Toniolo and Linder, Equation (10+)

Percentage Accurate: 56.2% → 85.5%

Time: 17.7s

Alternatives: 22

Speedup: 1.9×

• 27.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 56.2% 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: 85.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\sqrt[3]{\ell}\right)}^{2}\\ \mathbf{if}\;\ell \cdot \ell \leq 10^{-313}:\\ \;\;\;\;\frac{2}{{\left(t \cdot \frac{\sqrt[3]{k}}{t\_1}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+300}:\\ \;\;\;\;\frac{2}{t \cdot \mathsf{fma}\left({k}^{2}, \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}, \frac{2}{{\ell}^{2}} \cdot \frac{{\left(t \cdot \sin k\right)}^{2}}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{t\_1} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin k}\right)\right)\right)}^{3} \cdot \left(2 \cdot \frac{\sin k}{\cos k}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (cbrt l) 2.0)))
   (if (<= (* l l) 1e-313)
     (/
      2.0
      (*
       (pow (* t (/ (cbrt k) t_1)) 3.0)
       (* (tan k) (+ 1.0 (+ 1.0 (pow (/ k t) 2.0))))))
     (if (<= (* l l) 2e+300)
       (/
        2.0
        (*
         t
         (fma
          (pow k 2.0)
          (/ (pow (sin k) 2.0) (* (pow l 2.0) (cos k)))
          (* (/ 2.0 (pow l 2.0)) (/ (pow (* t (sin k)) 2.0) (cos k))))))
       (/
        2.0
        (*
         (pow (* (/ t t_1) (expm1 (log1p (cbrt (sin k))))) 3.0)
         (* 2.0 (/ (sin k) (cos k)))))))))

C

double code(double t, double l, double k) {
	double t_1 = pow(cbrt(l), 2.0);
	double tmp;
	if ((l * l) <= 1e-313) {
		tmp = 2.0 / (pow((t * (cbrt(k) / t_1)), 3.0) * (tan(k) * (1.0 + (1.0 + pow((k / t), 2.0)))));
	} else if ((l * l) <= 2e+300) {
		tmp = 2.0 / (t * fma(pow(k, 2.0), (pow(sin(k), 2.0) / (pow(l, 2.0) * cos(k))), ((2.0 / pow(l, 2.0)) * (pow((t * sin(k)), 2.0) / cos(k)))));
	} else {
		tmp = 2.0 / (pow(((t / t_1) * expm1(log1p(cbrt(sin(k))))), 3.0) * (2.0 * (sin(k) / cos(k))));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = cbrt(l) ^ 2.0
	tmp = 0.0
	if (Float64(l * l) <= 1e-313)
		tmp = Float64(2.0 / Float64((Float64(t * Float64(cbrt(k) / t_1)) ^ 3.0) * Float64(tan(k) * Float64(1.0 + Float64(1.0 + (Float64(k / t) ^ 2.0))))));
	elseif (Float64(l * l) <= 2e+300)
		tmp = Float64(2.0 / Float64(t * fma((k ^ 2.0), Float64((sin(k) ^ 2.0) / Float64((l ^ 2.0) * cos(k))), Float64(Float64(2.0 / (l ^ 2.0)) * Float64((Float64(t * sin(k)) ^ 2.0) / cos(k))))));
	else
		tmp = Float64(2.0 / Float64((Float64(Float64(t / t_1) * expm1(log1p(cbrt(sin(k))))) ^ 3.0) * Float64(2.0 * Float64(sin(k) / cos(k)))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(l * l), $MachinePrecision], 1e-313], N[(2.0 / N[(N[Power[N[(t * N[(N[Power[k, 1/3], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 2e+300], N[(2.0 / N[(t * N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[(t / t$95$1), $MachinePrecision] * N[(Exp[N[Log[1 + N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * N[(N[Sin[k], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 1.00000000001e-313

   1. Initial program 55.5%

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

   3. Simplified 55.5%

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

   5. Taylor expanded in k around 0 56.6%

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

      1. add-cube-cbrt 56.6%

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

      2. pow2 56.6%

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

      3. cbrt-div 56.6%

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

      4. *-commutative 56.6%

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

      5. cbrt-prod 56.6%

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

      6. unpow3 56.6%

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

      7. add-cbrt-cube 56.6%

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

      8. unpow2 56.6%

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

      9. cbrt-prod 56.6%

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

      10. unpow2 56.6%

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

      11. cbrt-div 56.6%

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

      12. *-commutative 56.6%

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

      13. cbrt-prod 56.7%

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

      14. unpow3 56.7%

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

      15. add-cbrt-cube 77.0%

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

      16. unpow2 77.0%

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

      17. cbrt-prod 91.6%

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

      18. unpow2 91.6%

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

   7. Applied egg-rr 91.6%

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

      1. unpow2 91.6%

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

      2. unpow3 91.5%

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

      3. associate-/l* 91.6%

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

   9. Simplified 91.6%

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

   if 1.00000000001e-313 < (*.f64 l l) < 2.0000000000000001e300

   1. Initial program 62.6%

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

   3. Simplified 62.6%

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

      1. add-cube-cbrt 62.4%

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

      2. pow3 62.4%

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

      3. *-commutative 62.4%

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

      4. cbrt-prod 62.4%

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

      5. cbrt-div 63.9%

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

      6. rem-cbrt-cube 71.3%

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

      7. cbrt-prod 71.1%

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

      8. pow2 71.1%

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

   6. Applied egg-rr 71.1%

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

      1. *-commutative 71.1%

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

   8. Simplified 71.1%

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

      1. expm1-log1p-u 71.2%

         \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin k}\right)\right)}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      2. expm1-undefine 48.9%

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

   10. Applied egg-rr 48.9%

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

       1. expm1-define 71.2%

          \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin k}\right)\right)}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   12. Simplified 71.2%

       \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin k}\right)\right)}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   13. Taylor expanded in t around 0 81.6%

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

       1. +-commutative 81.6%

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

       2. associate-/l* 81.6%

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

       3. fma-define 81.6%

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

       4. associate-*r/ 81.6%

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

       5. times-frac 81.6%

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

   15. Simplified 87.4%

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

   if 2.0000000000000001e300 < (*.f64 l l)

   1. Initial program 29.9%

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

   3. Simplified 29.9%

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

      1. add-cube-cbrt 29.9%

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

      2. pow3 29.9%

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

      3. *-commutative 29.9%

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

      4. cbrt-prod 29.9%

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

      5. cbrt-div 29.9%

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

      6. rem-cbrt-cube 35.5%

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

      7. cbrt-prod 71.9%

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

      8. pow2 71.9%

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

   6. Applied egg-rr 71.9%

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

      1. *-commutative 71.9%

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

   8. Simplified 71.9%

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

      1. expm1-log1p-u 72.0%

         \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin k}\right)\right)}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      2. expm1-undefine 49.0%

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

   10. Applied egg-rr 49.0%

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

       1. expm1-define 72.0%

          \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin k}\right)\right)}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   12. Simplified 72.0%

       \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin k}\right)\right)}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   13. Taylor expanded in t around inf 80.6%

       \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin k}\right)\right)\right)}^{3} \cdot \color{blue}{\left(2 \cdot \frac{\sin k}{\cos k}\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 72.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 8.8 \cdot 10^{-69}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left(t \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin k}\right)\right)\right)}^{3} \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 8.8e-69)
   (/ 2.0 (* (pow k 2.0) (* t (/ (pow (sin k) 2.0) (* (pow l 2.0) (cos k))))))
   (/
    2.0
    (*
     (pow (* (/ t (pow (cbrt l) 2.0)) (expm1 (log1p (cbrt (sin k))))) 3.0)
     (* (tan k) (+ 2.0 (pow (/ k t) 2.0)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 8.8e-69) {
		tmp = 2.0 / (pow(k, 2.0) * (t * (pow(sin(k), 2.0) / (pow(l, 2.0) * cos(k)))));
	} else {
		tmp = 2.0 / (pow(((t / pow(cbrt(l), 2.0)) * expm1(log1p(cbrt(sin(k))))), 3.0) * (tan(k) * (2.0 + pow((k / t), 2.0))));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 8.8e-69) {
		tmp = 2.0 / (Math.pow(k, 2.0) * (t * (Math.pow(Math.sin(k), 2.0) / (Math.pow(l, 2.0) * Math.cos(k)))));
	} else {
		tmp = 2.0 / (Math.pow(((t / Math.pow(Math.cbrt(l), 2.0)) * Math.expm1(Math.log1p(Math.cbrt(Math.sin(k))))), 3.0) * (Math.tan(k) * (2.0 + Math.pow((k / t), 2.0))));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 8.8e-69)
		tmp = Float64(2.0 / Float64((k ^ 2.0) * Float64(t * Float64((sin(k) ^ 2.0) / Float64((l ^ 2.0) * cos(k))))));
	else
		tmp = Float64(2.0 / Float64((Float64(Float64(t / (cbrt(l) ^ 2.0)) * expm1(log1p(cbrt(sin(k))))) ^ 3.0) * Float64(tan(k) * Float64(2.0 + (Float64(k / t) ^ 2.0)))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 8.8e-69], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(Exp[N[Log[1 + N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 8.8000000000000001e-69

   1. Initial program 47.1%

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

   3. Simplified 47.1%

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

      1. add-cube-cbrt 47.1%

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

      2. pow3 47.1%

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

      3. *-commutative 47.1%

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

      4. cbrt-prod 47.0%

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

      5. cbrt-div 47.5%

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

      6. rem-cbrt-cube 58.8%

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

      7. cbrt-prod 66.8%

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

      8. pow2 66.8%

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

   6. Applied egg-rr 66.8%

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

      1. *-commutative 66.8%

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

   8. Simplified 66.8%

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

      1. expm1-log1p-u 66.8%

         \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin k}\right)\right)}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      2. expm1-undefine 48.4%

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

   10. Applied egg-rr 48.4%

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

       1. expm1-define 66.8%

          \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin k}\right)\right)}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   12. Simplified 66.8%

       \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin k}\right)\right)}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   13. Taylor expanded in t around 0 67.3%

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

       1. associate-/l* 66.5%

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

       2. associate-/l* 67.1%

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

   15. Simplified 67.1%

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

   if 8.8000000000000001e-69 < t

   1. Initial program 63.4%

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

   3. Simplified 63.4%

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

      1. add-cube-cbrt 63.2%

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

      2. pow3 63.2%

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

      3. *-commutative 63.2%

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

      4. cbrt-prod 63.2%

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

      5. cbrt-div 65.2%

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

      6. rem-cbrt-cube 72.6%

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

      7. cbrt-prod 93.1%

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

      8. pow2 93.1%

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

   6. Applied egg-rr 93.1%

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

      1. *-commutative 93.1%

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

   8. Simplified 93.1%

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

      1. expm1-log1p-u 93.2%

         \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin k}\right)\right)}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      2. expm1-undefine 56.0%

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

   10. Applied egg-rr 56.0%

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

       1. expm1-define 93.2%

          \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin k}\right)\right)}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   12. Simplified 93.2%

       \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin k}\right)\right)}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
   13. Step-by-step derivation

       1. associate-+r+ 93.2%

          \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin k}\right)\right)\right)}^{3} \cdot \left(\tan k \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}\right)} \]

       2. metadata-eval 93.2%

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

       3. distribute-lft-in 93.2%

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

   14. Applied egg-rr 93.2%

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

       1. distribute-lft-in 93.2%

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

   16. Simplified 93.2%

       \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin k}\right)\right)\right)}^{3} \cdot \color{blue}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 72.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 6.6 \cdot 10^{-69}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left(t \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right) \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 6.6e-69)
   (/ 2.0 (* (pow k 2.0) (* t (/ (pow (sin k) 2.0) (* (pow l 2.0) (cos k))))))
   (/
    2.0
    (*
     (* (tan k) (+ 1.0 (+ 1.0 (pow (/ k t) 2.0))))
     (pow (* (/ t (pow (cbrt l) 2.0)) (cbrt (sin k))) 3.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 6.6e-69) {
		tmp = 2.0 / (pow(k, 2.0) * (t * (pow(sin(k), 2.0) / (pow(l, 2.0) * cos(k)))));
	} else {
		tmp = 2.0 / ((tan(k) * (1.0 + (1.0 + pow((k / t), 2.0)))) * pow(((t / pow(cbrt(l), 2.0)) * cbrt(sin(k))), 3.0));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 6.6e-69) {
		tmp = 2.0 / (Math.pow(k, 2.0) * (t * (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 + Math.pow((k / t), 2.0)))) * Math.pow(((t / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt(Math.sin(k))), 3.0));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 6.6e-69)
		tmp = Float64(2.0 / Float64((k ^ 2.0) * Float64(t * Float64((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(k / t) ^ 2.0)))) * (Float64(Float64(t / (cbrt(l) ^ 2.0)) * cbrt(sin(k))) ^ 3.0)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 6.6e-69], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(t / 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]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 6.6000000000000001e-69

   1. Initial program 47.1%

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

   3. Simplified 47.1%

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

      1. add-cube-cbrt 47.1%

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

      2. pow3 47.1%

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

      3. *-commutative 47.1%

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

      4. cbrt-prod 47.0%

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

      5. cbrt-div 47.5%

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

      6. rem-cbrt-cube 58.8%

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

      7. cbrt-prod 66.8%

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

      8. pow2 66.8%

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

   6. Applied egg-rr 66.8%

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

      1. *-commutative 66.8%

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

   8. Simplified 66.8%

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

      1. expm1-log1p-u 66.8%

         \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin k}\right)\right)}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      2. expm1-undefine 48.4%

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

   10. Applied egg-rr 48.4%

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

       1. expm1-define 66.8%

          \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin k}\right)\right)}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   12. Simplified 66.8%

       \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin k}\right)\right)}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   13. Taylor expanded in t around 0 67.3%

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

       1. associate-/l* 66.5%

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

       2. associate-/l* 67.1%

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

   15. Simplified 67.1%

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

   if 6.6000000000000001e-69 < t

   1. Initial program 63.4%

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

   3. Simplified 63.4%

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

      1. add-cube-cbrt 63.2%

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

      2. pow3 63.2%

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

      3. *-commutative 63.2%

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

      4. cbrt-prod 63.2%

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

      5. cbrt-div 65.2%

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

      6. rem-cbrt-cube 72.6%

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

      7. cbrt-prod 93.1%

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

      8. pow2 93.1%

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

   6. Applied egg-rr 93.1%

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

      1. *-commutative 93.1%

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

   8. Simplified 93.1%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.6 \cdot 10^{-69}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left(t \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right) \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 7.6 \cdot 10^{-69}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left(t \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(t \cdot \left(\sqrt[3]{\sin k} \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 7.6e-69)
   (/ 2.0 (* (pow k 2.0) (* t (/ (pow (sin k) 2.0) (* (pow l 2.0) (cos k))))))
   (/
    2.0
    (*
     (* (tan k) (+ 2.0 (pow (/ k t) 2.0)))
     (pow (* t (* (cbrt (sin k)) (pow (cbrt l) -2.0))) 3.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 7.6e-69) {
		tmp = 2.0 / (pow(k, 2.0) * (t * (pow(sin(k), 2.0) / (pow(l, 2.0) * cos(k)))));
	} else {
		tmp = 2.0 / ((tan(k) * (2.0 + pow((k / t), 2.0))) * pow((t * (cbrt(sin(k)) * pow(cbrt(l), -2.0))), 3.0));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 7.6e-69) {
		tmp = 2.0 / (Math.pow(k, 2.0) * (t * (Math.pow(Math.sin(k), 2.0) / (Math.pow(l, 2.0) * Math.cos(k)))));
	} else {
		tmp = 2.0 / ((Math.tan(k) * (2.0 + Math.pow((k / t), 2.0))) * Math.pow((t * (Math.cbrt(Math.sin(k)) * Math.pow(Math.cbrt(l), -2.0))), 3.0));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 7.6e-69)
		tmp = Float64(2.0 / Float64((k ^ 2.0) * Float64(t * Float64((sin(k) ^ 2.0) / Float64((l ^ 2.0) * cos(k))))));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(2.0 + (Float64(k / t) ^ 2.0))) * (Float64(t * Float64(cbrt(sin(k)) * (cbrt(l) ^ -2.0))) ^ 3.0)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 7.6e-69], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(t * N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 7.5999999999999995e-69

   1. Initial program 47.1%

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

   3. Simplified 47.1%

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

      1. add-cube-cbrt 47.1%

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

      2. pow3 47.1%

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

      3. *-commutative 47.1%

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

      4. cbrt-prod 47.0%

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

      5. cbrt-div 47.5%

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

      6. rem-cbrt-cube 58.8%

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

      7. cbrt-prod 66.8%

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

      8. pow2 66.8%

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

   6. Applied egg-rr 66.8%

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

      1. *-commutative 66.8%

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

   8. Simplified 66.8%

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

      1. expm1-log1p-u 66.8%

         \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin k}\right)\right)}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      2. expm1-undefine 48.4%

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

   10. Applied egg-rr 48.4%

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

       1. expm1-define 66.8%

          \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin k}\right)\right)}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   12. Simplified 66.8%

       \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin k}\right)\right)}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   13. Taylor expanded in t around 0 67.3%

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

       1. associate-/l* 66.5%

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

       2. associate-/l* 67.1%

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

   15. Simplified 67.1%

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

   if 7.5999999999999995e-69 < t

   1. Initial program 63.4%

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

   3. Simplified 63.4%

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

      1. add-cube-cbrt 63.2%

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

      2. pow3 63.2%

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

      3. *-commutative 63.2%

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

      4. cbrt-prod 63.2%

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

      5. cbrt-div 65.2%

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

      6. rem-cbrt-cube 72.6%

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

      7. cbrt-prod 93.1%

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

      8. pow2 93.1%

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

   6. Applied egg-rr 93.1%

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

      1. *-commutative 93.1%

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

   8. Simplified 93.1%

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

      1. expm1-log1p-u 93.2%

         \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin k}\right)\right)}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      2. expm1-undefine 56.0%

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

   10. Applied egg-rr 56.0%

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

       1. expm1-define 93.2%

          \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin k}\right)\right)}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   12. Simplified 93.2%

       \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin k}\right)\right)}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
   13. Step-by-step derivation

       1. associate-+r+ 93.2%

          \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin k}\right)\right)\right)}^{3} \cdot \left(\tan k \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}\right)} \]

       2. metadata-eval 93.2%

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

       3. distribute-lft-in 93.2%

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

   14. Applied egg-rr 93.2%

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

       1. distribute-lft-in 93.2%

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

   16. Simplified 93.2%

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

       1. expm1-log1p-u 93.1%

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

       2. pow1 93.1%

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

       3. div-inv 93.1%

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

       4. pow-flip 93.1%

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

       5. metadata-eval 93.1%

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

   18. Applied egg-rr 93.1%

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

       1. unpow1 93.1%

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

       2. associate-*l* 93.0%

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

   20. Simplified 93.0%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.6 \cdot 10^{-69}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left(t \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(t \cdot \left(\sqrt[3]{\sin k} \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.15 \cdot 10^{-7}:\\ \;\;\;\;\frac{2}{{\left(t \cdot \frac{\sqrt[3]{k}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot {k}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 2.15e-7)
   (/
    2.0
    (*
     (pow (* t (/ (cbrt k) (pow (cbrt l) 2.0))) 3.0)
     (* (tan k) (+ 1.0 (+ 1.0 (pow (/ k t) 2.0))))))
   (*
    2.0
    (/ (* (pow l 2.0) (cos k)) (* (pow (sin k) 2.0) (* t (pow k 2.0)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.15e-7) {
		tmp = 2.0 / (pow((t * (cbrt(k) / pow(cbrt(l), 2.0))), 3.0) * (tan(k) * (1.0 + (1.0 + pow((k / t), 2.0)))));
	} else {
		tmp = 2.0 * ((pow(l, 2.0) * cos(k)) / (pow(sin(k), 2.0) * (t * pow(k, 2.0))));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.15e-7) {
		tmp = 2.0 / (Math.pow((t * (Math.cbrt(k) / Math.pow(Math.cbrt(l), 2.0))), 3.0) * (Math.tan(k) * (1.0 + (1.0 + Math.pow((k / t), 2.0)))));
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) * Math.cos(k)) / (Math.pow(Math.sin(k), 2.0) * (t * Math.pow(k, 2.0))));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 2.15e-7)
		tmp = Float64(2.0 / Float64((Float64(t * Float64(cbrt(k) / (cbrt(l) ^ 2.0))) ^ 3.0) * Float64(tan(k) * Float64(1.0 + Float64(1.0 + (Float64(k / t) ^ 2.0))))));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k)) / Float64((sin(k) ^ 2.0) * Float64(t * (k ^ 2.0)))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 2.15e-7], N[(2.0 / N[(N[Power[N[(t * N[(N[Power[k, 1/3], $MachinePrecision] / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.1500000000000001e-7

   1. Initial program 55.0%

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

   3. Simplified 55.0%

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

   5. Taylor expanded in k around 0 54.5%

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

      1. add-cube-cbrt 54.4%

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

      2. pow2 54.4%

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

      3. cbrt-div 54.4%

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

      4. *-commutative 54.4%

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

      5. cbrt-prod 54.4%

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

      6. unpow3 54.4%

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

      7. add-cbrt-cube 54.4%

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

      8. unpow2 54.4%

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

      9. cbrt-prod 54.4%

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

      10. unpow2 54.4%

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

      11. cbrt-div 54.4%

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

      12. *-commutative 54.4%

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

      13. cbrt-prod 54.8%

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

      14. unpow3 54.8%

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

      15. add-cbrt-cube 63.9%

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

      16. unpow2 63.9%

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

      17. cbrt-prod 75.7%

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

      18. unpow2 75.7%

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

   7. Applied egg-rr 75.7%

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

      1. unpow2 75.7%

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

      2. unpow3 75.7%

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

      3. associate-/l* 75.7%

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

   9. Simplified 75.7%

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

   if 2.1500000000000001e-7 < k

   1. Initial program 46.4%

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

   3. Simplified 45.1%

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

   5. Taylor expanded in t around 0 76.7%

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

      1. associate-*r* 76.7%

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

   7. Simplified 76.7%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.15 \cdot 10^{-7}:\\ \;\;\;\;\frac{2}{{\left(t \cdot \frac{\sqrt[3]{k}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot {k}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 8 \cdot 10^{-69}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left(t \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right) \cdot \left(\sin k \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 8e-69)
   (/ 2.0 (* (pow k 2.0) (* t (/ (pow (sin k) 2.0) (* (pow l 2.0) (cos k))))))
   (/
    2.0
    (*
     (* (tan k) (+ 1.0 (+ 1.0 (pow (/ k t) 2.0))))
     (* (sin k) (pow (/ t (pow (cbrt l) 2.0)) 3.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 8e-69) {
		tmp = 2.0 / (pow(k, 2.0) * (t * (pow(sin(k), 2.0) / (pow(l, 2.0) * cos(k)))));
	} else {
		tmp = 2.0 / ((tan(k) * (1.0 + (1.0 + pow((k / t), 2.0)))) * (sin(k) * pow((t / pow(cbrt(l), 2.0)), 3.0)));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 8e-69) {
		tmp = 2.0 / (Math.pow(k, 2.0) * (t * (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 + Math.pow((k / t), 2.0)))) * (Math.sin(k) * Math.pow((t / Math.pow(Math.cbrt(l), 2.0)), 3.0)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 8e-69)
		tmp = Float64(2.0 / Float64((k ^ 2.0) * Float64(t * Float64((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(k / t) ^ 2.0)))) * Float64(sin(k) * (Float64(t / (cbrt(l) ^ 2.0)) ^ 3.0))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 8e-69], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 7.9999999999999997e-69

   1. Initial program 47.1%

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

   3. Simplified 47.1%

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

      1. add-cube-cbrt 47.1%

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

      2. pow3 47.1%

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

      3. *-commutative 47.1%

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

      4. cbrt-prod 47.0%

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

      5. cbrt-div 47.5%

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

      6. rem-cbrt-cube 58.8%

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

      7. cbrt-prod 66.8%

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

      8. pow2 66.8%

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

   6. Applied egg-rr 66.8%

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

      1. *-commutative 66.8%

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

   8. Simplified 66.8%

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

      1. expm1-log1p-u 66.8%

         \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin k}\right)\right)}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      2. expm1-undefine 48.4%

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

   10. Applied egg-rr 48.4%

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

       1. expm1-define 66.8%

          \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin k}\right)\right)}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   12. Simplified 66.8%

       \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin k}\right)\right)}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   13. Taylor expanded in t around 0 67.3%

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

       1. associate-/l* 66.5%

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

       2. associate-/l* 67.1%

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

   15. Simplified 67.1%

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

   if 7.9999999999999997e-69 < t

   1. Initial program 63.4%

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

   3. Simplified 63.4%

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

      1. add-cube-cbrt 63.3%

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

      2. pow3 63.3%

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

      3. cbrt-div 63.2%

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

      4. rem-cbrt-cube 69.5%

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

      5. cbrt-prod 85.2%

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

      6. pow2 85.2%

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

   6. Applied egg-rr 85.2%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8 \cdot 10^{-69}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left(t \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right) \cdot \left(\sin k \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 49.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.55 \cdot 10^{-144}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{2} \cdot \sqrt{\frac{\cos k}{t}}}{k \cdot \sin k}\right)}^{2}\\ \mathbf{elif}\;t \leq 8.6 \cdot 10^{+137}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 1.55e-144)
   (pow (* l (/ (* (sqrt 2.0) (sqrt (/ (cos k) t))) (* k (sin k)))) 2.0)
   (if (<= t 8.6e+137)
     (/
      2.0
      (*
       (* (tan k) (+ 2.0 (pow (/ k t) 2.0)))
       (* (sin k) (* (/ (pow t 2.0) l) (/ t l)))))
     (/
      2.0
      (* (pow (* (/ t (pow (cbrt l) 2.0)) (cbrt (sin k))) 3.0) (* 2.0 k))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 1.55e-144) {
		tmp = pow((l * ((sqrt(2.0) * sqrt((cos(k) / t))) / (k * sin(k)))), 2.0);
	} else if (t <= 8.6e+137) {
		tmp = 2.0 / ((tan(k) * (2.0 + pow((k / t), 2.0))) * (sin(k) * ((pow(t, 2.0) / l) * (t / l))));
	} else {
		tmp = 2.0 / (pow(((t / pow(cbrt(l), 2.0)) * cbrt(sin(k))), 3.0) * (2.0 * k));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 1.55e-144) {
		tmp = Math.pow((l * ((Math.sqrt(2.0) * Math.sqrt((Math.cos(k) / t))) / (k * Math.sin(k)))), 2.0);
	} else if (t <= 8.6e+137) {
		tmp = 2.0 / ((Math.tan(k) * (2.0 + Math.pow((k / t), 2.0))) * (Math.sin(k) * ((Math.pow(t, 2.0) / l) * (t / l))));
	} else {
		tmp = 2.0 / (Math.pow(((t / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt(Math.sin(k))), 3.0) * (2.0 * k));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 1.55e-144)
		tmp = Float64(l * Float64(Float64(sqrt(2.0) * sqrt(Float64(cos(k) / t))) / Float64(k * sin(k)))) ^ 2.0;
	elseif (t <= 8.6e+137)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(2.0 + (Float64(k / t) ^ 2.0))) * Float64(sin(k) * Float64(Float64((t ^ 2.0) / l) * Float64(t / l)))));
	else
		tmp = Float64(2.0 / Float64((Float64(Float64(t / (cbrt(l) ^ 2.0)) * cbrt(sin(k))) ^ 3.0) * Float64(2.0 * k)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 1.55e-144], N[Power[N[(l * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[t, 8.6e+137], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[(t / 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]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.55e-144

   1. Initial program 47.8%

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

   3. Simplified 43.5%

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

      1. add-sqr-sqrt 28.7%

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

   6. Applied egg-rr 26.1%

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

      1. unpow2 26.1%

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

      2. associate-/l* 26.7%

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

      3. associate-/r* 26.7%

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

   8. Simplified 26.7%

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

   9. Taylor expanded in t around 0 38.1%

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

       1. associate-*l/ 38.2%

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

   11. Simplified 38.2%

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

   if 1.55e-144 < t < 8.59999999999999929e137

   1. Initial program 54.3%

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

   3. Simplified 54.3%

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

      1. unpow3 54.2%

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

      2. times-frac 73.0%

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

      3. pow2 73.0%

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

   6. Applied egg-rr 73.0%

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

      1. associate-+r+ 82.6%

         \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin k}\right)\right)\right)}^{3} \cdot \left(\tan k \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}\right)} \]

      2. metadata-eval 82.6%

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

      3. distribute-lft-in 82.6%

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

   8. Applied egg-rr 73.0%

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

      1. distribute-lft-in 82.6%

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

   10. Simplified 73.0%

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

   if 8.59999999999999929e137 < 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-cube-cbrt 67.1%

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

      2. pow3 67.1%

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

      3. *-commutative 67.1%

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

      4. cbrt-prod 67.1%

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

      5. cbrt-div 67.1%

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

      6. rem-cbrt-cube 72.0%

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

      7. cbrt-prod 94.8%

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

      8. pow2 94.8%

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

   6. Applied egg-rr 94.8%

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

      1. *-commutative 94.8%

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

   8. Simplified 94.8%

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

   9. Taylor expanded in k around 0 87.0%

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

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.55 \cdot 10^{-144}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{2} \cdot \sqrt{\frac{\cos k}{t}}}{k \cdot \sin k}\right)}^{2}\\ \mathbf{elif}\;t \leq 8.6 \cdot 10^{+137}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.3% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 3e-7)
   (/ 2.0 (* (pow (* (/ t (pow (cbrt l) 2.0)) (cbrt (sin k))) 3.0) (* 2.0 k)))
   (*
    2.0
    (/ (* (pow l 2.0) (cos k)) (* (pow (sin k) 2.0) (* t (pow k 2.0)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 3e-7) {
		tmp = 2.0 / (pow(((t / pow(cbrt(l), 2.0)) * cbrt(sin(k))), 3.0) * (2.0 * k));
	} else {
		tmp = 2.0 * ((pow(l, 2.0) * cos(k)) / (pow(sin(k), 2.0) * (t * pow(k, 2.0))));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 3e-7) {
		tmp = 2.0 / (Math.pow(((t / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt(Math.sin(k))), 3.0) * (2.0 * k));
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) * Math.cos(k)) / (Math.pow(Math.sin(k), 2.0) * (t * Math.pow(k, 2.0))));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 3e-7)
		tmp = Float64(2.0 / Float64((Float64(Float64(t / (cbrt(l) ^ 2.0)) * cbrt(sin(k))) ^ 3.0) * Float64(2.0 * k)));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k)) / Float64((sin(k) ^ 2.0) * Float64(t * (k ^ 2.0)))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 3e-7], N[(2.0 / N[(N[Power[N[(N[(t / 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[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.9999999999999999e-7

   1. Initial program 55.0%

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

   3. Simplified 55.0%

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

      1. add-cube-cbrt 54.9%

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

      2. pow3 54.9%

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

      3. *-commutative 54.9%

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

      4. cbrt-prod 54.9%

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

      5. cbrt-div 56.3%

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

      6. rem-cbrt-cube 64.4%

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

      7. cbrt-prod 78.1%

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

      8. pow2 78.1%

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

   6. Applied egg-rr 78.1%

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

      1. *-commutative 78.1%

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

   8. Simplified 78.1%

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

   9. Taylor expanded in k around 0 72.5%

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

   if 2.9999999999999999e-7 < k

   1. Initial program 46.4%

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

   3. Simplified 45.1%

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

   5. Taylor expanded in t around 0 76.7%

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

      1. associate-*r* 76.7%

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

   7. Simplified 76.7%

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

4. Final simplification 73.8%

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

Alternative 9: 49.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 4.7 \cdot 10^{-144}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{2} \cdot \sqrt{\frac{\cos k}{t}}}{k \cdot \sin k}\right)}^{2}\\ \mathbf{elif}\;t \leq 2.35 \cdot 10^{+138}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t \cdot \frac{\sqrt[3]{k}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 4.7e-144)
   (pow (* l (/ (* (sqrt 2.0) (sqrt (/ (cos k) t))) (* k (sin k)))) 2.0)
   (if (<= t 2.35e+138)
     (/
      2.0
      (*
       (* (tan k) (+ 2.0 (pow (/ k t) 2.0)))
       (* (sin k) (* (/ (pow t 2.0) l) (/ t l)))))
     (/ 2.0 (* (pow (* t (/ (cbrt k) (pow (cbrt l) 2.0))) 3.0) (* 2.0 k))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 4.7e-144) {
		tmp = pow((l * ((sqrt(2.0) * sqrt((cos(k) / t))) / (k * sin(k)))), 2.0);
	} else if (t <= 2.35e+138) {
		tmp = 2.0 / ((tan(k) * (2.0 + pow((k / t), 2.0))) * (sin(k) * ((pow(t, 2.0) / l) * (t / l))));
	} else {
		tmp = 2.0 / (pow((t * (cbrt(k) / pow(cbrt(l), 2.0))), 3.0) * (2.0 * k));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 4.7e-144) {
		tmp = Math.pow((l * ((Math.sqrt(2.0) * Math.sqrt((Math.cos(k) / t))) / (k * Math.sin(k)))), 2.0);
	} else if (t <= 2.35e+138) {
		tmp = 2.0 / ((Math.tan(k) * (2.0 + Math.pow((k / t), 2.0))) * (Math.sin(k) * ((Math.pow(t, 2.0) / l) * (t / l))));
	} else {
		tmp = 2.0 / (Math.pow((t * (Math.cbrt(k) / Math.pow(Math.cbrt(l), 2.0))), 3.0) * (2.0 * k));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 4.7e-144)
		tmp = Float64(l * Float64(Float64(sqrt(2.0) * sqrt(Float64(cos(k) / t))) / Float64(k * sin(k)))) ^ 2.0;
	elseif (t <= 2.35e+138)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(2.0 + (Float64(k / t) ^ 2.0))) * Float64(sin(k) * Float64(Float64((t ^ 2.0) / l) * Float64(t / l)))));
	else
		tmp = Float64(2.0 / Float64((Float64(t * Float64(cbrt(k) / (cbrt(l) ^ 2.0))) ^ 3.0) * Float64(2.0 * k)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 4.7e-144], N[Power[N[(l * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[t, 2.35e+138], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(t * N[(N[Power[k, 1/3], $MachinePrecision] / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 4.7000000000000002e-144

   1. Initial program 47.8%

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

   3. Simplified 43.5%

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

      1. add-sqr-sqrt 28.7%

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

   6. Applied egg-rr 26.1%

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

      1. unpow2 26.1%

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

      2. associate-/l* 26.7%

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

      3. associate-/r* 26.7%

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

   8. Simplified 26.7%

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

   9. Taylor expanded in t around 0 38.1%

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

       1. associate-*l/ 38.2%

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

   11. Simplified 38.2%

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

   if 4.7000000000000002e-144 < t < 2.3499999999999999e138

   1. Initial program 54.3%

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

   3. Simplified 54.3%

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

      1. unpow3 54.2%

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

      2. times-frac 73.0%

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

      3. pow2 73.0%

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

   6. Applied egg-rr 73.0%

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

      1. associate-+r+ 82.6%

         \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin k}\right)\right)\right)}^{3} \cdot \left(\tan k \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}\right)} \]

      2. metadata-eval 82.6%

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

      3. distribute-lft-in 82.6%

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

   8. Applied egg-rr 73.0%

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

      1. distribute-lft-in 82.6%

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

   10. Simplified 73.0%

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

   if 2.3499999999999999e138 < 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. Taylor expanded in k around 0 67.1%

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

   6. Taylor expanded in k around 0 67.1%

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

      1. add-cube-cbrt 67.1%

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

      2. pow2 67.1%

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

      3. cbrt-div 67.1%

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

      4. *-commutative 67.1%

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

      5. cbrt-prod 67.1%

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

      6. unpow3 67.1%

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

      7. add-cbrt-cube 67.1%

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

      8. unpow2 67.1%

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

      9. cbrt-prod 67.1%

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

      10. unpow2 67.1%

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

      11. cbrt-div 67.1%

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

      12. *-commutative 67.1%

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

      13. cbrt-prod 67.1%

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

      14. unpow3 67.1%

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

      15. add-cbrt-cube 71.9%

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

      16. unpow2 71.9%

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

      17. cbrt-prod 86.7%

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

      18. unpow2 86.7%

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

   8. Applied egg-rr 86.7%

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

      1. unpow2 86.7%

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

      2. unpow3 86.7%

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

      3. associate-/l* 86.7%

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

   10. Simplified 86.7%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.7 \cdot 10^{-144}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{2} \cdot \sqrt{\frac{\cos k}{t}}}{k \cdot \sin k}\right)}^{2}\\ \mathbf{elif}\;t \leq 2.35 \cdot 10^{+138}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t \cdot \frac{\sqrt[3]{k}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 49.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 6 \cdot 10^{-144}:\\ \;\;\;\;{\left(\ell \cdot \left(\sqrt{\frac{\cos k}{t}} \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)\right)}^{2}\\ \mathbf{elif}\;t \leq 2.35 \cdot 10^{+138}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t \cdot \frac{\sqrt[3]{k}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 6e-144)
   (pow (* l (* (sqrt (/ (cos k) t)) (/ (sqrt 2.0) (* k (sin k))))) 2.0)
   (if (<= t 2.35e+138)
     (/
      2.0
      (*
       (* (tan k) (+ 2.0 (pow (/ k t) 2.0)))
       (* (sin k) (* (/ (pow t 2.0) l) (/ t l)))))
     (/ 2.0 (* (pow (* t (/ (cbrt k) (pow (cbrt l) 2.0))) 3.0) (* 2.0 k))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 6e-144) {
		tmp = pow((l * (sqrt((cos(k) / t)) * (sqrt(2.0) / (k * sin(k))))), 2.0);
	} else if (t <= 2.35e+138) {
		tmp = 2.0 / ((tan(k) * (2.0 + pow((k / t), 2.0))) * (sin(k) * ((pow(t, 2.0) / l) * (t / l))));
	} else {
		tmp = 2.0 / (pow((t * (cbrt(k) / pow(cbrt(l), 2.0))), 3.0) * (2.0 * k));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 6e-144) {
		tmp = Math.pow((l * (Math.sqrt((Math.cos(k) / t)) * (Math.sqrt(2.0) / (k * Math.sin(k))))), 2.0);
	} else if (t <= 2.35e+138) {
		tmp = 2.0 / ((Math.tan(k) * (2.0 + Math.pow((k / t), 2.0))) * (Math.sin(k) * ((Math.pow(t, 2.0) / l) * (t / l))));
	} else {
		tmp = 2.0 / (Math.pow((t * (Math.cbrt(k) / Math.pow(Math.cbrt(l), 2.0))), 3.0) * (2.0 * k));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 6e-144)
		tmp = Float64(l * Float64(sqrt(Float64(cos(k) / t)) * Float64(sqrt(2.0) / Float64(k * sin(k))))) ^ 2.0;
	elseif (t <= 2.35e+138)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(2.0 + (Float64(k / t) ^ 2.0))) * Float64(sin(k) * Float64(Float64((t ^ 2.0) / l) * Float64(t / l)))));
	else
		tmp = Float64(2.0 / Float64((Float64(t * Float64(cbrt(k) / (cbrt(l) ^ 2.0))) ^ 3.0) * Float64(2.0 * k)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 6e-144], N[Power[N[(l * N[(N[Sqrt[N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[t, 2.35e+138], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(t * N[(N[Power[k, 1/3], $MachinePrecision] / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 5.9999999999999997e-144

   1. Initial program 47.8%

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

   3. Simplified 43.5%

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

      1. add-sqr-sqrt 28.7%

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

   6. Applied egg-rr 26.1%

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

      1. unpow2 26.1%

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

      2. associate-/l* 26.7%

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

      3. associate-/r* 26.7%

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

   8. Simplified 26.7%

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

   9. Taylor expanded in t around 0 38.1%

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

   if 5.9999999999999997e-144 < t < 2.3499999999999999e138

   1. Initial program 54.3%

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

   3. Simplified 54.3%

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

      1. unpow3 54.2%

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

      2. times-frac 73.0%

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

      3. pow2 73.0%

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

   6. Applied egg-rr 73.0%

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

      1. associate-+r+ 82.6%

         \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin k}\right)\right)\right)}^{3} \cdot \left(\tan k \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}\right)} \]

      2. metadata-eval 82.6%

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

      3. distribute-lft-in 82.6%

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

   8. Applied egg-rr 73.0%

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

      1. distribute-lft-in 82.6%

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

   10. Simplified 73.0%

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

   if 2.3499999999999999e138 < 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. Taylor expanded in k around 0 67.1%

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

   6. Taylor expanded in k around 0 67.1%

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

      1. add-cube-cbrt 67.1%

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

      2. pow2 67.1%

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

      3. cbrt-div 67.1%

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

      4. *-commutative 67.1%

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

      5. cbrt-prod 67.1%

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

      6. unpow3 67.1%

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

      7. add-cbrt-cube 67.1%

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

      8. unpow2 67.1%

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

      9. cbrt-prod 67.1%

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

      10. unpow2 67.1%

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

      11. cbrt-div 67.1%

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

      12. *-commutative 67.1%

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

      13. cbrt-prod 67.1%

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

      14. unpow3 67.1%

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

      15. add-cbrt-cube 71.9%

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

      16. unpow2 71.9%

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

      17. cbrt-prod 86.7%

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

      18. unpow2 86.7%

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

   8. Applied egg-rr 86.7%

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

      1. unpow2 86.7%

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

      2. unpow3 86.7%

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

      3. associate-/l* 86.7%

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

   10. Simplified 86.7%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6 \cdot 10^{-144}:\\ \;\;\;\;{\left(\ell \cdot \left(\sqrt{\frac{\cos k}{t}} \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)\right)}^{2}\\ \mathbf{elif}\;t \leq 2.35 \cdot 10^{+138}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t \cdot \frac{\sqrt[3]{k}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2.1 \cdot 10^{-73}:\\ \;\;\;\;\frac{2}{\frac{t \cdot k}{{\ell}^{2}} \cdot \frac{{k}^{3}}{\cos k}}\\ \mathbf{elif}\;t \leq 2.35 \cdot 10^{+138}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t \cdot \frac{\sqrt[3]{k}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 2.1e-73)
   (/ 2.0 (* (/ (* t k) (pow l 2.0)) (/ (pow k 3.0) (cos k))))
   (if (<= t 2.35e+138)
     (/
      2.0
      (*
       (* (tan k) (+ 2.0 (pow (/ k t) 2.0)))
       (* (sin k) (* (/ (pow t 2.0) l) (/ t l)))))
     (/ 2.0 (* (pow (* t (/ (cbrt k) (pow (cbrt l) 2.0))) 3.0) (* 2.0 k))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 2.1e-73) {
		tmp = 2.0 / (((t * k) / pow(l, 2.0)) * (pow(k, 3.0) / cos(k)));
	} else if (t <= 2.35e+138) {
		tmp = 2.0 / ((tan(k) * (2.0 + pow((k / t), 2.0))) * (sin(k) * ((pow(t, 2.0) / l) * (t / l))));
	} else {
		tmp = 2.0 / (pow((t * (cbrt(k) / pow(cbrt(l), 2.0))), 3.0) * (2.0 * k));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 2.1e-73) {
		tmp = 2.0 / (((t * k) / Math.pow(l, 2.0)) * (Math.pow(k, 3.0) / Math.cos(k)));
	} else if (t <= 2.35e+138) {
		tmp = 2.0 / ((Math.tan(k) * (2.0 + Math.pow((k / t), 2.0))) * (Math.sin(k) * ((Math.pow(t, 2.0) / l) * (t / l))));
	} else {
		tmp = 2.0 / (Math.pow((t * (Math.cbrt(k) / Math.pow(Math.cbrt(l), 2.0))), 3.0) * (2.0 * k));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 2.1e-73)
		tmp = Float64(2.0 / Float64(Float64(Float64(t * k) / (l ^ 2.0)) * Float64((k ^ 3.0) / cos(k))));
	elseif (t <= 2.35e+138)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(2.0 + (Float64(k / t) ^ 2.0))) * Float64(sin(k) * Float64(Float64((t ^ 2.0) / l) * Float64(t / l)))));
	else
		tmp = Float64(2.0 / Float64((Float64(t * Float64(cbrt(k) / (cbrt(l) ^ 2.0))) ^ 3.0) * Float64(2.0 * k)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 2.1e-73], N[(2.0 / N[(N[(N[(t * k), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[k, 3.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.35e+138], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(t * N[(N[Power[k, 1/3], $MachinePrecision] / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 2.0999999999999999e-73

   1. Initial program 47.3%

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

   3. Simplified 47.3%

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

   5. Taylor expanded in k around 0 43.6%

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

   6. Taylor expanded in k around inf 51.9%

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

      1. *-commutative 51.9%

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

      2. times-frac 53.0%

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

   8. Simplified 53.0%

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

   9. Taylor expanded in k around 0 56.8%

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

   if 2.0999999999999999e-73 < t < 2.3499999999999999e138

   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. Step-by-step derivation

      1. unpow3 57.3%

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

      2. times-frac 76.4%

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

      3. pow2 76.4%

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

   6. Applied egg-rr 76.4%

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

      1. associate-+r+ 87.3%

         \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin k}\right)\right)\right)}^{3} \cdot \left(\tan k \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}\right)} \]

      2. metadata-eval 87.3%

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

      3. distribute-lft-in 87.3%

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

   8. Applied egg-rr 76.4%

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

      1. distribute-lft-in 87.3%

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

   10. Simplified 76.4%

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

   if 2.3499999999999999e138 < 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. Taylor expanded in k around 0 67.1%

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

   6. Taylor expanded in k around 0 67.1%

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

      1. add-cube-cbrt 67.1%

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

      2. pow2 67.1%

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

      3. cbrt-div 67.1%

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

      4. *-commutative 67.1%

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

      5. cbrt-prod 67.1%

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

      6. unpow3 67.1%

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

      7. add-cbrt-cube 67.1%

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

      8. unpow2 67.1%

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

      9. cbrt-prod 67.1%

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

      10. unpow2 67.1%

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

      11. cbrt-div 67.1%

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

      12. *-commutative 67.1%

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

      13. cbrt-prod 67.1%

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

      14. unpow3 67.1%

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

      15. add-cbrt-cube 71.9%

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

      16. unpow2 71.9%

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

      17. cbrt-prod 86.7%

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

      18. unpow2 86.7%

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

   8. Applied egg-rr 86.7%

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

      1. unpow2 86.7%

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

      2. unpow3 86.7%

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

      3. associate-/l* 86.7%

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

   10. Simplified 86.7%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.1 \cdot 10^{-73}:\\ \;\;\;\;\frac{2}{\frac{t \cdot k}{{\ell}^{2}} \cdot \frac{{k}^{3}}{\cos k}}\\ \mathbf{elif}\;t \leq 2.35 \cdot 10^{+138}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t \cdot \frac{\sqrt[3]{k}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 64.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 8.5 \cdot 10^{-87}:\\ \;\;\;\;\frac{2}{\frac{t \cdot k}{{\ell}^{2}} \cdot \frac{{k}^{3}}{\cos k}}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-27}:\\ \;\;\;\;\frac{2}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \sin k\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t \cdot \frac{\sqrt[3]{k}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 8.5e-87)
   (/ 2.0 (* (/ (* t k) (pow l 2.0)) (/ (pow k 3.0) (cos k))))
   (if (<= t 4.2e-27)
     (/
      2.0
      (/
       (* (/ (pow t 3.0) l) (* (+ 2.0 (pow (/ k t) 2.0)) (* (tan k) (sin k))))
       l))
     (/ 2.0 (* (pow (* t (/ (cbrt k) (pow (cbrt l) 2.0))) 3.0) (* 2.0 k))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 8.5e-87) {
		tmp = 2.0 / (((t * k) / pow(l, 2.0)) * (pow(k, 3.0) / cos(k)));
	} else if (t <= 4.2e-27) {
		tmp = 2.0 / (((pow(t, 3.0) / l) * ((2.0 + pow((k / t), 2.0)) * (tan(k) * sin(k)))) / l);
	} else {
		tmp = 2.0 / (pow((t * (cbrt(k) / pow(cbrt(l), 2.0))), 3.0) * (2.0 * k));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 8.5e-87) {
		tmp = 2.0 / (((t * k) / Math.pow(l, 2.0)) * (Math.pow(k, 3.0) / Math.cos(k)));
	} else if (t <= 4.2e-27) {
		tmp = 2.0 / (((Math.pow(t, 3.0) / l) * ((2.0 + Math.pow((k / t), 2.0)) * (Math.tan(k) * Math.sin(k)))) / l);
	} else {
		tmp = 2.0 / (Math.pow((t * (Math.cbrt(k) / Math.pow(Math.cbrt(l), 2.0))), 3.0) * (2.0 * k));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 8.5e-87)
		tmp = Float64(2.0 / Float64(Float64(Float64(t * k) / (l ^ 2.0)) * Float64((k ^ 3.0) / cos(k))));
	elseif (t <= 4.2e-27)
		tmp = Float64(2.0 / Float64(Float64(Float64((t ^ 3.0) / l) * Float64(Float64(2.0 + (Float64(k / t) ^ 2.0)) * Float64(tan(k) * sin(k)))) / l));
	else
		tmp = Float64(2.0 / Float64((Float64(t * Float64(cbrt(k) / (cbrt(l) ^ 2.0))) ^ 3.0) * Float64(2.0 * k)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 8.5e-87], N[(2.0 / N[(N[(N[(t * k), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[k, 3.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.2e-27], N[(2.0 / N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(t * N[(N[Power[k, 1/3], $MachinePrecision] / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 8.5000000000000001e-87

   1. Initial program 46.9%

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

   3. Simplified 46.9%

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

   5. Taylor expanded in k around 0 43.2%

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

   6. Taylor expanded in k around inf 51.7%

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

      1. *-commutative 51.7%

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

      2. times-frac 52.7%

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

   8. Simplified 52.7%

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

   9. Taylor expanded in k around 0 56.6%

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

   if 8.5000000000000001e-87 < t < 4.20000000000000031e-27

   1. Initial program 67.8%

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

   3. Simplified 67.8%

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

      1. associate-*l* 67.6%

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

      2. associate-/r* 67.6%

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

      3. associate-+r+ 67.6%

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

      4. metadata-eval 67.6%

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

      5. associate-*l* 67.5%

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

      6. associate-*l/ 74.1%

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

   6. Applied egg-rr 74.1%

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

   if 4.20000000000000031e-27 < t

   1. Initial program 61.5%

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

   3. Simplified 61.5%

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

   5. Taylor expanded in k around 0 62.4%

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

   6. Taylor expanded in k around 0 62.5%

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

      1. add-cube-cbrt 62.4%

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

      2. pow2 62.4%

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

      3. cbrt-div 62.4%

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

      4. *-commutative 62.4%

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

      5. cbrt-prod 62.4%

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

      6. unpow3 62.4%

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

      7. add-cbrt-cube 62.4%

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

      8. unpow2 62.4%

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

      9. cbrt-prod 62.5%

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

      10. unpow2 62.5%

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

      11. cbrt-div 62.4%

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

      12. *-commutative 62.4%

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

      13. cbrt-prod 62.4%

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

      14. unpow3 62.4%

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

      15. add-cbrt-cube 69.4%

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

      16. unpow2 69.4%

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

      17. cbrt-prod 82.6%

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

      18. unpow2 82.6%

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

   8. Applied egg-rr 82.7%

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

      1. unpow2 82.6%

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

      2. unpow3 82.6%

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

      3. associate-/l* 82.5%

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

   10. Simplified 82.6%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.5 \cdot 10^{-87}:\\ \;\;\;\;\frac{2}{\frac{t \cdot k}{{\ell}^{2}} \cdot \frac{{k}^{3}}{\cos k}}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-27}:\\ \;\;\;\;\frac{2}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \sin k\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t \cdot \frac{\sqrt[3]{k}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 64.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 7.5 \cdot 10^{-74}:\\ \;\;\;\;\frac{2}{\frac{t \cdot k}{{\ell}^{2}} \cdot \frac{{k}^{3}}{\cos k}}\\ \mathbf{elif}\;t \leq 9.78 \cdot 10^{-44}:\\ \;\;\;\;\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left(2 + \frac{k}{t} \cdot \frac{k}{t}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t \cdot \frac{\sqrt[3]{k}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 7.5e-74)
   (/ 2.0 (* (/ (* t k) (pow l 2.0)) (/ (pow k 3.0) (cos k))))
   (if (<= t 9.78e-44)
     (/
      2.0
      (*
       (/ (/ (pow t 3.0) l) l)
       (* (* (tan k) (sin k)) (+ 2.0 (* (/ k t) (/ k t))))))
     (/ 2.0 (* (pow (* t (/ (cbrt k) (pow (cbrt l) 2.0))) 3.0) (* 2.0 k))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 7.5e-74) {
		tmp = 2.0 / (((t * k) / pow(l, 2.0)) * (pow(k, 3.0) / cos(k)));
	} else if (t <= 9.78e-44) {
		tmp = 2.0 / (((pow(t, 3.0) / l) / l) * ((tan(k) * sin(k)) * (2.0 + ((k / t) * (k / t)))));
	} else {
		tmp = 2.0 / (pow((t * (cbrt(k) / pow(cbrt(l), 2.0))), 3.0) * (2.0 * k));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 7.5e-74) {
		tmp = 2.0 / (((t * k) / Math.pow(l, 2.0)) * (Math.pow(k, 3.0) / Math.cos(k)));
	} else if (t <= 9.78e-44) {
		tmp = 2.0 / (((Math.pow(t, 3.0) / l) / l) * ((Math.tan(k) * Math.sin(k)) * (2.0 + ((k / t) * (k / t)))));
	} else {
		tmp = 2.0 / (Math.pow((t * (Math.cbrt(k) / Math.pow(Math.cbrt(l), 2.0))), 3.0) * (2.0 * k));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 7.5e-74)
		tmp = Float64(2.0 / Float64(Float64(Float64(t * k) / (l ^ 2.0)) * Float64((k ^ 3.0) / cos(k))));
	elseif (t <= 9.78e-44)
		tmp = Float64(2.0 / Float64(Float64(Float64((t ^ 3.0) / l) / l) * Float64(Float64(tan(k) * sin(k)) * Float64(2.0 + Float64(Float64(k / t) * Float64(k / t))))));
	else
		tmp = Float64(2.0 / Float64((Float64(t * Float64(cbrt(k) / (cbrt(l) ^ 2.0))) ^ 3.0) * Float64(2.0 * k)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 7.5e-74], N[(2.0 / N[(N[(N[(t * k), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[k, 3.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 9.78e-44], N[(2.0 / N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(N[(k / t), $MachinePrecision] * N[(k / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(t * N[(N[Power[k, 1/3], $MachinePrecision] / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 7.5e-74

   1. Initial program 47.3%

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

   3. Simplified 47.3%

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

   5. Taylor expanded in k around 0 43.6%

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

   6. Taylor expanded in k around inf 51.9%

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

      1. *-commutative 51.9%

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

      2. times-frac 53.0%

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

   8. Simplified 53.0%

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

   9. Taylor expanded in k around 0 56.8%

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

   if 7.5e-74 < t < 9.77999999999999994e-44

   1. Initial program 61.8%

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

   3. Simplified 61.5%

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

      1. unpow2 61.5%

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

   6. Applied egg-rr 61.5%

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

   if 9.77999999999999994e-44 < t

   1. Initial program 62.5%

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

   3. Simplified 62.5%

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

   5. Taylor expanded in k around 0 62.2%

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

   6. Taylor expanded in k around 0 62.1%

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

      1. add-cube-cbrt 62.2%

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

      2. pow2 62.2%

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

      3. cbrt-div 62.2%

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

      4. *-commutative 62.2%

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

      5. cbrt-prod 62.2%

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

      6. unpow3 62.2%

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

      7. add-cbrt-cube 62.2%

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

      8. unpow2 62.2%

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

      9. cbrt-prod 62.2%

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

      10. unpow2 62.2%

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

      11. cbrt-div 62.2%

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

      12. *-commutative 62.2%

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

      13. cbrt-prod 62.2%

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

      14. unpow3 62.2%

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

      15. add-cbrt-cube 69.0%

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

      16. unpow2 69.0%

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

      17. cbrt-prod 81.8%

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

      18. unpow2 81.8%

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

   8. Applied egg-rr 81.8%

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

      1. unpow2 81.8%

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

      2. unpow3 81.8%

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

      3. associate-/l* 81.8%

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

   10. Simplified 81.7%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.5 \cdot 10^{-74}:\\ \;\;\;\;\frac{2}{\frac{t \cdot k}{{\ell}^{2}} \cdot \frac{{k}^{3}}{\cos k}}\\ \mathbf{elif}\;t \leq 9.78 \cdot 10^{-44}:\\ \;\;\;\;\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left(2 + \frac{k}{t} \cdot \frac{k}{t}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t \cdot \frac{\sqrt[3]{k}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 63.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 7 \cdot 10^{-74}:\\ \;\;\;\;\frac{2}{\frac{t \cdot k}{{\ell}^{2}} \cdot \frac{{k}^{3}}{\cos k}}\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{+43}:\\ \;\;\;\;\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left(2 + \frac{k}{t} \cdot \frac{k}{t}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right) \cdot \left(2 \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 7e-74)
   (/ 2.0 (* (/ (* t k) (pow l 2.0)) (/ (pow k 3.0) (cos k))))
   (if (<= t 3.5e+43)
     (/
      2.0
      (*
       (/ (/ (pow t 3.0) l) l)
       (* (* (tan k) (sin k)) (+ 2.0 (* (/ k t) (/ k t))))))
     (/ 2.0 (* (* (sin k) (pow (/ t (pow (cbrt l) 2.0)) 3.0)) (* 2.0 k))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 7e-74) {
		tmp = 2.0 / (((t * k) / pow(l, 2.0)) * (pow(k, 3.0) / cos(k)));
	} else if (t <= 3.5e+43) {
		tmp = 2.0 / (((pow(t, 3.0) / l) / l) * ((tan(k) * sin(k)) * (2.0 + ((k / t) * (k / t)))));
	} else {
		tmp = 2.0 / ((sin(k) * pow((t / pow(cbrt(l), 2.0)), 3.0)) * (2.0 * k));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 7e-74) {
		tmp = 2.0 / (((t * k) / Math.pow(l, 2.0)) * (Math.pow(k, 3.0) / Math.cos(k)));
	} else if (t <= 3.5e+43) {
		tmp = 2.0 / (((Math.pow(t, 3.0) / l) / l) * ((Math.tan(k) * Math.sin(k)) * (2.0 + ((k / t) * (k / t)))));
	} else {
		tmp = 2.0 / ((Math.sin(k) * Math.pow((t / Math.pow(Math.cbrt(l), 2.0)), 3.0)) * (2.0 * k));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 7e-74)
		tmp = Float64(2.0 / Float64(Float64(Float64(t * k) / (l ^ 2.0)) * Float64((k ^ 3.0) / cos(k))));
	elseif (t <= 3.5e+43)
		tmp = Float64(2.0 / Float64(Float64(Float64((t ^ 3.0) / l) / l) * Float64(Float64(tan(k) * sin(k)) * Float64(2.0 + Float64(Float64(k / t) * Float64(k / t))))));
	else
		tmp = Float64(2.0 / Float64(Float64(sin(k) * (Float64(t / (cbrt(l) ^ 2.0)) ^ 3.0)) * Float64(2.0 * k)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 7e-74], N[(2.0 / N[(N[(N[(t * k), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[k, 3.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.5e+43], N[(2.0 / N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(N[(k / t), $MachinePrecision] * N[(k / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[Power[N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 7.00000000000000029e-74

   1. Initial program 47.3%

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

   3. Simplified 47.3%

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

   5. Taylor expanded in k around 0 43.6%

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

   6. Taylor expanded in k around inf 51.9%

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

      1. *-commutative 51.9%

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

      2. times-frac 53.0%

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

   8. Simplified 53.0%

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

   9. Taylor expanded in k around 0 56.8%

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

   if 7.00000000000000029e-74 < t < 3.5000000000000001e43

   1. Initial program 66.5%

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

   3. Simplified 65.8%

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

      1. unpow2 65.8%

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

   6. Applied egg-rr 65.8%

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

   if 3.5000000000000001e43 < t

   1. Initial program 61.0%

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

   3. Simplified 61.0%

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

      1. add-cube-cbrt 61.0%

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

      2. pow3 61.0%

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

      3. cbrt-div 61.0%

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

      4. rem-cbrt-cube 69.2%

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

      5. cbrt-prod 89.9%

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

      6. pow2 89.9%

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

   6. Applied egg-rr 89.9%

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

   7. Taylor expanded in k around 0 80.0%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7 \cdot 10^{-74}:\\ \;\;\;\;\frac{2}{\frac{t \cdot k}{{\ell}^{2}} \cdot \frac{{k}^{3}}{\cos k}}\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{+43}:\\ \;\;\;\;\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left(2 + \frac{k}{t} \cdot \frac{k}{t}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right) \cdot \left(2 \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 62.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 7 \cdot 10^{-74}:\\ \;\;\;\;\frac{2}{\frac{t \cdot k}{{\ell}^{2}} \cdot \frac{{k}^{3}}{\cos k}}\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{+43}:\\ \;\;\;\;\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left(2 + \frac{k}{t} \cdot \frac{k}{t}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 7e-74)
   (/ 2.0 (* (/ (* t k) (pow l 2.0)) (/ (pow k 3.0) (cos k))))
   (if (<= t 3.5e+43)
     (/
      2.0
      (*
       (/ (/ (pow t 3.0) l) l)
       (* (* (tan k) (sin k)) (+ 2.0 (* (/ k t) (/ k t))))))
     (/ 2.0 (* (* 2.0 k) (* (sin k) (* (/ (pow t 2.0) l) (/ t l))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 7e-74) {
		tmp = 2.0 / (((t * k) / pow(l, 2.0)) * (pow(k, 3.0) / cos(k)));
	} else if (t <= 3.5e+43) {
		tmp = 2.0 / (((pow(t, 3.0) / l) / l) * ((tan(k) * sin(k)) * (2.0 + ((k / t) * (k / t)))));
	} else {
		tmp = 2.0 / ((2.0 * k) * (sin(k) * ((pow(t, 2.0) / l) * (t / l))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= 7d-74) then
        tmp = 2.0d0 / (((t * k) / (l ** 2.0d0)) * ((k ** 3.0d0) / cos(k)))
    else if (t <= 3.5d+43) then
        tmp = 2.0d0 / ((((t ** 3.0d0) / l) / l) * ((tan(k) * sin(k)) * (2.0d0 + ((k / t) * (k / t)))))
    else
        tmp = 2.0d0 / ((2.0d0 * k) * (sin(k) * (((t ** 2.0d0) / l) * (t / l))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 7e-74) {
		tmp = 2.0 / (((t * k) / Math.pow(l, 2.0)) * (Math.pow(k, 3.0) / Math.cos(k)));
	} else if (t <= 3.5e+43) {
		tmp = 2.0 / (((Math.pow(t, 3.0) / l) / l) * ((Math.tan(k) * Math.sin(k)) * (2.0 + ((k / t) * (k / t)))));
	} else {
		tmp = 2.0 / ((2.0 * k) * (Math.sin(k) * ((Math.pow(t, 2.0) / l) * (t / l))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 7e-74:
		tmp = 2.0 / (((t * k) / math.pow(l, 2.0)) * (math.pow(k, 3.0) / math.cos(k)))
	elif t <= 3.5e+43:
		tmp = 2.0 / (((math.pow(t, 3.0) / l) / l) * ((math.tan(k) * math.sin(k)) * (2.0 + ((k / t) * (k / t)))))
	else:
		tmp = 2.0 / ((2.0 * k) * (math.sin(k) * ((math.pow(t, 2.0) / l) * (t / l))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 7e-74)
		tmp = Float64(2.0 / Float64(Float64(Float64(t * k) / (l ^ 2.0)) * Float64((k ^ 3.0) / cos(k))));
	elseif (t <= 3.5e+43)
		tmp = Float64(2.0 / Float64(Float64(Float64((t ^ 3.0) / l) / l) * Float64(Float64(tan(k) * sin(k)) * Float64(2.0 + Float64(Float64(k / t) * Float64(k / t))))));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * Float64(sin(k) * Float64(Float64((t ^ 2.0) / l) * Float64(t / l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 7e-74)
		tmp = 2.0 / (((t * k) / (l ^ 2.0)) * ((k ^ 3.0) / cos(k)));
	elseif (t <= 3.5e+43)
		tmp = 2.0 / ((((t ^ 3.0) / l) / l) * ((tan(k) * sin(k)) * (2.0 + ((k / t) * (k / t)))));
	else
		tmp = 2.0 / ((2.0 * k) * (sin(k) * (((t ^ 2.0) / l) * (t / l))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 7e-74], N[(2.0 / N[(N[(N[(t * k), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[k, 3.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.5e+43], N[(2.0 / N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(N[(k / t), $MachinePrecision] * N[(k / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 7.00000000000000029e-74

   1. Initial program 47.3%

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

   3. Simplified 47.3%

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

   5. Taylor expanded in k around 0 43.6%

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

   6. Taylor expanded in k around inf 51.9%

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

      1. *-commutative 51.9%

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

      2. times-frac 53.0%

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

   8. Simplified 53.0%

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

   9. Taylor expanded in k around 0 56.8%

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

   if 7.00000000000000029e-74 < t < 3.5000000000000001e43

   1. Initial program 66.5%

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

   3. Simplified 65.8%

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

      1. unpow2 65.8%

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

   6. Applied egg-rr 65.8%

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

   if 3.5000000000000001e43 < t

   1. Initial program 61.0%

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

   3. Simplified 61.0%

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

      1. unpow3 61.0%

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

      2. times-frac 78.6%

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

      3. pow2 78.6%

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

   6. Applied egg-rr 78.6%

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

   7. Taylor expanded in k around 0 74.0%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7 \cdot 10^{-74}:\\ \;\;\;\;\frac{2}{\frac{t \cdot k}{{\ell}^{2}} \cdot \frac{{k}^{3}}{\cos k}}\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{+43}:\\ \;\;\;\;\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left(2 + \frac{k}{t} \cdot \frac{k}{t}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 61.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 3.1 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\frac{t \cdot k}{{\ell}^{2}} \cdot \frac{{k}^{3}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 3.1e-18)
   (/ 2.0 (* (/ (* t k) (pow l 2.0)) (/ (pow k 3.0) (cos k))))
   (/ 2.0 (* (* 2.0 k) (* (sin k) (* (/ (pow t 2.0) l) (/ t l)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 3.1e-18) {
		tmp = 2.0 / (((t * k) / pow(l, 2.0)) * (pow(k, 3.0) / cos(k)));
	} else {
		tmp = 2.0 / ((2.0 * k) * (sin(k) * ((pow(t, 2.0) / l) * (t / l))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= 3.1d-18) then
        tmp = 2.0d0 / (((t * k) / (l ** 2.0d0)) * ((k ** 3.0d0) / cos(k)))
    else
        tmp = 2.0d0 / ((2.0d0 * k) * (sin(k) * (((t ** 2.0d0) / l) * (t / l))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 3.1e-18) {
		tmp = 2.0 / (((t * k) / Math.pow(l, 2.0)) * (Math.pow(k, 3.0) / Math.cos(k)));
	} else {
		tmp = 2.0 / ((2.0 * k) * (Math.sin(k) * ((Math.pow(t, 2.0) / l) * (t / l))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 3.1e-18:
		tmp = 2.0 / (((t * k) / math.pow(l, 2.0)) * (math.pow(k, 3.0) / math.cos(k)))
	else:
		tmp = 2.0 / ((2.0 * k) * (math.sin(k) * ((math.pow(t, 2.0) / l) * (t / l))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 3.1e-18)
		tmp = Float64(2.0 / Float64(Float64(Float64(t * k) / (l ^ 2.0)) * Float64((k ^ 3.0) / cos(k))));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * Float64(sin(k) * Float64(Float64((t ^ 2.0) / l) * Float64(t / l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 3.1e-18)
		tmp = 2.0 / (((t * k) / (l ^ 2.0)) * ((k ^ 3.0) / cos(k)));
	else
		tmp = 2.0 / ((2.0 * k) * (sin(k) * (((t ^ 2.0) / l) * (t / l))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 3.1e-18], N[(2.0 / N[(N[(N[(t * k), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[k, 3.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.10000000000000007e-18

   1. Initial program 49.2%

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

   3. Simplified 49.2%

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

   5. Taylor expanded in k around 0 45.3%

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

   6. Taylor expanded in k around inf 52.5%

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

      1. *-commutative 52.5%

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

      2. times-frac 53.5%

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

   8. Simplified 53.5%

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

   9. Taylor expanded in k around 0 57.0%

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

   if 3.10000000000000007e-18 < t

   1. Initial program 60.5%

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

   3. Simplified 60.5%

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

      1. unpow3 60.5%

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

      2. times-frac 75.8%

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

      3. pow2 75.8%

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

   6. Applied egg-rr 75.8%

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

   7. Taylor expanded in k around 0 70.6%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.1 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\frac{t \cdot k}{{\ell}^{2}} \cdot \frac{{k}^{3}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 61.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.85 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\frac{{k}^{3}}{\cos k} \cdot \left(k \cdot \frac{t}{{\ell}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 1.85e-18)
   (/ 2.0 (* (/ (pow k 3.0) (cos k)) (* k (/ t (pow l 2.0)))))
   (/ 2.0 (* (* 2.0 k) (* (sin k) (* (/ (pow t 2.0) l) (/ t l)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 1.85e-18) {
		tmp = 2.0 / ((pow(k, 3.0) / cos(k)) * (k * (t / pow(l, 2.0))));
	} else {
		tmp = 2.0 / ((2.0 * k) * (sin(k) * ((pow(t, 2.0) / l) * (t / l))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= 1.85d-18) then
        tmp = 2.0d0 / (((k ** 3.0d0) / cos(k)) * (k * (t / (l ** 2.0d0))))
    else
        tmp = 2.0d0 / ((2.0d0 * k) * (sin(k) * (((t ** 2.0d0) / l) * (t / l))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 1.85e-18) {
		tmp = 2.0 / ((Math.pow(k, 3.0) / Math.cos(k)) * (k * (t / Math.pow(l, 2.0))));
	} else {
		tmp = 2.0 / ((2.0 * k) * (Math.sin(k) * ((Math.pow(t, 2.0) / l) * (t / l))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 1.85e-18:
		tmp = 2.0 / ((math.pow(k, 3.0) / math.cos(k)) * (k * (t / math.pow(l, 2.0))))
	else:
		tmp = 2.0 / ((2.0 * k) * (math.sin(k) * ((math.pow(t, 2.0) / l) * (t / l))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 1.85e-18)
		tmp = Float64(2.0 / Float64(Float64((k ^ 3.0) / cos(k)) * Float64(k * Float64(t / (l ^ 2.0)))));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * Float64(sin(k) * Float64(Float64((t ^ 2.0) / l) * Float64(t / l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 1.85e-18)
		tmp = 2.0 / (((k ^ 3.0) / cos(k)) * (k * (t / (l ^ 2.0))));
	else
		tmp = 2.0 / ((2.0 * k) * (sin(k) * (((t ^ 2.0) / l) * (t / l))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 1.85e-18], N[(2.0 / N[(N[(N[Power[k, 3.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(k * N[(t / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.8500000000000002e-18

   1. Initial program 49.2%

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

   3. Simplified 49.2%

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

   5. Taylor expanded in k around 0 45.3%

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

   6. Taylor expanded in k around inf 52.5%

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

      1. *-commutative 52.5%

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

      2. times-frac 53.5%

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

   8. Simplified 53.5%

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

   9. Taylor expanded in k around 0 57.0%

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

       1. associate-/l* 56.9%

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

   11. Simplified 56.9%

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

   if 1.8500000000000002e-18 < t

   1. Initial program 60.5%

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

   3. Simplified 60.5%

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

      1. unpow3 60.5%

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

      2. times-frac 75.8%

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

      3. pow2 75.8%

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

   6. Applied egg-rr 75.8%

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

   7. Taylor expanded in k around 0 70.6%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.85 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\frac{{k}^{3}}{\cos k} \cdot \left(k \cdot \frac{t}{{\ell}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 63.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4.4 \cdot 10^{+73}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left({\ell}^{2} \cdot {k}^{-4}\right) \cdot \frac{2}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 4.4e+73)
   (/ 2.0 (* (* 2.0 k) (* (sin k) (* (/ (pow t 2.0) l) (/ t l)))))
   (* (* (pow l 2.0) (pow k -4.0)) (/ 2.0 t))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 4.4e+73) {
		tmp = 2.0 / ((2.0 * k) * (sin(k) * ((pow(t, 2.0) / l) * (t / l))));
	} else {
		tmp = (pow(l, 2.0) * pow(k, -4.0)) * (2.0 / t);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 4.4d+73) then
        tmp = 2.0d0 / ((2.0d0 * k) * (sin(k) * (((t ** 2.0d0) / l) * (t / l))))
    else
        tmp = ((l ** 2.0d0) * (k ** (-4.0d0))) * (2.0d0 / t)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 4.4e+73) {
		tmp = 2.0 / ((2.0 * k) * (Math.sin(k) * ((Math.pow(t, 2.0) / l) * (t / l))));
	} else {
		tmp = (Math.pow(l, 2.0) * Math.pow(k, -4.0)) * (2.0 / t);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 4.4e+73:
		tmp = 2.0 / ((2.0 * k) * (math.sin(k) * ((math.pow(t, 2.0) / l) * (t / l))))
	else:
		tmp = (math.pow(l, 2.0) * math.pow(k, -4.0)) * (2.0 / t)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 4.4e+73)
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * Float64(sin(k) * Float64(Float64((t ^ 2.0) / l) * Float64(t / l)))));
	else
		tmp = Float64(Float64((l ^ 2.0) * (k ^ -4.0)) * Float64(2.0 / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 4.4e+73)
		tmp = 2.0 / ((2.0 * k) * (sin(k) * (((t ^ 2.0) / l) * (t / l))));
	else
		tmp = ((l ^ 2.0) * (k ^ -4.0)) * (2.0 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 4.4e+73], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Power[k, -4.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.4e73

   1. Initial program 54.5%

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

   3. Simplified 54.5%

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

      1. unpow3 54.6%

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

      2. times-frac 64.7%

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

      3. pow2 64.7%

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

   6. Applied egg-rr 64.7%

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

   7. Taylor expanded in k around 0 60.9%

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

   if 4.4e73 < k

   1. Initial program 45.0%

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

   3. Simplified 45.0%

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

   5. Taylor expanded in k around 0 40.2%

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

   6. Taylor expanded in k around inf 62.2%

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

      1. *-commutative 62.2%

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

      2. times-frac 62.0%

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

   8. Simplified 62.0%

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

   9. Taylor expanded in k around 0 62.2%

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

       1. associate-*r/ 62.2%

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

       2. *-commutative 62.2%

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

       3. times-frac 63.9%

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

   11. Simplified 63.9%

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

       1. pow2 63.9%

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

       2. div-inv 63.9%

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

       3. pow2 63.9%

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

       4. pow-flip 63.9%

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

       5. metadata-eval 63.9%

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

   13. Applied egg-rr 63.9%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.4 \cdot 10^{+73}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left({\ell}^{2} \cdot {k}^{-4}\right) \cdot \frac{2}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 57.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.7 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{2 \cdot {\ell}^{2}}{{k}^{4}}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \frac{k \cdot {t}^{3}}{{\ell}^{2}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 1.7e-32)
   (/ (/ (* 2.0 (pow l 2.0)) (pow k 4.0)) t)
   (/ 2.0 (* (* 2.0 k) (/ (* k (pow t 3.0)) (pow l 2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 1.7e-32) {
		tmp = ((2.0 * pow(l, 2.0)) / pow(k, 4.0)) / t;
	} else {
		tmp = 2.0 / ((2.0 * k) * ((k * pow(t, 3.0)) / pow(l, 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= 1.7d-32) then
        tmp = ((2.0d0 * (l ** 2.0d0)) / (k ** 4.0d0)) / t
    else
        tmp = 2.0d0 / ((2.0d0 * k) * ((k * (t ** 3.0d0)) / (l ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 1.7e-32) {
		tmp = ((2.0 * Math.pow(l, 2.0)) / Math.pow(k, 4.0)) / t;
	} else {
		tmp = 2.0 / ((2.0 * k) * ((k * Math.pow(t, 3.0)) / Math.pow(l, 2.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 1.7e-32:
		tmp = ((2.0 * math.pow(l, 2.0)) / math.pow(k, 4.0)) / t
	else:
		tmp = 2.0 / ((2.0 * k) * ((k * math.pow(t, 3.0)) / math.pow(l, 2.0)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 1.7e-32)
		tmp = Float64(Float64(Float64(2.0 * (l ^ 2.0)) / (k ^ 4.0)) / t);
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * Float64(Float64(k * (t ^ 3.0)) / (l ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 1.7e-32)
		tmp = ((2.0 * (l ^ 2.0)) / (k ^ 4.0)) / t;
	else
		tmp = 2.0 / ((2.0 * k) * ((k * (t ^ 3.0)) / (l ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 1.7e-32], N[(N[(N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[(N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.69999999999999989e-32

   1. Initial program 48.4%

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

   3. Simplified 48.4%

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

   5. Taylor expanded in k around 0 44.4%

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

   6. Taylor expanded in k around inf 51.7%

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

      1. *-commutative 51.7%

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

      2. times-frac 52.7%

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

   8. Simplified 52.7%

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

   9. Taylor expanded in k around 0 51.0%

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

       1. associate-*r/ 51.0%

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

       2. *-commutative 51.0%

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

       3. times-frac 52.7%

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

   11. Simplified 52.7%

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

   12. Taylor expanded in l around 0 51.0%

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

       1. associate-*r/ 51.0%

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

       2. associate-/r* 52.7%

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

   14. Simplified 52.7%

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

   if 1.69999999999999989e-32 < t

   1. Initial program 62.1%

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

   3. Simplified 62.1%

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

   5. Taylor expanded in k around 0 62.9%

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

   6. Taylor expanded in k around 0 63.0%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.7 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{2 \cdot {\ell}^{2}}{{k}^{4}}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \frac{k \cdot {t}^{3}}{{\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 57.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2.05 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{2 \cdot {\ell}^{2}}{{k}^{4}}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot {t}^{3}\right) \cdot \frac{2 \cdot k}{{\ell}^{2}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 2.05e-32)
   (/ (/ (* 2.0 (pow l 2.0)) (pow k 4.0)) t)
   (/ 2.0 (* (* k (pow t 3.0)) (/ (* 2.0 k) (pow l 2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 2.05e-32) {
		tmp = ((2.0 * pow(l, 2.0)) / pow(k, 4.0)) / t;
	} else {
		tmp = 2.0 / ((k * pow(t, 3.0)) * ((2.0 * k) / pow(l, 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= 2.05d-32) then
        tmp = ((2.0d0 * (l ** 2.0d0)) / (k ** 4.0d0)) / t
    else
        tmp = 2.0d0 / ((k * (t ** 3.0d0)) * ((2.0d0 * k) / (l ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 2.05e-32) {
		tmp = ((2.0 * Math.pow(l, 2.0)) / Math.pow(k, 4.0)) / t;
	} else {
		tmp = 2.0 / ((k * Math.pow(t, 3.0)) * ((2.0 * k) / Math.pow(l, 2.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 2.05e-32:
		tmp = ((2.0 * math.pow(l, 2.0)) / math.pow(k, 4.0)) / t
	else:
		tmp = 2.0 / ((k * math.pow(t, 3.0)) * ((2.0 * k) / math.pow(l, 2.0)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 2.05e-32)
		tmp = Float64(Float64(Float64(2.0 * (l ^ 2.0)) / (k ^ 4.0)) / t);
	else
		tmp = Float64(2.0 / Float64(Float64(k * (t ^ 3.0)) * Float64(Float64(2.0 * k) / (l ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 2.05e-32)
		tmp = ((2.0 * (l ^ 2.0)) / (k ^ 4.0)) / t;
	else
		tmp = 2.0 / ((k * (t ^ 3.0)) * ((2.0 * k) / (l ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 2.05e-32], N[(N[(N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], N[(2.0 / N[(N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * k), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.04999999999999988e-32

   1. Initial program 48.4%

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

   3. Simplified 48.4%

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

   5. Taylor expanded in k around 0 44.4%

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

   6. Taylor expanded in k around inf 51.7%

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

      1. *-commutative 51.7%

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

      2. times-frac 52.7%

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

   8. Simplified 52.7%

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

   9. Taylor expanded in k around 0 51.0%

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

       1. associate-*r/ 51.0%

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

       2. *-commutative 51.0%

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

       3. times-frac 52.7%

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

   11. Simplified 52.7%

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

   12. Taylor expanded in l around 0 51.0%

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

       1. associate-*r/ 51.0%

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

       2. associate-/r* 52.7%

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

   14. Simplified 52.7%

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

   if 2.04999999999999988e-32 < t

   1. Initial program 62.1%

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

   3. Simplified 62.1%

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

   5. Taylor expanded in k around 0 62.9%

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

   6. Taylor expanded in k around 0 63.0%

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

      1. associate-*l/ 62.0%

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

   8. Applied egg-rr 62.0%

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

      1. associate-/l* 61.5%

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

      2. *-commutative 61.5%

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

   10. Simplified 61.5%

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

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.05 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{2 \cdot {\ell}^{2}}{{k}^{4}}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot {t}^{3}\right) \cdot \frac{2 \cdot k}{{\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 53.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\frac{t}{{\ell}^{2}} \cdot {k}^{4}} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (/ 2.0 (* (/ t (pow l 2.0)) (pow k 4.0))))

C

double code(double t, double l, double k) {
	return 2.0 / ((t / pow(l, 2.0)) * pow(k, 4.0));
}

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 / (l ** 2.0d0)) * (k ** 4.0d0))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / ((t / Math.pow(l, 2.0)) * Math.pow(k, 4.0));
}

Python

def code(t, l, k):
	return 2.0 / ((t / math.pow(l, 2.0)) * math.pow(k, 4.0))

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(t / (l ^ 2.0)) * (k ^ 4.0)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / ((t / (l ^ 2.0)) * (k ^ 4.0));
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(t / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.4%

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

3. Simplified 52.4%

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

5. Taylor expanded in k around 0 49.8%

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

6. Taylor expanded in k around inf 50.2%

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

   1. *-commutative 50.2%

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

   2. times-frac 50.9%

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

8. Simplified 50.9%

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

9. Taylor expanded in k around 0 49.7%

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

    1. associate-/l* 51.2%

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

11. Simplified 51.2%

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

12. Final simplification 51.2%

    \[\leadsto \frac{2}{\frac{t}{{\ell}^{2}} \cdot {k}^{4}} \]
13. Add Preprocessing

Alternative 22: 53.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ {k}^{-4} \cdot \left({\ell}^{2} \cdot \frac{2}{t}\right) \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* (pow k -4.0) (* (pow l 2.0) (/ 2.0 t))))

C

double code(double t, double l, double k) {
	return pow(k, -4.0) * (pow(l, 2.0) * (2.0 / t));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (k ** (-4.0d0)) * ((l ** 2.0d0) * (2.0d0 / t))
end function

Java

public static double code(double t, double l, double k) {
	return Math.pow(k, -4.0) * (Math.pow(l, 2.0) * (2.0 / t));
}

Python

def code(t, l, k):
	return math.pow(k, -4.0) * (math.pow(l, 2.0) * (2.0 / t))

Julia

function code(t, l, k)
	return Float64((k ^ -4.0) * Float64((l ^ 2.0) * Float64(2.0 / t)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (k ^ -4.0) * ((l ^ 2.0) * (2.0 / t));
end

Wolfram

code[t_, l_, k_] := N[(N[Power[k, -4.0], $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] * N[(2.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.4%

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

3. Simplified 52.4%

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

5. Taylor expanded in k around 0 49.8%

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

6. Taylor expanded in k around inf 50.2%

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

   1. *-commutative 50.2%

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

   2. times-frac 50.9%

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

8. Simplified 50.9%

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

9. Taylor expanded in k around 0 49.7%

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

    1. associate-*r/ 49.7%

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

    2. *-commutative 49.7%

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

    3. times-frac 50.6%

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

11. Simplified 50.6%

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

    1. associate-*r/ 50.6%

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

    2. pow2 50.6%

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

    3. div-inv 50.6%

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

    4. pow2 50.6%

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

    5. pow-flip 50.6%

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

    6. metadata-eval 50.6%

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

13. Applied egg-rr 50.6%

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

    1. associate-/l* 50.6%

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

    2. *-commutative 50.6%

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

    3. associate-*r* 50.8%

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

    4. *-commutative 50.8%

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

15. Simplified 50.8%

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

16. Final simplification 50.8%

    \[\leadsto {k}^{-4} \cdot \left({\ell}^{2} \cdot \frac{2}{t}\right) \]
17. Add Preprocessing

Reproduce

herbie shell --seed 2024110 
(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))))