.1%

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

      2. clear-num 67.1%

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

      3. un-div-inv 67.1%

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

   6. Applied egg-rr 67.1%

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

      1. add-cube-cbrt 67.1%

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

      2. pow3 67.1%

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

      3. associate-/r* 70.4%

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

      4. *-commutative 70.4%

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

      5. cbrt-prod 70.4%

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

      6. associate-/r* 67.1%

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

      7. div-inv 67.1%

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

      8. cbrt-prod 67.1%

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

      9. rem-cbrt-cube 80.8%

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

      10. pow2 80.8%

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

      11. pow-flip 81.5%

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

      12. metadata-eval 81.5%

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

   8. Applied egg-rr 81.5%

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

      1. pow1 81.5%

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

      2. associate-*r* 81.6%

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

   10. Applied egg-rr 81.6%

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

       1. unpow1 81.6%

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

       2. associate-*l* 81.5%

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

       3. *-commutative 81.5%

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

       4. unpow1/3 80.6%

          \[\leadsto \frac{2}{{\left(\left(t \cdot \color{blue}{{\left({\ell}^{-2}\right)}^{0.3333333333333333}}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)} \]

       5. exp-to-pow 44.9%

          \[\leadsto \frac{2}{{\left(\left(t \cdot {\color{blue}{\left(e^{\log \ell \cdot -2}\right)}}^{0.3333333333333333}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)} \]

       6. *-commutative 44.9%

          \[\leadsto \frac{2}{{\left(\left(t \cdot {\left(e^{\color{blue}{-2 \cdot \log \ell}}\right)}^{0.3333333333333333}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)} \]

       7. exp-prod 51.1%

          \[\leadsto \frac{2}{{\left(\left(t \cdot \color{blue}{e^{\left(-2 \cdot \log \ell\right) \cdot 0.3333333333333333}}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)} \]

       8. associate-*l* 51.1%

          \[\leadsto \frac{2}{{\left(\left(t \cdot e^{\color{blue}{-2 \cdot \left(\log \ell \cdot 0.3333333333333333\right)}}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)} \]

       9. rem-log-exp 51.1%

          \[\leadsto \frac{2}{{\left(\left(t \cdot e^{-2 \cdot \color{blue}{\log \left(e^{\log \ell \cdot 0.3333333333333333}\right)}}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)} \]

       10. exp-to-pow 51.2%

           \[\leadsto \frac{2}{{\left(\left(t \cdot e^{-2 \cdot \log \color{blue}{\left({\ell}^{0.3333333333333333}\right)}}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)} \]

       11. unpow1/3 51.2%

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

       12. *-commutative 51.2%

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

       13. exp-to-pow 96.4%

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

   12. Simplified 96.4%

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

   13. Taylor expanded in k around 0 91.3%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.8 \cdot 10^{-71}:\\ \;\;\;\;\frac{2}{k \cdot k} \cdot \left(\frac{{\ell}^{2}}{{\sin k}^{2}} \cdot \frac{\cos k}{t}\right)\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{2 \cdot \ell}{\sin k \cdot \left(\tan k \cdot {t}^{3}\right)} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.5 \cdot 10^{-70}:\\ \;\;\;\;\frac{2}{k \cdot k} \cdot \left(\frac{{\ell}^{2}}{{\sin k}^{2}} \cdot \frac{\cos k}{t\_m}\right)\\ \mathbf{elif}\;t\_m \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{2 \cdot \ell}{\sin k \cdot \left(\tan k \cdot {t\_m}^{3}\right)} \cdot \frac{\ell}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left(\sqrt[3]{\sin k} \cdot \left(t\_m \cdot \sqrt[3]{{\ell}^{-2}}\right)\right)}^{3}}\\ \end{array} \end{array} \]

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 4.5e-70)
    (* (/ 2.0 (* k k)) (* (/ (pow l 2.0) (pow (sin k) 2.0)) (/ (cos k) t_m)))
    (if (<= t_m 5.6e+102)
      (*
       (/ (* 2.0 l) (* (sin k) (* (tan k) (pow t_m 3.0))))
       (/ l (+ 2.0 (pow (/ k t_m) 2.0))))
      (/
       2.0
       (*
        (* 2.0 k)
        (pow (* (cbrt (sin k)) (* t_m (cbrt (pow l -2.0)))) 3.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.5e-70) {
		tmp = (2.0 / (k * k)) * ((pow(l, 2.0) / pow(sin(k), 2.0)) * (cos(k) / t_m));
	} else if (t_m <= 5.6e+102) {
		tmp = ((2.0 * l) / (sin(k) * (tan(k) * pow(t_m, 3.0)))) * (l / (2.0 + pow((k / t_m), 2.0)));
	} else {
		tmp = 2.0 / ((2.0 * k) * pow((cbrt(sin(k)) * (t_m * cbrt(pow(l, -2.0)))), 3.0));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.5e-70) {
		tmp = (2.0 / (k * k)) * ((Math.pow(l, 2.0) / Math.pow(Math.sin(k), 2.0)) * (Math.cos(k) / t_m));
	} else if (t_m <= 5.6e+102) {
		tmp = ((2.0 * l) / (Math.sin(k) * (Math.tan(k) * Math.pow(t_m, 3.0)))) * (l / (2.0 + Math.pow((k / t_m), 2.0)));
	} else {
		tmp = 2.0 / ((2.0 * k) * Math.pow((Math.cbrt(Math.sin(k)) * (t_m * Math.cbrt(Math.pow(l, -2.0)))), 3.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 4.5e-70)
		tmp = Float64(Float64(2.0 / Float64(k * k)) * Float64(Float64((l ^ 2.0) / (sin(k) ^ 2.0)) * Float64(cos(k) / t_m)));
	elseif (t_m <= 5.6e+102)
		tmp = Float64(Float64(Float64(2.0 * l) / Float64(sin(k) * Float64(tan(k) * (t_m ^ 3.0)))) * Float64(l / Float64(2.0 + (Float64(k / t_m) ^ 2.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * (Float64(cbrt(sin(k)) * Float64(t_m * cbrt((l ^ -2.0)))) ^ 3.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 4.5e-70], N[(N[(2.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.6e+102], N[(N[(N[(2.0 * l), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[Power[N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(t$95$m * N[Power[N[Power[l, -2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 4.50000000000000022e-70

   1. Initial program 50.0%

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

   3. Simplified 50.0%

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

   5. Taylor expanded in t around 0 69.8%

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

      1. associate-*r/ 69.8%

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

      2. times-frac 68.7%

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

      3. *-commutative 68.7%

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

      4. times-frac 68.8%

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

   7. Simplified 68.8%

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

      1. unpow2 68.8%

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

   9. Applied egg-rr 68.8%

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

   if 4.50000000000000022e-70 < t < 5.60000000000000037e102

   1. Initial program 73.7%

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

   3. Simplified 76.7%

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

      1. associate-*r* 79.7%

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

      2. *-un-lft-identity 79.7%

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

      3. times-frac 82.4%

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

      4. associate-*r* 83.7%

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

   6. Applied egg-rr 83.7%

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

      1. /-rgt-identity 83.7%

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

      2. *-commutative 83.7%

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

      3. *-commutative 83.7%

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

   8. Simplified 83.7%

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

      1. associate-*r/ 84.8%

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

      2. associate-*l* 84.8%

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

   10. Applied egg-rr 84.8%

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

   if 5.60000000000000037e102 < t

   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 67.1%

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

      1. unpow2 67.1%

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

      2. clear-num 67.1%

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

      3. un-div-inv 67.1%

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

   6. Applied egg-rr 67.1%

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

      1. add-cube-cbrt 67.1%

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

      2. pow3 67.1%

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

      3. associate-/r* 70.4%

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

      4. *-commutative 70.4%

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

      5. cbrt-prod 70.4%

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

      6. associate-/r* 67.1%

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

      7. div-inv 67.1%

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

      8. cbrt-prod 67.1%

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

      9. rem-cbrt-cube 80.8%

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

      10. pow2 80.8%

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

      11. pow-flip 81.5%

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

      12. metadata-eval 81.5%

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

   8. Applied egg-rr 81.5%

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

   9. Taylor expanded in k around 0 81.4%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.5 \cdot 10^{-70}:\\ \;\;\;\;\frac{2}{k \cdot k} \cdot \left(\frac{{\ell}^{2}}{{\sin k}^{2}} \cdot \frac{\cos k}{t}\right)\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{2 \cdot \ell}{\sin k \cdot \left(\tan k \cdot {t}^{3}\right)} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left(\sqrt[3]{\sin k} \cdot \left(t \cdot \sqrt[3]{{\ell}^{-2}}\right)\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.05 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{{\left(\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(t\_m \cdot {k}^{2}\right) \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\\ \end{array} \end{array} \]

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 (<= k 1.05e-18)
    (/
     2.0
     (* (pow (* (* t_m (pow (cbrt l) -2.0)) (cbrt (sin k))) 3.0) (* 2.0 k)))
    (/
     2.0
     (/ (* (* t_m (pow k 2.0)) (pow (sin k) 2.0)) (* (pow l 2.0) (cos k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.05e-18) {
		tmp = 2.0 / (pow(((t_m * pow(cbrt(l), -2.0)) * cbrt(sin(k))), 3.0) * (2.0 * k));
	} else {
		tmp = 2.0 / (((t_m * pow(k, 2.0)) * pow(sin(k), 2.0)) / (pow(l, 2.0) * cos(k)));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.05e-18) {
		tmp = 2.0 / (Math.pow(((t_m * Math.pow(Math.cbrt(l), -2.0)) * Math.cbrt(Math.sin(k))), 3.0) * (2.0 * k));
	} else {
		tmp = 2.0 / (((t_m * Math.pow(k, 2.0)) * Math.pow(Math.sin(k), 2.0)) / (Math.pow(l, 2.0) * Math.cos(k)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.05e-18)
		tmp = Float64(2.0 / Float64((Float64(Float64(t_m * (cbrt(l) ^ -2.0)) * cbrt(sin(k))) ^ 3.0) * Float64(2.0 * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k ^ 2.0)) * (sin(k) ^ 2.0)) / Float64((l ^ 2.0) * cos(k))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1.05e-18], N[(2.0 / N[(N[Power[N[(N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.05e-18

   1. Initial program 57.1%

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

   3. Simplified 57.1%

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

      1. unpow2 57.1%

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

      2. clear-num 57.1%

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

      3. un-div-inv 57.1%

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

   6. Applied egg-rr 57.1%

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

      1. add-cube-cbrt 57.0%

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

      2. pow3 57.0%

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

      3. associate-/r* 62.4%

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

      4. *-commutative 62.4%

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

      5. cbrt-prod 62.3%

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

      6. associate-/r* 57.0%

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

      7. div-inv 56.5%

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

      8. cbrt-prod 57.4%

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

      9. rem-cbrt-cube 69.7%

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

      10. pow2 69.7%

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

      11. pow-flip 69.9%

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

      12. metadata-eval 69.9%

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

   8. Applied egg-rr 69.9%

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

      1. pow1 69.9%

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

      2. associate-*r* 69.9%

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

   10. Applied egg-rr 69.9%

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

       1. unpow1 69.9%

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

       2. associate-*l* 69.9%

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

       3. *-commutative 69.9%

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

       4. unpow1/3 69.1%

          \[\leadsto \frac{2}{{\left(\left(t \cdot \color{blue}{{\left({\ell}^{-2}\right)}^{0.3333333333333333}}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)} \]

       5. exp-to-pow 34.2%

          \[\leadsto \frac{2}{{\left(\left(t \cdot {\color{blue}{\left(e^{\log \ell \cdot -2}\right)}}^{0.3333333333333333}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)} \]

       6. *-commutative 34.2%

          \[\leadsto \frac{2}{{\left(\left(t \cdot {\left(e^{\color{blue}{-2 \cdot \log \ell}}\right)}^{0.3333333333333333}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)} \]

       7. exp-prod 40.0%

          \[\leadsto \frac{2}{{\left(\left(t \cdot \color{blue}{e^{\left(-2 \cdot \log \ell\right) \cdot 0.3333333333333333}}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)} \]

       8. associate-*l* 40.0%

          \[\leadsto \frac{2}{{\left(\left(t \cdot e^{\color{blue}{-2 \cdot \left(\log \ell \cdot 0.3333333333333333\right)}}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)} \]

       9. rem-log-exp 40.0%

          \[\leadsto \frac{2}{{\left(\left(t \cdot e^{-2 \cdot \color{blue}{\log \left(e^{\log \ell \cdot 0.3333333333333333}\right)}}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)} \]

       10. exp-to-pow 40.0%

           \[\leadsto \frac{2}{{\left(\left(t \cdot e^{-2 \cdot \log \color{blue}{\left({\ell}^{0.3333333333333333}\right)}}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)} \]

       11. unpow1/3 40.1%

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

       12. *-commutative 40.1%

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

       13. exp-to-pow 80.5%

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

   12. Simplified 80.5%

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

   13. Taylor expanded in k around 0 71.6%

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

   if 1.05e-18 < k

   1. Initial program 51.6%

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

   3. Simplified 51.6%

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

   5. Taylor expanded in t around 0 74.8%

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

      1. associate-*r* 74.8%

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

   7. Simplified 74.8%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.05 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot {k}^{2}\right) \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.1 \cdot 10^{-70}:\\ \;\;\;\;\frac{2}{k \cdot k} \cdot \left(\frac{{\ell}^{2}}{{\sin k}^{2}} \cdot \frac{\cos k}{t\_m}\right)\\ \mathbf{elif}\;t\_m \leq 5.8 \cdot 10^{+102}:\\ \;\;\;\;\frac{2 \cdot \ell}{\sin k \cdot \left(\tan k \cdot {t\_m}^{3}\right)} \cdot \frac{\ell}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot {\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(2 \cdot \tan k\right)}\\ \end{array} \end{array} \]

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 (<= t_m 1.1e-70)
    (* (/ 2.0 (* k k)) (* (/ (pow l 2.0) (pow (sin k) 2.0)) (/ (cos k) t_m)))
    (if (<= t_m 5.8e+102)
      (*
       (/ (* 2.0 l) (* (sin k) (* (tan k) (pow t_m 3.0))))
       (/ l (+ 2.0 (pow (/ k t_m) 2.0))))
      (/ 2.0 (* (* (sin k) (pow (/ (pow t_m 1.5) l) 2.0)) (* 2.0 (tan k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.1e-70) {
		tmp = (2.0 / (k * k)) * ((pow(l, 2.0) / pow(sin(k), 2.0)) * (cos(k) / t_m));
	} else if (t_m <= 5.8e+102) {
		tmp = ((2.0 * l) / (sin(k) * (tan(k) * pow(t_m, 3.0)))) * (l / (2.0 + pow((k / t_m), 2.0)));
	} else {
		tmp = 2.0 / ((sin(k) * pow((pow(t_m, 1.5) / l), 2.0)) * (2.0 * tan(k)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.1d-70) then
        tmp = (2.0d0 / (k * k)) * (((l ** 2.0d0) / (sin(k) ** 2.0d0)) * (cos(k) / t_m))
    else if (t_m <= 5.8d+102) then
        tmp = ((2.0d0 * l) / (sin(k) * (tan(k) * (t_m ** 3.0d0)))) * (l / (2.0d0 + ((k / t_m) ** 2.0d0)))
    else
        tmp = 2.0d0 / ((sin(k) * (((t_m ** 1.5d0) / l) ** 2.0d0)) * (2.0d0 * tan(k)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.1e-70) {
		tmp = (2.0 / (k * k)) * ((Math.pow(l, 2.0) / Math.pow(Math.sin(k), 2.0)) * (Math.cos(k) / t_m));
	} else if (t_m <= 5.8e+102) {
		tmp = ((2.0 * l) / (Math.sin(k) * (Math.tan(k) * Math.pow(t_m, 3.0)))) * (l / (2.0 + Math.pow((k / t_m), 2.0)));
	} else {
		tmp = 2.0 / ((Math.sin(k) * Math.pow((Math.pow(t_m, 1.5) / l), 2.0)) * (2.0 * Math.tan(k)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.1e-70:
		tmp = (2.0 / (k * k)) * ((math.pow(l, 2.0) / math.pow(math.sin(k), 2.0)) * (math.cos(k) / t_m))
	elif t_m <= 5.8e+102:
		tmp = ((2.0 * l) / (math.sin(k) * (math.tan(k) * math.pow(t_m, 3.0)))) * (l / (2.0 + math.pow((k / t_m), 2.0)))
	else:
		tmp = 2.0 / ((math.sin(k) * math.pow((math.pow(t_m, 1.5) / l), 2.0)) * (2.0 * math.tan(k)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.1e-70)
		tmp = Float64(Float64(2.0 / Float64(k * k)) * Float64(Float64((l ^ 2.0) / (sin(k) ^ 2.0)) * Float64(cos(k) / t_m)));
	elseif (t_m <= 5.8e+102)
		tmp = Float64(Float64(Float64(2.0 * l) / Float64(sin(k) * Float64(tan(k) * (t_m ^ 3.0)))) * Float64(l / Float64(2.0 + (Float64(k / t_m) ^ 2.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(sin(k) * (Float64((t_m ^ 1.5) / l) ^ 2.0)) * Float64(2.0 * tan(k))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.1e-70)
		tmp = (2.0 / (k * k)) * (((l ^ 2.0) / (sin(k) ^ 2.0)) * (cos(k) / t_m));
	elseif (t_m <= 5.8e+102)
		tmp = ((2.0 * l) / (sin(k) * (tan(k) * (t_m ^ 3.0)))) * (l / (2.0 + ((k / t_m) ^ 2.0)));
	else
		tmp = 2.0 / ((sin(k) * (((t_m ^ 1.5) / l) ^ 2.0)) * (2.0 * tan(k)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.1e-70], N[(N[(2.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.8e+102], N[(N[(N[(2.0 * l), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.0999999999999999e-70

   1. Initial program 50.0%

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

   3. Simplified 50.0%

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

   5. Taylor expanded in t around 0 69.8%

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

      1. associate-*r/ 69.8%

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

      2. times-frac 68.7%

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

      3. *-commutative 68.7%

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

      4. times-frac 68.8%

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

   7. Simplified 68.8%

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

      1. unpow2 68.8%

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

   9. Applied egg-rr 68.8%

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

   if 1.0999999999999999e-70 < t < 5.8000000000000005e102

   1. Initial program 73.7%

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

   3. Simplified 76.7%

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

      1. associate-*r* 79.7%

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

      2. *-un-lft-identity 79.7%

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

      3. times-frac 82.4%

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

      4. associate-*r* 83.7%

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

   6. Applied egg-rr 83.7%

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

      1. /-rgt-identity 83.7%

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

      2. *-commutative 83.7%

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

      3. *-commutative 83.7%

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

   8. Simplified 83.7%

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

      1. associate-*r/ 84.8%

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

      2. associate-*l* 84.8%

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

   10. Applied egg-rr 84.8%

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

   if 5.8000000000000005e102 < t

   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 67.1%

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

      1. add-sqr-sqrt 47.5%

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

      2. pow2 47.5%

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

      3. associate-/r* 50.7%

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

      4. *-commutative 50.7%

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

      5. sqrt-prod 50.7%

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

      6. associate-/r* 47.5%

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

      7. sqrt-div 47.5%

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

      8. sqrt-pow1 50.3%

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

      9. metadata-eval 50.3%

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

      10. sqrt-prod 25.1%

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

      11. add-sqr-sqrt 58.7%

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

   6. Applied egg-rr 58.7%

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

      1. *-commutative 58.7%

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

   8. Simplified 58.7%

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

      1. unpow-prod-down 55.9%

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

      2. pow2 55.9%

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

      3. add-sqr-sqrt 81.1%

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

   10. Applied egg-rr 81.1%

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

   11. Taylor expanded in k around 0 81.1%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.1 \cdot 10^{-70}:\\ \;\;\;\;\frac{2}{k \cdot k} \cdot \left(\frac{{\ell}^{2}}{{\sin k}^{2}} \cdot \frac{\cos k}{t}\right)\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{+102}:\\ \;\;\;\;\frac{2 \cdot \ell}{\sin k \cdot \left(\tan k \cdot {t}^{3}\right)} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(2 \cdot \tan k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 72.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \ell \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot {t\_m}^{3}\right)}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.45 \cdot 10^{-215}:\\ \;\;\;\;t\_2 \cdot \frac{\ell}{2}\\ \mathbf{elif}\;t\_m \leq 10^{-72}:\\ \;\;\;\;\frac{2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}}}{t\_m}}{{k}^{2}}\\ \mathbf{elif}\;t\_m \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;t\_2 \cdot \frac{\ell}{2 + \frac{k}{t\_m \cdot \frac{t\_m}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot {\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(2 \cdot \tan k\right)}\\ \end{array} \end{array} \end{array} \]

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 (* l (/ 2.0 (* (tan k) (* (sin k) (pow t_m 3.0)))))))
   (*
    t_s
    (if (<= t_m 1.45e-215)
      (* t_2 (/ l 2.0))
      (if (<= t_m 1e-72)
        (/ (* 2.0 (/ (/ (pow l 2.0) (pow k 2.0)) t_m)) (pow k 2.0))
        (if (<= t_m 5.6e+102)
          (* t_2 (/ l (+ 2.0 (/ k (* t_m (/ t_m k))))))
          (/
           2.0
           (* (* (sin k) (pow (/ (pow t_m 1.5) l) 2.0)) (* 2.0 (tan 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 = l * (2.0 / (tan(k) * (sin(k) * pow(t_m, 3.0))));
	double tmp;
	if (t_m <= 1.45e-215) {
		tmp = t_2 * (l / 2.0);
	} else if (t_m <= 1e-72) {
		tmp = (2.0 * ((pow(l, 2.0) / pow(k, 2.0)) / t_m)) / pow(k, 2.0);
	} else if (t_m <= 5.6e+102) {
		tmp = t_2 * (l / (2.0 + (k / (t_m * (t_m / k)))));
	} else {
		tmp = 2.0 / ((sin(k) * pow((pow(t_m, 1.5) / l), 2.0)) * (2.0 * tan(k)));
	}
	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 = l * (2.0d0 / (tan(k) * (sin(k) * (t_m ** 3.0d0))))
    if (t_m <= 1.45d-215) then
        tmp = t_2 * (l / 2.0d0)
    else if (t_m <= 1d-72) then
        tmp = (2.0d0 * (((l ** 2.0d0) / (k ** 2.0d0)) / t_m)) / (k ** 2.0d0)
    else if (t_m <= 5.6d+102) then
        tmp = t_2 * (l / (2.0d0 + (k / (t_m * (t_m / k)))))
    else
        tmp = 2.0d0 / ((sin(k) * (((t_m ** 1.5d0) / l) ** 2.0d0)) * (2.0d0 * tan(k)))
    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 = l * (2.0 / (Math.tan(k) * (Math.sin(k) * Math.pow(t_m, 3.0))));
	double tmp;
	if (t_m <= 1.45e-215) {
		tmp = t_2 * (l / 2.0);
	} else if (t_m <= 1e-72) {
		tmp = (2.0 * ((Math.pow(l, 2.0) / Math.pow(k, 2.0)) / t_m)) / Math.pow(k, 2.0);
	} else if (t_m <= 5.6e+102) {
		tmp = t_2 * (l / (2.0 + (k / (t_m * (t_m / k)))));
	} else {
		tmp = 2.0 / ((Math.sin(k) * Math.pow((Math.pow(t_m, 1.5) / l), 2.0)) * (2.0 * Math.tan(k)));
	}
	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 = l * (2.0 / (math.tan(k) * (math.sin(k) * math.pow(t_m, 3.0))))
	tmp = 0
	if t_m <= 1.45e-215:
		tmp = t_2 * (l / 2.0)
	elif t_m <= 1e-72:
		tmp = (2.0 * ((math.pow(l, 2.0) / math.pow(k, 2.0)) / t_m)) / math.pow(k, 2.0)
	elif t_m <= 5.6e+102:
		tmp = t_2 * (l / (2.0 + (k / (t_m * (t_m / k)))))
	else:
		tmp = 2.0 / ((math.sin(k) * math.pow((math.pow(t_m, 1.5) / l), 2.0)) * (2.0 * math.tan(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(l * Float64(2.0 / Float64(tan(k) * Float64(sin(k) * (t_m ^ 3.0)))))
	tmp = 0.0
	if (t_m <= 1.45e-215)
		tmp = Float64(t_2 * Float64(l / 2.0));
	elseif (t_m <= 1e-72)
		tmp = Float64(Float64(2.0 * Float64(Float64((l ^ 2.0) / (k ^ 2.0)) / t_m)) / (k ^ 2.0));
	elseif (t_m <= 5.6e+102)
		tmp = Float64(t_2 * Float64(l / Float64(2.0 + Float64(k / Float64(t_m * Float64(t_m / k))))));
	else
		tmp = Float64(2.0 / Float64(Float64(sin(k) * (Float64((t_m ^ 1.5) / l) ^ 2.0)) * Float64(2.0 * tan(k))));
	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 = l * (2.0 / (tan(k) * (sin(k) * (t_m ^ 3.0))));
	tmp = 0.0;
	if (t_m <= 1.45e-215)
		tmp = t_2 * (l / 2.0);
	elseif (t_m <= 1e-72)
		tmp = (2.0 * (((l ^ 2.0) / (k ^ 2.0)) / t_m)) / (k ^ 2.0);
	elseif (t_m <= 5.6e+102)
		tmp = t_2 * (l / (2.0 + (k / (t_m * (t_m / k)))));
	else
		tmp = 2.0 / ((sin(k) * (((t_m ^ 1.5) / l) ^ 2.0)) * (2.0 * tan(k)));
	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[(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[(t$95$s * If[LessEqual[t$95$m, 1.45e-215], N[(t$95$2 * N[(l / 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1e-72], N[(N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.6e+102], N[(t$95$2 * N[(l / N[(2.0 + N[(k / N[(t$95$m * N[(t$95$m / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 1.45e-215

   1. Initial program 48.5%

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

   3. Simplified 46.4%

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

      1. associate-*r* 51.8%

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

      2. *-un-lft-identity 51.8%

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

      3. times-frac 52.4%

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

      4. associate-*r* 55.2%

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

   6. Applied egg-rr 55.2%

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

      1. /-rgt-identity 55.2%

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

      2. *-commutative 55.2%

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

      3. *-commutative 55.2%

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

   8. Simplified 55.2%

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

   9. Taylor expanded in k around 0 56.2%

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

   if 1.45e-215 < t < 9.9999999999999997e-73

   1. Initial program 57.3%

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

   3. Simplified 57.3%

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

   5. Taylor expanded in t around 0 87.2%

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

      1. associate-*r/ 87.2%

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

      2. times-frac 87.2%

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

      3. *-commutative 87.2%

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

      4. times-frac 87.3%

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

   7. Simplified 87.3%

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

   8. Taylor expanded in k around 0 80.8%

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

      1. associate-*l/ 80.8%

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

      2. associate-/r* 80.8%

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

   10. Applied egg-rr 80.8%

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

   if 9.9999999999999997e-73 < t < 5.60000000000000037e102

   1. Initial program 73.7%

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

   3. Simplified 76.7%

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

      1. associate-*r* 79.7%

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

      2. *-un-lft-identity 79.7%

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

      3. times-frac 82.4%

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

      4. associate-*r* 83.7%

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

   6. Applied egg-rr 83.7%

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

      1. /-rgt-identity 83.7%

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

      2. *-commutative 83.7%

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

      3. *-commutative 83.7%

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

   8. Simplified 83.7%

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

      1. unpow2 83.7%

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

      2. clear-num 83.7%

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

      3. frac-times 83.7%

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

      4. *-un-lft-identity 83.7%

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

   10. Applied egg-rr 83.7%

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

   if 5.60000000000000037e102 < t

   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 67.1%

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

      1. add-sqr-sqrt 47.5%

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

      2. pow2 47.5%

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

      3. associate-/r* 50.7%

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

      4. *-commutative 50.7%

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

      5. sqrt-prod 50.7%

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

      6. associate-/r* 47.5%

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

      7. sqrt-div 47.5%

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

      8. sqrt-pow1 50.3%

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

      9. metadata-eval 50.3%

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

      10. sqrt-prod 25.1%

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

      11. add-sqr-sqrt 58.7%

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

   6. Applied egg-rr 58.7%

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

      1. *-commutative 58.7%

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

   8. Simplified 58.7%

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

      1. unpow-prod-down 55.9%

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

      2. pow2 55.9%

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

      3. add-sqr-sqrt 81.1%

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

   10. Applied egg-rr 81.1%

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

   11. Taylor expanded in k around 0 81.1%

       \[\leadsto \frac{2}{\left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{2}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.45 \cdot 10^{-215}:\\ \;\;\;\;\left(\ell \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot {t}^{3}\right)}\right) \cdot \frac{\ell}{2}\\ \mathbf{elif}\;t \leq 10^{-72}:\\ \;\;\;\;\frac{2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}}}{t}}{{k}^{2}}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\left(\ell \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot {t}^{3}\right)}\right) \cdot \frac{\ell}{2 + \frac{k}{t \cdot \frac{t}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(2 \cdot \tan k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 43.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6.5 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left(\frac{{t\_m}^{1.5}}{\ell} \cdot \sqrt{\sin k}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot k} \cdot \left(\frac{{\ell}^{2}}{{\sin k}^{2}} \cdot \frac{\cos k}{t\_m}\right)\\ \end{array} \end{array} \]

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 (<= k 6.5e-18)
    (/ 2.0 (* (* 2.0 k) (pow (* (/ (pow t_m 1.5) l) (sqrt (sin k))) 2.0)))
    (*
     (/ 2.0 (* k k))
     (* (/ (pow l 2.0) (pow (sin k) 2.0)) (/ (cos 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 (k <= 6.5e-18) {
		tmp = 2.0 / ((2.0 * k) * pow(((pow(t_m, 1.5) / l) * sqrt(sin(k))), 2.0));
	} else {
		tmp = (2.0 / (k * k)) * ((pow(l, 2.0) / pow(sin(k), 2.0)) * (cos(k) / t_m));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 6.5d-18) then
        tmp = 2.0d0 / ((2.0d0 * k) * ((((t_m ** 1.5d0) / l) * sqrt(sin(k))) ** 2.0d0))
    else
        tmp = (2.0d0 / (k * k)) * (((l ** 2.0d0) / (sin(k) ** 2.0d0)) * (cos(k) / t_m))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 6.5e-18) {
		tmp = 2.0 / ((2.0 * k) * Math.pow(((Math.pow(t_m, 1.5) / l) * Math.sqrt(Math.sin(k))), 2.0));
	} else {
		tmp = (2.0 / (k * k)) * ((Math.pow(l, 2.0) / Math.pow(Math.sin(k), 2.0)) * (Math.cos(k) / t_m));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 6.5e-18:
		tmp = 2.0 / ((2.0 * k) * math.pow(((math.pow(t_m, 1.5) / l) * math.sqrt(math.sin(k))), 2.0))
	else:
		tmp = (2.0 / (k * k)) * ((math.pow(l, 2.0) / math.pow(math.sin(k), 2.0)) * (math.cos(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 (k <= 6.5e-18)
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * (Float64(Float64((t_m ^ 1.5) / l) * sqrt(sin(k))) ^ 2.0)));
	else
		tmp = Float64(Float64(2.0 / Float64(k * k)) * Float64(Float64((l ^ 2.0) / (sin(k) ^ 2.0)) * Float64(cos(k) / t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 6.5e-18)
		tmp = 2.0 / ((2.0 * k) * ((((t_m ^ 1.5) / l) * sqrt(sin(k))) ^ 2.0));
	else
		tmp = (2.0 / (k * k)) * (((l ^ 2.0) / (sin(k) ^ 2.0)) * (cos(k) / t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 6.5e-18], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[Sin[k], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 6.50000000000000008e-18

   1. Initial program 57.1%

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

   3. Simplified 57.1%

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

      1. add-sqr-sqrt 27.8%

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

      2. pow2 27.8%

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

      3. associate-/r* 30.4%

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

      4. *-commutative 30.4%

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

      5. sqrt-prod 17.2%

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

      6. associate-/r* 15.6%

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

      7. sqrt-div 15.6%

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

      8. sqrt-pow1 16.6%

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

      9. metadata-eval 16.6%

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

      10. sqrt-prod 5.7%

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

      11. add-sqr-sqrt 19.1%

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

   6. Applied egg-rr 19.1%

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

      1. *-commutative 19.1%

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

   8. Simplified 19.1%

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

   9. Taylor expanded in k around 0 15.8%

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

   if 6.50000000000000008e-18 < k

   1. Initial program 51.6%

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

   3. Simplified 51.6%

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

   5. Taylor expanded in t around 0 74.8%

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

      1. associate-*r/ 74.8%

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

      2. times-frac 70.7%

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

      3. *-commutative 70.7%

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

      4. times-frac 70.7%

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

   7. Simplified 70.7%

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

      1. unpow2 70.7%

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

   9. Applied egg-rr 70.7%

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

4. Final simplification 29.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.5 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot k} \cdot \left(\frac{{\ell}^{2}}{{\sin k}^{2}} \cdot \frac{\cos k}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 41.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.06 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left(\frac{{t\_m}^{1.5}}{\ell} \cdot \sqrt{\sin k}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \left(\frac{\cos k}{t\_m} \cdot \frac{{\ell}^{2}}{{k}^{2}}\right)\\ \end{array} \end{array} \]

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 (<= k 1.06e-18)
    (/ 2.0 (* (* 2.0 k) (pow (* (/ (pow t_m 1.5) l) (sqrt (sin k))) 2.0)))
    (* (/ 2.0 (pow k 2.0)) (* (/ (cos k) t_m) (/ (pow l 2.0) (pow k 2.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.06e-18) {
		tmp = 2.0 / ((2.0 * k) * pow(((pow(t_m, 1.5) / l) * sqrt(sin(k))), 2.0));
	} else {
		tmp = (2.0 / pow(k, 2.0)) * ((cos(k) / t_m) * (pow(l, 2.0) / pow(k, 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) :: tmp
    if (k <= 1.06d-18) then
        tmp = 2.0d0 / ((2.0d0 * k) * ((((t_m ** 1.5d0) / l) * sqrt(sin(k))) ** 2.0d0))
    else
        tmp = (2.0d0 / (k ** 2.0d0)) * ((cos(k) / t_m) * ((l ** 2.0d0) / (k ** 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 tmp;
	if (k <= 1.06e-18) {
		tmp = 2.0 / ((2.0 * k) * Math.pow(((Math.pow(t_m, 1.5) / l) * Math.sqrt(Math.sin(k))), 2.0));
	} else {
		tmp = (2.0 / Math.pow(k, 2.0)) * ((Math.cos(k) / t_m) * (Math.pow(l, 2.0) / Math.pow(k, 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):
	tmp = 0
	if k <= 1.06e-18:
		tmp = 2.0 / ((2.0 * k) * math.pow(((math.pow(t_m, 1.5) / l) * math.sqrt(math.sin(k))), 2.0))
	else:
		tmp = (2.0 / math.pow(k, 2.0)) * ((math.cos(k) / t_m) * (math.pow(l, 2.0) / math.pow(k, 2.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.06e-18)
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * (Float64(Float64((t_m ^ 1.5) / l) * sqrt(sin(k))) ^ 2.0)));
	else
		tmp = Float64(Float64(2.0 / (k ^ 2.0)) * Float64(Float64(cos(k) / t_m) * Float64((l ^ 2.0) / (k ^ 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)
	tmp = 0.0;
	if (k <= 1.06e-18)
		tmp = 2.0 / ((2.0 * k) * ((((t_m ^ 1.5) / l) * sqrt(sin(k))) ^ 2.0));
	else
		tmp = (2.0 / (k ^ 2.0)) * ((cos(k) / t_m) * ((l ^ 2.0) / (k ^ 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_] := N[(t$95$s * If[LessEqual[k, 1.06e-18], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[Sin[k], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.05999999999999994e-18

   1. Initial program 57.1%

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

   3. Simplified 57.1%

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

      1. add-sqr-sqrt 27.8%

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

      2. pow2 27.8%

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

      3. associate-/r* 30.4%

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

      4. *-commutative 30.4%

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

      5. sqrt-prod 17.2%

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

      6. associate-/r* 15.6%

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

      7. sqrt-div 15.6%

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

      8. sqrt-pow1 16.6%

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

      9. metadata-eval 16.6%

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

      10. sqrt-prod 5.7%

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

      11. add-sqr-sqrt 19.1%

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

   6. Applied egg-rr 19.1%

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

      1. *-commutative 19.1%

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

   8. Simplified 19.1%

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

   9. Taylor expanded in k around 0 15.8%

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

   if 1.05999999999999994e-18 < k

   1. Initial program 51.6%

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

   3. Simplified 51.6%

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

   5. Taylor expanded in t around 0 74.8%

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

      1. associate-*r/ 74.8%

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

      2. times-frac 70.7%

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

      3. *-commutative 70.7%

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

      4. times-frac 70.7%

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

   7. Simplified 70.7%

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

   8. Taylor expanded in k around 0 64.6%

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

4. Final simplification 27.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.06 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \left(\frac{\cos k}{t} \cdot \frac{{\ell}^{2}}{{k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 40.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.7 \cdot 10^{-17}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left(\frac{{t\_m}^{1.5}}{\ell} \cdot \sqrt{\sin k}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t\_m \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

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 (<= k 2.7e-17)
    (/ 2.0 (* (* 2.0 k) (pow (* (/ (pow t_m 1.5) l) (sqrt (sin k))) 2.0)))
    (* (/ 2.0 (pow k 2.0)) (/ (pow l 2.0) (* t_m (* 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 tmp;
	if (k <= 2.7e-17) {
		tmp = 2.0 / ((2.0 * k) * pow(((pow(t_m, 1.5) / l) * sqrt(sin(k))), 2.0));
	} else {
		tmp = (2.0 / pow(k, 2.0)) * (pow(l, 2.0) / (t_m * (k * k)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.7d-17) then
        tmp = 2.0d0 / ((2.0d0 * k) * ((((t_m ** 1.5d0) / l) * sqrt(sin(k))) ** 2.0d0))
    else
        tmp = (2.0d0 / (k ** 2.0d0)) * ((l ** 2.0d0) / (t_m * (k * k)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.7e-17) {
		tmp = 2.0 / ((2.0 * k) * Math.pow(((Math.pow(t_m, 1.5) / l) * Math.sqrt(Math.sin(k))), 2.0));
	} else {
		tmp = (2.0 / Math.pow(k, 2.0)) * (Math.pow(l, 2.0) / (t_m * (k * k)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 2.7e-17:
		tmp = 2.0 / ((2.0 * k) * math.pow(((math.pow(t_m, 1.5) / l) * math.sqrt(math.sin(k))), 2.0))
	else:
		tmp = (2.0 / math.pow(k, 2.0)) * (math.pow(l, 2.0) / (t_m * (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)
	tmp = 0.0
	if (k <= 2.7e-17)
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * (Float64(Float64((t_m ^ 1.5) / l) * sqrt(sin(k))) ^ 2.0)));
	else
		tmp = Float64(Float64(2.0 / (k ^ 2.0)) * Float64((l ^ 2.0) / Float64(t_m * Float64(k * k))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 2.7e-17)
		tmp = 2.0 / ((2.0 * k) * ((((t_m ^ 1.5) / l) * sqrt(sin(k))) ^ 2.0));
	else
		tmp = (2.0 / (k ^ 2.0)) * ((l ^ 2.0) / (t_m * (k * k)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 2.7e-17], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[Sin[k], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.7000000000000001e-17

   1. Initial program 57.1%

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

   3. Simplified 57.1%

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

      1. add-sqr-sqrt 27.8%

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

      2. pow2 27.8%

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

      3. associate-/r* 30.4%

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

      4. *-commutative 30.4%

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

      5. sqrt-prod 17.2%

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

      6. associate-/r* 15.6%

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

      7. sqrt-div 15.6%

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

      8. sqrt-pow1 16.6%

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

      9. metadata-eval 16.6%

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

      10. sqrt-prod 5.7%

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

      11. add-sqr-sqrt 19.1%

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

   6. Applied egg-rr 19.1%

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

      1. *-commutative 19.1%

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

   8. Simplified 19.1%

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

   9. Taylor expanded in k around 0 15.8%

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

   if 2.7000000000000001e-17 < k

   1. Initial program 51.6%

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

   3. Simplified 51.6%

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

   5. Taylor expanded in t around 0 74.8%

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

      1. associate-*r/ 74.8%

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

      2. times-frac 70.7%

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

      3. *-commutative 70.7%

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

      4. times-frac 70.7%

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

   7. Simplified 70.7%

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

   8. Taylor expanded in k around 0 61.1%

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

      1. unpow2 70.7%

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

   10. Applied egg-rr 61.1%

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

4. Final simplification 27.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.7 \cdot 10^{-17}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t \cdot \left(k \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 72.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \ell \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot {t\_m}^{3}\right)}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.2 \cdot 10^{-213}:\\ \;\;\;\;t\_2 \cdot \frac{\ell}{2}\\ \mathbf{elif}\;t\_m \leq 2.15 \cdot 10^{-72}:\\ \;\;\;\;\frac{2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}}}{t\_m}}{{k}^{2}}\\ \mathbf{elif}\;t\_m \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;t\_2 \cdot \frac{\ell}{2 + \frac{k}{t\_m \cdot \frac{t\_m}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot {\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2}\right)}\\ \end{array} \end{array} \end{array} \]

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 (* l (/ 2.0 (* (tan k) (* (sin k) (pow t_m 3.0)))))))
   (*
    t_s
    (if (<= t_m 2.2e-213)
      (* t_2 (/ l 2.0))
      (if (<= t_m 2.15e-72)
        (/ (* 2.0 (/ (/ (pow l 2.0) (pow k 2.0)) t_m)) (pow k 2.0))
        (if (<= t_m 5.6e+102)
          (* t_2 (/ l (+ 2.0 (/ k (* t_m (/ t_m k))))))
          (/ 2.0 (* (* 2.0 k) (* (sin k) (pow (/ (pow t_m 1.5) l) 2.0))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = l * (2.0 / (tan(k) * (sin(k) * pow(t_m, 3.0))));
	double tmp;
	if (t_m <= 2.2e-213) {
		tmp = t_2 * (l / 2.0);
	} else if (t_m <= 2.15e-72) {
		tmp = (2.0 * ((pow(l, 2.0) / pow(k, 2.0)) / t_m)) / pow(k, 2.0);
	} else if (t_m <= 5.6e+102) {
		tmp = t_2 * (l / (2.0 + (k / (t_m * (t_m / k)))));
	} else {
		tmp = 2.0 / ((2.0 * k) * (sin(k) * pow((pow(t_m, 1.5) / l), 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 = l * (2.0d0 / (tan(k) * (sin(k) * (t_m ** 3.0d0))))
    if (t_m <= 2.2d-213) then
        tmp = t_2 * (l / 2.0d0)
    else if (t_m <= 2.15d-72) then
        tmp = (2.0d0 * (((l ** 2.0d0) / (k ** 2.0d0)) / t_m)) / (k ** 2.0d0)
    else if (t_m <= 5.6d+102) then
        tmp = t_2 * (l / (2.0d0 + (k / (t_m * (t_m / k)))))
    else
        tmp = 2.0d0 / ((2.0d0 * k) * (sin(k) * (((t_m ** 1.5d0) / l) ** 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 = l * (2.0 / (Math.tan(k) * (Math.sin(k) * Math.pow(t_m, 3.0))));
	double tmp;
	if (t_m <= 2.2e-213) {
		tmp = t_2 * (l / 2.0);
	} else if (t_m <= 2.15e-72) {
		tmp = (2.0 * ((Math.pow(l, 2.0) / Math.pow(k, 2.0)) / t_m)) / Math.pow(k, 2.0);
	} else if (t_m <= 5.6e+102) {
		tmp = t_2 * (l / (2.0 + (k / (t_m * (t_m / k)))));
	} else {
		tmp = 2.0 / ((2.0 * k) * (Math.sin(k) * Math.pow((Math.pow(t_m, 1.5) / l), 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 = l * (2.0 / (math.tan(k) * (math.sin(k) * math.pow(t_m, 3.0))))
	tmp = 0
	if t_m <= 2.2e-213:
		tmp = t_2 * (l / 2.0)
	elif t_m <= 2.15e-72:
		tmp = (2.0 * ((math.pow(l, 2.0) / math.pow(k, 2.0)) / t_m)) / math.pow(k, 2.0)
	elif t_m <= 5.6e+102:
		tmp = t_2 * (l / (2.0 + (k / (t_m * (t_m / k)))))
	else:
		tmp = 2.0 / ((2.0 * k) * (math.sin(k) * math.pow((math.pow(t_m, 1.5) / l), 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(l * Float64(2.0 / Float64(tan(k) * Float64(sin(k) * (t_m ^ 3.0)))))
	tmp = 0.0
	if (t_m <= 2.2e-213)
		tmp = Float64(t_2 * Float64(l / 2.0));
	elseif (t_m <= 2.15e-72)
		tmp = Float64(Float64(2.0 * Float64(Float64((l ^ 2.0) / (k ^ 2.0)) / t_m)) / (k ^ 2.0));
	elseif (t_m <= 5.6e+102)
		tmp = Float64(t_2 * Float64(l / Float64(2.0 + Float64(k / Float64(t_m * Float64(t_m / k))))));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * Float64(sin(k) * (Float64((t_m ^ 1.5) / l) ^ 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 = l * (2.0 / (tan(k) * (sin(k) * (t_m ^ 3.0))));
	tmp = 0.0;
	if (t_m <= 2.2e-213)
		tmp = t_2 * (l / 2.0);
	elseif (t_m <= 2.15e-72)
		tmp = (2.0 * (((l ^ 2.0) / (k ^ 2.0)) / t_m)) / (k ^ 2.0);
	elseif (t_m <= 5.6e+102)
		tmp = t_2 * (l / (2.0 + (k / (t_m * (t_m / k)))));
	else
		tmp = 2.0 / ((2.0 * k) * (sin(k) * (((t_m ^ 1.5) / l) ^ 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[(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[(t$95$s * If[LessEqual[t$95$m, 2.2e-213], N[(t$95$2 * N[(l / 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.15e-72], N[(N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.6e+102], N[(t$95$2 * N[(l / N[(2.0 + N[(k / N[(t$95$m * N[(t$95$m / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 2.2000000000000001e-213

   1. Initial program 48.5%

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

   3. Simplified 46.4%

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

      1. associate-*r* 51.8%

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

      2. *-un-lft-identity 51.8%

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

      3. times-frac 52.4%

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

      4. associate-*r* 55.2%

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

   6. Applied egg-rr 55.2%

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

      1. /-rgt-identity 55.2%

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

      2. *-commutative 55.2%

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

      3. *-commutative 55.2%

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

   8. Simplified 55.2%

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

   9. Taylor expanded in k around 0 56.2%

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

   if 2.2000000000000001e-213 < t < 2.1499999999999999e-72

   1. Initial program 57.3%

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

   3. Simplified 57.3%

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

   5. Taylor expanded in t around 0 87.2%

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

      1. associate-*r/ 87.2%

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

      2. times-frac 87.2%

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

      3. *-commutative 87.2%

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

      4. times-frac 87.3%

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

   7. Simplified 87.3%

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

   8. Taylor expanded in k around 0 80.8%

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

      1. associate-*l/ 80.8%

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

      2. associate-/r* 80.8%

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

   10. Applied egg-rr 80.8%

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

   if 2.1499999999999999e-72 < t < 5.60000000000000037e102

   1. Initial program 73.7%

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

   3. Simplified 76.7%

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

      1. associate-*r* 79.7%

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

      2. *-un-lft-identity 79.7%

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

      3. times-frac 82.4%

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

      4. associate-*r* 83.7%

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

   6. Applied egg-rr 83.7%

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

      1. /-rgt-identity 83.7%

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

      2. *-commutative 83.7%

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

      3. *-commutative 83.7%

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

   8. Simplified 83.7%

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

      1. unpow2 83.7%

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

      2. clear-num 83.7%

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

      3. frac-times 83.7%

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

      4. *-un-lft-identity 83.7%

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

   10. Applied egg-rr 83.7%

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

   if 5.60000000000000037e102 < t

   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 67.1%

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

      1. add-sqr-sqrt 47.5%

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

      2. pow2 47.5%

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

      3. associate-/r* 50.7%

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

      4. *-commutative 50.7%

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

      5. sqrt-prod 50.7%

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

      6. associate-/r* 47.5%

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

      7. sqrt-div 47.5%

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

      8. sqrt-pow1 50.3%

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

      9. metadata-eval 50.3%

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

      10. sqrt-prod 25.1%

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

      11. add-sqr-sqrt 58.7%

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

   6. Applied egg-rr 58.7%

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

      1. *-commutative 58.7%

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

   8. Simplified 58.7%

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

      1. unpow-prod-down 55.9%

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

      2. pow2 55.9%

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

      3. add-sqr-sqrt 81.1%

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

   10. Applied egg-rr 81.1%

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

   11. Taylor expanded in k around 0 75.8%

       \[\leadsto \frac{2}{\left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.2 \cdot 10^{-213}:\\ \;\;\;\;\left(\ell \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot {t}^{3}\right)}\right) \cdot \frac{\ell}{2}\\ \mathbf{elif}\;t \leq 2.15 \cdot 10^{-72}:\\ \;\;\;\;\frac{2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}}}{t}}{{k}^{2}}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\left(\ell \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot {t}^{3}\right)}\right) \cdot \frac{\ell}{2 + \frac{k}{t \cdot \frac{t}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 64.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.6 \cdot 10^{-17}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot {\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t\_m \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

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 (<= k 2.6e-17)
    (/ 2.0 (* (* 2.0 k) (* (sin k) (pow (/ (pow t_m 1.5) l) 2.0))))
    (* (/ 2.0 (pow k 2.0)) (/ (pow l 2.0) (* t_m (* 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 tmp;
	if (k <= 2.6e-17) {
		tmp = 2.0 / ((2.0 * k) * (sin(k) * pow((pow(t_m, 1.5) / l), 2.0)));
	} else {
		tmp = (2.0 / pow(k, 2.0)) * (pow(l, 2.0) / (t_m * (k * k)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.6d-17) then
        tmp = 2.0d0 / ((2.0d0 * k) * (sin(k) * (((t_m ** 1.5d0) / l) ** 2.0d0)))
    else
        tmp = (2.0d0 / (k ** 2.0d0)) * ((l ** 2.0d0) / (t_m * (k * k)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.6e-17) {
		tmp = 2.0 / ((2.0 * k) * (Math.sin(k) * Math.pow((Math.pow(t_m, 1.5) / l), 2.0)));
	} else {
		tmp = (2.0 / Math.pow(k, 2.0)) * (Math.pow(l, 2.0) / (t_m * (k * k)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 2.6e-17:
		tmp = 2.0 / ((2.0 * k) * (math.sin(k) * math.pow((math.pow(t_m, 1.5) / l), 2.0)))
	else:
		tmp = (2.0 / math.pow(k, 2.0)) * (math.pow(l, 2.0) / (t_m * (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)
	tmp = 0.0
	if (k <= 2.6e-17)
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * Float64(sin(k) * (Float64((t_m ^ 1.5) / l) ^ 2.0))));
	else
		tmp = Float64(Float64(2.0 / (k ^ 2.0)) * Float64((l ^ 2.0) / Float64(t_m * Float64(k * k))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 2.6e-17)
		tmp = 2.0 / ((2.0 * k) * (sin(k) * (((t_m ^ 1.5) / l) ^ 2.0)));
	else
		tmp = (2.0 / (k ^ 2.0)) * ((l ^ 2.0) / (t_m * (k * k)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 2.6e-17], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.60000000000000003e-17

   1. Initial program 57.1%

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

   3. Simplified 57.1%

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

      1. add-sqr-sqrt 27.8%

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

      2. pow2 27.8%

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

      3. associate-/r* 30.4%

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

      4. *-commutative 30.4%

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

      5. sqrt-prod 17.2%

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

      6. associate-/r* 15.6%

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

      7. sqrt-div 15.6%

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

      8. sqrt-pow1 16.6%

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

      9. metadata-eval 16.6%

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

      10. sqrt-prod 5.7%

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

      11. add-sqr-sqrt 19.1%

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

   6. Applied egg-rr 19.1%

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

      1. *-commutative 19.1%

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

   8. Simplified 19.1%

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

      1. unpow-prod-down 18.1%

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

      2. pow2 18.1%

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

      3. add-sqr-sqrt 32.8%

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

   10. Applied egg-rr 32.8%

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

   11. Taylor expanded in k around 0 29.0%

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

   if 2.60000000000000003e-17 < k

   1. Initial program 51.6%

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

   3. Simplified 51.6%

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

   5. Taylor expanded in t around 0 74.8%

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

      1. associate-*r/ 74.8%

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

      2. times-frac 70.7%

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

      3. *-commutative 70.7%

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

      4. times-frac 70.7%

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

   7. Simplified 70.7%

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

   8. Taylor expanded in k around 0 61.1%

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

      1. unpow2 70.7%

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

   10. Applied egg-rr 61.1%

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

4. Final simplification 36.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.6 \cdot 10^{-17}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t \cdot \left(k \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 61.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{-17}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2} \cdot \left(2 \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t\_m \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

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 (<= k 1.5e-17)
    (/ 2.0 (* (pow (/ (pow t_m 1.5) l) 2.0) (* 2.0 (* k k))))
    (* (/ 2.0 (pow k 2.0)) (/ (pow l 2.0) (* t_m (* 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 tmp;
	if (k <= 1.5e-17) {
		tmp = 2.0 / (pow((pow(t_m, 1.5) / l), 2.0) * (2.0 * (k * k)));
	} else {
		tmp = (2.0 / pow(k, 2.0)) * (pow(l, 2.0) / (t_m * (k * k)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.5d-17) then
        tmp = 2.0d0 / ((((t_m ** 1.5d0) / l) ** 2.0d0) * (2.0d0 * (k * k)))
    else
        tmp = (2.0d0 / (k ** 2.0d0)) * ((l ** 2.0d0) / (t_m * (k * k)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.5e-17) {
		tmp = 2.0 / (Math.pow((Math.pow(t_m, 1.5) / l), 2.0) * (2.0 * (k * k)));
	} else {
		tmp = (2.0 / Math.pow(k, 2.0)) * (Math.pow(l, 2.0) / (t_m * (k * k)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 1.5e-17:
		tmp = 2.0 / (math.pow((math.pow(t_m, 1.5) / l), 2.0) * (2.0 * (k * k)))
	else:
		tmp = (2.0 / math.pow(k, 2.0)) * (math.pow(l, 2.0) / (t_m * (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)
	tmp = 0.0
	if (k <= 1.5e-17)
		tmp = Float64(2.0 / Float64((Float64((t_m ^ 1.5) / l) ^ 2.0) * Float64(2.0 * Float64(k * k))));
	else
		tmp = Float64(Float64(2.0 / (k ^ 2.0)) * Float64((l ^ 2.0) / Float64(t_m * Float64(k * k))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 1.5e-17)
		tmp = 2.0 / ((((t_m ^ 1.5) / l) ^ 2.0) * (2.0 * (k * k)));
	else
		tmp = (2.0 / (k ^ 2.0)) * ((l ^ 2.0) / (t_m * (k * k)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1.5e-17], 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], N[(N[(2.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.50000000000000003e-17

   1. Initial program 57.1%

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

   3. Simplified 57.0%

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

   5. Taylor expanded in k around 0 56.3%

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

      1. unpow2 60.5%

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

   7. Applied egg-rr 56.3%

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

      1. add-sqr-sqrt 24.8%

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

      2. pow2 24.8%

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

      3. associate-/r* 23.6%

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

      4. sqrt-div 23.6%

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

      5. sqrt-pow1 25.2%

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

      6. metadata-eval 25.2%

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

      7. sqrt-prod 12.0%

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

      8. add-sqr-sqrt 25.8%

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

   9. Applied egg-rr 25.8%

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

   if 1.50000000000000003e-17 < k

   1. Initial program 51.6%

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

   3. Simplified 51.6%

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

   5. Taylor expanded in t around 0 74.8%

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

      1. associate-*r/ 74.8%

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

      2. times-frac 70.7%

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

      3. *-commutative 70.7%

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

      4. times-frac 70.7%

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

   7. Simplified 70.7%

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

   8. Taylor expanded in k around 0 61.1%

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

      1. unpow2 70.7%

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

   10. Applied egg-rr 61.1%

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

4. Final simplification 34.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{-17}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(2 \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t \cdot \left(k \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 59.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.18 \cdot 10^{-24}:\\ \;\;\;\;\frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(\left({t\_m}^{2} \cdot \frac{1}{\ell}\right) \cdot \frac{t\_m}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t\_m \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

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 (<= k 1.18e-24)
    (/ 2.0 (* (* 2.0 (* k k)) (* (* (pow t_m 2.0) (/ 1.0 l)) (/ t_m l))))
    (* (/ 2.0 (pow k 2.0)) (/ (pow l 2.0) (* t_m (* 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 tmp;
	if (k <= 1.18e-24) {
		tmp = 2.0 / ((2.0 * (k * k)) * ((pow(t_m, 2.0) * (1.0 / l)) * (t_m / l)));
	} else {
		tmp = (2.0 / pow(k, 2.0)) * (pow(l, 2.0) / (t_m * (k * k)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.18d-24) then
        tmp = 2.0d0 / ((2.0d0 * (k * k)) * (((t_m ** 2.0d0) * (1.0d0 / l)) * (t_m / l)))
    else
        tmp = (2.0d0 / (k ** 2.0d0)) * ((l ** 2.0d0) / (t_m * (k * k)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.18e-24) {
		tmp = 2.0 / ((2.0 * (k * k)) * ((Math.pow(t_m, 2.0) * (1.0 / l)) * (t_m / l)));
	} else {
		tmp = (2.0 / Math.pow(k, 2.0)) * (Math.pow(l, 2.0) / (t_m * (k * k)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 1.18e-24:
		tmp = 2.0 / ((2.0 * (k * k)) * ((math.pow(t_m, 2.0) * (1.0 / l)) * (t_m / l)))
	else:
		tmp = (2.0 / math.pow(k, 2.0)) * (math.pow(l, 2.0) / (t_m * (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)
	tmp = 0.0
	if (k <= 1.18e-24)
		tmp = Float64(2.0 / Float64(Float64(2.0 * Float64(k * k)) * Float64(Float64((t_m ^ 2.0) * Float64(1.0 / l)) * Float64(t_m / l))));
	else
		tmp = Float64(Float64(2.0 / (k ^ 2.0)) * Float64((l ^ 2.0) / Float64(t_m * Float64(k * k))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 1.18e-24)
		tmp = 2.0 / ((2.0 * (k * k)) * (((t_m ^ 2.0) * (1.0 / l)) * (t_m / l)));
	else
		tmp = (2.0 / (k ^ 2.0)) * ((l ^ 2.0) / (t_m * (k * k)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1.18e-24], N[(2.0 / N[(N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.18e-24

   1. Initial program 56.7%

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

   3. Simplified 56.6%

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

   5. Taylor expanded in k around 0 56.4%

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

      1. unpow2 60.1%

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

   7. Applied egg-rr 56.4%

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

      1. associate-/r* 51.9%

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

      2. unpow3 51.9%

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

      3. times-frac 60.2%

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

      4. pow2 60.2%

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

   9. Applied egg-rr 60.2%

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

       1. div-inv 60.2%

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

   11. Applied egg-rr 60.2%

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

   if 1.18e-24 < k

   1. Initial program 53.1%

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

   3. Simplified 53.1%

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

   5. Taylor expanded in t around 0 74.8%

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

      1. associate-*r/ 74.8%

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

      2. times-frac 71.0%

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

      3. *-commutative 71.0%

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

      4. times-frac 71.0%

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

   7. Simplified 71.0%

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

   8. Taylor expanded in k around 0 62.0%

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

      1. unpow2 71.0%

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

   10. Applied egg-rr 62.0%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.18 \cdot 10^{-24}:\\ \;\;\;\;\frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(\left({t}^{2} \cdot \frac{1}{\ell}\right) \cdot \frac{t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t \cdot \left(k \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 59.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 7 \cdot 10^{-25}:\\ \;\;\;\;\frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(\left({t\_m}^{2} \cdot \frac{1}{\ell}\right) \cdot \frac{t\_m}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot {\ell}^{2}}{t\_m \cdot {k}^{4}}\\ \end{array} \end{array} \]

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 (<= k 7e-25)
    (/ 2.0 (* (* 2.0 (* k k)) (* (* (pow t_m 2.0) (/ 1.0 l)) (/ t_m l))))
    (/ (* 2.0 (pow l 2.0)) (* t_m (pow k 4.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 7e-25) {
		tmp = 2.0 / ((2.0 * (k * k)) * ((pow(t_m, 2.0) * (1.0 / l)) * (t_m / l)));
	} else {
		tmp = (2.0 * pow(l, 2.0)) / (t_m * pow(k, 4.0));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 7d-25) then
        tmp = 2.0d0 / ((2.0d0 * (k * k)) * (((t_m ** 2.0d0) * (1.0d0 / l)) * (t_m / l)))
    else
        tmp = (2.0d0 * (l ** 2.0d0)) / (t_m * (k ** 4.0d0))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 7e-25) {
		tmp = 2.0 / ((2.0 * (k * k)) * ((Math.pow(t_m, 2.0) * (1.0 / l)) * (t_m / l)));
	} else {
		tmp = (2.0 * Math.pow(l, 2.0)) / (t_m * Math.pow(k, 4.0));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 7e-25:
		tmp = 2.0 / ((2.0 * (k * k)) * ((math.pow(t_m, 2.0) * (1.0 / l)) * (t_m / l)))
	else:
		tmp = (2.0 * math.pow(l, 2.0)) / (t_m * math.pow(k, 4.0))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 7e-25)
		tmp = Float64(2.0 / Float64(Float64(2.0 * Float64(k * k)) * Float64(Float64((t_m ^ 2.0) * Float64(1.0 / l)) * Float64(t_m / l))));
	else
		tmp = Float64(Float64(2.0 * (l ^ 2.0)) / Float64(t_m * (k ^ 4.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 7e-25)
		tmp = 2.0 / ((2.0 * (k * k)) * (((t_m ^ 2.0) * (1.0 / l)) * (t_m / l)));
	else
		tmp = (2.0 * (l ^ 2.0)) / (t_m * (k ^ 4.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 7e-25], N[(2.0 / N[(N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 7.0000000000000004e-25

   1. Initial program 56.7%

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

   3. Simplified 56.6%

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

   5. Taylor expanded in k around 0 56.4%

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

      1. unpow2 60.1%

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

   7. Applied egg-rr 56.4%

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

      1. associate-/r* 51.9%

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

      2. unpow3 51.9%

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

      3. times-frac 60.2%

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

      4. pow2 60.2%

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

   9. Applied egg-rr 60.2%

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

       1. div-inv 60.2%

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

   11. Applied egg-rr 60.2%

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

   if 7.0000000000000004e-25 < k

   1. Initial program 53.1%

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

   3. Simplified 53.1%

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

   5. Taylor expanded in t around 0 74.8%

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

      1. associate-*r/ 74.8%

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

      2. times-frac 71.0%

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

      3. *-commutative 71.0%

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

      4. times-frac 71.0%

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

   7. Simplified 71.0%

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

   8. Taylor expanded in k around 0 59.0%

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

      1. associate-*r/ 59.0%

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

      2. *-commutative 59.0%

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

   10. Simplified 59.0%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7 \cdot 10^{-25}:\\ \;\;\;\;\frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(\left({t}^{2} \cdot \frac{1}{\ell}\right) \cdot \frac{t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot {\ell}^{2}}{t \cdot {k}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 59.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.1 \cdot 10^{-24}:\\ \;\;\;\;\frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(\left({t\_m}^{2} \cdot \frac{1}{\ell}\right) \cdot \frac{t\_m}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{t\_m}\\ \end{array} \end{array} \]

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 (<= k 1.1e-24)
    (/ 2.0 (* (* 2.0 (* k k)) (* (* (pow t_m 2.0) (/ 1.0 l)) (/ t_m l))))
    (* 2.0 (/ (/ (pow l 2.0) (pow k 4.0)) t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.1e-24) {
		tmp = 2.0 / ((2.0 * (k * k)) * ((pow(t_m, 2.0) * (1.0 / l)) * (t_m / l)));
	} else {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k, 4.0)) / t_m);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.1d-24) then
        tmp = 2.0d0 / ((2.0d0 * (k * k)) * (((t_m ** 2.0d0) * (1.0d0 / l)) * (t_m / l)))
    else
        tmp = 2.0d0 * (((l ** 2.0d0) / (k ** 4.0d0)) / t_m)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.1e-24) {
		tmp = 2.0 / ((2.0 * (k * k)) * ((Math.pow(t_m, 2.0) * (1.0 / l)) * (t_m / l)));
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k, 4.0)) / t_m);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 1.1e-24:
		tmp = 2.0 / ((2.0 * (k * k)) * ((math.pow(t_m, 2.0) * (1.0 / l)) * (t_m / l)))
	else:
		tmp = 2.0 * ((math.pow(l, 2.0) / math.pow(k, 4.0)) / t_m)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.1e-24)
		tmp = Float64(2.0 / Float64(Float64(2.0 * Float64(k * k)) * Float64(Float64((t_m ^ 2.0) * Float64(1.0 / l)) * Float64(t_m / l))));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k ^ 4.0)) / t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 1.1e-24)
		tmp = 2.0 / ((2.0 * (k * k)) * (((t_m ^ 2.0) * (1.0 / l)) * (t_m / l)));
	else
		tmp = 2.0 * (((l ^ 2.0) / (k ^ 4.0)) / t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1.1e-24], N[(2.0 / N[(N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.10000000000000001e-24

   1. Initial program 56.7%

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

   3. Simplified 56.6%

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

   5. Taylor expanded in k around 0 56.4%

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

      1. unpow2 60.1%

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

   7. Applied egg-rr 56.4%

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

      1. associate-/r* 51.9%

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

      2. unpow3 51.9%

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

      3. times-frac 60.2%

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

      4. pow2 60.2%

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

   9. Applied egg-rr 60.2%

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

       1. div-inv 60.2%

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

   11. Applied egg-rr 60.2%

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

   if 1.10000000000000001e-24 < k

   1. Initial program 53.1%

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

   3. Simplified 53.1%

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

   5. Taylor expanded in t around 0 74.8%

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

      1. associate-*r/ 74.8%

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

      2. times-frac 71.0%

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

      3. *-commutative 71.0%

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

      4. times-frac 71.0%

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

   7. Simplified 71.0%

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

   8. Taylor expanded in k around 0 62.0%

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

   9. Taylor expanded in k around 0 59.0%

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

       1. associate-/r* 59.0%

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

   11. Simplified 59.0%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.1 \cdot 10^{-24}:\\ \;\;\;\;\frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(\left({t}^{2} \cdot \frac{1}{\ell}\right) \cdot \frac{t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 58.6% accurate, 3.6× speedup

Math

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

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) (/ 1.0 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) * (1.0 / 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) * (1.0d0 / 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) * (1.0 / 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) * (1.0 / 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) * Float64(1.0 / 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) * (1.0 / 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] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.7%

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

3. Simplified 56.4%

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

5. Taylor expanded in k around 0 55.0%

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

   1. unpow2 63.0%

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

7. Applied egg-rr 55.0%

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

   1. associate-/r* 51.6%

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

   2. unpow3 51.6%

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

   3. times-frac 58.1%

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

   4. pow2 58.1%

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

9. Applied egg-rr 58.1%

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

    1. div-inv 58.2%

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

11. Applied egg-rr 58.2%

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

12. Final simplification 58.2%

    \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(\left({t}^{2} \cdot \frac{1}{\ell}\right) \cdot \frac{t}{\ell}\right)} \]
13. Add Preprocessing

Alternative 21: 58.6% accurate, 24.8× speedup

Math

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

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)) (* (/ t_m l) (/ (* t_m 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)) * ((t_m / l) * ((t_m * 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 / l) * ((t_m * 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)) * ((t_m / l) * ((t_m * 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)) * ((t_m / l) * ((t_m * 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 / l) * Float64(Float64(t_m * 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 / l) * ((t_m * 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[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.7%

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

3. Simplified 56.4%

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

5. Taylor expanded in k around 0 55.0%

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

   1. unpow2 63.0%

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

7. Applied egg-rr 55.0%

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

   1. associate-/r* 51.6%

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

   2. unpow3 51.6%

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

   3. times-frac 58.1%

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

   4. pow2 58.1%

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

9. Applied egg-rr 58.1%

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

    1. unpow2 58.1%

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

11. Applied egg-rr 58.1%

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

12. Final simplification 58.1%

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

Reproduce

herbie shell --seed 2024137 
(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))))