Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.2% → 87.3%

Time: 17.9s

Alternatives: 18

Speedup: 3.8×

• 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: 55.2% 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: 87.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k\_m}{t}\right)\right)}\\ \mathbf{if}\;k\_m \leq 9.5 \cdot 10^{-16}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot {\left(\sqrt[3]{k\_m}\right)}^{2}}\right)}^{3}\\ \mathbf{elif}\;k\_m \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \frac{\ell \cdot \cos k\_m}{{k\_m}^{2} \cdot \left(t \cdot {\sin k\_m}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot {\left(\frac{\sqrt[3]{2 \cdot t\_1}}{t \cdot \sqrt[3]{\sin k\_m \cdot \tan k\_m}}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ l (hypot 1.0 (hypot 1.0 (/ k_m t))))))
   (if (<= k_m 9.5e-16)
     (pow (/ (pow (cbrt l) 2.0) (* t (pow (cbrt k_m) 2.0))) 3.0)
     (if (<= k_m 1.32e+154)
       (*
        l
        (*
         2.0
         (/ (* l (cos k_m)) (* (pow k_m 2.0) (* t (pow (sin k_m) 2.0))))))
       (*
        t_1
        (pow
         (/ (cbrt (* 2.0 t_1)) (* t (cbrt (* (sin k_m) (tan k_m)))))
         3.0))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = l / hypot(1.0, hypot(1.0, (k_m / t)));
	double tmp;
	if (k_m <= 9.5e-16) {
		tmp = pow((pow(cbrt(l), 2.0) / (t * pow(cbrt(k_m), 2.0))), 3.0);
	} else if (k_m <= 1.32e+154) {
		tmp = l * (2.0 * ((l * cos(k_m)) / (pow(k_m, 2.0) * (t * pow(sin(k_m), 2.0)))));
	} else {
		tmp = t_1 * pow((cbrt((2.0 * t_1)) / (t * cbrt((sin(k_m) * tan(k_m))))), 3.0);
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = l / Math.hypot(1.0, Math.hypot(1.0, (k_m / t)));
	double tmp;
	if (k_m <= 9.5e-16) {
		tmp = Math.pow((Math.pow(Math.cbrt(l), 2.0) / (t * Math.pow(Math.cbrt(k_m), 2.0))), 3.0);
	} else if (k_m <= 1.32e+154) {
		tmp = l * (2.0 * ((l * Math.cos(k_m)) / (Math.pow(k_m, 2.0) * (t * Math.pow(Math.sin(k_m), 2.0)))));
	} else {
		tmp = t_1 * Math.pow((Math.cbrt((2.0 * t_1)) / (t * Math.cbrt((Math.sin(k_m) * Math.tan(k_m))))), 3.0);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(l / hypot(1.0, hypot(1.0, Float64(k_m / t))))
	tmp = 0.0
	if (k_m <= 9.5e-16)
		tmp = Float64((cbrt(l) ^ 2.0) / Float64(t * (cbrt(k_m) ^ 2.0))) ^ 3.0;
	elseif (k_m <= 1.32e+154)
		tmp = Float64(l * Float64(2.0 * Float64(Float64(l * cos(k_m)) / Float64((k_m ^ 2.0) * Float64(t * (sin(k_m) ^ 2.0))))));
	else
		tmp = Float64(t_1 * (Float64(cbrt(Float64(2.0 * t_1)) / Float64(t * cbrt(Float64(sin(k_m) * tan(k_m))))) ^ 3.0));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(l / N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k$95$m / t), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 9.5e-16], N[Power[N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / N[(t * N[Power[N[Power[k$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], If[LessEqual[k$95$m, 1.32e+154], N[(l * N[(2.0 * N[(N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Power[N[(N[Power[N[(2.0 * t$95$1), $MachinePrecision], 1/3], $MachinePrecision] / N[(t * N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 9.5000000000000005e-16

   1. Initial program 57.1%

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

      1. associate-/r* 57.1%

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

      2. unpow2 57.1%

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

      3. sqr-neg 57.1%

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

      4. distribute-frac-neg2 57.1%

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

      5. distribute-frac-neg2 57.1%

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

      6. unpow2 57.1%

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

      7. +-commutative 57.1%

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

      8. +-commutative 57.1%

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

      9. associate-/l/ 57.1%

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

      10. associate-*l/ 58.0%

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

      11. associate-/r/ 56.5%

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

   3. Simplified 56.5%

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

      1. associate-*r* 64.4%

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

      2. *-un-lft-identity 64.4%

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

      3. times-frac 65.0%

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

      4. associate-/l/ 65.0%

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

   6. Applied egg-rr 65.0%

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

      1. times-frac 64.5%

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

      2. *-commutative 64.5%

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

      3. times-frac 65.0%

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

      4. associate-*l/ 65.5%

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

      5. associate-*l* 60.1%

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

      6. times-frac 60.0%

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

      7. /-rgt-identity 60.0%

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

   8. Simplified 60.0%

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

   9. Taylor expanded in k around 0 58.8%

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

       1. add-cube-cbrt 58.7%

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

       2. pow3 58.7%

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

       3. associate-*l/ 51.8%

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

       4. cbrt-div 51.8%

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

       5. cbrt-prod 58.7%

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

       6. unpow2 58.7%

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

       7. *-commutative 58.7%

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

       8. cbrt-prod 59.2%

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

       9. unpow3 59.2%

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

       10. add-cbrt-cube 71.2%

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

       11. unpow2 71.2%

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

       12. cbrt-prod 82.6%

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

       13. pow2 82.6%

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

   11. Applied egg-rr 82.6%

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

   if 9.5000000000000005e-16 < k < 1.31999999999999998e154

   1. Initial program 39.0%

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

      1. associate-/r* 39.0%

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

      2. unpow2 39.0%

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

      3. sqr-neg 39.0%

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

      4. distribute-frac-neg2 39.0%

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

      5. distribute-frac-neg2 39.0%

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

      6. unpow2 39.0%

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

      7. +-commutative 39.0%

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

      8. +-commutative 39.0%

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

      9. associate-/l/ 39.0%

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

      10. associate-*l/ 39.1%

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

      11. associate-/r/ 39.0%

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

   3. Simplified 39.0%

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

      1. associate-*r* 39.3%

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

      2. *-un-lft-identity 39.3%

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

      3. times-frac 41.8%

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

      4. associate-/l/ 41.8%

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

   6. Applied egg-rr 41.8%

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

      1. times-frac 39.3%

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

      2. *-commutative 39.3%

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

      3. times-frac 41.8%

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

      4. associate-*l/ 41.8%

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

      5. associate-*l* 41.9%

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

      6. times-frac 41.9%

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

      7. /-rgt-identity 41.9%

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

   8. Simplified 41.9%

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

   9. Taylor expanded in t around 0 86.4%

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

   if 1.31999999999999998e154 < k

   1. Initial program 36.1%

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

      1. associate-/r* 36.1%

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

      2. unpow2 36.1%

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

      3. sqr-neg 36.1%

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

      4. distribute-frac-neg2 36.1%

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

      5. distribute-frac-neg2 36.1%

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

      6. unpow2 36.1%

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

      7. +-commutative 36.1%

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

      8. +-commutative 36.1%

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

      9. associate-/l/ 36.1%

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

      10. associate-*l/ 36.1%

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

      11. associate-/r/ 36.1%

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

   3. Simplified 36.1%

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

      1. associate-*r* 41.3%

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

      2. add-sqr-sqrt 41.3%

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

      3. times-frac 41.5%

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

   6. Applied egg-rr 52.7%

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

      1. associate-/l* 52.7%

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

      2. associate-*l* 52.7%

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

   8. Simplified 52.7%

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

      1. add-cube-cbrt 52.7%

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

      2. pow3 52.7%

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

      3. associate-*l/ 52.6%

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

      4. cbrt-div 52.6%

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

      5. cbrt-prod 52.5%

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

      6. unpow3 52.4%

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

      7. add-cbrt-cube 83.1%

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

   10. Applied egg-rr 83.1%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.5 \cdot 10^{-16}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot {\left(\sqrt[3]{k}\right)}^{2}}\right)}^{3}\\ \mathbf{elif}\;k \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \frac{\ell \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot {\left(\frac{\sqrt[3]{2 \cdot \frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}}}{t \cdot \sqrt[3]{\sin k \cdot \tan k}}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k\_m}{t}\right)\right)}\\ \mathbf{if}\;t \leq 5.5 \cdot 10^{-72}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \left(\frac{\ell}{{k\_m}^{2}} \cdot \frac{\cos k\_m}{t \cdot {\sin k\_m}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot {\left(\frac{\sqrt[3]{2 \cdot t\_1}}{t \cdot \left(\sqrt[3]{\tan k\_m} \cdot \sqrt[3]{\sin k\_m}\right)}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ l (hypot 1.0 (hypot 1.0 (/ k_m t))))))
   (if (<= t 5.5e-72)
     (*
      l
      (* 2.0 (* (/ l (pow k_m 2.0)) (/ (cos k_m) (* t (pow (sin k_m) 2.0))))))
     (*
      t_1
      (pow
       (/ (cbrt (* 2.0 t_1)) (* t (* (cbrt (tan k_m)) (cbrt (sin k_m)))))
       3.0)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = l / hypot(1.0, hypot(1.0, (k_m / t)));
	double tmp;
	if (t <= 5.5e-72) {
		tmp = l * (2.0 * ((l / pow(k_m, 2.0)) * (cos(k_m) / (t * pow(sin(k_m), 2.0)))));
	} else {
		tmp = t_1 * pow((cbrt((2.0 * t_1)) / (t * (cbrt(tan(k_m)) * cbrt(sin(k_m))))), 3.0);
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = l / Math.hypot(1.0, Math.hypot(1.0, (k_m / t)));
	double tmp;
	if (t <= 5.5e-72) {
		tmp = l * (2.0 * ((l / Math.pow(k_m, 2.0)) * (Math.cos(k_m) / (t * Math.pow(Math.sin(k_m), 2.0)))));
	} else {
		tmp = t_1 * Math.pow((Math.cbrt((2.0 * t_1)) / (t * (Math.cbrt(Math.tan(k_m)) * Math.cbrt(Math.sin(k_m))))), 3.0);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(l / hypot(1.0, hypot(1.0, Float64(k_m / t))))
	tmp = 0.0
	if (t <= 5.5e-72)
		tmp = Float64(l * Float64(2.0 * Float64(Float64(l / (k_m ^ 2.0)) * Float64(cos(k_m) / Float64(t * (sin(k_m) ^ 2.0))))));
	else
		tmp = Float64(t_1 * (Float64(cbrt(Float64(2.0 * t_1)) / Float64(t * Float64(cbrt(tan(k_m)) * cbrt(sin(k_m))))) ^ 3.0));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(l / N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k$95$m / t), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 5.5e-72], N[(l * N[(2.0 * N[(N[(l / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[(t * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Power[N[(N[Power[N[(2.0 * t$95$1), $MachinePrecision], 1/3], $MachinePrecision] / N[(t * N[(N[Power[N[Tan[k$95$m], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[Sin[k$95$m], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 5.49999999999999994e-72

   1. Initial program 49.2%

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

      1. associate-/r* 49.1%

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

      2. unpow2 49.1%

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

      3. sqr-neg 49.1%

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

      4. distribute-frac-neg2 49.1%

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

      5. distribute-frac-neg2 49.1%

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

      6. unpow2 49.1%

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

      7. +-commutative 49.1%

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

      8. +-commutative 49.1%

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

      9. associate-/l/ 49.2%

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

      10. associate-*l/ 49.5%

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

      11. associate-/r/ 48.2%

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

   3. Simplified 48.2%

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

      1. associate-*r* 54.4%

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

      2. *-un-lft-identity 54.4%

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

      3. times-frac 54.9%

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

      4. associate-/l/ 54.9%

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

   6. Applied egg-rr 54.9%

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

      1. times-frac 54.4%

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

      2. *-commutative 54.4%

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

      3. times-frac 54.9%

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

      4. associate-*l/ 55.4%

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

      5. associate-*l* 52.6%

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

      6. times-frac 52.5%

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

      7. /-rgt-identity 52.5%

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

   8. Simplified 52.5%

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

   9. Taylor expanded in t around 0 68.0%

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

       1. times-frac 68.2%

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

   11. Simplified 68.2%

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

   if 5.49999999999999994e-72 < t

   1. Initial program 57.6%

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

      1. associate-/r* 57.6%

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

      2. unpow2 57.6%

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

      3. sqr-neg 57.6%

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

      4. distribute-frac-neg2 57.6%

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

      5. distribute-frac-neg2 57.6%

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

      6. unpow2 57.6%

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

      7. +-commutative 57.6%

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

      8. +-commutative 57.6%

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

      9. associate-/l/ 57.6%

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

      10. associate-*l/ 59.2%

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

      11. associate-/r/ 59.0%

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

   3. Simplified 59.0%

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

      1. associate-*r* 66.0%

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

      2. add-sqr-sqrt 65.9%

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

      3. times-frac 67.6%

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

   6. Applied egg-rr 69.3%

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

      1. associate-/l* 70.9%

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

      2. associate-*l* 64.0%

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

   8. Simplified 64.0%

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

      1. add-cube-cbrt 63.8%

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

      2. pow3 63.8%

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

      3. associate-*l/ 63.8%

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

      4. cbrt-div 63.8%

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

      5. cbrt-prod 63.7%

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

      6. unpow3 63.8%

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

      7. add-cbrt-cube 76.3%

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

   10. Applied egg-rr 76.3%

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

       1. *-commutative 76.3%

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

       2. cbrt-prod 93.0%

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

   12. Applied egg-rr 93.0%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.5 \cdot 10^{-72}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \left(\frac{\ell}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot {\left(\frac{\sqrt[3]{2 \cdot \frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}}}{t \cdot \left(\sqrt[3]{\tan k} \cdot \sqrt[3]{\sin k}\right)}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k\_m}{t}\right)\right)\\ t_2 := {\left(\sqrt[3]{\ell}\right)}^{2}\\ \mathbf{if}\;t \leq 2.9 \cdot 10^{-62}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \left(\frac{\ell}{{k\_m}^{2}} \cdot \frac{\cos k\_m}{t \cdot {\sin k\_m}^{2}}\right)\right)\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+96}:\\ \;\;\;\;\frac{\ell}{t\_1} \cdot \frac{\frac{2}{{t}^{3}} \cdot \frac{\frac{\ell}{\sin k\_m}}{\tan k\_m}}{t\_1}\\ \mathbf{elif}\;t \leq 2.15 \cdot 10^{+216}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k\_m} \cdot \frac{t}{t\_2}\right)}^{3} \cdot \left(\tan k\_m \cdot {t\_1}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{t\_2}{t \cdot {\left(\sqrt[3]{k\_m}\right)}^{2}}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (hypot 1.0 (hypot 1.0 (/ k_m t)))) (t_2 (pow (cbrt l) 2.0)))
   (if (<= t 2.9e-62)
     (*
      l
      (* 2.0 (* (/ l (pow k_m 2.0)) (/ (cos k_m) (* t (pow (sin k_m) 2.0))))))
     (if (<= t 6e+96)
       (*
        (/ l t_1)
        (/ (* (/ 2.0 (pow t 3.0)) (/ (/ l (sin k_m)) (tan k_m))) t_1))
       (if (<= t 2.15e+216)
         (/
          2.0
          (*
           (pow (* (cbrt (sin k_m)) (/ t t_2)) 3.0)
           (* (tan k_m) (pow t_1 2.0))))
         (pow (/ t_2 (* t (pow (cbrt k_m) 2.0))) 3.0))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = hypot(1.0, hypot(1.0, (k_m / t)));
	double t_2 = pow(cbrt(l), 2.0);
	double tmp;
	if (t <= 2.9e-62) {
		tmp = l * (2.0 * ((l / pow(k_m, 2.0)) * (cos(k_m) / (t * pow(sin(k_m), 2.0)))));
	} else if (t <= 6e+96) {
		tmp = (l / t_1) * (((2.0 / pow(t, 3.0)) * ((l / sin(k_m)) / tan(k_m))) / t_1);
	} else if (t <= 2.15e+216) {
		tmp = 2.0 / (pow((cbrt(sin(k_m)) * (t / t_2)), 3.0) * (tan(k_m) * pow(t_1, 2.0)));
	} else {
		tmp = pow((t_2 / (t * pow(cbrt(k_m), 2.0))), 3.0);
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = Math.hypot(1.0, Math.hypot(1.0, (k_m / t)));
	double t_2 = Math.pow(Math.cbrt(l), 2.0);
	double tmp;
	if (t <= 2.9e-62) {
		tmp = l * (2.0 * ((l / Math.pow(k_m, 2.0)) * (Math.cos(k_m) / (t * Math.pow(Math.sin(k_m), 2.0)))));
	} else if (t <= 6e+96) {
		tmp = (l / t_1) * (((2.0 / Math.pow(t, 3.0)) * ((l / Math.sin(k_m)) / Math.tan(k_m))) / t_1);
	} else if (t <= 2.15e+216) {
		tmp = 2.0 / (Math.pow((Math.cbrt(Math.sin(k_m)) * (t / t_2)), 3.0) * (Math.tan(k_m) * Math.pow(t_1, 2.0)));
	} else {
		tmp = Math.pow((t_2 / (t * Math.pow(Math.cbrt(k_m), 2.0))), 3.0);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = hypot(1.0, hypot(1.0, Float64(k_m / t)))
	t_2 = cbrt(l) ^ 2.0
	tmp = 0.0
	if (t <= 2.9e-62)
		tmp = Float64(l * Float64(2.0 * Float64(Float64(l / (k_m ^ 2.0)) * Float64(cos(k_m) / Float64(t * (sin(k_m) ^ 2.0))))));
	elseif (t <= 6e+96)
		tmp = Float64(Float64(l / t_1) * Float64(Float64(Float64(2.0 / (t ^ 3.0)) * Float64(Float64(l / sin(k_m)) / tan(k_m))) / t_1));
	elseif (t <= 2.15e+216)
		tmp = Float64(2.0 / Float64((Float64(cbrt(sin(k_m)) * Float64(t / t_2)) ^ 3.0) * Float64(tan(k_m) * (t_1 ^ 2.0))));
	else
		tmp = Float64(t_2 / Float64(t * (cbrt(k_m) ^ 2.0))) ^ 3.0;
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k$95$m / t), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t, 2.9e-62], N[(l * N[(2.0 * N[(N[(l / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[(t * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6e+96], N[(N[(l / t$95$1), $MachinePrecision] * N[(N[(N[(2.0 / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] / N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.15e+216], N[(2.0 / N[(N[Power[N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 1/3], $MachinePrecision] * N[(t / t$95$2), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(t$95$2 / N[(t * N[Power[N[Power[k$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < 2.89999999999999986e-62

   1. Initial program 48.9%

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

      1. associate-/r* 48.9%

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

      2. unpow2 48.9%

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

      3. sqr-neg 48.9%

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

      4. distribute-frac-neg2 48.9%

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

      5. distribute-frac-neg2 48.9%

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

      6. unpow2 48.9%

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

      7. +-commutative 48.9%

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

      8. +-commutative 48.9%

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

      9. associate-/l/ 48.9%

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

      10. associate-*l/ 49.2%

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

      11. associate-/r/ 47.9%

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

   3. Simplified 47.9%

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

      1. associate-*r* 54.1%

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

      2. *-un-lft-identity 54.1%

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

      3. times-frac 54.6%

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

      4. associate-/l/ 54.6%

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

   6. Applied egg-rr 54.6%

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

      1. times-frac 54.1%

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

      2. *-commutative 54.1%

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

      3. times-frac 54.6%

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

      4. associate-*l/ 55.1%

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

      5. associate-*l* 52.3%

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

      6. times-frac 52.3%

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

      7. /-rgt-identity 52.3%

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

   8. Simplified 52.3%

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

   9. Taylor expanded in t around 0 68.1%

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

       1. times-frac 68.3%

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

   11. Simplified 68.3%

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

   if 2.89999999999999986e-62 < t < 6.0000000000000001e96

   1. Initial program 65.4%

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

      1. associate-/r* 65.4%

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

      2. unpow2 65.4%

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

      3. sqr-neg 65.4%

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

      4. distribute-frac-neg2 65.4%

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

      5. distribute-frac-neg2 65.4%

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

      6. unpow2 65.4%

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

      7. +-commutative 65.4%

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

      8. +-commutative 65.4%

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

      9. associate-/l/ 65.4%

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

      10. associate-*l/ 69.2%

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

      11. associate-/r/ 68.8%

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

   3. Simplified 68.8%

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

      1. associate-*r* 76.6%

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

      2. add-sqr-sqrt 76.4%

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

      3. times-frac 80.5%

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

   6. Applied egg-rr 84.4%

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

      1. associate-/l* 84.3%

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

      2. associate-*l* 76.7%

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

   8. Simplified 76.7%

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

      1. associate-*r/ 76.7%

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

      2. frac-times 76.7%

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

   10. Applied egg-rr 76.7%

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

       1. associate-/l* 76.7%

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

       2. associate-/r* 76.7%

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

       3. times-frac 76.8%

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

       4. associate-/r* 92.1%

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

   12. Simplified 92.1%

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

   if 6.0000000000000001e96 < t < 2.14999999999999985e216

   1. Initial program 44.4%

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

   3. Simplified 44.4%

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

      1. add-cube-cbrt 44.4%

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

      2. pow3 44.4%

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

      3. associate-/r* 49.5%

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

      4. *-commutative 49.5%

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

      5. cbrt-prod 49.5%

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

      6. associate-/r* 44.4%

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

      7. cbrt-div 44.4%

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

      8. rem-cbrt-cube 74.8%

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

      9. cbrt-prod 91.3%

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

      10. pow2 91.3%

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

   6. Applied egg-rr 91.3%

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

      1. add-sqr-sqrt 47.8%

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

      2. pow2 47.8%

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

      3. associate-+r+ 47.8%

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

      4. metadata-eval 47.8%

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

      5. sqrt-prod 47.8%

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

      6. metadata-eval 47.8%

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

      7. associate-+r+ 47.8%

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

      8. add-sqr-sqrt 47.8%

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

      9. hypot-1-def 47.8%

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

      10. unpow2 47.8%

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

      11. hypot-1-def 47.8%

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

   8. Applied egg-rr 47.8%

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

      1. unpow2 47.8%

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

      2. swap-sqr 47.8%

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

      3. rem-square-sqrt 91.3%

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

      4. unpow2 91.3%

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

   10. Simplified 91.3%

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

   if 2.14999999999999985e216 < t

   1. Initial program 72.7%

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

      1. associate-/r* 72.7%

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

      2. unpow2 72.7%

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

      3. sqr-neg 72.7%

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

      4. distribute-frac-neg2 72.7%

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

      5. distribute-frac-neg2 72.7%

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

      6. unpow2 72.7%

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

      7. +-commutative 72.7%

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

      8. +-commutative 72.7%

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

      9. associate-/l/ 72.7%

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

      10. associate-*l/ 72.7%

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

      11. associate-/r/ 72.7%

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

   3. Simplified 72.7%

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

      1. associate-*r* 82.3%

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

      2. *-un-lft-identity 82.3%

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

      3. times-frac 82.3%

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

      4. associate-/l/ 82.3%

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

   6. Applied egg-rr 82.3%

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

      1. times-frac 82.3%

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

      2. *-commutative 82.3%

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

      3. times-frac 82.3%

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

      4. associate-*l/ 82.3%

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

      5. associate-*l* 73.0%

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

      6. times-frac 72.7%

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

      7. /-rgt-identity 72.7%

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

   8. Simplified 72.7%

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

   9. Taylor expanded in k around 0 73.0%

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

       1. add-cube-cbrt 73.0%

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

       2. pow3 73.0%

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

       3. associate-*l/ 63.6%

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

       4. cbrt-div 63.6%

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

       5. cbrt-prod 73.0%

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

       6. unpow2 73.0%

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

       7. *-commutative 73.0%

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

       8. cbrt-prod 73.0%

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

       9. unpow3 73.0%

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

       10. add-cbrt-cube 82.4%

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

       11. unpow2 82.4%

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

       12. cbrt-prod 99.2%

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

       13. pow2 99.2%

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

   11. Applied egg-rr 99.2%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.9 \cdot 10^{-62}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \left(\frac{\ell}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\right)\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+96}:\\ \;\;\;\;\frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \frac{\frac{2}{{t}^{3}} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\\ \mathbf{elif}\;t \leq 2.15 \cdot 10^{+216}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(\tan k \cdot {\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot {\left(\sqrt[3]{k}\right)}^{2}}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k\_m}{t}\right)\right)\\ t_2 := {\left(\sqrt[3]{\ell}\right)}^{2}\\ \mathbf{if}\;t \leq 1.3 \cdot 10^{-62}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \left(\frac{\ell}{{k\_m}^{2}} \cdot \frac{\cos k\_m}{t \cdot {\sin k\_m}^{2}}\right)\right)\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{+96}:\\ \;\;\;\;\frac{\ell}{t\_1} \cdot \frac{\frac{2}{{t}^{3}} \cdot \frac{\frac{\ell}{\sin k\_m}}{\tan k\_m}}{t\_1}\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{+216}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k\_m} \cdot \frac{t}{t\_2}\right)}^{3} \cdot \left(\tan k\_m \cdot \left(1 + \left(1 + {\left(\frac{k\_m}{t}\right)}^{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{t\_2}{t \cdot {\left(\sqrt[3]{k\_m}\right)}^{2}}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (hypot 1.0 (hypot 1.0 (/ k_m t)))) (t_2 (pow (cbrt l) 2.0)))
   (if (<= t 1.3e-62)
     (*
      l
      (* 2.0 (* (/ l (pow k_m 2.0)) (/ (cos k_m) (* t (pow (sin k_m) 2.0))))))
     (if (<= t 1.7e+96)
       (*
        (/ l t_1)
        (/ (* (/ 2.0 (pow t 3.0)) (/ (/ l (sin k_m)) (tan k_m))) t_1))
       (if (<= t 1.8e+216)
         (/
          2.0
          (*
           (pow (* (cbrt (sin k_m)) (/ t t_2)) 3.0)
           (* (tan k_m) (+ 1.0 (+ 1.0 (pow (/ k_m t) 2.0))))))
         (pow (/ t_2 (* t (pow (cbrt k_m) 2.0))) 3.0))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = hypot(1.0, hypot(1.0, (k_m / t)));
	double t_2 = pow(cbrt(l), 2.0);
	double tmp;
	if (t <= 1.3e-62) {
		tmp = l * (2.0 * ((l / pow(k_m, 2.0)) * (cos(k_m) / (t * pow(sin(k_m), 2.0)))));
	} else if (t <= 1.7e+96) {
		tmp = (l / t_1) * (((2.0 / pow(t, 3.0)) * ((l / sin(k_m)) / tan(k_m))) / t_1);
	} else if (t <= 1.8e+216) {
		tmp = 2.0 / (pow((cbrt(sin(k_m)) * (t / t_2)), 3.0) * (tan(k_m) * (1.0 + (1.0 + pow((k_m / t), 2.0)))));
	} else {
		tmp = pow((t_2 / (t * pow(cbrt(k_m), 2.0))), 3.0);
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = Math.hypot(1.0, Math.hypot(1.0, (k_m / t)));
	double t_2 = Math.pow(Math.cbrt(l), 2.0);
	double tmp;
	if (t <= 1.3e-62) {
		tmp = l * (2.0 * ((l / Math.pow(k_m, 2.0)) * (Math.cos(k_m) / (t * Math.pow(Math.sin(k_m), 2.0)))));
	} else if (t <= 1.7e+96) {
		tmp = (l / t_1) * (((2.0 / Math.pow(t, 3.0)) * ((l / Math.sin(k_m)) / Math.tan(k_m))) / t_1);
	} else if (t <= 1.8e+216) {
		tmp = 2.0 / (Math.pow((Math.cbrt(Math.sin(k_m)) * (t / t_2)), 3.0) * (Math.tan(k_m) * (1.0 + (1.0 + Math.pow((k_m / t), 2.0)))));
	} else {
		tmp = Math.pow((t_2 / (t * Math.pow(Math.cbrt(k_m), 2.0))), 3.0);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = hypot(1.0, hypot(1.0, Float64(k_m / t)))
	t_2 = cbrt(l) ^ 2.0
	tmp = 0.0
	if (t <= 1.3e-62)
		tmp = Float64(l * Float64(2.0 * Float64(Float64(l / (k_m ^ 2.0)) * Float64(cos(k_m) / Float64(t * (sin(k_m) ^ 2.0))))));
	elseif (t <= 1.7e+96)
		tmp = Float64(Float64(l / t_1) * Float64(Float64(Float64(2.0 / (t ^ 3.0)) * Float64(Float64(l / sin(k_m)) / tan(k_m))) / t_1));
	elseif (t <= 1.8e+216)
		tmp = Float64(2.0 / Float64((Float64(cbrt(sin(k_m)) * Float64(t / t_2)) ^ 3.0) * Float64(tan(k_m) * Float64(1.0 + Float64(1.0 + (Float64(k_m / t) ^ 2.0))))));
	else
		tmp = Float64(t_2 / Float64(t * (cbrt(k_m) ^ 2.0))) ^ 3.0;
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k$95$m / t), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t, 1.3e-62], N[(l * N[(2.0 * N[(N[(l / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[(t * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.7e+96], N[(N[(l / t$95$1), $MachinePrecision] * N[(N[(N[(2.0 / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] / N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.8e+216], N[(2.0 / N[(N[Power[N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 1/3], $MachinePrecision] * N[(t / t$95$2), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(t$95$2 / N[(t * N[Power[N[Power[k$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < 1.3e-62

   1. Initial program 48.9%

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

      1. associate-/r* 48.9%

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

      2. unpow2 48.9%

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

      3. sqr-neg 48.9%

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

      4. distribute-frac-neg2 48.9%

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

      5. distribute-frac-neg2 48.9%

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

      6. unpow2 48.9%

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

      7. +-commutative 48.9%

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

      8. +-commutative 48.9%

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

      9. associate-/l/ 48.9%

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

      10. associate-*l/ 49.2%

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

      11. associate-/r/ 47.9%

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

   3. Simplified 47.9%

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

      1. associate-*r* 54.1%

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

      2. *-un-lft-identity 54.1%

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

      3. times-frac 54.6%

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

      4. associate-/l/ 54.6%

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

   6. Applied egg-rr 54.6%

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

      1. times-frac 54.1%

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

      2. *-commutative 54.1%

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

      3. times-frac 54.6%

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

      4. associate-*l/ 55.1%

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

      5. associate-*l* 52.3%

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

      6. times-frac 52.3%

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

      7. /-rgt-identity 52.3%

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

   8. Simplified 52.3%

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

   9. Taylor expanded in t around 0 68.1%

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

       1. times-frac 68.3%

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

   11. Simplified 68.3%

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

   if 1.3e-62 < t < 1.7e96

   1. Initial program 65.4%

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

      1. associate-/r* 65.4%

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

      2. unpow2 65.4%

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

      3. sqr-neg 65.4%

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

      4. distribute-frac-neg2 65.4%

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

      5. distribute-frac-neg2 65.4%

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

      6. unpow2 65.4%

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

      7. +-commutative 65.4%

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

      8. +-commutative 65.4%

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

      9. associate-/l/ 65.4%

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

      10. associate-*l/ 69.2%

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

      11. associate-/r/ 68.8%

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

   3. Simplified 68.8%

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

      1. associate-*r* 76.6%

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

      2. add-sqr-sqrt 76.4%

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

      3. times-frac 80.5%

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

   6. Applied egg-rr 84.4%

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

      1. associate-/l* 84.3%

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

      2. associate-*l* 76.7%

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

   8. Simplified 76.7%

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

      1. associate-*r/ 76.7%

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

      2. frac-times 76.7%

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

   10. Applied egg-rr 76.7%

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

       1. associate-/l* 76.7%

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

       2. associate-/r* 76.7%

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

       3. times-frac 76.8%

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

       4. associate-/r* 92.1%

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

   12. Simplified 92.1%

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

   if 1.7e96 < t < 1.8000000000000001e216

   1. Initial program 44.4%

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

   3. Simplified 44.4%

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

      1. add-cube-cbrt 44.4%

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

      2. pow3 44.4%

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

      3. associate-/r* 49.5%

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

      4. *-commutative 49.5%

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

      5. cbrt-prod 49.5%

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

      6. associate-/r* 44.4%

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

      7. cbrt-div 44.4%

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

      8. rem-cbrt-cube 74.8%

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

      9. cbrt-prod 91.3%

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

      10. pow2 91.3%

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

   6. Applied egg-rr 91.3%

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

   if 1.8000000000000001e216 < t

   1. Initial program 72.7%

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

      1. associate-/r* 72.7%

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

      2. unpow2 72.7%

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

      3. sqr-neg 72.7%

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

      4. distribute-frac-neg2 72.7%

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

      5. distribute-frac-neg2 72.7%

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

      6. unpow2 72.7%

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

      7. +-commutative 72.7%

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

      8. +-commutative 72.7%

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

      9. associate-/l/ 72.7%

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

      10. associate-*l/ 72.7%

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

      11. associate-/r/ 72.7%

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

   3. Simplified 72.7%

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

      1. associate-*r* 82.3%

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

      2. *-un-lft-identity 82.3%

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

      3. times-frac 82.3%

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

      4. associate-/l/ 82.3%

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

   6. Applied egg-rr 82.3%

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

      1. times-frac 82.3%

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

      2. *-commutative 82.3%

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

      3. times-frac 82.3%

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

      4. associate-*l/ 82.3%

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

      5. associate-*l* 73.0%

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

      6. times-frac 72.7%

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

      7. /-rgt-identity 72.7%

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

   8. Simplified 72.7%

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

   9. Taylor expanded in k around 0 73.0%

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

       1. add-cube-cbrt 73.0%

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

       2. pow3 73.0%

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

       3. associate-*l/ 63.6%

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

       4. cbrt-div 63.6%

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

       5. cbrt-prod 73.0%

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

       6. unpow2 73.0%

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

       7. *-commutative 73.0%

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

       8. cbrt-prod 73.0%

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

       9. unpow3 73.0%

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

       10. add-cbrt-cube 82.4%

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

       11. unpow2 82.4%

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

       12. cbrt-prod 99.2%

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

       13. pow2 99.2%

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

   11. Applied egg-rr 99.2%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.3 \cdot 10^{-62}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \left(\frac{\ell}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\right)\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{+96}:\\ \;\;\;\;\frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \frac{\frac{2}{{t}^{3}} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{+216}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot {\left(\sqrt[3]{k}\right)}^{2}}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := {\left(\sqrt[3]{\ell}\right)}^{2}\\ \mathbf{if}\;k\_m \leq 1.26 \cdot 10^{-15}:\\ \;\;\;\;{\left(\frac{t\_1}{t \cdot {\left(\sqrt[3]{k\_m}\right)}^{2}}\right)}^{3}\\ \mathbf{elif}\;k\_m \leq 1.6 \cdot 10^{+154}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \frac{\ell \cdot \cos k\_m}{{k\_m}^{2} \cdot \left(t \cdot {\sin k\_m}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{t\_1} \cdot \sqrt[3]{\sin k\_m \cdot \left(\tan k\_m \cdot \left(2 + {\left(\frac{k\_m}{t}\right)}^{2}\right)\right)}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (pow (cbrt l) 2.0)))
   (if (<= k_m 1.26e-15)
     (pow (/ t_1 (* t (pow (cbrt k_m) 2.0))) 3.0)
     (if (<= k_m 1.6e+154)
       (*
        l
        (*
         2.0
         (/ (* l (cos k_m)) (* (pow k_m 2.0) (* t (pow (sin k_m) 2.0))))))
       (/
        2.0
        (pow
         (*
          (/ t t_1)
          (cbrt (* (sin k_m) (* (tan k_m) (+ 2.0 (pow (/ k_m t) 2.0))))))
         3.0))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = pow(cbrt(l), 2.0);
	double tmp;
	if (k_m <= 1.26e-15) {
		tmp = pow((t_1 / (t * pow(cbrt(k_m), 2.0))), 3.0);
	} else if (k_m <= 1.6e+154) {
		tmp = l * (2.0 * ((l * cos(k_m)) / (pow(k_m, 2.0) * (t * pow(sin(k_m), 2.0)))));
	} else {
		tmp = 2.0 / pow(((t / t_1) * cbrt((sin(k_m) * (tan(k_m) * (2.0 + pow((k_m / t), 2.0)))))), 3.0);
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = Math.pow(Math.cbrt(l), 2.0);
	double tmp;
	if (k_m <= 1.26e-15) {
		tmp = Math.pow((t_1 / (t * Math.pow(Math.cbrt(k_m), 2.0))), 3.0);
	} else if (k_m <= 1.6e+154) {
		tmp = l * (2.0 * ((l * Math.cos(k_m)) / (Math.pow(k_m, 2.0) * (t * Math.pow(Math.sin(k_m), 2.0)))));
	} else {
		tmp = 2.0 / Math.pow(((t / t_1) * Math.cbrt((Math.sin(k_m) * (Math.tan(k_m) * (2.0 + Math.pow((k_m / t), 2.0)))))), 3.0);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = cbrt(l) ^ 2.0
	tmp = 0.0
	if (k_m <= 1.26e-15)
		tmp = Float64(t_1 / Float64(t * (cbrt(k_m) ^ 2.0))) ^ 3.0;
	elseif (k_m <= 1.6e+154)
		tmp = Float64(l * Float64(2.0 * Float64(Float64(l * cos(k_m)) / Float64((k_m ^ 2.0) * Float64(t * (sin(k_m) ^ 2.0))))));
	else
		tmp = Float64(2.0 / (Float64(Float64(t / t_1) * cbrt(Float64(sin(k_m) * Float64(tan(k_m) * Float64(2.0 + (Float64(k_m / t) ^ 2.0)))))) ^ 3.0));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[k$95$m, 1.26e-15], N[Power[N[(t$95$1 / N[(t * N[Power[N[Power[k$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], If[LessEqual[k$95$m, 1.6e+154], N[(l * N[(2.0 * N[(N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(t / t$95$1), $MachinePrecision] * N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[(2.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.26e-15

   1. Initial program 57.1%

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

      1. associate-/r* 57.1%

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

      2. unpow2 57.1%

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

      3. sqr-neg 57.1%

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

      4. distribute-frac-neg2 57.1%

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

      5. distribute-frac-neg2 57.1%

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

      6. unpow2 57.1%

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

      7. +-commutative 57.1%

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

      8. +-commutative 57.1%

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

      9. associate-/l/ 57.1%

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

      10. associate-*l/ 58.0%

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

      11. associate-/r/ 56.5%

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

   3. Simplified 56.5%

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

      1. associate-*r* 64.4%

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

      2. *-un-lft-identity 64.4%

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

      3. times-frac 65.0%

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

      4. associate-/l/ 65.0%

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

   6. Applied egg-rr 65.0%

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

      1. times-frac 64.5%

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

      2. *-commutative 64.5%

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

      3. times-frac 65.0%

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

      4. associate-*l/ 65.5%

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

      5. associate-*l* 60.1%

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

      6. times-frac 60.0%

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

      7. /-rgt-identity 60.0%

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

   8. Simplified 60.0%

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

   9. Taylor expanded in k around 0 58.8%

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

       1. add-cube-cbrt 58.7%

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

       2. pow3 58.7%

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

       3. associate-*l/ 51.8%

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

       4. cbrt-div 51.8%

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

       5. cbrt-prod 58.7%

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

       6. unpow2 58.7%

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

       7. *-commutative 58.7%

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

       8. cbrt-prod 59.2%

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

       9. unpow3 59.2%

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

       10. add-cbrt-cube 71.2%

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

       11. unpow2 71.2%

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

       12. cbrt-prod 82.6%

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

       13. pow2 82.6%

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

   11. Applied egg-rr 82.6%

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

   if 1.26e-15 < k < 1.6e154

   1. Initial program 39.0%

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

      1. associate-/r* 39.0%

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

      2. unpow2 39.0%

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

      3. sqr-neg 39.0%

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

      4. distribute-frac-neg2 39.0%

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

      5. distribute-frac-neg2 39.0%

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

      6. unpow2 39.0%

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

      7. +-commutative 39.0%

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

      8. +-commutative 39.0%

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

      9. associate-/l/ 39.0%

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

      10. associate-*l/ 39.1%

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

      11. associate-/r/ 39.0%

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

   3. Simplified 39.0%

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

      1. associate-*r* 39.3%

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

      2. *-un-lft-identity 39.3%

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

      3. times-frac 41.8%

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

      4. associate-/l/ 41.8%

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

   6. Applied egg-rr 41.8%

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

      1. times-frac 39.3%

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

      2. *-commutative 39.3%

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

      3. times-frac 41.8%

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

      4. associate-*l/ 41.8%

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

      5. associate-*l* 41.9%

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

      6. times-frac 41.9%

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

      7. /-rgt-identity 41.9%

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

   8. Simplified 41.9%

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

   9. Taylor expanded in t around 0 86.4%

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

   if 1.6e154 < k

   1. Initial program 36.1%

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

   3. Simplified 36.1%

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

      1. associate-*l* 36.1%

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

      2. associate-/r* 41.3%

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

      3. associate-+r+ 41.3%

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

      4. metadata-eval 41.3%

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

      5. associate-*l* 41.3%

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

      6. add-cube-cbrt 41.3%

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

      7. pow3 41.3%

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

   6. Applied egg-rr 71.7%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.26 \cdot 10^{-15}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot {\left(\sqrt[3]{k}\right)}^{2}}\right)}^{3}\\ \mathbf{elif}\;k \leq 1.6 \cdot 10^{+154}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \frac{\ell \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.25 \cdot 10^{-15}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot {\left(\sqrt[3]{k\_m}\right)}^{2}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \frac{\ell \cdot \cos k\_m}{{k\_m}^{2} \cdot \left(t \cdot {\sin k\_m}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.25e-15)
   (pow (/ (pow (cbrt l) 2.0) (* t (pow (cbrt k_m) 2.0))) 3.0)
   (*
    l
    (* 2.0 (/ (* l (cos k_m)) (* (pow k_m 2.0) (* t (pow (sin k_m) 2.0))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.25e-15) {
		tmp = pow((pow(cbrt(l), 2.0) / (t * pow(cbrt(k_m), 2.0))), 3.0);
	} else {
		tmp = l * (2.0 * ((l * cos(k_m)) / (pow(k_m, 2.0) * (t * pow(sin(k_m), 2.0)))));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.25e-15) {
		tmp = Math.pow((Math.pow(Math.cbrt(l), 2.0) / (t * Math.pow(Math.cbrt(k_m), 2.0))), 3.0);
	} else {
		tmp = l * (2.0 * ((l * Math.cos(k_m)) / (Math.pow(k_m, 2.0) * (t * Math.pow(Math.sin(k_m), 2.0)))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.25e-15)
		tmp = Float64((cbrt(l) ^ 2.0) / Float64(t * (cbrt(k_m) ^ 2.0))) ^ 3.0;
	else
		tmp = Float64(l * Float64(2.0 * Float64(Float64(l * cos(k_m)) / Float64((k_m ^ 2.0) * Float64(t * (sin(k_m) ^ 2.0))))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.25e-15], N[Power[N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / N[(t * N[Power[N[Power[k$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], N[(l * N[(2.0 * N[(N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.25e-15

   1. Initial program 57.1%

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

      1. associate-/r* 57.1%

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

      2. unpow2 57.1%

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

      3. sqr-neg 57.1%

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

      4. distribute-frac-neg2 57.1%

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

      5. distribute-frac-neg2 57.1%

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

      6. unpow2 57.1%

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

      7. +-commutative 57.1%

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

      8. +-commutative 57.1%

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

      9. associate-/l/ 57.1%

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

      10. associate-*l/ 58.0%

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

      11. associate-/r/ 56.5%

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

   3. Simplified 56.5%

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

      1. associate-*r* 64.4%

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

      2. *-un-lft-identity 64.4%

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

      3. times-frac 65.0%

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

      4. associate-/l/ 65.0%

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

   6. Applied egg-rr 65.0%

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

      1. times-frac 64.5%

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

      2. *-commutative 64.5%

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

      3. times-frac 65.0%

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

      4. associate-*l/ 65.5%

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

      5. associate-*l* 60.1%

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

      6. times-frac 60.0%

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

      7. /-rgt-identity 60.0%

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

   8. Simplified 60.0%

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

   9. Taylor expanded in k around 0 58.8%

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

       1. add-cube-cbrt 58.7%

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

       2. pow3 58.7%

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

       3. associate-*l/ 51.8%

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

       4. cbrt-div 51.8%

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

       5. cbrt-prod 58.7%

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

       6. unpow2 58.7%

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

       7. *-commutative 58.7%

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

       8. cbrt-prod 59.2%

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

       9. unpow3 59.2%

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

       10. add-cbrt-cube 71.2%

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

       11. unpow2 71.2%

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

       12. cbrt-prod 82.6%

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

       13. pow2 82.6%

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

   11. Applied egg-rr 82.6%

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

   if 1.25e-15 < k

   1. Initial program 37.5%

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

      1. associate-/r* 37.5%

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

      2. unpow2 37.5%

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

      3. sqr-neg 37.5%

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

      4. distribute-frac-neg2 37.5%

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

      5. distribute-frac-neg2 37.5%

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

      6. unpow2 37.5%

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

      7. +-commutative 37.5%

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

      8. +-commutative 37.5%

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

      9. associate-/l/ 37.5%

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

      10. associate-*l/ 37.5%

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

      11. associate-/r/ 37.5%

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

   3. Simplified 37.5%

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

      1. associate-*r* 40.4%

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

      2. *-un-lft-identity 40.4%

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

      3. times-frac 41.6%

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

      4. associate-/l/ 41.6%

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

   6. Applied egg-rr 41.6%

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

      1. times-frac 40.4%

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

      2. *-commutative 40.4%

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

      3. times-frac 41.6%

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

      4. associate-*l/ 41.6%

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

      5. associate-*l* 41.6%

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

      6. times-frac 41.6%

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

      7. /-rgt-identity 41.6%

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

   8. Simplified 41.6%

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

   9. Taylor expanded in t around 0 72.7%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.25 \cdot 10^{-15}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot {\left(\sqrt[3]{k}\right)}^{2}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \frac{\ell \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.16 \cdot 10^{-15}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{k\_m} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot k\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \frac{\ell \cdot \cos k\_m}{{k\_m}^{2} \cdot \left(t \cdot {\sin k\_m}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.16e-15)
   (/ 2.0 (* (pow (* (cbrt k_m) (/ t (pow (cbrt l) 2.0))) 3.0) (* 2.0 k_m)))
   (*
    l
    (* 2.0 (/ (* l (cos k_m)) (* (pow k_m 2.0) (* t (pow (sin k_m) 2.0))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.16e-15) {
		tmp = 2.0 / (pow((cbrt(k_m) * (t / pow(cbrt(l), 2.0))), 3.0) * (2.0 * k_m));
	} else {
		tmp = l * (2.0 * ((l * cos(k_m)) / (pow(k_m, 2.0) * (t * pow(sin(k_m), 2.0)))));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.16e-15) {
		tmp = 2.0 / (Math.pow((Math.cbrt(k_m) * (t / Math.pow(Math.cbrt(l), 2.0))), 3.0) * (2.0 * k_m));
	} else {
		tmp = l * (2.0 * ((l * Math.cos(k_m)) / (Math.pow(k_m, 2.0) * (t * Math.pow(Math.sin(k_m), 2.0)))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.16e-15)
		tmp = Float64(2.0 / Float64((Float64(cbrt(k_m) * Float64(t / (cbrt(l) ^ 2.0))) ^ 3.0) * Float64(2.0 * k_m)));
	else
		tmp = Float64(l * Float64(2.0 * Float64(Float64(l * cos(k_m)) / Float64((k_m ^ 2.0) * Float64(t * (sin(k_m) ^ 2.0))))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.16e-15], N[(2.0 / N[(N[Power[N[(N[Power[k$95$m, 1/3], $MachinePrecision] * N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(2.0 * N[(N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.1599999999999999e-15

   1. Initial program 57.1%

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

   3. Simplified 57.1%

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

      1. add-cube-cbrt 57.1%

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

      2. pow3 57.1%

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

      3. associate-/r* 62.8%

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

      4. *-commutative 62.8%

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

      5. cbrt-prod 62.7%

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

      6. associate-/r* 57.1%

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

      7. cbrt-div 57.9%

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

      8. rem-cbrt-cube 69.2%

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

      9. cbrt-prod 81.5%

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

      10. pow2 81.5%

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

   6. Applied egg-rr 81.5%

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

   7. Taylor expanded in k around 0 75.9%

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

   8. Taylor expanded in k around 0 77.2%

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

   if 1.1599999999999999e-15 < k

   1. Initial program 37.5%

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

      1. associate-/r* 37.5%

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

      2. unpow2 37.5%

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

      3. sqr-neg 37.5%

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

      4. distribute-frac-neg2 37.5%

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

      5. distribute-frac-neg2 37.5%

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

      6. unpow2 37.5%

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

      7. +-commutative 37.5%

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

      8. +-commutative 37.5%

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

      9. associate-/l/ 37.5%

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

      10. associate-*l/ 37.5%

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

      11. associate-/r/ 37.5%

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

   3. Simplified 37.5%

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

      1. associate-*r* 40.4%

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

      2. *-un-lft-identity 40.4%

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

      3. times-frac 41.6%

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

      4. associate-/l/ 41.6%

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

   6. Applied egg-rr 41.6%

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

      1. times-frac 40.4%

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

      2. *-commutative 40.4%

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

      3. times-frac 41.6%

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

      4. associate-*l/ 41.6%

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

      5. associate-*l* 41.6%

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

      6. times-frac 41.6%

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

      7. /-rgt-identity 41.6%

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

   8. Simplified 41.6%

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

   9. Taylor expanded in t around 0 72.7%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.16 \cdot 10^{-15}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \frac{\ell \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.08 \cdot 10^{-15}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{k\_m} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot k\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \left(\frac{\ell}{{k\_m}^{2}} \cdot \frac{\cos k\_m}{t \cdot {\sin k\_m}^{2}}\right)\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.08e-15)
   (/ 2.0 (* (pow (* (cbrt k_m) (/ t (pow (cbrt l) 2.0))) 3.0) (* 2.0 k_m)))
   (*
    l
    (* 2.0 (* (/ l (pow k_m 2.0)) (/ (cos k_m) (* t (pow (sin k_m) 2.0))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.08e-15) {
		tmp = 2.0 / (pow((cbrt(k_m) * (t / pow(cbrt(l), 2.0))), 3.0) * (2.0 * k_m));
	} else {
		tmp = l * (2.0 * ((l / pow(k_m, 2.0)) * (cos(k_m) / (t * pow(sin(k_m), 2.0)))));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.08e-15) {
		tmp = 2.0 / (Math.pow((Math.cbrt(k_m) * (t / Math.pow(Math.cbrt(l), 2.0))), 3.0) * (2.0 * k_m));
	} else {
		tmp = l * (2.0 * ((l / Math.pow(k_m, 2.0)) * (Math.cos(k_m) / (t * Math.pow(Math.sin(k_m), 2.0)))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.08e-15)
		tmp = Float64(2.0 / Float64((Float64(cbrt(k_m) * Float64(t / (cbrt(l) ^ 2.0))) ^ 3.0) * Float64(2.0 * k_m)));
	else
		tmp = Float64(l * Float64(2.0 * Float64(Float64(l / (k_m ^ 2.0)) * Float64(cos(k_m) / Float64(t * (sin(k_m) ^ 2.0))))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.08e-15], N[(2.0 / N[(N[Power[N[(N[Power[k$95$m, 1/3], $MachinePrecision] * N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(2.0 * N[(N[(l / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[(t * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.08e-15

   1. Initial program 57.1%

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

   3. Simplified 57.1%

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

      1. add-cube-cbrt 57.1%

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

      2. pow3 57.1%

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

      3. associate-/r* 62.8%

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

      4. *-commutative 62.8%

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

      5. cbrt-prod 62.7%

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

      6. associate-/r* 57.1%

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

      7. cbrt-div 57.9%

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

      8. rem-cbrt-cube 69.2%

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

      9. cbrt-prod 81.5%

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

      10. pow2 81.5%

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

   6. Applied egg-rr 81.5%

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

   7. Taylor expanded in k around 0 75.9%

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

   8. Taylor expanded in k around 0 77.2%

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

   if 1.08e-15 < k

   1. Initial program 37.5%

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

      1. associate-/r* 37.5%

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

      2. unpow2 37.5%

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

      3. sqr-neg 37.5%

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

      4. distribute-frac-neg2 37.5%

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

      5. distribute-frac-neg2 37.5%

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

      6. unpow2 37.5%

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

      7. +-commutative 37.5%

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

      8. +-commutative 37.5%

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

      9. associate-/l/ 37.5%

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

      10. associate-*l/ 37.5%

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

      11. associate-/r/ 37.5%

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

   3. Simplified 37.5%

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

      1. associate-*r* 40.4%

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

      2. *-un-lft-identity 40.4%

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

      3. times-frac 41.6%

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

      4. associate-/l/ 41.6%

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

   6. Applied egg-rr 41.6%

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

      1. times-frac 40.4%

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

      2. *-commutative 40.4%

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

      3. times-frac 41.6%

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

      4. associate-*l/ 41.6%

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

      5. associate-*l* 41.6%

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

      6. times-frac 41.6%

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

      7. /-rgt-identity 41.6%

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

   8. Simplified 41.6%

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

   9. Taylor expanded in t around 0 72.7%

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

       1. times-frac 72.0%

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

   11. Simplified 72.0%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.08 \cdot 10^{-15}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \left(\frac{\ell}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 0.32:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{k\_m} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot k\_m\right)}\\ \mathbf{elif}\;k\_m \leq 1.08 \cdot 10^{+105}:\\ \;\;\;\;\frac{2}{\left(\sin k\_m \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(2 \cdot \tan k\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot {\left(\frac{\sqrt[3]{\frac{\ell}{{k\_m}^{2}}}}{t}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 0.32)
   (/ 2.0 (* (pow (* (cbrt k_m) (/ t (pow (cbrt l) 2.0))) 3.0) (* 2.0 k_m)))
   (if (<= k_m 1.08e+105)
     (/ 2.0 (* (* (sin k_m) (/ (pow t 3.0) (* l l))) (* 2.0 (tan k_m))))
     (* l (pow (/ (cbrt (/ l (pow k_m 2.0))) t) 3.0)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 0.32) {
		tmp = 2.0 / (pow((cbrt(k_m) * (t / pow(cbrt(l), 2.0))), 3.0) * (2.0 * k_m));
	} else if (k_m <= 1.08e+105) {
		tmp = 2.0 / ((sin(k_m) * (pow(t, 3.0) / (l * l))) * (2.0 * tan(k_m)));
	} else {
		tmp = l * pow((cbrt((l / pow(k_m, 2.0))) / t), 3.0);
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 0.32) {
		tmp = 2.0 / (Math.pow((Math.cbrt(k_m) * (t / Math.pow(Math.cbrt(l), 2.0))), 3.0) * (2.0 * k_m));
	} else if (k_m <= 1.08e+105) {
		tmp = 2.0 / ((Math.sin(k_m) * (Math.pow(t, 3.0) / (l * l))) * (2.0 * Math.tan(k_m)));
	} else {
		tmp = l * Math.pow((Math.cbrt((l / Math.pow(k_m, 2.0))) / t), 3.0);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 0.32)
		tmp = Float64(2.0 / Float64((Float64(cbrt(k_m) * Float64(t / (cbrt(l) ^ 2.0))) ^ 3.0) * Float64(2.0 * k_m)));
	elseif (k_m <= 1.08e+105)
		tmp = Float64(2.0 / Float64(Float64(sin(k_m) * Float64((t ^ 3.0) / Float64(l * l))) * Float64(2.0 * tan(k_m))));
	else
		tmp = Float64(l * (Float64(cbrt(Float64(l / (k_m ^ 2.0))) / t) ^ 3.0));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 0.32], N[(2.0 / N[(N[Power[N[(N[Power[k$95$m, 1/3], $MachinePrecision] * N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1.08e+105], N[(2.0 / N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[Power[N[(N[Power[N[(l / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / t), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 0.320000000000000007

   1. Initial program 56.5%

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

   3. Simplified 56.5%

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

      1. add-cube-cbrt 56.4%

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

      2. pow3 56.4%

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

      3. associate-/r* 62.1%

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

      4. *-commutative 62.1%

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

      5. cbrt-prod 62.0%

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

      6. associate-/r* 56.4%

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

      7. cbrt-div 57.3%

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

      8. rem-cbrt-cube 68.5%

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

      9. cbrt-prod 80.6%

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

      10. pow2 80.6%

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

   6. Applied egg-rr 80.6%

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

   7. Taylor expanded in k around 0 75.6%

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

   8. Taylor expanded in k around 0 76.9%

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

   if 0.320000000000000007 < k < 1.07999999999999994e105

   1. Initial program 35.0%

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

   3. Simplified 35.0%

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

   5. Taylor expanded in k around 0 49.2%

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

   if 1.07999999999999994e105 < k

   1. Initial program 40.0%

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

      1. associate-/r* 40.0%

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

      2. unpow2 40.0%

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

      3. sqr-neg 40.0%

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

      4. distribute-frac-neg2 40.0%

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

      5. distribute-frac-neg2 40.0%

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

      6. unpow2 40.0%

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

      7. +-commutative 40.0%

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

      8. +-commutative 40.0%

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

      9. associate-/l/ 40.0%

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

      10. associate-*l/ 40.0%

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

      11. associate-/r/ 40.0%

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

   3. Simplified 40.0%

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

      1. associate-*r* 44.1%

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

      2. *-un-lft-identity 44.1%

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

      3. times-frac 44.3%

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

      4. associate-/l/ 44.3%

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

   6. Applied egg-rr 44.3%

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

      1. times-frac 44.1%

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

      2. *-commutative 44.1%

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

      3. times-frac 44.3%

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

      4. associate-*l/ 44.2%

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

      5. associate-*l* 44.2%

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

      6. times-frac 44.2%

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

      7. /-rgt-identity 44.2%

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

   8. Simplified 44.2%

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

   9. Taylor expanded in k around 0 46.3%

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

       1. add-cube-cbrt 46.3%

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

       2. pow3 46.3%

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

       3. associate-/r* 46.2%

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

       4. cbrt-div 46.2%

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

       5. unpow3 46.2%

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

       6. add-cbrt-cube 63.5%

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

   11. Applied egg-rr 63.5%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.32:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \mathbf{elif}\;k \leq 1.08 \cdot 10^{+105}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(2 \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot {\left(\frac{\sqrt[3]{\frac{\ell}{{k}^{2}}}}{t}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 0.42:\\ \;\;\;\;\ell \cdot \frac{\ell}{{\left(t \cdot {\left(\sqrt[3]{k\_m}\right)}^{2}\right)}^{3}}\\ \mathbf{elif}\;k\_m \leq 1.8 \cdot 10^{+105}:\\ \;\;\;\;\frac{2}{\left(\sin k\_m \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(2 \cdot \tan k\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot {\left(\frac{\sqrt[3]{\frac{\ell}{{k\_m}^{2}}}}{t}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 0.42)
   (* l (/ l (pow (* t (pow (cbrt k_m) 2.0)) 3.0)))
   (if (<= k_m 1.8e+105)
     (/ 2.0 (* (* (sin k_m) (/ (pow t 3.0) (* l l))) (* 2.0 (tan k_m))))
     (* l (pow (/ (cbrt (/ l (pow k_m 2.0))) t) 3.0)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 0.42) {
		tmp = l * (l / pow((t * pow(cbrt(k_m), 2.0)), 3.0));
	} else if (k_m <= 1.8e+105) {
		tmp = 2.0 / ((sin(k_m) * (pow(t, 3.0) / (l * l))) * (2.0 * tan(k_m)));
	} else {
		tmp = l * pow((cbrt((l / pow(k_m, 2.0))) / t), 3.0);
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 0.42) {
		tmp = l * (l / Math.pow((t * Math.pow(Math.cbrt(k_m), 2.0)), 3.0));
	} else if (k_m <= 1.8e+105) {
		tmp = 2.0 / ((Math.sin(k_m) * (Math.pow(t, 3.0) / (l * l))) * (2.0 * Math.tan(k_m)));
	} else {
		tmp = l * Math.pow((Math.cbrt((l / Math.pow(k_m, 2.0))) / t), 3.0);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 0.42)
		tmp = Float64(l * Float64(l / (Float64(t * (cbrt(k_m) ^ 2.0)) ^ 3.0)));
	elseif (k_m <= 1.8e+105)
		tmp = Float64(2.0 / Float64(Float64(sin(k_m) * Float64((t ^ 3.0) / Float64(l * l))) * Float64(2.0 * tan(k_m))));
	else
		tmp = Float64(l * (Float64(cbrt(Float64(l / (k_m ^ 2.0))) / t) ^ 3.0));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 0.42], N[(l * N[(l / N[Power[N[(t * N[Power[N[Power[k$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1.8e+105], N[(2.0 / N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[Power[N[(N[Power[N[(l / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / t), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 0.419999999999999984

   1. Initial program 56.2%

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

      1. associate-/r* 56.2%

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

      2. unpow2 56.2%

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

      3. sqr-neg 56.2%

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

      4. distribute-frac-neg2 56.2%

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

      5. distribute-frac-neg2 56.2%

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

      6. unpow2 56.2%

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

      7. +-commutative 56.2%

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

      8. +-commutative 56.2%

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

      9. associate-/l/ 56.2%

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

      10. associate-*l/ 57.1%

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

      11. associate-/r/ 55.7%

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

   3. Simplified 55.7%

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

      1. associate-*r* 63.4%

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

      2. *-un-lft-identity 63.4%

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

      3. times-frac 64.5%

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

      4. associate-/l/ 64.5%

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

   6. Applied egg-rr 64.5%

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

      1. times-frac 63.4%

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

      2. *-commutative 63.4%

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

      3. times-frac 64.5%

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

      4. associate-*l/ 65.0%

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

      5. associate-*l* 59.7%

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

      6. times-frac 59.6%

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

      7. /-rgt-identity 59.6%

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

   8. Simplified 59.6%

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

   9. Taylor expanded in k around 0 58.4%

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

       1. add-cube-cbrt 58.4%

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

       2. pow3 58.4%

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

       3. *-commutative 58.4%

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

       4. cbrt-prod 58.3%

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

       5. unpow3 58.3%

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

       6. add-cbrt-cube 67.4%

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

       7. unpow2 67.4%

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

       8. cbrt-prod 75.2%

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

       9. pow2 75.2%

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

   11. Applied egg-rr 75.2%

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

   if 0.419999999999999984 < k < 1.7999999999999999e105

   1. Initial program 36.2%

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

   3. Simplified 36.2%

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

   5. Taylor expanded in k around 0 51.1%

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

   if 1.7999999999999999e105 < k

   1. Initial program 40.0%

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

      1. associate-/r* 40.0%

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

      2. unpow2 40.0%

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

      3. sqr-neg 40.0%

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

      4. distribute-frac-neg2 40.0%

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

      5. distribute-frac-neg2 40.0%

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

      6. unpow2 40.0%

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

      7. +-commutative 40.0%

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

      8. +-commutative 40.0%

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

      9. associate-/l/ 40.0%

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

      10. associate-*l/ 40.0%

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

      11. associate-/r/ 40.0%

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

   3. Simplified 40.0%

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

      1. associate-*r* 44.1%

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

      2. *-un-lft-identity 44.1%

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

      3. times-frac 44.3%

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

      4. associate-/l/ 44.3%

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

   6. Applied egg-rr 44.3%

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

      1. times-frac 44.1%

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

      2. *-commutative 44.1%

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

      3. times-frac 44.3%

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

      4. associate-*l/ 44.2%

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

      5. associate-*l* 44.2%

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

      6. times-frac 44.2%

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

      7. /-rgt-identity 44.2%

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

   8. Simplified 44.2%

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

   9. Taylor expanded in k around 0 46.3%

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

       1. add-cube-cbrt 46.3%

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

       2. pow3 46.3%

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

       3. associate-/r* 46.2%

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

       4. cbrt-div 46.2%

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

       5. unpow3 46.2%

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

       6. add-cbrt-cube 63.5%

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

   11. Applied egg-rr 63.5%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.42:\\ \;\;\;\;\ell \cdot \frac{\ell}{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \mathbf{elif}\;k \leq 1.8 \cdot 10^{+105}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(2 \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot {\left(\frac{\sqrt[3]{\frac{\ell}{{k}^{2}}}}{t}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 71.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.5 \cdot 10^{-123}:\\ \;\;\;\;\ell \cdot \frac{\ell}{{\left(t \cdot {\left(\sqrt[3]{k\_m}\right)}^{2}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot {\left(\frac{\sqrt[3]{\frac{\ell}{{k\_m}^{2}}}}{t}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.5e-123)
   (* l (/ l (pow (* t (pow (cbrt k_m) 2.0)) 3.0)))
   (* l (pow (/ (cbrt (/ l (pow k_m 2.0))) t) 3.0))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.5e-123) {
		tmp = l * (l / pow((t * pow(cbrt(k_m), 2.0)), 3.0));
	} else {
		tmp = l * pow((cbrt((l / pow(k_m, 2.0))) / t), 3.0);
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.5e-123) {
		tmp = l * (l / Math.pow((t * Math.pow(Math.cbrt(k_m), 2.0)), 3.0));
	} else {
		tmp = l * Math.pow((Math.cbrt((l / Math.pow(k_m, 2.0))) / t), 3.0);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.5e-123)
		tmp = Float64(l * Float64(l / (Float64(t * (cbrt(k_m) ^ 2.0)) ^ 3.0)));
	else
		tmp = Float64(l * (Float64(cbrt(Float64(l / (k_m ^ 2.0))) / t) ^ 3.0));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.5e-123], N[(l * N[(l / N[Power[N[(t * N[Power[N[Power[k$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[Power[N[(N[Power[N[(l / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / t), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.49999999999999992e-123

   1. Initial program 55.8%

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

      1. associate-/r* 55.8%

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

      2. unpow2 55.8%

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

      3. sqr-neg 55.8%

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

      4. distribute-frac-neg2 55.8%

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

      5. distribute-frac-neg2 55.8%

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

      6. unpow2 55.8%

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

      7. +-commutative 55.8%

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

      8. +-commutative 55.8%

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

      9. associate-/l/ 55.8%

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

      10. associate-*l/ 56.8%

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

      11. associate-/r/ 55.8%

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

   3. Simplified 55.8%

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

      1. associate-*r* 64.4%

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

      2. *-un-lft-identity 64.4%

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

      3. times-frac 65.0%

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

      4. associate-/l/ 65.0%

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

   6. Applied egg-rr 65.0%

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

      1. times-frac 64.4%

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

      2. *-commutative 64.4%

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

      3. times-frac 65.0%

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

      4. associate-*l/ 65.1%

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

      5. associate-*l* 59.2%

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

      6. times-frac 59.0%

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

      7. /-rgt-identity 59.0%

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

   8. Simplified 59.0%

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

   9. Taylor expanded in k around 0 56.5%

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

       1. add-cube-cbrt 56.5%

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

       2. pow3 56.5%

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

       3. *-commutative 56.5%

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

       4. cbrt-prod 56.4%

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

       5. unpow3 56.4%

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

       6. add-cbrt-cube 65.9%

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

       7. unpow2 65.9%

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

       8. cbrt-prod 74.5%

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

       9. pow2 74.5%

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

   11. Applied egg-rr 74.5%

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

   if 1.49999999999999992e-123 < k

   1. Initial program 42.7%

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

      1. associate-/r* 42.7%

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

      2. unpow2 42.7%

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

      3. sqr-neg 42.7%

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

      4. distribute-frac-neg2 42.7%

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

      5. distribute-frac-neg2 42.7%

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

      6. unpow2 42.7%

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

      7. +-commutative 42.7%

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

      8. +-commutative 42.7%

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

      9. associate-/l/ 42.7%

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

      10. associate-*l/ 42.7%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1} \]

      11. associate-/r/ 41.7%

          \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1} \]

   3. Simplified 41.7%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* 44.2%

         \[\leadsto \frac{\color{blue}{\left(\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell\right) \cdot \ell}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. *-un-lft-identity 44.2%

         \[\leadsto \frac{\left(\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell\right) \cdot \ell}{\color{blue}{1 \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]

      3. times-frac 45.2%

         \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell}{1} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

      4. associate-/l/ 45.2%

         \[\leadsto \frac{\color{blue}{\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \cdot \ell}{1} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   6. Applied egg-rr 45.2%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \ell}{1} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   7. Step-by-step derivation

      1. times-frac 44.2%

         \[\leadsto \color{blue}{\frac{\left(\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \ell\right) \cdot \ell}{1 \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]

      2. *-commutative 44.2%

         \[\leadsto \frac{\left(\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \ell\right) \cdot \ell}{\color{blue}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot 1}} \]

      3. times-frac 45.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{1}} \]

      4. associate-*l/ 46.1%

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot \ell}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{1} \]

      5. associate-*l* 46.1%

         \[\leadsto \frac{\frac{2 \cdot \ell}{\color{blue}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{1} \]

      6. times-frac 46.3%

         \[\leadsto \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \frac{\ell}{\sin k \cdot \tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{1} \]

      7. /-rgt-identity 46.3%

         \[\leadsto \frac{\frac{2}{{t}^{3}} \cdot \frac{\ell}{\sin k \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\ell} \]

   8. Simplified 46.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}} \cdot \frac{\ell}{\sin k \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \ell} \]

   9. Taylor expanded in k around 0 45.7%

      \[\leadsto \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \cdot \ell \]
   10. Step-by-step derivation

       1. add-cube-cbrt 45.7%

          \[\leadsto \color{blue}{\left(\left(\sqrt[3]{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \cdot \sqrt[3]{\frac{\ell}{{k}^{2} \cdot {t}^{3}}}\right) \cdot \sqrt[3]{\frac{\ell}{{k}^{2} \cdot {t}^{3}}}\right)} \cdot \ell \]

       2. pow3 45.7%

          \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{\ell}{{k}^{2} \cdot {t}^{3}}}\right)}^{3}} \cdot \ell \]

       3. associate-/r* 45.8%

          \[\leadsto {\left(\sqrt[3]{\color{blue}{\frac{\frac{\ell}{{k}^{2}}}{{t}^{3}}}}\right)}^{3} \cdot \ell \]

       4. cbrt-div 45.8%

          \[\leadsto {\color{blue}{\left(\frac{\sqrt[3]{\frac{\ell}{{k}^{2}}}}{\sqrt[3]{{t}^{3}}}\right)}}^{3} \cdot \ell \]

       5. unpow3 45.8%

          \[\leadsto {\left(\frac{\sqrt[3]{\frac{\ell}{{k}^{2}}}}{\sqrt[3]{\color{blue}{\left(t \cdot t\right) \cdot t}}}\right)}^{3} \cdot \ell \]

       6. add-cbrt-cube 57.8%

          \[\leadsto {\left(\frac{\sqrt[3]{\frac{\ell}{{k}^{2}}}}{\color{blue}{t}}\right)}^{3} \cdot \ell \]

   11. Applied egg-rr 57.8%

       \[\leadsto \color{blue}{{\left(\frac{\sqrt[3]{\frac{\ell}{{k}^{2}}}}{t}\right)}^{3}} \cdot \ell \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{-123}:\\ \;\;\;\;\ell \cdot \frac{\ell}{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot {\left(\frac{\sqrt[3]{\frac{\ell}{{k}^{2}}}}{t}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 2.2 \cdot 10^{-88}:\\ \;\;\;\;\frac{2}{{k\_m}^{4} \cdot \frac{t}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{{\left(t \cdot {\left(\sqrt[3]{k\_m}\right)}^{2}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 2.2e-88)
   (/ 2.0 (* (pow k_m 4.0) (/ t (pow l 2.0))))
   (* l (/ l (pow (* t (pow (cbrt k_m) 2.0)) 3.0)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 2.2e-88) {
		tmp = 2.0 / (pow(k_m, 4.0) * (t / pow(l, 2.0)));
	} else {
		tmp = l * (l / pow((t * pow(cbrt(k_m), 2.0)), 3.0));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 2.2e-88) {
		tmp = 2.0 / (Math.pow(k_m, 4.0) * (t / Math.pow(l, 2.0)));
	} else {
		tmp = l * (l / Math.pow((t * Math.pow(Math.cbrt(k_m), 2.0)), 3.0));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 2.2e-88)
		tmp = Float64(2.0 / Float64((k_m ^ 4.0) * Float64(t / (l ^ 2.0))));
	else
		tmp = Float64(l * Float64(l / (Float64(t * (cbrt(k_m) ^ 2.0)) ^ 3.0)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 2.2e-88], N[(2.0 / N[(N[Power[k$95$m, 4.0], $MachinePrecision] * N[(t / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(l / N[Power[N[(t * N[Power[N[Power[k$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.20000000000000005e-88

   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 47.8%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around inf 23.1%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\frac{{k}^{2}}{{t}^{2}}}\right)} \]
   6. Step-by-step derivation

      1. unpow2 23.1%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \frac{\color{blue}{k \cdot k}}{{t}^{2}}\right)} \]

      2. unpow2 23.1%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \frac{k \cdot k}{\color{blue}{t \cdot t}}\right)} \]

      3. times-frac 37.2%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(\frac{k}{t} \cdot \frac{k}{t}\right)}\right)} \]

      4. unpow2 37.2%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right)} \]

   7. Simplified 37.2%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right)} \]

   8. Taylor expanded in k around 0 50.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
   9. Step-by-step derivation

      1. associate-/l* 50.5%

         \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}} \]

   10. Simplified 50.5%

       \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}} \]

   if 2.20000000000000005e-88 < t

   1. Initial program 60.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 60.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. unpow2 60.9%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) + 1} \]

      3. sqr-neg 60.9%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + \color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)}\right) + 1} \]

      4. distribute-frac-neg2 60.9%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + \color{blue}{\frac{k}{-t}} \cdot \left(-\frac{k}{t}\right)\right) + 1} \]

      5. distribute-frac-neg2 60.9%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + \frac{k}{-t} \cdot \color{blue}{\frac{k}{-t}}\right) + 1} \]

      6. unpow2 60.9%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) + 1} \]

      7. +-commutative 60.9%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 1\right)} + 1} \]

      8. +-commutative 60.9%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)} + 1} \]

      9. associate-/l/ 60.9%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1} \]

      10. associate-*l/ 62.3%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1} \]

      11. associate-/r/ 62.2%

          \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1} \]

   3. Simplified 62.2%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* 68.6%

         \[\leadsto \frac{\color{blue}{\left(\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell\right) \cdot \ell}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. *-un-lft-identity 68.6%

         \[\leadsto \frac{\left(\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell\right) \cdot \ell}{\color{blue}{1 \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]

      3. times-frac 70.1%

         \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell}{1} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

      4. associate-/l/ 70.2%

         \[\leadsto \frac{\color{blue}{\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \cdot \ell}{1} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   6. Applied egg-rr 70.2%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \ell}{1} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   7. Step-by-step derivation

      1. times-frac 68.7%

         \[\leadsto \color{blue}{\frac{\left(\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \ell\right) \cdot \ell}{1 \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]

      2. *-commutative 68.7%

         \[\leadsto \frac{\left(\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \ell\right) \cdot \ell}{\color{blue}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot 1}} \]

      3. times-frac 70.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{1}} \]

      4. associate-*l/ 70.2%

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot \ell}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{1} \]

      5. associate-*l* 63.8%

         \[\leadsto \frac{\frac{2 \cdot \ell}{\color{blue}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{1} \]

      6. times-frac 63.7%

         \[\leadsto \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \frac{\ell}{\sin k \cdot \tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{1} \]

      7. /-rgt-identity 63.7%

         \[\leadsto \frac{\frac{2}{{t}^{3}} \cdot \frac{\ell}{\sin k \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\ell} \]

   8. Simplified 63.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}} \cdot \frac{\ell}{\sin k \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \ell} \]

   9. Taylor expanded in k around 0 56.6%

      \[\leadsto \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \cdot \ell \]
   10. Step-by-step derivation

       1. add-cube-cbrt 56.5%

          \[\leadsto \frac{\ell}{\color{blue}{\left(\sqrt[3]{{k}^{2} \cdot {t}^{3}} \cdot \sqrt[3]{{k}^{2} \cdot {t}^{3}}\right) \cdot \sqrt[3]{{k}^{2} \cdot {t}^{3}}}} \cdot \ell \]

       2. pow3 56.6%

          \[\leadsto \frac{\ell}{\color{blue}{{\left(\sqrt[3]{{k}^{2} \cdot {t}^{3}}\right)}^{3}}} \cdot \ell \]

       3. *-commutative 56.6%

          \[\leadsto \frac{\ell}{{\left(\sqrt[3]{\color{blue}{{t}^{3} \cdot {k}^{2}}}\right)}^{3}} \cdot \ell \]

       4. cbrt-prod 56.5%

          \[\leadsto \frac{\ell}{{\color{blue}{\left(\sqrt[3]{{t}^{3}} \cdot \sqrt[3]{{k}^{2}}\right)}}^{3}} \cdot \ell \]

       5. unpow3 56.5%

          \[\leadsto \frac{\ell}{{\left(\sqrt[3]{\color{blue}{\left(t \cdot t\right) \cdot t}} \cdot \sqrt[3]{{k}^{2}}\right)}^{3}} \cdot \ell \]

       6. add-cbrt-cube 66.3%

          \[\leadsto \frac{\ell}{{\left(\color{blue}{t} \cdot \sqrt[3]{{k}^{2}}\right)}^{3}} \cdot \ell \]

       7. unpow2 66.3%

          \[\leadsto \frac{\ell}{{\left(t \cdot \sqrt[3]{\color{blue}{k \cdot k}}\right)}^{3}} \cdot \ell \]

       8. cbrt-prod 77.6%

          \[\leadsto \frac{\ell}{{\left(t \cdot \color{blue}{\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)}\right)}^{3}} \cdot \ell \]

       9. pow2 77.6%

          \[\leadsto \frac{\ell}{{\left(t \cdot \color{blue}{{\left(\sqrt[3]{k}\right)}^{2}}\right)}^{3}} \cdot \ell \]

   11. Applied egg-rr 77.6%

       \[\leadsto \frac{\ell}{\color{blue}{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}} \cdot \ell \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.2 \cdot 10^{-88}:\\ \;\;\;\;\frac{2}{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 61.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.7 \cdot 10^{-162}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\_m\right) \cdot \frac{k\_m \cdot {t}^{3}}{{\ell}^{2}}}\\ \mathbf{elif}\;k\_m \leq 9.8 \cdot 10^{-16}:\\ \;\;\;\;\ell \cdot \frac{1}{{k\_m}^{2} \cdot \frac{{t}^{3}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k\_m}^{4}}{{\ell}^{2}}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.7e-162)
   (/ 2.0 (* (* 2.0 k_m) (/ (* k_m (pow t 3.0)) (pow l 2.0))))
   (if (<= k_m 9.8e-16)
     (* l (/ 1.0 (* (pow k_m 2.0) (/ (pow t 3.0) l))))
     (/ 2.0 (/ (* t (pow k_m 4.0)) (pow l 2.0))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.7e-162) {
		tmp = 2.0 / ((2.0 * k_m) * ((k_m * pow(t, 3.0)) / pow(l, 2.0)));
	} else if (k_m <= 9.8e-16) {
		tmp = l * (1.0 / (pow(k_m, 2.0) * (pow(t, 3.0) / l)));
	} else {
		tmp = 2.0 / ((t * pow(k_m, 4.0)) / pow(l, 2.0));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.7d-162) then
        tmp = 2.0d0 / ((2.0d0 * k_m) * ((k_m * (t ** 3.0d0)) / (l ** 2.0d0)))
    else if (k_m <= 9.8d-16) then
        tmp = l * (1.0d0 / ((k_m ** 2.0d0) * ((t ** 3.0d0) / l)))
    else
        tmp = 2.0d0 / ((t * (k_m ** 4.0d0)) / (l ** 2.0d0))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.7e-162) {
		tmp = 2.0 / ((2.0 * k_m) * ((k_m * Math.pow(t, 3.0)) / Math.pow(l, 2.0)));
	} else if (k_m <= 9.8e-16) {
		tmp = l * (1.0 / (Math.pow(k_m, 2.0) * (Math.pow(t, 3.0) / l)));
	} else {
		tmp = 2.0 / ((t * Math.pow(k_m, 4.0)) / Math.pow(l, 2.0));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1.7e-162:
		tmp = 2.0 / ((2.0 * k_m) * ((k_m * math.pow(t, 3.0)) / math.pow(l, 2.0)))
	elif k_m <= 9.8e-16:
		tmp = l * (1.0 / (math.pow(k_m, 2.0) * (math.pow(t, 3.0) / l)))
	else:
		tmp = 2.0 / ((t * math.pow(k_m, 4.0)) / math.pow(l, 2.0))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.7e-162)
		tmp = Float64(2.0 / Float64(Float64(2.0 * k_m) * Float64(Float64(k_m * (t ^ 3.0)) / (l ^ 2.0))));
	elseif (k_m <= 9.8e-16)
		tmp = Float64(l * Float64(1.0 / Float64((k_m ^ 2.0) * Float64((t ^ 3.0) / l))));
	else
		tmp = Float64(2.0 / Float64(Float64(t * (k_m ^ 4.0)) / (l ^ 2.0)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.7e-162)
		tmp = 2.0 / ((2.0 * k_m) * ((k_m * (t ^ 3.0)) / (l ^ 2.0)));
	elseif (k_m <= 9.8e-16)
		tmp = l * (1.0 / ((k_m ^ 2.0) * ((t ^ 3.0) / l)));
	else
		tmp = 2.0 / ((t * (k_m ^ 4.0)) / (l ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.7e-162], N[(2.0 / N[(N[(2.0 * k$95$m), $MachinePrecision] * N[(N[(k$95$m * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 9.8e-16], N[(l * N[(1.0 / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.7e-162

   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. add-cube-cbrt 54.2%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}\right)} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      2. pow3 54.2%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}\right)}^{3}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      3. associate-/r* 61.2%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      4. *-commutative 61.2%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\color{blue}{\sin k \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      5. cbrt-prod 61.1%

         \[\leadsto \frac{2}{{\color{blue}{\left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\frac{\frac{{t}^{3}}{\ell}}{\ell}}\right)}}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      6. associate-/r* 54.2%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      7. cbrt-div 55.3%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \color{blue}{\frac{\sqrt[3]{{t}^{3}}}{\sqrt[3]{\ell \cdot \ell}}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      8. rem-cbrt-cube 66.9%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{\color{blue}{t}}{\sqrt[3]{\ell \cdot \ell}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      9. cbrt-prod 80.5%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{\color{blue}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      10. pow2 80.5%

          \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   6. Applied egg-rr 80.5%

      \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   7. Taylor expanded in k around 0 73.1%

      \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \color{blue}{\left(2 \cdot k\right)}} \]

   8. Taylor expanded in k around 0 53.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot {t}^{3}}{{\ell}^{2}}} \cdot \left(2 \cdot k\right)} \]

   if 1.7e-162 < k < 9.7999999999999995e-16

   1. Initial program 71.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 71.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. unpow2 71.1%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) + 1} \]

      3. sqr-neg 71.1%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + \color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)}\right) + 1} \]

      4. distribute-frac-neg2 71.1%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + \color{blue}{\frac{k}{-t}} \cdot \left(-\frac{k}{t}\right)\right) + 1} \]

      5. distribute-frac-neg2 71.1%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + \frac{k}{-t} \cdot \color{blue}{\frac{k}{-t}}\right) + 1} \]

      6. unpow2 71.1%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) + 1} \]

      7. +-commutative 71.1%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 1\right)} + 1} \]

      8. +-commutative 71.1%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)} + 1} \]

      9. associate-/l/ 71.1%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1} \]

      10. associate-*l/ 71.0%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1} \]

      11. associate-/r/ 68.0%

          \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1} \]

   3. Simplified 68.0%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* 77.4%

         \[\leadsto \frac{\color{blue}{\left(\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell\right) \cdot \ell}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. *-un-lft-identity 77.4%

         \[\leadsto \frac{\left(\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell\right) \cdot \ell}{\color{blue}{1 \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]

      3. times-frac 77.4%

         \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell}{1} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

      4. associate-/l/ 77.4%

         \[\leadsto \frac{\color{blue}{\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \cdot \ell}{1} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   6. Applied egg-rr 77.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \ell}{1} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   7. Step-by-step derivation

      1. times-frac 77.4%

         \[\leadsto \color{blue}{\frac{\left(\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \ell\right) \cdot \ell}{1 \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]

      2. *-commutative 77.4%

         \[\leadsto \frac{\left(\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \ell\right) \cdot \ell}{\color{blue}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot 1}} \]

      3. times-frac 77.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{1}} \]

      4. associate-*l/ 80.2%

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot \ell}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{1} \]

      5. associate-*l* 80.1%

         \[\leadsto \frac{\frac{2 \cdot \ell}{\color{blue}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{1} \]

      6. times-frac 80.4%

         \[\leadsto \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \frac{\ell}{\sin k \cdot \tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{1} \]

      7. /-rgt-identity 80.4%

         \[\leadsto \frac{\frac{2}{{t}^{3}} \cdot \frac{\ell}{\sin k \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\ell} \]

   8. Simplified 80.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}} \cdot \frac{\ell}{\sin k \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \ell} \]

   9. Taylor expanded in k around 0 86.8%

      \[\leadsto \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \cdot \ell \]
   10. Step-by-step derivation

       1. clear-num 86.8%

          \[\leadsto \color{blue}{\frac{1}{\frac{{k}^{2} \cdot {t}^{3}}{\ell}}} \cdot \ell \]

       2. inv-pow 86.8%

          \[\leadsto \color{blue}{{\left(\frac{{k}^{2} \cdot {t}^{3}}{\ell}\right)}^{-1}} \cdot \ell \]

   11. Applied egg-rr 86.8%

       \[\leadsto \color{blue}{{\left(\frac{{k}^{2} \cdot {t}^{3}}{\ell}\right)}^{-1}} \cdot \ell \]
   12. Step-by-step derivation

       1. unpow-1 86.8%

          \[\leadsto \color{blue}{\frac{1}{\frac{{k}^{2} \cdot {t}^{3}}{\ell}}} \cdot \ell \]

       2. associate-/l* 87.2%

          \[\leadsto \frac{1}{\color{blue}{{k}^{2} \cdot \frac{{t}^{3}}{\ell}}} \cdot \ell \]

   13. Simplified 87.2%

       \[\leadsto \color{blue}{\frac{1}{{k}^{2} \cdot \frac{{t}^{3}}{\ell}}} \cdot \ell \]

   if 9.7999999999999995e-16 < k

   1. Initial program 37.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 37.5%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around inf 22.3%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\frac{{k}^{2}}{{t}^{2}}}\right)} \]
   6. Step-by-step derivation

      1. unpow2 22.3%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \frac{\color{blue}{k \cdot k}}{{t}^{2}}\right)} \]

      2. unpow2 22.3%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \frac{k \cdot k}{\color{blue}{t \cdot t}}\right)} \]

      3. times-frac 33.8%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(\frac{k}{t} \cdot \frac{k}{t}\right)}\right)} \]

      4. unpow2 33.8%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right)} \]

   7. Simplified 33.8%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right)} \]

   8. Taylor expanded in k around 0 47.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.7 \cdot 10^{-162}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \frac{k \cdot {t}^{3}}{{\ell}^{2}}}\\ \mathbf{elif}\;k \leq 9.8 \cdot 10^{-16}:\\ \;\;\;\;\ell \cdot \frac{1}{{k}^{2} \cdot \frac{{t}^{3}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{4}}{{\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 61.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 3.2 \cdot 10^{-162}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\_m\right) \cdot \left(\left(k\_m \cdot {t}^{3}\right) \cdot {\ell}^{-2}\right)}\\ \mathbf{elif}\;k\_m \leq 1.08 \cdot 10^{-15}:\\ \;\;\;\;\ell \cdot \frac{1}{{k\_m}^{2} \cdot \frac{{t}^{3}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k\_m}^{4}}{{\ell}^{2}}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 3.2e-162)
   (/ 2.0 (* (* 2.0 k_m) (* (* k_m (pow t 3.0)) (pow l -2.0))))
   (if (<= k_m 1.08e-15)
     (* l (/ 1.0 (* (pow k_m 2.0) (/ (pow t 3.0) l))))
     (/ 2.0 (/ (* t (pow k_m 4.0)) (pow l 2.0))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 3.2e-162) {
		tmp = 2.0 / ((2.0 * k_m) * ((k_m * pow(t, 3.0)) * pow(l, -2.0)));
	} else if (k_m <= 1.08e-15) {
		tmp = l * (1.0 / (pow(k_m, 2.0) * (pow(t, 3.0) / l)));
	} else {
		tmp = 2.0 / ((t * pow(k_m, 4.0)) / pow(l, 2.0));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 3.2d-162) then
        tmp = 2.0d0 / ((2.0d0 * k_m) * ((k_m * (t ** 3.0d0)) * (l ** (-2.0d0))))
    else if (k_m <= 1.08d-15) then
        tmp = l * (1.0d0 / ((k_m ** 2.0d0) * ((t ** 3.0d0) / l)))
    else
        tmp = 2.0d0 / ((t * (k_m ** 4.0d0)) / (l ** 2.0d0))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 3.2e-162) {
		tmp = 2.0 / ((2.0 * k_m) * ((k_m * Math.pow(t, 3.0)) * Math.pow(l, -2.0)));
	} else if (k_m <= 1.08e-15) {
		tmp = l * (1.0 / (Math.pow(k_m, 2.0) * (Math.pow(t, 3.0) / l)));
	} else {
		tmp = 2.0 / ((t * Math.pow(k_m, 4.0)) / Math.pow(l, 2.0));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 3.2e-162:
		tmp = 2.0 / ((2.0 * k_m) * ((k_m * math.pow(t, 3.0)) * math.pow(l, -2.0)))
	elif k_m <= 1.08e-15:
		tmp = l * (1.0 / (math.pow(k_m, 2.0) * (math.pow(t, 3.0) / l)))
	else:
		tmp = 2.0 / ((t * math.pow(k_m, 4.0)) / math.pow(l, 2.0))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 3.2e-162)
		tmp = Float64(2.0 / Float64(Float64(2.0 * k_m) * Float64(Float64(k_m * (t ^ 3.0)) * (l ^ -2.0))));
	elseif (k_m <= 1.08e-15)
		tmp = Float64(l * Float64(1.0 / Float64((k_m ^ 2.0) * Float64((t ^ 3.0) / l))));
	else
		tmp = Float64(2.0 / Float64(Float64(t * (k_m ^ 4.0)) / (l ^ 2.0)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 3.2e-162)
		tmp = 2.0 / ((2.0 * k_m) * ((k_m * (t ^ 3.0)) * (l ^ -2.0)));
	elseif (k_m <= 1.08e-15)
		tmp = l * (1.0 / ((k_m ^ 2.0) * ((t ^ 3.0) / l)));
	else
		tmp = 2.0 / ((t * (k_m ^ 4.0)) / (l ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 3.2e-162], N[(2.0 / N[(N[(2.0 * k$95$m), $MachinePrecision] * N[(N[(k$95$m * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] * N[Power[l, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1.08e-15], N[(l * N[(1.0 / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 3.19999999999999975e-162

   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. add-cube-cbrt 54.2%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}\right)} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      2. pow3 54.2%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}\right)}^{3}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      3. associate-/r* 61.2%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      4. *-commutative 61.2%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\color{blue}{\sin k \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      5. cbrt-prod 61.1%

         \[\leadsto \frac{2}{{\color{blue}{\left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\frac{\frac{{t}^{3}}{\ell}}{\ell}}\right)}}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      6. associate-/r* 54.2%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      7. cbrt-div 55.3%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \color{blue}{\frac{\sqrt[3]{{t}^{3}}}{\sqrt[3]{\ell \cdot \ell}}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      8. rem-cbrt-cube 66.9%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{\color{blue}{t}}{\sqrt[3]{\ell \cdot \ell}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      9. cbrt-prod 80.5%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{\color{blue}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      10. pow2 80.5%

          \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   6. Applied egg-rr 80.5%

      \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   7. Taylor expanded in k around 0 73.1%

      \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \color{blue}{\left(2 \cdot k\right)}} \]

   8. Taylor expanded in k around 0 65.0%

      \[\leadsto \frac{2}{{\color{blue}{\left(\sqrt[3]{\frac{k}{{\ell}^{2}}} \cdot t\right)}}^{3} \cdot \left(2 \cdot k\right)} \]
   9. Step-by-step derivation

      1. div-inv 65.0%

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{\left(\sqrt[3]{\frac{k}{{\ell}^{2}}} \cdot t\right)}^{3} \cdot \left(2 \cdot k\right)}} \]

      2. *-commutative 65.0%

         \[\leadsto 2 \cdot \frac{1}{{\color{blue}{\left(t \cdot \sqrt[3]{\frac{k}{{\ell}^{2}}}\right)}}^{3} \cdot \left(2 \cdot k\right)} \]

      3. unpow-prod-down 53.5%

         \[\leadsto 2 \cdot \frac{1}{\color{blue}{\left({t}^{3} \cdot {\left(\sqrt[3]{\frac{k}{{\ell}^{2}}}\right)}^{3}\right)} \cdot \left(2 \cdot k\right)} \]

      4. pow1/3 26.1%

         \[\leadsto 2 \cdot \frac{1}{\left({t}^{3} \cdot {\color{blue}{\left({\left(\frac{k}{{\ell}^{2}}\right)}^{0.3333333333333333}\right)}}^{3}\right) \cdot \left(2 \cdot k\right)} \]

      5. pow2 26.1%

         \[\leadsto 2 \cdot \frac{1}{\left({t}^{3} \cdot {\left({\left(\frac{k}{\color{blue}{\ell \cdot \ell}}\right)}^{0.3333333333333333}\right)}^{3}\right) \cdot \left(2 \cdot k\right)} \]

      6. pow-pow 53.6%

         \[\leadsto 2 \cdot \frac{1}{\left({t}^{3} \cdot \color{blue}{{\left(\frac{k}{\ell \cdot \ell}\right)}^{\left(0.3333333333333333 \cdot 3\right)}}\right) \cdot \left(2 \cdot k\right)} \]

      7. metadata-eval 53.6%

         \[\leadsto 2 \cdot \frac{1}{\left({t}^{3} \cdot {\left(\frac{k}{\ell \cdot \ell}\right)}^{\color{blue}{1}}\right) \cdot \left(2 \cdot k\right)} \]

      8. pow1 53.6%

         \[\leadsto 2 \cdot \frac{1}{\left({t}^{3} \cdot \color{blue}{\frac{k}{\ell \cdot \ell}}\right) \cdot \left(2 \cdot k\right)} \]

      9. div-inv 53.0%

         \[\leadsto 2 \cdot \frac{1}{\left({t}^{3} \cdot \color{blue}{\left(k \cdot \frac{1}{\ell \cdot \ell}\right)}\right) \cdot \left(2 \cdot k\right)} \]

      10. pow2 53.0%

          \[\leadsto 2 \cdot \frac{1}{\left({t}^{3} \cdot \left(k \cdot \frac{1}{\color{blue}{{\ell}^{2}}}\right)\right) \cdot \left(2 \cdot k\right)} \]

      11. pow-flip 53.1%

          \[\leadsto 2 \cdot \frac{1}{\left({t}^{3} \cdot \left(k \cdot \color{blue}{{\ell}^{\left(-2\right)}}\right)\right) \cdot \left(2 \cdot k\right)} \]

      12. metadata-eval 53.1%

          \[\leadsto 2 \cdot \frac{1}{\left({t}^{3} \cdot \left(k \cdot {\ell}^{\color{blue}{-2}}\right)\right) \cdot \left(2 \cdot k\right)} \]

   10. Applied egg-rr 53.1%

       \[\leadsto \color{blue}{2 \cdot \frac{1}{\left({t}^{3} \cdot \left(k \cdot {\ell}^{-2}\right)\right) \cdot \left(2 \cdot k\right)}} \]
   11. Step-by-step derivation

       1. associate-*r/ 53.1%

          \[\leadsto \color{blue}{\frac{2 \cdot 1}{\left({t}^{3} \cdot \left(k \cdot {\ell}^{-2}\right)\right) \cdot \left(2 \cdot k\right)}} \]

       2. metadata-eval 53.1%

          \[\leadsto \frac{\color{blue}{2}}{\left({t}^{3} \cdot \left(k \cdot {\ell}^{-2}\right)\right) \cdot \left(2 \cdot k\right)} \]

       3. associate-*r* 53.2%

          \[\leadsto \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot k\right) \cdot {\ell}^{-2}\right)} \cdot \left(2 \cdot k\right)} \]

       4. *-commutative 53.2%

          \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot {t}^{3}\right)} \cdot {\ell}^{-2}\right) \cdot \left(2 \cdot k\right)} \]

   12. Simplified 53.2%

       \[\leadsto \color{blue}{\frac{2}{\left(\left(k \cdot {t}^{3}\right) \cdot {\ell}^{-2}\right) \cdot \left(2 \cdot k\right)}} \]

   if 3.19999999999999975e-162 < k < 1.08e-15

   1. Initial program 71.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 71.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. unpow2 71.1%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) + 1} \]

      3. sqr-neg 71.1%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + \color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)}\right) + 1} \]

      4. distribute-frac-neg2 71.1%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + \color{blue}{\frac{k}{-t}} \cdot \left(-\frac{k}{t}\right)\right) + 1} \]

      5. distribute-frac-neg2 71.1%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + \frac{k}{-t} \cdot \color{blue}{\frac{k}{-t}}\right) + 1} \]

      6. unpow2 71.1%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) + 1} \]

      7. +-commutative 71.1%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 1\right)} + 1} \]

      8. +-commutative 71.1%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)} + 1} \]

      9. associate-/l/ 71.1%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1} \]

      10. associate-*l/ 71.0%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1} \]

      11. associate-/r/ 68.0%

          \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1} \]

   3. Simplified 68.0%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* 77.4%

         \[\leadsto \frac{\color{blue}{\left(\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell\right) \cdot \ell}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. *-un-lft-identity 77.4%

         \[\leadsto \frac{\left(\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell\right) \cdot \ell}{\color{blue}{1 \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]

      3. times-frac 77.4%

         \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell}{1} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

      4. associate-/l/ 77.4%

         \[\leadsto \frac{\color{blue}{\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \cdot \ell}{1} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   6. Applied egg-rr 77.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \ell}{1} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   7. Step-by-step derivation

      1. times-frac 77.4%

         \[\leadsto \color{blue}{\frac{\left(\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \ell\right) \cdot \ell}{1 \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]

      2. *-commutative 77.4%

         \[\leadsto \frac{\left(\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \ell\right) \cdot \ell}{\color{blue}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot 1}} \]

      3. times-frac 77.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{1}} \]

      4. associate-*l/ 80.2%

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot \ell}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{1} \]

      5. associate-*l* 80.1%

         \[\leadsto \frac{\frac{2 \cdot \ell}{\color{blue}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{1} \]

      6. times-frac 80.4%

         \[\leadsto \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \frac{\ell}{\sin k \cdot \tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{1} \]

      7. /-rgt-identity 80.4%

         \[\leadsto \frac{\frac{2}{{t}^{3}} \cdot \frac{\ell}{\sin k \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\ell} \]

   8. Simplified 80.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}} \cdot \frac{\ell}{\sin k \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \ell} \]

   9. Taylor expanded in k around 0 86.8%

      \[\leadsto \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \cdot \ell \]
   10. Step-by-step derivation

       1. clear-num 86.8%

          \[\leadsto \color{blue}{\frac{1}{\frac{{k}^{2} \cdot {t}^{3}}{\ell}}} \cdot \ell \]

       2. inv-pow 86.8%

          \[\leadsto \color{blue}{{\left(\frac{{k}^{2} \cdot {t}^{3}}{\ell}\right)}^{-1}} \cdot \ell \]

   11. Applied egg-rr 86.8%

       \[\leadsto \color{blue}{{\left(\frac{{k}^{2} \cdot {t}^{3}}{\ell}\right)}^{-1}} \cdot \ell \]
   12. Step-by-step derivation

       1. unpow-1 86.8%

          \[\leadsto \color{blue}{\frac{1}{\frac{{k}^{2} \cdot {t}^{3}}{\ell}}} \cdot \ell \]

       2. associate-/l* 87.2%

          \[\leadsto \frac{1}{\color{blue}{{k}^{2} \cdot \frac{{t}^{3}}{\ell}}} \cdot \ell \]

   13. Simplified 87.2%

       \[\leadsto \color{blue}{\frac{1}{{k}^{2} \cdot \frac{{t}^{3}}{\ell}}} \cdot \ell \]

   if 1.08e-15 < k

   1. Initial program 37.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 37.5%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around inf 22.3%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\frac{{k}^{2}}{{t}^{2}}}\right)} \]
   6. Step-by-step derivation

      1. unpow2 22.3%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \frac{\color{blue}{k \cdot k}}{{t}^{2}}\right)} \]

      2. unpow2 22.3%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \frac{k \cdot k}{\color{blue}{t \cdot t}}\right)} \]

      3. times-frac 33.8%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(\frac{k}{t} \cdot \frac{k}{t}\right)}\right)} \]

      4. unpow2 33.8%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right)} \]

   7. Simplified 33.8%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right)} \]

   8. Taylor expanded in k around 0 47.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.2 \cdot 10^{-162}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\left(k \cdot {t}^{3}\right) \cdot {\ell}^{-2}\right)}\\ \mathbf{elif}\;k \leq 1.08 \cdot 10^{-15}:\\ \;\;\;\;\ell \cdot \frac{1}{{k}^{2} \cdot \frac{{t}^{3}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{4}}{{\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 61.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.3 \cdot 10^{-162}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\_m\right) \cdot \left(k\_m \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}\\ \mathbf{elif}\;k\_m \leq 1.16 \cdot 10^{-15}:\\ \;\;\;\;\ell \cdot \frac{1}{{k\_m}^{2} \cdot \frac{{t}^{3}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k\_m}^{4}}{{\ell}^{2}}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.3e-162)
   (/ 2.0 (* (* 2.0 k_m) (* k_m (/ (pow t 3.0) (pow l 2.0)))))
   (if (<= k_m 1.16e-15)
     (* l (/ 1.0 (* (pow k_m 2.0) (/ (pow t 3.0) l))))
     (/ 2.0 (/ (* t (pow k_m 4.0)) (pow l 2.0))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.3e-162) {
		tmp = 2.0 / ((2.0 * k_m) * (k_m * (pow(t, 3.0) / pow(l, 2.0))));
	} else if (k_m <= 1.16e-15) {
		tmp = l * (1.0 / (pow(k_m, 2.0) * (pow(t, 3.0) / l)));
	} else {
		tmp = 2.0 / ((t * pow(k_m, 4.0)) / pow(l, 2.0));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.3d-162) then
        tmp = 2.0d0 / ((2.0d0 * k_m) * (k_m * ((t ** 3.0d0) / (l ** 2.0d0))))
    else if (k_m <= 1.16d-15) then
        tmp = l * (1.0d0 / ((k_m ** 2.0d0) * ((t ** 3.0d0) / l)))
    else
        tmp = 2.0d0 / ((t * (k_m ** 4.0d0)) / (l ** 2.0d0))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.3e-162) {
		tmp = 2.0 / ((2.0 * k_m) * (k_m * (Math.pow(t, 3.0) / Math.pow(l, 2.0))));
	} else if (k_m <= 1.16e-15) {
		tmp = l * (1.0 / (Math.pow(k_m, 2.0) * (Math.pow(t, 3.0) / l)));
	} else {
		tmp = 2.0 / ((t * Math.pow(k_m, 4.0)) / Math.pow(l, 2.0));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1.3e-162:
		tmp = 2.0 / ((2.0 * k_m) * (k_m * (math.pow(t, 3.0) / math.pow(l, 2.0))))
	elif k_m <= 1.16e-15:
		tmp = l * (1.0 / (math.pow(k_m, 2.0) * (math.pow(t, 3.0) / l)))
	else:
		tmp = 2.0 / ((t * math.pow(k_m, 4.0)) / math.pow(l, 2.0))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.3e-162)
		tmp = Float64(2.0 / Float64(Float64(2.0 * k_m) * Float64(k_m * Float64((t ^ 3.0) / (l ^ 2.0)))));
	elseif (k_m <= 1.16e-15)
		tmp = Float64(l * Float64(1.0 / Float64((k_m ^ 2.0) * Float64((t ^ 3.0) / l))));
	else
		tmp = Float64(2.0 / Float64(Float64(t * (k_m ^ 4.0)) / (l ^ 2.0)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.3e-162)
		tmp = 2.0 / ((2.0 * k_m) * (k_m * ((t ^ 3.0) / (l ^ 2.0))));
	elseif (k_m <= 1.16e-15)
		tmp = l * (1.0 / ((k_m ^ 2.0) * ((t ^ 3.0) / l)));
	else
		tmp = 2.0 / ((t * (k_m ^ 4.0)) / (l ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.3e-162], N[(2.0 / N[(N[(2.0 * k$95$m), $MachinePrecision] * N[(k$95$m * N[(N[Power[t, 3.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1.16e-15], N[(l * N[(1.0 / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.3e-162

   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. add-cube-cbrt 54.2%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}\right)} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      2. pow3 54.2%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}\right)}^{3}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      3. associate-/r* 61.2%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      4. *-commutative 61.2%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\color{blue}{\sin k \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      5. cbrt-prod 61.1%

         \[\leadsto \frac{2}{{\color{blue}{\left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\frac{\frac{{t}^{3}}{\ell}}{\ell}}\right)}}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      6. associate-/r* 54.2%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      7. cbrt-div 55.3%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \color{blue}{\frac{\sqrt[3]{{t}^{3}}}{\sqrt[3]{\ell \cdot \ell}}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      8. rem-cbrt-cube 66.9%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{\color{blue}{t}}{\sqrt[3]{\ell \cdot \ell}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      9. cbrt-prod 80.5%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{\color{blue}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      10. pow2 80.5%

          \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   6. Applied egg-rr 80.5%

      \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   7. Taylor expanded in k around 0 73.1%

      \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \color{blue}{\left(2 \cdot k\right)}} \]

   8. Taylor expanded in k around 0 53.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot {t}^{3}}{{\ell}^{2}}} \cdot \left(2 \cdot k\right)} \]
   9. Step-by-step derivation

      1. associate-/l* 52.6%

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \cdot \left(2 \cdot k\right)} \]

   10. Simplified 52.6%

       \[\leadsto \frac{2}{\color{blue}{\left(k \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \cdot \left(2 \cdot k\right)} \]

   if 1.3e-162 < k < 1.1599999999999999e-15

   1. Initial program 71.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 71.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. unpow2 71.1%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) + 1} \]

      3. sqr-neg 71.1%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + \color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)}\right) + 1} \]

      4. distribute-frac-neg2 71.1%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + \color{blue}{\frac{k}{-t}} \cdot \left(-\frac{k}{t}\right)\right) + 1} \]

      5. distribute-frac-neg2 71.1%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + \frac{k}{-t} \cdot \color{blue}{\frac{k}{-t}}\right) + 1} \]

      6. unpow2 71.1%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) + 1} \]

      7. +-commutative 71.1%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 1\right)} + 1} \]

      8. +-commutative 71.1%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)} + 1} \]

      9. associate-/l/ 71.1%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1} \]

      10. associate-*l/ 71.0%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1} \]

      11. associate-/r/ 68.0%

          \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1} \]

   3. Simplified 68.0%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* 77.4%

         \[\leadsto \frac{\color{blue}{\left(\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell\right) \cdot \ell}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. *-un-lft-identity 77.4%

         \[\leadsto \frac{\left(\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell\right) \cdot \ell}{\color{blue}{1 \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]

      3. times-frac 77.4%

         \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell}{1} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

      4. associate-/l/ 77.4%

         \[\leadsto \frac{\color{blue}{\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \cdot \ell}{1} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   6. Applied egg-rr 77.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \ell}{1} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   7. Step-by-step derivation

      1. times-frac 77.4%

         \[\leadsto \color{blue}{\frac{\left(\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \ell\right) \cdot \ell}{1 \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]

      2. *-commutative 77.4%

         \[\leadsto \frac{\left(\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \ell\right) \cdot \ell}{\color{blue}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot 1}} \]

      3. times-frac 77.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{1}} \]

      4. associate-*l/ 80.2%

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot \ell}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{1} \]

      5. associate-*l* 80.1%

         \[\leadsto \frac{\frac{2 \cdot \ell}{\color{blue}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{1} \]

      6. times-frac 80.4%

         \[\leadsto \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \frac{\ell}{\sin k \cdot \tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{1} \]

      7. /-rgt-identity 80.4%

         \[\leadsto \frac{\frac{2}{{t}^{3}} \cdot \frac{\ell}{\sin k \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\ell} \]

   8. Simplified 80.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}} \cdot \frac{\ell}{\sin k \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \ell} \]

   9. Taylor expanded in k around 0 86.8%

      \[\leadsto \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \cdot \ell \]
   10. Step-by-step derivation

       1. clear-num 86.8%

          \[\leadsto \color{blue}{\frac{1}{\frac{{k}^{2} \cdot {t}^{3}}{\ell}}} \cdot \ell \]

       2. inv-pow 86.8%

          \[\leadsto \color{blue}{{\left(\frac{{k}^{2} \cdot {t}^{3}}{\ell}\right)}^{-1}} \cdot \ell \]

   11. Applied egg-rr 86.8%

       \[\leadsto \color{blue}{{\left(\frac{{k}^{2} \cdot {t}^{3}}{\ell}\right)}^{-1}} \cdot \ell \]
   12. Step-by-step derivation

       1. unpow-1 86.8%

          \[\leadsto \color{blue}{\frac{1}{\frac{{k}^{2} \cdot {t}^{3}}{\ell}}} \cdot \ell \]

       2. associate-/l* 87.2%

          \[\leadsto \frac{1}{\color{blue}{{k}^{2} \cdot \frac{{t}^{3}}{\ell}}} \cdot \ell \]

   13. Simplified 87.2%

       \[\leadsto \color{blue}{\frac{1}{{k}^{2} \cdot \frac{{t}^{3}}{\ell}}} \cdot \ell \]

   if 1.1599999999999999e-15 < k

   1. Initial program 37.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 37.5%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around inf 22.3%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\frac{{k}^{2}}{{t}^{2}}}\right)} \]
   6. Step-by-step derivation

      1. unpow2 22.3%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \frac{\color{blue}{k \cdot k}}{{t}^{2}}\right)} \]

      2. unpow2 22.3%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \frac{k \cdot k}{\color{blue}{t \cdot t}}\right)} \]

      3. times-frac 33.8%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(\frac{k}{t} \cdot \frac{k}{t}\right)}\right)} \]

      4. unpow2 33.8%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right)} \]

   7. Simplified 33.8%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right)} \]

   8. Taylor expanded in k around 0 47.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.3 \cdot 10^{-162}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(k \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}\\ \mathbf{elif}\;k \leq 1.16 \cdot 10^{-15}:\\ \;\;\;\;\ell \cdot \frac{1}{{k}^{2} \cdot \frac{{t}^{3}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{4}}{{\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 56.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 3.7 \cdot 10^{-88}:\\ \;\;\;\;\frac{2}{{k\_m}^{4} \cdot \frac{t}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{1}{{k\_m}^{2} \cdot \frac{{t}^{3}}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 3.7e-88)
   (/ 2.0 (* (pow k_m 4.0) (/ t (pow l 2.0))))
   (* l (/ 1.0 (* (pow k_m 2.0) (/ (pow t 3.0) l))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 3.7e-88) {
		tmp = 2.0 / (pow(k_m, 4.0) * (t / pow(l, 2.0)));
	} else {
		tmp = l * (1.0 / (pow(k_m, 2.0) * (pow(t, 3.0) / l)));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t <= 3.7d-88) then
        tmp = 2.0d0 / ((k_m ** 4.0d0) * (t / (l ** 2.0d0)))
    else
        tmp = l * (1.0d0 / ((k_m ** 2.0d0) * ((t ** 3.0d0) / l)))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 3.7e-88) {
		tmp = 2.0 / (Math.pow(k_m, 4.0) * (t / Math.pow(l, 2.0)));
	} else {
		tmp = l * (1.0 / (Math.pow(k_m, 2.0) * (Math.pow(t, 3.0) / l)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 3.7e-88:
		tmp = 2.0 / (math.pow(k_m, 4.0) * (t / math.pow(l, 2.0)))
	else:
		tmp = l * (1.0 / (math.pow(k_m, 2.0) * (math.pow(t, 3.0) / l)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 3.7e-88)
		tmp = Float64(2.0 / Float64((k_m ^ 4.0) * Float64(t / (l ^ 2.0))));
	else
		tmp = Float64(l * Float64(1.0 / Float64((k_m ^ 2.0) * Float64((t ^ 3.0) / l))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 3.7e-88)
		tmp = 2.0 / ((k_m ^ 4.0) * (t / (l ^ 2.0)));
	else
		tmp = l * (1.0 / ((k_m ^ 2.0) * ((t ^ 3.0) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 3.7e-88], N[(2.0 / N[(N[Power[k$95$m, 4.0], $MachinePrecision] * N[(t / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(1.0 / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.6999999999999997e-88

   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 47.8%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around inf 23.1%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\frac{{k}^{2}}{{t}^{2}}}\right)} \]
   6. Step-by-step derivation

      1. unpow2 23.1%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \frac{\color{blue}{k \cdot k}}{{t}^{2}}\right)} \]

      2. unpow2 23.1%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \frac{k \cdot k}{\color{blue}{t \cdot t}}\right)} \]

      3. times-frac 37.2%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(\frac{k}{t} \cdot \frac{k}{t}\right)}\right)} \]

      4. unpow2 37.2%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right)} \]

   7. Simplified 37.2%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right)} \]

   8. Taylor expanded in k around 0 50.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
   9. Step-by-step derivation

      1. associate-/l* 50.5%

         \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}} \]

   10. Simplified 50.5%

       \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}} \]

   if 3.6999999999999997e-88 < t

   1. Initial program 60.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 60.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. unpow2 60.9%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) + 1} \]

      3. sqr-neg 60.9%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + \color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)}\right) + 1} \]

      4. distribute-frac-neg2 60.9%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + \color{blue}{\frac{k}{-t}} \cdot \left(-\frac{k}{t}\right)\right) + 1} \]

      5. distribute-frac-neg2 60.9%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + \frac{k}{-t} \cdot \color{blue}{\frac{k}{-t}}\right) + 1} \]

      6. unpow2 60.9%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) + 1} \]

      7. +-commutative 60.9%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 1\right)} + 1} \]

      8. +-commutative 60.9%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)} + 1} \]

      9. associate-/l/ 60.9%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1} \]

      10. associate-*l/ 62.3%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1} \]

      11. associate-/r/ 62.2%

          \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1} \]

   3. Simplified 62.2%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* 68.6%

         \[\leadsto \frac{\color{blue}{\left(\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell\right) \cdot \ell}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. *-un-lft-identity 68.6%

         \[\leadsto \frac{\left(\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell\right) \cdot \ell}{\color{blue}{1 \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]

      3. times-frac 70.1%

         \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell}{1} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

      4. associate-/l/ 70.2%

         \[\leadsto \frac{\color{blue}{\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \cdot \ell}{1} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   6. Applied egg-rr 70.2%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \ell}{1} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   7. Step-by-step derivation

      1. times-frac 68.7%

         \[\leadsto \color{blue}{\frac{\left(\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \ell\right) \cdot \ell}{1 \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]

      2. *-commutative 68.7%

         \[\leadsto \frac{\left(\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \ell\right) \cdot \ell}{\color{blue}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot 1}} \]

      3. times-frac 70.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{1}} \]

      4. associate-*l/ 70.2%

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot \ell}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{1} \]

      5. associate-*l* 63.8%

         \[\leadsto \frac{\frac{2 \cdot \ell}{\color{blue}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{1} \]

      6. times-frac 63.7%

         \[\leadsto \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \frac{\ell}{\sin k \cdot \tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{1} \]

      7. /-rgt-identity 63.7%

         \[\leadsto \frac{\frac{2}{{t}^{3}} \cdot \frac{\ell}{\sin k \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\ell} \]

   8. Simplified 63.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}} \cdot \frac{\ell}{\sin k \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \ell} \]

   9. Taylor expanded in k around 0 56.6%

      \[\leadsto \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \cdot \ell \]
   10. Step-by-step derivation

       1. clear-num 56.6%

          \[\leadsto \color{blue}{\frac{1}{\frac{{k}^{2} \cdot {t}^{3}}{\ell}}} \cdot \ell \]

       2. inv-pow 56.6%

          \[\leadsto \color{blue}{{\left(\frac{{k}^{2} \cdot {t}^{3}}{\ell}\right)}^{-1}} \cdot \ell \]

   11. Applied egg-rr 56.6%

       \[\leadsto \color{blue}{{\left(\frac{{k}^{2} \cdot {t}^{3}}{\ell}\right)}^{-1}} \cdot \ell \]
   12. Step-by-step derivation

       1. unpow-1 56.6%

          \[\leadsto \color{blue}{\frac{1}{\frac{{k}^{2} \cdot {t}^{3}}{\ell}}} \cdot \ell \]

       2. associate-/l* 55.1%

          \[\leadsto \frac{1}{\color{blue}{{k}^{2} \cdot \frac{{t}^{3}}{\ell}}} \cdot \ell \]

   13. Simplified 55.1%

       \[\leadsto \color{blue}{\frac{1}{{k}^{2} \cdot \frac{{t}^{3}}{\ell}}} \cdot \ell \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.7 \cdot 10^{-88}:\\ \;\;\;\;\frac{2}{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{1}{{k}^{2} \cdot \frac{{t}^{3}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 55.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 9.5 \cdot 10^{-88}:\\ \;\;\;\;\frac{2}{{k\_m}^{4} \cdot \frac{t}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k\_m \cdot k\_m\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 9.5e-88)
   (/ 2.0 (* (pow k_m 4.0) (/ t (pow l 2.0))))
   (* l (/ l (* (pow t 3.0) (* k_m k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 9.5e-88) {
		tmp = 2.0 / (pow(k_m, 4.0) * (t / pow(l, 2.0)));
	} else {
		tmp = l * (l / (pow(t, 3.0) * (k_m * k_m)));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t <= 9.5d-88) then
        tmp = 2.0d0 / ((k_m ** 4.0d0) * (t / (l ** 2.0d0)))
    else
        tmp = l * (l / ((t ** 3.0d0) * (k_m * k_m)))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 9.5e-88) {
		tmp = 2.0 / (Math.pow(k_m, 4.0) * (t / Math.pow(l, 2.0)));
	} else {
		tmp = l * (l / (Math.pow(t, 3.0) * (k_m * k_m)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 9.5e-88:
		tmp = 2.0 / (math.pow(k_m, 4.0) * (t / math.pow(l, 2.0)))
	else:
		tmp = l * (l / (math.pow(t, 3.0) * (k_m * k_m)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 9.5e-88)
		tmp = Float64(2.0 / Float64((k_m ^ 4.0) * Float64(t / (l ^ 2.0))));
	else
		tmp = Float64(l * Float64(l / Float64((t ^ 3.0) * Float64(k_m * k_m))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 9.5e-88)
		tmp = 2.0 / ((k_m ^ 4.0) * (t / (l ^ 2.0)));
	else
		tmp = l * (l / ((t ^ 3.0) * (k_m * k_m)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 9.5e-88], N[(2.0 / N[(N[Power[k$95$m, 4.0], $MachinePrecision] * N[(t / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(l / N[(N[Power[t, 3.0], $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 9.5e-88

   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 47.8%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around inf 23.1%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\frac{{k}^{2}}{{t}^{2}}}\right)} \]
   6. Step-by-step derivation

      1. unpow2 23.1%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \frac{\color{blue}{k \cdot k}}{{t}^{2}}\right)} \]

      2. unpow2 23.1%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \frac{k \cdot k}{\color{blue}{t \cdot t}}\right)} \]

      3. times-frac 37.2%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(\frac{k}{t} \cdot \frac{k}{t}\right)}\right)} \]

      4. unpow2 37.2%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right)} \]

   7. Simplified 37.2%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right)} \]

   8. Taylor expanded in k around 0 50.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
   9. Step-by-step derivation

      1. associate-/l* 50.5%

         \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}} \]

   10. Simplified 50.5%

       \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}} \]

   if 9.5e-88 < t

   1. Initial program 60.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 60.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. unpow2 60.9%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) + 1} \]

      3. sqr-neg 60.9%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + \color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)}\right) + 1} \]

      4. distribute-frac-neg2 60.9%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + \color{blue}{\frac{k}{-t}} \cdot \left(-\frac{k}{t}\right)\right) + 1} \]

      5. distribute-frac-neg2 60.9%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + \frac{k}{-t} \cdot \color{blue}{\frac{k}{-t}}\right) + 1} \]

      6. unpow2 60.9%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) + 1} \]

      7. +-commutative 60.9%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 1\right)} + 1} \]

      8. +-commutative 60.9%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)} + 1} \]

      9. associate-/l/ 60.9%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1} \]

      10. associate-*l/ 62.3%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1} \]

      11. associate-/r/ 62.2%

          \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1} \]

   3. Simplified 62.2%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* 68.6%

         \[\leadsto \frac{\color{blue}{\left(\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell\right) \cdot \ell}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. *-un-lft-identity 68.6%

         \[\leadsto \frac{\left(\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell\right) \cdot \ell}{\color{blue}{1 \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]

      3. times-frac 70.1%

         \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell}{1} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

      4. associate-/l/ 70.2%

         \[\leadsto \frac{\color{blue}{\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \cdot \ell}{1} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   6. Applied egg-rr 70.2%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \ell}{1} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   7. Step-by-step derivation

      1. times-frac 68.7%

         \[\leadsto \color{blue}{\frac{\left(\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \ell\right) \cdot \ell}{1 \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]

      2. *-commutative 68.7%

         \[\leadsto \frac{\left(\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \ell\right) \cdot \ell}{\color{blue}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot 1}} \]

      3. times-frac 70.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{1}} \]

      4. associate-*l/ 70.2%

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot \ell}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{1} \]

      5. associate-*l* 63.8%

         \[\leadsto \frac{\frac{2 \cdot \ell}{\color{blue}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{1} \]

      6. times-frac 63.7%

         \[\leadsto \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \frac{\ell}{\sin k \cdot \tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{1} \]

      7. /-rgt-identity 63.7%

         \[\leadsto \frac{\frac{2}{{t}^{3}} \cdot \frac{\ell}{\sin k \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\ell} \]

   8. Simplified 63.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}} \cdot \frac{\ell}{\sin k \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \ell} \]

   9. Taylor expanded in k around 0 56.6%

      \[\leadsto \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \cdot \ell \]
   10. Step-by-step derivation

       1. unpow2 56.6%

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \cdot \ell \]

   11. Applied egg-rr 56.6%

       \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \cdot \ell \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9.5 \cdot 10^{-88}:\\ \;\;\;\;\frac{2}{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 55.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k\_m \cdot k\_m\right)} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m) :precision binary64 (* l (/ l (* (pow t 3.0) (* k_m k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return l * (l / (pow(t, 3.0) * (k_m * k_m)));
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = l * (l / ((t ** 3.0d0) * (k_m * k_m)))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return l * (l / (Math.pow(t, 3.0) * (k_m * k_m)));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return l * (l / (math.pow(t, 3.0) * (k_m * k_m)))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(l * Float64(l / Float64((t ^ 3.0) * Float64(k_m * k_m))))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = l * (l / ((t ^ 3.0) * (k_m * k_m)));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(l * N[(l / N[(N[Power[t, 3.0], $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.1%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/r* 51.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

   2. unpow2 51.1%

      \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) + 1} \]

   3. sqr-neg 51.1%

      \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + \color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)}\right) + 1} \]

   4. distribute-frac-neg2 51.1%

      \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + \color{blue}{\frac{k}{-t}} \cdot \left(-\frac{k}{t}\right)\right) + 1} \]

   5. distribute-frac-neg2 51.1%

      \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + \frac{k}{-t} \cdot \color{blue}{\frac{k}{-t}}\right) + 1} \]

   6. unpow2 51.1%

      \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) + 1} \]

   7. +-commutative 51.1%

      \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 1\right)} + 1} \]

   8. +-commutative 51.1%

      \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)} + 1} \]

   9. associate-/l/ 51.1%

      \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1} \]

   10. associate-*l/ 51.8%

       \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1} \]

   11. associate-/r/ 50.7%

       \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1} \]

3. Simplified 50.7%

   \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*r* 57.1%

      \[\leadsto \frac{\color{blue}{\left(\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell\right) \cdot \ell}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   2. *-un-lft-identity 57.1%

      \[\leadsto \frac{\left(\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell\right) \cdot \ell}{\color{blue}{1 \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]

   3. times-frac 57.9%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell}{1} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. associate-/l/ 57.9%

      \[\leadsto \frac{\color{blue}{\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \cdot \ell}{1} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

6. Applied egg-rr 57.9%

   \[\leadsto \color{blue}{\frac{\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \ell}{1} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
7. Step-by-step derivation

   1. times-frac 57.1%

      \[\leadsto \color{blue}{\frac{\left(\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \ell\right) \cdot \ell}{1 \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]

   2. *-commutative 57.1%

      \[\leadsto \frac{\left(\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \ell\right) \cdot \ell}{\color{blue}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot 1}} \]

   3. times-frac 57.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{1}} \]

   4. associate-*l/ 58.3%

      \[\leadsto \frac{\color{blue}{\frac{2 \cdot \ell}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{1} \]

   5. associate-*l* 54.5%

      \[\leadsto \frac{\frac{2 \cdot \ell}{\color{blue}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{1} \]

   6. times-frac 54.4%

      \[\leadsto \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \frac{\ell}{\sin k \cdot \tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{1} \]

   7. /-rgt-identity 54.4%

      \[\leadsto \frac{\frac{2}{{t}^{3}} \cdot \frac{\ell}{\sin k \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\ell} \]

8. Simplified 54.4%

   \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}} \cdot \frac{\ell}{\sin k \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \ell} \]

9. Taylor expanded in k around 0 52.7%

   \[\leadsto \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \cdot \ell \]
10. Step-by-step derivation

    1. unpow2 52.7%

       \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \cdot \ell \]

11. Applied egg-rr 52.7%

    \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \cdot \ell \]

12. Final simplification 52.7%

    \[\leadsto \ell \cdot \frac{\ell}{{t}^{3} \cdot \left(k \cdot k\right)} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024151 
(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))))