Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.6% → 86.7%

Time: 20.4s

Alternatives: 23

Speedup: 2.0×

• 27.0% 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: 54.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\frac{k}{t\_m}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.35 \cdot 10^{-45}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{2} \cdot \left(t\_m \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}{\ell}}\\ \mathbf{elif}\;t\_m \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\left(\frac{\frac{2}{{t\_m}^{3}}}{\sin k} \cdot \frac{\ell}{\tan k}\right) \cdot \frac{\ell}{2 + t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t\_m \cdot \frac{\sqrt[3]{\sin k}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(t\_2 + 1\right)\right)\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (/ k t_m) 2.0)))
   (*
    t_s
    (if (<= t_m 1.35e-45)
      (/ 2.0 (/ (/ (* (pow k 2.0) (* t_m (pow (sin k) 2.0))) (* l (cos k))) l))
      (if (<= t_m 5.6e+102)
        (*
         (* (/ (/ 2.0 (pow t_m 3.0)) (sin k)) (/ l (tan k)))
         (/ l (+ 2.0 t_2)))
        (/
         2.0
         (*
          (pow (* t_m (/ (cbrt (sin k)) (pow (cbrt l) 2.0))) 3.0)
          (* (tan k) (+ 1.0 (+ t_2 1.0))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 1.35e-45) {
		tmp = 2.0 / (((pow(k, 2.0) * (t_m * pow(sin(k), 2.0))) / (l * cos(k))) / l);
	} else if (t_m <= 5.6e+102) {
		tmp = (((2.0 / pow(t_m, 3.0)) / sin(k)) * (l / tan(k))) * (l / (2.0 + t_2));
	} else {
		tmp = 2.0 / (pow((t_m * (cbrt(sin(k)) / pow(cbrt(l), 2.0))), 3.0) * (tan(k) * (1.0 + (t_2 + 1.0))));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 1.35e-45) {
		tmp = 2.0 / (((Math.pow(k, 2.0) * (t_m * Math.pow(Math.sin(k), 2.0))) / (l * Math.cos(k))) / l);
	} else if (t_m <= 5.6e+102) {
		tmp = (((2.0 / Math.pow(t_m, 3.0)) / Math.sin(k)) * (l / Math.tan(k))) * (l / (2.0 + t_2));
	} else {
		tmp = 2.0 / (Math.pow((t_m * (Math.cbrt(Math.sin(k)) / Math.pow(Math.cbrt(l), 2.0))), 3.0) * (Math.tan(k) * (1.0 + (t_2 + 1.0))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(k / t_m) ^ 2.0
	tmp = 0.0
	if (t_m <= 1.35e-45)
		tmp = Float64(2.0 / Float64(Float64(Float64((k ^ 2.0) * Float64(t_m * (sin(k) ^ 2.0))) / Float64(l * cos(k))) / l));
	elseif (t_m <= 5.6e+102)
		tmp = Float64(Float64(Float64(Float64(2.0 / (t_m ^ 3.0)) / sin(k)) * Float64(l / tan(k))) * Float64(l / Float64(2.0 + t_2)));
	else
		tmp = Float64(2.0 / Float64((Float64(t_m * Float64(cbrt(sin(k)) / (cbrt(l) ^ 2.0))) ^ 3.0) * Float64(tan(k) * Float64(1.0 + Float64(t_2 + 1.0)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.35e-45], N[(2.0 / N[(N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.6e+102], N[(N[(N[(N[(2.0 / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(t$95$m * N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.34999999999999992e-45

   1. Initial program 51.6%

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

   3. Simplified 51.7%

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

      1. add-cube-cbrt 51.7%

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

      2. pow3 51.7%

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

      3. cbrt-div 51.6%

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

      4. rem-cbrt-cube 56.6%

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

   6. Applied egg-rr 56.6%

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

      1. associate-*l/ 58.8%

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

      2. add-cbrt-cube 53.8%

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

      3. unpow3 53.8%

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

      4. cbrt-div 53.8%

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

      5. pow3 53.8%

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

      6. add-cube-cbrt 53.9%

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

      7. metadata-eval 53.9%

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

      8. associate-+r+ 53.9%

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

      9. associate-*l* 53.9%

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

      10. associate-+r+ 53.9%

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

      11. metadata-eval 53.9%

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

   8. Applied egg-rr 53.9%

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

   9. Taylor expanded in t around 0 64.7%

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

   if 1.34999999999999992e-45 < t < 5.60000000000000037e102

   1. Initial program 85.5%

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

   3. Simplified 85.5%

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

      1. associate-*r* 91.4%

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

      2. *-un-lft-identity 91.4%

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

      3. times-frac 94.2%

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

   6. Applied egg-rr 94.2%

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

      1. /-rgt-identity 94.2%

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

      2. associate-*l/ 94.2%

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

      3. associate-*r* 97.1%

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

      4. times-frac 97.0%

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

      5. associate-/r* 97.1%

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

   8. Simplified 97.1%

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

   if 5.60000000000000037e102 < t

   1. Initial program 61.2%

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

   3. Simplified 61.2%

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

      1. add-cube-cbrt 61.2%

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

      2. pow3 61.2%

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

      3. *-commutative 61.2%

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

      4. cbrt-prod 61.2%

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

      5. cbrt-div 61.2%

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

      6. rem-cbrt-cube 77.7%

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

      7. cbrt-prod 93.5%

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

      8. pow2 93.5%

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

   6. Applied egg-rr 93.5%

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

      1. *-commutative 93.5%

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

   8. Simplified 93.5%

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

      1. associate-*l/ 93.5%

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

   10. Applied egg-rr 93.5%

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

       1. associate-/l* 93.6%

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

   12. Simplified 93.6%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.35 \cdot 10^{-45}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}{\ell}}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\left(\frac{\frac{2}{{t}^{3}}}{\sin k} \cdot \frac{\ell}{\tan k}\right) \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t \cdot \frac{\sqrt[3]{\sin k}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\frac{k}{t\_m}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.5 \cdot 10^{-46}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{2} \cdot \left(t\_m \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}{\ell}}\\ \mathbf{elif}\;t\_m \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\left(\frac{\frac{2}{{t\_m}^{3}}}{\sin k} \cdot \frac{\ell}{\tan k}\right) \cdot \frac{\ell}{2 + t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(t\_2 + 1\right)\right)\right) \cdot \left(\sin k \cdot {\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (/ k t_m) 2.0)))
   (*
    t_s
    (if (<= t_m 7.5e-46)
      (/ 2.0 (/ (/ (* (pow k 2.0) (* t_m (pow (sin k) 2.0))) (* l (cos k))) l))
      (if (<= t_m 5.6e+102)
        (*
         (* (/ (/ 2.0 (pow t_m 3.0)) (sin k)) (/ l (tan k)))
         (/ l (+ 2.0 t_2)))
        (/
         2.0
         (*
          (* (tan k) (+ 1.0 (+ t_2 1.0)))
          (* (sin k) (pow (/ t_m (pow (cbrt l) 2.0)) 3.0)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 7.5e-46) {
		tmp = 2.0 / (((pow(k, 2.0) * (t_m * pow(sin(k), 2.0))) / (l * cos(k))) / l);
	} else if (t_m <= 5.6e+102) {
		tmp = (((2.0 / pow(t_m, 3.0)) / sin(k)) * (l / tan(k))) * (l / (2.0 + t_2));
	} else {
		tmp = 2.0 / ((tan(k) * (1.0 + (t_2 + 1.0))) * (sin(k) * pow((t_m / pow(cbrt(l), 2.0)), 3.0)));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 7.5e-46) {
		tmp = 2.0 / (((Math.pow(k, 2.0) * (t_m * Math.pow(Math.sin(k), 2.0))) / (l * Math.cos(k))) / l);
	} else if (t_m <= 5.6e+102) {
		tmp = (((2.0 / Math.pow(t_m, 3.0)) / Math.sin(k)) * (l / Math.tan(k))) * (l / (2.0 + t_2));
	} else {
		tmp = 2.0 / ((Math.tan(k) * (1.0 + (t_2 + 1.0))) * (Math.sin(k) * Math.pow((t_m / Math.pow(Math.cbrt(l), 2.0)), 3.0)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(k / t_m) ^ 2.0
	tmp = 0.0
	if (t_m <= 7.5e-46)
		tmp = Float64(2.0 / Float64(Float64(Float64((k ^ 2.0) * Float64(t_m * (sin(k) ^ 2.0))) / Float64(l * cos(k))) / l));
	elseif (t_m <= 5.6e+102)
		tmp = Float64(Float64(Float64(Float64(2.0 / (t_m ^ 3.0)) / sin(k)) * Float64(l / tan(k))) * Float64(l / Float64(2.0 + t_2)));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(1.0 + Float64(t_2 + 1.0))) * Float64(sin(k) * (Float64(t_m / (cbrt(l) ^ 2.0)) ^ 3.0))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 7.5e-46], N[(2.0 / N[(N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.6e+102], N[(N[(N[(N[(2.0 / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 7.50000000000000027e-46

   1. Initial program 51.6%

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

   3. Simplified 51.7%

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

      1. add-cube-cbrt 51.7%

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

      2. pow3 51.7%

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

      3. cbrt-div 51.6%

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

      4. rem-cbrt-cube 56.6%

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

   6. Applied egg-rr 56.6%

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

      1. associate-*l/ 58.8%

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

      2. add-cbrt-cube 53.8%

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

      3. unpow3 53.8%

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

      4. cbrt-div 53.8%

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

      5. pow3 53.8%

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

      6. add-cube-cbrt 53.9%

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

      7. metadata-eval 53.9%

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

      8. associate-+r+ 53.9%

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

      9. associate-*l* 53.9%

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

      10. associate-+r+ 53.9%

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

      11. metadata-eval 53.9%

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

   8. Applied egg-rr 53.9%

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

   9. Taylor expanded in t around 0 64.7%

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

   if 7.50000000000000027e-46 < t < 5.60000000000000037e102

   1. Initial program 85.5%

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

   3. Simplified 85.5%

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

      1. associate-*r* 91.4%

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

      2. *-un-lft-identity 91.4%

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

      3. times-frac 94.2%

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

   6. Applied egg-rr 94.2%

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

      1. /-rgt-identity 94.2%

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

      2. associate-*l/ 94.2%

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

      3. associate-*r* 97.1%

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

      4. times-frac 97.0%

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

      5. associate-/r* 97.1%

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

   8. Simplified 97.1%

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

   if 5.60000000000000037e102 < t

   1. Initial program 61.2%

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

   3. Simplified 61.2%

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

      1. add-cube-cbrt 61.2%

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

      2. pow3 61.2%

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

      3. cbrt-div 61.2%

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

      4. rem-cbrt-cube 69.8%

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

      5. cbrt-prod 81.6%

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

      6. pow2 81.6%

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

   6. Applied egg-rr 81.6%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.5 \cdot 10^{-46}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}{\ell}}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\left(\frac{\frac{2}{{t}^{3}}}{\sin k} \cdot \frac{\ell}{\tan k}\right) \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right) \cdot \left(\sin k \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 + {\left(\frac{k}{t\_m}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.8 \cdot 10^{-46}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{2} \cdot \left(t\_m \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}{\ell}}\\ \mathbf{elif}\;t\_m \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\left(\frac{\frac{2}{{t\_m}^{3}}}{\sin k} \cdot \frac{\ell}{\tan k}\right) \cdot \frac{\ell}{t\_2}\\ \mathbf{elif}\;t\_m \leq 2.6 \cdot 10^{+202}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot t\_2\right) \cdot \left(\sin k \cdot {\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{k}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (+ 2.0 (pow (/ k t_m) 2.0))))
   (*
    t_s
    (if (<= t_m 5.8e-46)
      (/ 2.0 (/ (/ (* (pow k 2.0) (* t_m (pow (sin k) 2.0))) (* l (cos k))) l))
      (if (<= t_m 5.6e+102)
        (* (* (/ (/ 2.0 (pow t_m 3.0)) (sin k)) (/ l (tan k))) (/ l t_2))
        (if (<= t_m 2.6e+202)
          (/ 2.0 (* (* (tan k) t_2) (* (sin k) (pow (/ (pow t_m 1.5) l) 2.0))))
          (/
           2.0
           (*
            (pow (* (/ t_m (pow (cbrt l) 2.0)) (cbrt k)) 3.0)
            (* 2.0 k)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = 2.0 + pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 5.8e-46) {
		tmp = 2.0 / (((pow(k, 2.0) * (t_m * pow(sin(k), 2.0))) / (l * cos(k))) / l);
	} else if (t_m <= 5.6e+102) {
		tmp = (((2.0 / pow(t_m, 3.0)) / sin(k)) * (l / tan(k))) * (l / t_2);
	} else if (t_m <= 2.6e+202) {
		tmp = 2.0 / ((tan(k) * t_2) * (sin(k) * pow((pow(t_m, 1.5) / l), 2.0)));
	} else {
		tmp = 2.0 / (pow(((t_m / pow(cbrt(l), 2.0)) * cbrt(k)), 3.0) * (2.0 * k));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = 2.0 + Math.pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 5.8e-46) {
		tmp = 2.0 / (((Math.pow(k, 2.0) * (t_m * Math.pow(Math.sin(k), 2.0))) / (l * Math.cos(k))) / l);
	} else if (t_m <= 5.6e+102) {
		tmp = (((2.0 / Math.pow(t_m, 3.0)) / Math.sin(k)) * (l / Math.tan(k))) * (l / t_2);
	} else if (t_m <= 2.6e+202) {
		tmp = 2.0 / ((Math.tan(k) * t_2) * (Math.sin(k) * Math.pow((Math.pow(t_m, 1.5) / l), 2.0)));
	} else {
		tmp = 2.0 / (Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt(k)), 3.0) * (2.0 * k));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(2.0 + (Float64(k / t_m) ^ 2.0))
	tmp = 0.0
	if (t_m <= 5.8e-46)
		tmp = Float64(2.0 / Float64(Float64(Float64((k ^ 2.0) * Float64(t_m * (sin(k) ^ 2.0))) / Float64(l * cos(k))) / l));
	elseif (t_m <= 5.6e+102)
		tmp = Float64(Float64(Float64(Float64(2.0 / (t_m ^ 3.0)) / sin(k)) * Float64(l / tan(k))) * Float64(l / t_2));
	elseif (t_m <= 2.6e+202)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * t_2) * Float64(sin(k) * (Float64((t_m ^ 1.5) / l) ^ 2.0))));
	else
		tmp = Float64(2.0 / Float64((Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * cbrt(k)) ^ 3.0) * Float64(2.0 * k)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 5.8e-46], N[(2.0 / N[(N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.6e+102], N[(N[(N[(N[(2.0 / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.6e+202], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * t$95$2), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[k, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 5.80000000000000009e-46

   1. Initial program 51.6%

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

   3. Simplified 51.7%

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

      1. add-cube-cbrt 51.7%

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

      2. pow3 51.7%

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

      3. cbrt-div 51.6%

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

      4. rem-cbrt-cube 56.6%

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

   6. Applied egg-rr 56.6%

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

      1. associate-*l/ 58.8%

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

      2. add-cbrt-cube 53.8%

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

      3. unpow3 53.8%

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

      4. cbrt-div 53.8%

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

      5. pow3 53.8%

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

      6. add-cube-cbrt 53.9%

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

      7. metadata-eval 53.9%

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

      8. associate-+r+ 53.9%

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

      9. associate-*l* 53.9%

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

      10. associate-+r+ 53.9%

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

      11. metadata-eval 53.9%

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

   8. Applied egg-rr 53.9%

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

   9. Taylor expanded in t around 0 64.7%

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

   if 5.80000000000000009e-46 < t < 5.60000000000000037e102

   1. Initial program 85.5%

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

   3. Simplified 85.5%

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

      1. associate-*r* 91.4%

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

      2. *-un-lft-identity 91.4%

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

      3. times-frac 94.2%

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

   6. Applied egg-rr 94.2%

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

      1. /-rgt-identity 94.2%

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

      2. associate-*l/ 94.2%

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

      3. associate-*r* 97.1%

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

      4. times-frac 97.0%

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

      5. associate-/r* 97.1%

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

   8. Simplified 97.1%

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

   if 5.60000000000000037e102 < t < 2.6000000000000002e202

   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-sqr-sqrt 29.6%

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

      2. pow2 29.6%

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

      3. *-commutative 29.6%

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

      4. sqrt-prod 29.6%

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

      5. sqrt-div 29.6%

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

      6. sqrt-pow1 43.7%

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

      7. metadata-eval 43.7%

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

      8. sqrt-prod 24.4%

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

      9. add-sqr-sqrt 52.9%

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

   6. Applied egg-rr 52.9%

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

      1. *-commutative 52.9%

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

   8. Simplified 52.9%

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

      1. pow1 52.9%

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

      2. *-commutative 52.9%

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

      3. associate-+r+ 52.9%

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

      4. metadata-eval 52.9%

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

      5. *-commutative 52.9%

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

      6. unpow-prod-down 52.9%

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

      7. pow2 52.9%

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

      8. add-sqr-sqrt 81.8%

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

   10. Applied egg-rr 81.8%

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

       1. unpow1 81.8%

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

   12. Simplified 81.8%

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

   if 2.6000000000000002e202 < t

   1. Initial program 74.2%

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

   3. Simplified 74.2%

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

      1. add-cube-cbrt 74.2%

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

      2. pow3 74.2%

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

      3. *-commutative 74.2%

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

      4. cbrt-prod 74.2%

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

      5. cbrt-div 74.2%

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

      6. rem-cbrt-cube 81.8%

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

      7. cbrt-prod 96.3%

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

      8. pow2 96.3%

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

   6. Applied egg-rr 96.3%

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

      1. *-commutative 96.3%

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

   8. Simplified 96.3%

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

   9. Taylor expanded in k around 0 92.9%

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

   10. Taylor expanded in k around 0 92.9%

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

Alternative 4: 83.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.8 \cdot 10^{-47}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{2} \cdot \left(t\_m \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}{\ell}}\\ \mathbf{elif}\;t\_m \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\left(\frac{\frac{2}{{t\_m}^{3}}}{\sin k} \cdot \frac{\ell}{\tan k}\right) \cdot \frac{\ell}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}\\ \mathbf{elif}\;t\_m \leq 6.2 \cdot 10^{+202}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot {\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2}\right) \cdot \frac{2 \cdot \sin k}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{k}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 7.8e-47)
    (/ 2.0 (/ (/ (* (pow k 2.0) (* t_m (pow (sin k) 2.0))) (* l (cos k))) l))
    (if (<= t_m 5.6e+102)
      (*
       (* (/ (/ 2.0 (pow t_m 3.0)) (sin k)) (/ l (tan k)))
       (/ l (+ 2.0 (pow (/ k t_m) 2.0))))
      (if (<= t_m 6.2e+202)
        (/
         2.0
         (*
          (* (sin k) (pow (/ (pow t_m 1.5) l) 2.0))
          (/ (* 2.0 (sin k)) (cos k))))
        (/
         2.0
         (* (pow (* (/ t_m (pow (cbrt l) 2.0)) (cbrt k)) 3.0) (* 2.0 k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 7.8e-47) {
		tmp = 2.0 / (((pow(k, 2.0) * (t_m * pow(sin(k), 2.0))) / (l * cos(k))) / l);
	} else if (t_m <= 5.6e+102) {
		tmp = (((2.0 / pow(t_m, 3.0)) / sin(k)) * (l / tan(k))) * (l / (2.0 + pow((k / t_m), 2.0)));
	} else if (t_m <= 6.2e+202) {
		tmp = 2.0 / ((sin(k) * pow((pow(t_m, 1.5) / l), 2.0)) * ((2.0 * sin(k)) / cos(k)));
	} else {
		tmp = 2.0 / (pow(((t_m / pow(cbrt(l), 2.0)) * cbrt(k)), 3.0) * (2.0 * k));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 7.8e-47) {
		tmp = 2.0 / (((Math.pow(k, 2.0) * (t_m * Math.pow(Math.sin(k), 2.0))) / (l * Math.cos(k))) / l);
	} else if (t_m <= 5.6e+102) {
		tmp = (((2.0 / Math.pow(t_m, 3.0)) / Math.sin(k)) * (l / Math.tan(k))) * (l / (2.0 + Math.pow((k / t_m), 2.0)));
	} else if (t_m <= 6.2e+202) {
		tmp = 2.0 / ((Math.sin(k) * Math.pow((Math.pow(t_m, 1.5) / l), 2.0)) * ((2.0 * Math.sin(k)) / Math.cos(k)));
	} else {
		tmp = 2.0 / (Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt(k)), 3.0) * (2.0 * k));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 7.8e-47)
		tmp = Float64(2.0 / Float64(Float64(Float64((k ^ 2.0) * Float64(t_m * (sin(k) ^ 2.0))) / Float64(l * cos(k))) / l));
	elseif (t_m <= 5.6e+102)
		tmp = Float64(Float64(Float64(Float64(2.0 / (t_m ^ 3.0)) / sin(k)) * Float64(l / tan(k))) * Float64(l / Float64(2.0 + (Float64(k / t_m) ^ 2.0))));
	elseif (t_m <= 6.2e+202)
		tmp = Float64(2.0 / Float64(Float64(sin(k) * (Float64((t_m ^ 1.5) / l) ^ 2.0)) * Float64(Float64(2.0 * sin(k)) / cos(k))));
	else
		tmp = Float64(2.0 / Float64((Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * cbrt(k)) ^ 3.0) * Float64(2.0 * k)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 7.8e-47], N[(2.0 / N[(N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.6e+102], N[(N[(N[(N[(2.0 / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6.2e+202], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[k, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if t < 7.79999999999999956e-47

   1. Initial program 51.6%

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

   3. Simplified 51.7%

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

      1. add-cube-cbrt 51.7%

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

      2. pow3 51.7%

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

      3. cbrt-div 51.6%

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

      4. rem-cbrt-cube 56.6%

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

   6. Applied egg-rr 56.6%

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

      1. associate-*l/ 58.8%

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

      2. add-cbrt-cube 53.8%

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

      3. unpow3 53.8%

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

      4. cbrt-div 53.8%

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

      5. pow3 53.8%

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

      6. add-cube-cbrt 53.9%

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

      7. metadata-eval 53.9%

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

      8. associate-+r+ 53.9%

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

      9. associate-*l* 53.9%

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

      10. associate-+r+ 53.9%

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

      11. metadata-eval 53.9%

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

   8. Applied egg-rr 53.9%

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

   9. Taylor expanded in t around 0 64.7%

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

   if 7.79999999999999956e-47 < t < 5.60000000000000037e102

   1. Initial program 85.5%

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

   3. Simplified 85.5%

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

      1. associate-*r* 91.4%

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

      2. *-un-lft-identity 91.4%

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

      3. times-frac 94.2%

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

   6. Applied egg-rr 94.2%

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

      1. /-rgt-identity 94.2%

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

      2. associate-*l/ 94.2%

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

      3. associate-*r* 97.1%

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

      4. times-frac 97.0%

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

      5. associate-/r* 97.1%

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

   8. Simplified 97.1%

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

   if 5.60000000000000037e102 < t < 6.19999999999999983e202

   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-sqr-sqrt 29.6%

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

      2. pow2 29.6%

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

      3. *-commutative 29.6%

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

      4. sqrt-prod 29.6%

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

      5. sqrt-div 29.6%

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

      6. sqrt-pow1 43.7%

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

      7. metadata-eval 43.7%

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

      8. sqrt-prod 24.4%

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

      9. add-sqr-sqrt 52.9%

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

   6. Applied egg-rr 52.9%

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

      1. *-commutative 52.9%

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

   8. Simplified 52.9%

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

      1. pow1 52.9%

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

      2. *-commutative 52.9%

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

      3. associate-+r+ 52.9%

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

      4. metadata-eval 52.9%

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

      5. *-commutative 52.9%

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

      6. unpow-prod-down 52.9%

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

      7. pow2 52.9%

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

      8. add-sqr-sqrt 81.8%

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

   10. Applied egg-rr 81.8%

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

       1. unpow1 81.8%

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

   12. Simplified 81.8%

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

   13. Taylor expanded in t around inf 72.9%

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

       1. associate-*r/ 72.9%

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

   15. Simplified 72.9%

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

   if 6.19999999999999983e202 < t

   1. Initial program 74.2%

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

   3. Simplified 74.2%

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

      1. add-cube-cbrt 74.2%

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

      2. pow3 74.2%

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

      3. *-commutative 74.2%

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

      4. cbrt-prod 74.2%

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

      5. cbrt-div 74.2%

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

      6. rem-cbrt-cube 81.8%

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

      7. cbrt-prod 96.3%

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

      8. pow2 96.3%

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

   6. Applied egg-rr 96.3%

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

      1. *-commutative 96.3%

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

   8. Simplified 96.3%

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

   9. Taylor expanded in k around 0 92.9%

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

   10. Taylor expanded in k around 0 92.9%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.8 \cdot 10^{-47}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}{\ell}}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\left(\frac{\frac{2}{{t}^{3}}}{\sin k} \cdot \frac{\ell}{\tan k}\right) \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{+202}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \frac{2 \cdot \sin k}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{k}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\frac{k}{t\_m}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.35 \cdot 10^{-45}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{2} \cdot \left(t\_m \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}{\ell}}\\ \mathbf{elif}\;t\_m \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\left(\frac{\frac{2}{{t\_m}^{3}}}{\sin k} \cdot \frac{\ell}{\tan k}\right) \cdot \frac{\ell}{2 + t\_2}\\ \mathbf{elif}\;t\_m \leq 1.08 \cdot 10^{+130}:\\ \;\;\;\;\frac{2}{\left(1 + \left(t\_2 + 1\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \left(\frac{{t\_m}^{2}}{\ell} \cdot \frac{t\_m}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{k}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (/ k t_m) 2.0)))
   (*
    t_s
    (if (<= t_m 1.35e-45)
      (/ 2.0 (/ (/ (* (pow k 2.0) (* t_m (pow (sin k) 2.0))) (* l (cos k))) l))
      (if (<= t_m 5.6e+102)
        (*
         (* (/ (/ 2.0 (pow t_m 3.0)) (sin k)) (/ l (tan k)))
         (/ l (+ 2.0 t_2)))
        (if (<= t_m 1.08e+130)
          (/
           2.0
           (*
            (+ 1.0 (+ t_2 1.0))
            (* (tan k) (* (sin k) (* (/ (pow t_m 2.0) l) (/ t_m l))))))
          (/
           2.0
           (*
            (pow (* (/ t_m (pow (cbrt l) 2.0)) (cbrt k)) 3.0)
            (* 2.0 k)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 1.35e-45) {
		tmp = 2.0 / (((pow(k, 2.0) * (t_m * pow(sin(k), 2.0))) / (l * cos(k))) / l);
	} else if (t_m <= 5.6e+102) {
		tmp = (((2.0 / pow(t_m, 3.0)) / sin(k)) * (l / tan(k))) * (l / (2.0 + t_2));
	} else if (t_m <= 1.08e+130) {
		tmp = 2.0 / ((1.0 + (t_2 + 1.0)) * (tan(k) * (sin(k) * ((pow(t_m, 2.0) / l) * (t_m / l)))));
	} else {
		tmp = 2.0 / (pow(((t_m / pow(cbrt(l), 2.0)) * cbrt(k)), 3.0) * (2.0 * k));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 1.35e-45) {
		tmp = 2.0 / (((Math.pow(k, 2.0) * (t_m * Math.pow(Math.sin(k), 2.0))) / (l * Math.cos(k))) / l);
	} else if (t_m <= 5.6e+102) {
		tmp = (((2.0 / Math.pow(t_m, 3.0)) / Math.sin(k)) * (l / Math.tan(k))) * (l / (2.0 + t_2));
	} else if (t_m <= 1.08e+130) {
		tmp = 2.0 / ((1.0 + (t_2 + 1.0)) * (Math.tan(k) * (Math.sin(k) * ((Math.pow(t_m, 2.0) / l) * (t_m / l)))));
	} else {
		tmp = 2.0 / (Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt(k)), 3.0) * (2.0 * k));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(k / t_m) ^ 2.0
	tmp = 0.0
	if (t_m <= 1.35e-45)
		tmp = Float64(2.0 / Float64(Float64(Float64((k ^ 2.0) * Float64(t_m * (sin(k) ^ 2.0))) / Float64(l * cos(k))) / l));
	elseif (t_m <= 5.6e+102)
		tmp = Float64(Float64(Float64(Float64(2.0 / (t_m ^ 3.0)) / sin(k)) * Float64(l / tan(k))) * Float64(l / Float64(2.0 + t_2)));
	elseif (t_m <= 1.08e+130)
		tmp = Float64(2.0 / Float64(Float64(1.0 + Float64(t_2 + 1.0)) * Float64(tan(k) * Float64(sin(k) * Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l))))));
	else
		tmp = Float64(2.0 / Float64((Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * cbrt(k)) ^ 3.0) * Float64(2.0 * k)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.35e-45], N[(2.0 / N[(N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.6e+102], N[(N[(N[(N[(2.0 / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.08e+130], N[(2.0 / N[(N[(1.0 + N[(t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[k, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 1.34999999999999992e-45

   1. Initial program 51.6%

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

   3. Simplified 51.7%

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

      1. add-cube-cbrt 51.7%

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

      2. pow3 51.7%

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

      3. cbrt-div 51.6%

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

      4. rem-cbrt-cube 56.6%

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

   6. Applied egg-rr 56.6%

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

      1. associate-*l/ 58.8%

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

      2. add-cbrt-cube 53.8%

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

      3. unpow3 53.8%

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

      4. cbrt-div 53.8%

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

      5. pow3 53.8%

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

      6. add-cube-cbrt 53.9%

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

      7. metadata-eval 53.9%

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

      8. associate-+r+ 53.9%

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

      9. associate-*l* 53.9%

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

      10. associate-+r+ 53.9%

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

      11. metadata-eval 53.9%

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

   8. Applied egg-rr 53.9%

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

   9. Taylor expanded in t around 0 64.7%

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

   if 1.34999999999999992e-45 < t < 5.60000000000000037e102

   1. Initial program 85.5%

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

   3. Simplified 85.5%

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

      1. associate-*r* 91.4%

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

      2. *-un-lft-identity 91.4%

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

      3. times-frac 94.2%

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

   6. Applied egg-rr 94.2%

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

      1. /-rgt-identity 94.2%

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

      2. associate-*l/ 94.2%

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

      3. associate-*r* 97.1%

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

      4. times-frac 97.0%

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

      5. associate-/r* 97.1%

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

   8. Simplified 97.1%

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

   if 5.60000000000000037e102 < t < 1.08e130

   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. Add Preprocessing
   3. Step-by-step derivation

      1. unpow3 60.8%

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

      2. times-frac 99.7%

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

      3. pow2 99.7%

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

   4. Applied egg-rr 99.7%

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

   if 1.08e130 < t

   1. Initial program 61.2%

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

   3. Simplified 61.2%

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

      1. add-cube-cbrt 61.2%

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

      2. pow3 61.2%

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

      3. *-commutative 61.2%

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

      4. cbrt-prod 61.2%

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

      5. cbrt-div 61.2%

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

      6. rem-cbrt-cube 77.4%

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

      7. cbrt-prod 93.0%

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

      8. pow2 93.0%

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

   6. Applied egg-rr 93.0%

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

      1. *-commutative 93.0%

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

   8. Simplified 93.0%

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

   9. Taylor expanded in k around 0 84.4%

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

   10. Taylor expanded in k around 0 84.5%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.35 \cdot 10^{-45}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}{\ell}}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\left(\frac{\frac{2}{{t}^{3}}}{\sin k} \cdot \frac{\ell}{\tan k}\right) \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;t \leq 1.08 \cdot 10^{+130}:\\ \;\;\;\;\frac{2}{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{k}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8 \cdot 10^{-47}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{2} \cdot \left(t\_m \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}{\ell}}\\ \mathbf{elif}\;t\_m \leq 7.5 \cdot 10^{+118}:\\ \;\;\;\;\left(\frac{\frac{2}{{t\_m}^{3}}}{\sin k} \cdot \frac{\ell}{\tan k}\right) \cdot \frac{\ell}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{k}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 8e-47)
    (/ 2.0 (/ (/ (* (pow k 2.0) (* t_m (pow (sin k) 2.0))) (* l (cos k))) l))
    (if (<= t_m 7.5e+118)
      (*
       (* (/ (/ 2.0 (pow t_m 3.0)) (sin k)) (/ l (tan k)))
       (/ l (+ 2.0 (pow (/ k t_m) 2.0))))
      (/
       2.0
       (* (pow (* (/ t_m (pow (cbrt l) 2.0)) (cbrt k)) 3.0) (* 2.0 k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 8e-47) {
		tmp = 2.0 / (((pow(k, 2.0) * (t_m * pow(sin(k), 2.0))) / (l * cos(k))) / l);
	} else if (t_m <= 7.5e+118) {
		tmp = (((2.0 / pow(t_m, 3.0)) / sin(k)) * (l / tan(k))) * (l / (2.0 + pow((k / t_m), 2.0)));
	} else {
		tmp = 2.0 / (pow(((t_m / pow(cbrt(l), 2.0)) * cbrt(k)), 3.0) * (2.0 * k));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 8e-47) {
		tmp = 2.0 / (((Math.pow(k, 2.0) * (t_m * Math.pow(Math.sin(k), 2.0))) / (l * Math.cos(k))) / l);
	} else if (t_m <= 7.5e+118) {
		tmp = (((2.0 / Math.pow(t_m, 3.0)) / Math.sin(k)) * (l / Math.tan(k))) * (l / (2.0 + Math.pow((k / t_m), 2.0)));
	} else {
		tmp = 2.0 / (Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt(k)), 3.0) * (2.0 * k));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 8e-47)
		tmp = Float64(2.0 / Float64(Float64(Float64((k ^ 2.0) * Float64(t_m * (sin(k) ^ 2.0))) / Float64(l * cos(k))) / l));
	elseif (t_m <= 7.5e+118)
		tmp = Float64(Float64(Float64(Float64(2.0 / (t_m ^ 3.0)) / sin(k)) * Float64(l / tan(k))) * Float64(l / Float64(2.0 + (Float64(k / t_m) ^ 2.0))));
	else
		tmp = Float64(2.0 / Float64((Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * cbrt(k)) ^ 3.0) * Float64(2.0 * k)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 8e-47], N[(2.0 / N[(N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 7.5e+118], N[(N[(N[(N[(2.0 / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[k, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 7.9999999999999998e-47

   1. Initial program 51.6%

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

   3. Simplified 51.7%

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

      1. add-cube-cbrt 51.7%

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

      2. pow3 51.7%

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

      3. cbrt-div 51.6%

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

      4. rem-cbrt-cube 56.6%

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

   6. Applied egg-rr 56.6%

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

      1. associate-*l/ 58.8%

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

      2. add-cbrt-cube 53.8%

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

      3. unpow3 53.8%

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

      4. cbrt-div 53.8%

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

      5. pow3 53.8%

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

      6. add-cube-cbrt 53.9%

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

      7. metadata-eval 53.9%

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

      8. associate-+r+ 53.9%

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

      9. associate-*l* 53.9%

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

      10. associate-+r+ 53.9%

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

      11. metadata-eval 53.9%

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

   8. Applied egg-rr 53.9%

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

   9. Taylor expanded in t around 0 64.7%

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

   if 7.9999999999999998e-47 < t < 7.50000000000000003e118

   1. Initial program 83.5%

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

   3. Simplified 83.5%

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

      1. associate-*r* 89.1%

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

      2. *-un-lft-identity 89.1%

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

      3. times-frac 91.8%

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

   6. Applied egg-rr 91.8%

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

      1. /-rgt-identity 91.8%

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

      2. associate-*l/ 91.8%

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

      3. associate-*r* 94.5%

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

      4. times-frac 94.4%

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

      5. associate-/r* 94.5%

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

   8. Simplified 94.5%

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

   if 7.50000000000000003e118 < t

   1. Initial program 61.7%

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

   3. Simplified 61.7%

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

      1. add-cube-cbrt 61.7%

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

      2. pow3 61.7%

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

      3. *-commutative 61.7%

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

      4. cbrt-prod 61.7%

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

      5. cbrt-div 61.7%

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

      6. rem-cbrt-cube 78.8%

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

      7. cbrt-prod 93.4%

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

      8. pow2 93.4%

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

   6. Applied egg-rr 93.4%

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

      1. *-commutative 93.4%

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

   8. Simplified 93.4%

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

   9. Taylor expanded in k around 0 83.4%

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

   10. Taylor expanded in k around 0 83.5%

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

Alternative 7: 72.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.9 \cdot 10^{-115}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{\sqrt[3]{\ell}} \cdot \sqrt[3]{\frac{2 \cdot {k}^{2}}{\ell}}\right)}^{3}}\\ \mathbf{elif}\;t\_m \leq 3400000000:\\ \;\;\;\;\frac{2}{\frac{\frac{{t\_m}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{1}{\frac{t\_m}{k} \cdot \frac{t\_m}{k}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{k}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.9e-115)
    (/ 2.0 (pow (* (/ t_m (cbrt l)) (cbrt (/ (* 2.0 (pow k 2.0)) l))) 3.0))
    (if (<= t_m 3400000000.0)
      (/
       2.0
       (*
        (/ (/ (pow t_m 3.0) l) l)
        (* (* (sin k) (tan k)) (+ 2.0 (/ 1.0 (* (/ t_m k) (/ t_m k)))))))
      (/
       2.0
       (* (pow (* (/ t_m (pow (cbrt l) 2.0)) (cbrt k)) 3.0) (* 2.0 k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.9e-115) {
		tmp = 2.0 / pow(((t_m / cbrt(l)) * cbrt(((2.0 * pow(k, 2.0)) / l))), 3.0);
	} else if (t_m <= 3400000000.0) {
		tmp = 2.0 / (((pow(t_m, 3.0) / l) / l) * ((sin(k) * tan(k)) * (2.0 + (1.0 / ((t_m / k) * (t_m / k))))));
	} else {
		tmp = 2.0 / (pow(((t_m / pow(cbrt(l), 2.0)) * cbrt(k)), 3.0) * (2.0 * k));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.9e-115) {
		tmp = 2.0 / Math.pow(((t_m / Math.cbrt(l)) * Math.cbrt(((2.0 * Math.pow(k, 2.0)) / l))), 3.0);
	} else if (t_m <= 3400000000.0) {
		tmp = 2.0 / (((Math.pow(t_m, 3.0) / l) / l) * ((Math.sin(k) * Math.tan(k)) * (2.0 + (1.0 / ((t_m / k) * (t_m / k))))));
	} else {
		tmp = 2.0 / (Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt(k)), 3.0) * (2.0 * k));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 3.9e-115)
		tmp = Float64(2.0 / (Float64(Float64(t_m / cbrt(l)) * cbrt(Float64(Float64(2.0 * (k ^ 2.0)) / l))) ^ 3.0));
	elseif (t_m <= 3400000000.0)
		tmp = Float64(2.0 / Float64(Float64(Float64((t_m ^ 3.0) / l) / l) * Float64(Float64(sin(k) * tan(k)) * Float64(2.0 + Float64(1.0 / Float64(Float64(t_m / k) * Float64(t_m / k)))))));
	else
		tmp = Float64(2.0 / Float64((Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * cbrt(k)) ^ 3.0) * Float64(2.0 * k)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 3.9e-115], N[(2.0 / N[Power[N[(N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3400000000.0], N[(2.0 / N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(1.0 / N[(N[(t$95$m / k), $MachinePrecision] * N[(t$95$m / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[k, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 3.8999999999999998e-115

   1. Initial program 48.4%

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

   3. Simplified 47.9%

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

   5. Taylor expanded in k around 0 51.4%

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

      1. associate-*l/ 52.6%

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

   7. Applied egg-rr 52.6%

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

      1. add-cube-cbrt 52.6%

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

      2. pow3 52.6%

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

      3. associate-/l* 52.5%

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

      4. cbrt-prod 52.5%

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

      5. cbrt-div 52.5%

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

      6. unpow3 52.5%

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

      7. add-cbrt-cube 58.5%

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

   9. Applied egg-rr 58.5%

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

   if 3.8999999999999998e-115 < t < 3.4e9

   1. Initial program 83.5%

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

   3. Simplified 88.5%

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

      1. unpow2 88.5%

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

      2. clear-num 88.5%

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

      3. clear-num 88.5%

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

      4. frac-times 88.6%

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

      5. metadata-eval 88.6%

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

   6. Applied egg-rr 88.6%

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

   if 3.4e9 < t

   1. Initial program 65.9%

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

   3. Simplified 65.9%

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

      1. add-cube-cbrt 65.8%

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

      2. pow3 65.8%

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

      3. *-commutative 65.8%

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

      4. cbrt-prod 65.9%

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

      5. cbrt-div 67.4%

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

      6. rem-cbrt-cube 80.0%

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

      7. cbrt-prod 93.5%

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

      8. pow2 93.5%

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

   6. Applied egg-rr 93.5%

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

      1. *-commutative 93.5%

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

   8. Simplified 93.5%

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

   9. Taylor expanded in k around 0 80.3%

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

   10. Taylor expanded in k around 0 82.0%

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

Alternative 8: 69.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.35 \cdot 10^{-138}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot {\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}}\\ \mathbf{elif}\;t\_m \leq 2200000:\\ \;\;\;\;\frac{2}{\frac{\frac{{t\_m}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{1}{\frac{t\_m}{k} \cdot \frac{t\_m}{k}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{k}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.35e-138)
    (/ 2.0 (* (* 2.0 (pow k 2.0)) (pow (* t_m (pow (cbrt l) -2.0)) 3.0)))
    (if (<= t_m 2200000.0)
      (/
       2.0
       (*
        (/ (/ (pow t_m 3.0) l) l)
        (* (* (sin k) (tan k)) (+ 2.0 (/ 1.0 (* (/ t_m k) (/ t_m k)))))))
      (/
       2.0
       (* (pow (* (/ t_m (pow (cbrt l) 2.0)) (cbrt k)) 3.0) (* 2.0 k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.35e-138) {
		tmp = 2.0 / ((2.0 * pow(k, 2.0)) * pow((t_m * pow(cbrt(l), -2.0)), 3.0));
	} else if (t_m <= 2200000.0) {
		tmp = 2.0 / (((pow(t_m, 3.0) / l) / l) * ((sin(k) * tan(k)) * (2.0 + (1.0 / ((t_m / k) * (t_m / k))))));
	} else {
		tmp = 2.0 / (pow(((t_m / pow(cbrt(l), 2.0)) * cbrt(k)), 3.0) * (2.0 * k));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.35e-138) {
		tmp = 2.0 / ((2.0 * Math.pow(k, 2.0)) * Math.pow((t_m * Math.pow(Math.cbrt(l), -2.0)), 3.0));
	} else if (t_m <= 2200000.0) {
		tmp = 2.0 / (((Math.pow(t_m, 3.0) / l) / l) * ((Math.sin(k) * Math.tan(k)) * (2.0 + (1.0 / ((t_m / k) * (t_m / k))))));
	} else {
		tmp = 2.0 / (Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt(k)), 3.0) * (2.0 * k));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.35e-138)
		tmp = Float64(2.0 / Float64(Float64(2.0 * (k ^ 2.0)) * (Float64(t_m * (cbrt(l) ^ -2.0)) ^ 3.0)));
	elseif (t_m <= 2200000.0)
		tmp = Float64(2.0 / Float64(Float64(Float64((t_m ^ 3.0) / l) / l) * Float64(Float64(sin(k) * tan(k)) * Float64(2.0 + Float64(1.0 / Float64(Float64(t_m / k) * Float64(t_m / k)))))));
	else
		tmp = Float64(2.0 / Float64((Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * cbrt(k)) ^ 3.0) * Float64(2.0 * k)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.35e-138], N[(2.0 / N[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2200000.0], N[(2.0 / N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(1.0 / N[(N[(t$95$m / k), $MachinePrecision] * N[(t$95$m / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[k, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.3500000000000001e-138

   1. Initial program 49.1%

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

   3. Simplified 48.5%

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

   5. Taylor expanded in k around 0 52.1%

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

      1. add-cube-cbrt 52.1%

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

      2. pow3 52.1%

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

      3. associate-/l/ 45.3%

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

      4. cbrt-div 45.3%

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

      5. unpow3 45.3%

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

      6. add-cbrt-cube 51.4%

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

      7. cbrt-unprod 57.4%

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

      8. unpow2 57.4%

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

      9. div-inv 57.4%

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

      10. unpow-prod-down 44.6%

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

      11. pow-flip 44.6%

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

      12. metadata-eval 44.6%

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

   7. Applied egg-rr 44.6%

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

      1. cube-prod 57.4%

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

   9. Simplified 57.4%

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

   if 2.3500000000000001e-138 < t < 2.2e6

   1. Initial program 76.5%

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

   3. Simplified 80.8%

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

      1. unpow2 80.8%

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

      2. clear-num 80.8%

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

      3. clear-num 80.8%

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

      4. frac-times 80.9%

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

      5. metadata-eval 80.9%

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

   6. Applied egg-rr 80.9%

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

   if 2.2e6 < t

   1. Initial program 65.9%

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

   3. Simplified 65.9%

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

      1. add-cube-cbrt 65.8%

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

      2. pow3 65.8%

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

      3. *-commutative 65.8%

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

      4. cbrt-prod 65.9%

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

      5. cbrt-div 67.4%

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

      6. rem-cbrt-cube 80.0%

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

      7. cbrt-prod 93.5%

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

      8. pow2 93.5%

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

   6. Applied egg-rr 93.5%

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

      1. *-commutative 93.5%

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

   8. Simplified 93.5%

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

   9. Taylor expanded in k around 0 80.3%

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

   10. Taylor expanded in k around 0 82.0%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.35 \cdot 10^{-138}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot {\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}}\\ \mathbf{elif}\;t \leq 2200000:\\ \;\;\;\;\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{1}{\frac{t}{k} \cdot \frac{t}{k}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{k}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 0.016:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{k}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{2} \cdot \left(t\_m \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 0.016)
    (/ 2.0 (* (pow (* (/ t_m (pow (cbrt l) 2.0)) (cbrt k)) 3.0) (* 2.0 k)))
    (/
     2.0
     (/ (/ (* (pow k 2.0) (* t_m (pow (sin k) 2.0))) (* l (cos k))) l)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 0.016) {
		tmp = 2.0 / (pow(((t_m / pow(cbrt(l), 2.0)) * cbrt(k)), 3.0) * (2.0 * k));
	} else {
		tmp = 2.0 / (((pow(k, 2.0) * (t_m * pow(sin(k), 2.0))) / (l * cos(k))) / l);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 0.016) {
		tmp = 2.0 / (Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt(k)), 3.0) * (2.0 * k));
	} else {
		tmp = 2.0 / (((Math.pow(k, 2.0) * (t_m * Math.pow(Math.sin(k), 2.0))) / (l * Math.cos(k))) / l);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 0.016)
		tmp = Float64(2.0 / Float64((Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * cbrt(k)) ^ 3.0) * Float64(2.0 * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((k ^ 2.0) * Float64(t_m * (sin(k) ^ 2.0))) / Float64(l * cos(k))) / l));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 0.016], N[(2.0 / N[(N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[k, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 0.016

   1. Initial program 58.8%

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

   3. Simplified 58.8%

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

      1. add-cube-cbrt 58.7%

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

      2. pow3 58.7%

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

      3. *-commutative 58.7%

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

      4. cbrt-prod 58.7%

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

      5. cbrt-div 59.6%

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

      6. rem-cbrt-cube 67.2%

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

      7. cbrt-prod 77.3%

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

      8. pow2 77.3%

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

   6. Applied egg-rr 77.3%

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

      1. *-commutative 77.3%

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

   8. Simplified 77.3%

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

   9. Taylor expanded in k around 0 71.7%

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

   10. Taylor expanded in k around 0 74.2%

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

   if 0.016 < k

   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 59.8%

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

      1. add-cube-cbrt 59.7%

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

      2. pow3 59.7%

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

      3. cbrt-div 59.6%

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

      4. rem-cbrt-cube 65.0%

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

   6. Applied egg-rr 65.0%

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

      1. associate-*l/ 63.5%

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

      2. add-cbrt-cube 58.1%

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

      3. unpow3 58.1%

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

      4. cbrt-div 58.2%

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

      5. pow3 58.2%

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

      6. add-cube-cbrt 58.2%

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

      7. metadata-eval 58.2%

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

      8. associate-+r+ 58.2%

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

      9. associate-*l* 58.2%

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

      10. associate-+r+ 58.2%

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

      11. metadata-eval 58.2%

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

   8. Applied egg-rr 58.2%

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

   9. Taylor expanded in t around 0 68.0%

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

Alternative 10: 73.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 0.014:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{k}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(\frac{t\_m}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 0.014)
    (/ 2.0 (* (pow (* (/ t_m (pow (cbrt l) 2.0)) (cbrt k)) 3.0) (* 2.0 k)))
    (/
     2.0
     (/ (* (pow k 2.0) (* (/ t_m l) (/ (pow (sin k) 2.0) (cos k)))) l)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 0.014) {
		tmp = 2.0 / (pow(((t_m / pow(cbrt(l), 2.0)) * cbrt(k)), 3.0) * (2.0 * k));
	} else {
		tmp = 2.0 / ((pow(k, 2.0) * ((t_m / l) * (pow(sin(k), 2.0) / cos(k)))) / l);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 0.014) {
		tmp = 2.0 / (Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt(k)), 3.0) * (2.0 * k));
	} else {
		tmp = 2.0 / ((Math.pow(k, 2.0) * ((t_m / l) * (Math.pow(Math.sin(k), 2.0) / Math.cos(k)))) / l);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 0.014)
		tmp = Float64(2.0 / Float64((Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * cbrt(k)) ^ 3.0) * Float64(2.0 * k)));
	else
		tmp = Float64(2.0 / Float64(Float64((k ^ 2.0) * Float64(Float64(t_m / l) * Float64((sin(k) ^ 2.0) / cos(k)))) / l));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 0.014], N[(2.0 / N[(N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[k, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 0.0140000000000000003

   1. Initial program 58.8%

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

   3. Simplified 58.8%

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

      1. add-cube-cbrt 58.7%

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

      2. pow3 58.7%

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

      3. *-commutative 58.7%

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

      4. cbrt-prod 58.7%

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

      5. cbrt-div 59.6%

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

      6. rem-cbrt-cube 67.2%

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

      7. cbrt-prod 77.3%

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

      8. pow2 77.3%

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

   6. Applied egg-rr 77.3%

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

      1. *-commutative 77.3%

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

   8. Simplified 77.3%

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

   9. Taylor expanded in k around 0 71.7%

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

   10. Taylor expanded in k around 0 74.2%

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

   if 0.0140000000000000003 < k

   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 59.8%

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

      1. add-cube-cbrt 59.7%

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

      2. pow3 59.7%

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

      3. cbrt-div 59.6%

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

      4. rem-cbrt-cube 65.0%

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

   6. Applied egg-rr 65.0%

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

      1. associate-*l/ 63.5%

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

      2. add-cbrt-cube 58.1%

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

      3. unpow3 58.1%

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

      4. cbrt-div 58.2%

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

      5. pow3 58.2%

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

      6. add-cube-cbrt 58.2%

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

      7. metadata-eval 58.2%

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

      8. associate-+r+ 58.2%

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

      9. associate-*l* 58.2%

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

      10. associate-+r+ 58.2%

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

      11. metadata-eval 58.2%

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

   8. Applied egg-rr 58.2%

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

   9. Taylor expanded in t around 0 68.0%

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

       1. associate-/l* 67.1%

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

       2. times-frac 67.2%

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

   11. Simplified 67.2%

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

Alternative 11: 63.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.35 \cdot 10^{-138}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot {\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}}\\ \mathbf{elif}\;t\_m \leq 108000000:\\ \;\;\;\;\frac{2}{\frac{\frac{{t\_m}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{1}{\frac{t\_m}{k} \cdot \frac{t\_m}{k}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left(t\_m \cdot \sqrt[3]{\frac{k}{{\ell}^{2}}}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.35e-138)
    (/ 2.0 (* (* 2.0 (pow k 2.0)) (pow (* t_m (pow (cbrt l) -2.0)) 3.0)))
    (if (<= t_m 108000000.0)
      (/
       2.0
       (*
        (/ (/ (pow t_m 3.0) l) l)
        (* (* (sin k) (tan k)) (+ 2.0 (/ 1.0 (* (/ t_m k) (/ t_m k)))))))
      (/ 2.0 (* (* 2.0 k) (pow (* t_m (cbrt (/ k (pow l 2.0)))) 3.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.35e-138) {
		tmp = 2.0 / ((2.0 * pow(k, 2.0)) * pow((t_m * pow(cbrt(l), -2.0)), 3.0));
	} else if (t_m <= 108000000.0) {
		tmp = 2.0 / (((pow(t_m, 3.0) / l) / l) * ((sin(k) * tan(k)) * (2.0 + (1.0 / ((t_m / k) * (t_m / k))))));
	} else {
		tmp = 2.0 / ((2.0 * k) * pow((t_m * cbrt((k / pow(l, 2.0)))), 3.0));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.35e-138) {
		tmp = 2.0 / ((2.0 * Math.pow(k, 2.0)) * Math.pow((t_m * Math.pow(Math.cbrt(l), -2.0)), 3.0));
	} else if (t_m <= 108000000.0) {
		tmp = 2.0 / (((Math.pow(t_m, 3.0) / l) / l) * ((Math.sin(k) * Math.tan(k)) * (2.0 + (1.0 / ((t_m / k) * (t_m / k))))));
	} else {
		tmp = 2.0 / ((2.0 * k) * Math.pow((t_m * Math.cbrt((k / Math.pow(l, 2.0)))), 3.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.35e-138)
		tmp = Float64(2.0 / Float64(Float64(2.0 * (k ^ 2.0)) * (Float64(t_m * (cbrt(l) ^ -2.0)) ^ 3.0)));
	elseif (t_m <= 108000000.0)
		tmp = Float64(2.0 / Float64(Float64(Float64((t_m ^ 3.0) / l) / l) * Float64(Float64(sin(k) * tan(k)) * Float64(2.0 + Float64(1.0 / Float64(Float64(t_m / k) * Float64(t_m / k)))))));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * (Float64(t_m * cbrt(Float64(k / (l ^ 2.0)))) ^ 3.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.35e-138], N[(2.0 / N[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 108000000.0], N[(2.0 / N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(1.0 / N[(N[(t$95$m / k), $MachinePrecision] * N[(t$95$m / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[Power[N[(t$95$m * N[Power[N[(k / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.3500000000000001e-138

   1. Initial program 49.1%

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

   3. Simplified 48.5%

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

   5. Taylor expanded in k around 0 52.1%

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

      1. add-cube-cbrt 52.1%

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

      2. pow3 52.1%

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

      3. associate-/l/ 45.3%

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

      4. cbrt-div 45.3%

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

      5. unpow3 45.3%

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

      6. add-cbrt-cube 51.4%

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

      7. cbrt-unprod 57.4%

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

      8. unpow2 57.4%

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

      9. div-inv 57.4%

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

      10. unpow-prod-down 44.6%

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

      11. pow-flip 44.6%

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

      12. metadata-eval 44.6%

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

   7. Applied egg-rr 44.6%

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

      1. cube-prod 57.4%

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

   9. Simplified 57.4%

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

   if 2.3500000000000001e-138 < t < 1.08e8

   1. Initial program 76.5%

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

   3. Simplified 80.8%

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

      1. unpow2 80.8%

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

      2. clear-num 80.8%

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

      3. clear-num 80.8%

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

      4. frac-times 80.9%

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

      5. metadata-eval 80.9%

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

   6. Applied egg-rr 80.9%

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

   if 1.08e8 < t

   1. Initial program 65.9%

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

   3. Simplified 65.9%

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

      1. add-cube-cbrt 65.8%

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

      2. pow3 65.8%

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

      3. *-commutative 65.8%

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

      4. cbrt-prod 65.9%

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

      5. cbrt-div 67.4%

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

      6. rem-cbrt-cube 80.0%

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

      7. cbrt-prod 93.5%

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

      8. pow2 93.5%

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

   6. Applied egg-rr 93.5%

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

      1. *-commutative 93.5%

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

   8. Simplified 93.5%

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

   9. Taylor expanded in k around 0 80.3%

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

   10. Taylor expanded in k around 0 72.6%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.35 \cdot 10^{-138}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot {\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}}\\ \mathbf{elif}\;t \leq 108000000:\\ \;\;\;\;\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{1}{\frac{t}{k} \cdot \frac{t}{k}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left(t \cdot \sqrt[3]{\frac{k}{{\ell}^{2}}}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 62.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.35 \cdot 10^{-138}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \left({t\_m}^{1.5} \cdot \frac{\frac{{t\_m}^{1.5}}{\ell}}{\ell}\right)}\\ \mathbf{elif}\;t\_m \leq 2900000000:\\ \;\;\;\;\frac{2}{\frac{\frac{{t\_m}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{1}{\frac{t\_m}{k} \cdot \frac{t\_m}{k}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left(t\_m \cdot \sqrt[3]{\frac{k}{{\ell}^{2}}}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.35e-138)
    (/ 2.0 (* (* 2.0 (pow k 2.0)) (* (pow t_m 1.5) (/ (/ (pow t_m 1.5) l) l))))
    (if (<= t_m 2900000000.0)
      (/
       2.0
       (*
        (/ (/ (pow t_m 3.0) l) l)
        (* (* (sin k) (tan k)) (+ 2.0 (/ 1.0 (* (/ t_m k) (/ t_m k)))))))
      (/ 2.0 (* (* 2.0 k) (pow (* t_m (cbrt (/ k (pow l 2.0)))) 3.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.35e-138) {
		tmp = 2.0 / ((2.0 * pow(k, 2.0)) * (pow(t_m, 1.5) * ((pow(t_m, 1.5) / l) / l)));
	} else if (t_m <= 2900000000.0) {
		tmp = 2.0 / (((pow(t_m, 3.0) / l) / l) * ((sin(k) * tan(k)) * (2.0 + (1.0 / ((t_m / k) * (t_m / k))))));
	} else {
		tmp = 2.0 / ((2.0 * k) * pow((t_m * cbrt((k / pow(l, 2.0)))), 3.0));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.35e-138) {
		tmp = 2.0 / ((2.0 * Math.pow(k, 2.0)) * (Math.pow(t_m, 1.5) * ((Math.pow(t_m, 1.5) / l) / l)));
	} else if (t_m <= 2900000000.0) {
		tmp = 2.0 / (((Math.pow(t_m, 3.0) / l) / l) * ((Math.sin(k) * Math.tan(k)) * (2.0 + (1.0 / ((t_m / k) * (t_m / k))))));
	} else {
		tmp = 2.0 / ((2.0 * k) * Math.pow((t_m * Math.cbrt((k / Math.pow(l, 2.0)))), 3.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.35e-138)
		tmp = Float64(2.0 / Float64(Float64(2.0 * (k ^ 2.0)) * Float64((t_m ^ 1.5) * Float64(Float64((t_m ^ 1.5) / l) / l))));
	elseif (t_m <= 2900000000.0)
		tmp = Float64(2.0 / Float64(Float64(Float64((t_m ^ 3.0) / l) / l) * Float64(Float64(sin(k) * tan(k)) * Float64(2.0 + Float64(1.0 / Float64(Float64(t_m / k) * Float64(t_m / k)))))));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * (Float64(t_m * cbrt(Float64(k / (l ^ 2.0)))) ^ 3.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.35e-138], N[(2.0 / N[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[t$95$m, 1.5], $MachinePrecision] * N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2900000000.0], N[(2.0 / N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(1.0 / N[(N[(t$95$m / k), $MachinePrecision] * N[(t$95$m / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[Power[N[(t$95$m * N[Power[N[(k / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.3500000000000001e-138

   1. Initial program 49.1%

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

   3. Simplified 48.5%

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

   5. Taylor expanded in k around 0 52.1%

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

      1. sqr-pow 10.7%

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

      2. *-un-lft-identity 10.7%

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

      3. times-frac 11.4%

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

      4. metadata-eval 11.4%

         \[\leadsto \frac{2}{\frac{\frac{{t}^{\color{blue}{1.5}}}{1} \cdot \frac{{t}^{\left(\frac{3}{2}\right)}}{\ell}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)} \]

      5. metadata-eval 11.4%

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

   7. Applied egg-rr 11.4%

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

      1. /-rgt-identity 11.4%

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

      2. associate-/l* 13.4%

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

   9. Applied egg-rr 13.4%

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

   if 2.3500000000000001e-138 < t < 2.9e9

   1. Initial program 76.5%

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

   3. Simplified 80.8%

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

      1. unpow2 80.8%

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

      2. clear-num 80.8%

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

      3. clear-num 80.8%

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

      4. frac-times 80.9%

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

      5. metadata-eval 80.9%

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

   6. Applied egg-rr 80.9%

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

   if 2.9e9 < t

   1. Initial program 65.9%

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

   3. Simplified 65.9%

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

      1. add-cube-cbrt 65.8%

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

      2. pow3 65.8%

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

      3. *-commutative 65.8%

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

      4. cbrt-prod 65.9%

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

      5. cbrt-div 67.4%

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

      6. rem-cbrt-cube 80.0%

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

      7. cbrt-prod 93.5%

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

      8. pow2 93.5%

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

   6. Applied egg-rr 93.5%

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

      1. *-commutative 93.5%

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

   8. Simplified 93.5%

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

   9. Taylor expanded in k around 0 80.3%

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

   10. Taylor expanded in k around 0 72.6%

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

4. Final simplification 39.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.35 \cdot 10^{-138}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \left({t}^{1.5} \cdot \frac{\frac{{t}^{1.5}}{\ell}}{\ell}\right)}\\ \mathbf{elif}\;t \leq 2900000000:\\ \;\;\;\;\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{1}{\frac{t}{k} \cdot \frac{t}{k}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left(t \cdot \sqrt[3]{\frac{k}{{\ell}^{2}}}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 62.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.35 \cdot 10^{-138}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \left({t\_m}^{1.5} \cdot \frac{\frac{{t\_m}^{1.5}}{\ell}}{\ell}\right)}\\ \mathbf{elif}\;t\_m \leq 6100000000:\\ \;\;\;\;\frac{2}{\frac{\frac{{t\_m}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{\frac{k}{t\_m}}{\frac{t\_m}{k}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left(t\_m \cdot \sqrt[3]{\frac{k}{{\ell}^{2}}}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.35e-138)
    (/ 2.0 (* (* 2.0 (pow k 2.0)) (* (pow t_m 1.5) (/ (/ (pow t_m 1.5) l) l))))
    (if (<= t_m 6100000000.0)
      (/
       2.0
       (*
        (/ (/ (pow t_m 3.0) l) l)
        (* (* (sin k) (tan k)) (+ 2.0 (/ (/ k t_m) (/ t_m k))))))
      (/ 2.0 (* (* 2.0 k) (pow (* t_m (cbrt (/ k (pow l 2.0)))) 3.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.35e-138) {
		tmp = 2.0 / ((2.0 * pow(k, 2.0)) * (pow(t_m, 1.5) * ((pow(t_m, 1.5) / l) / l)));
	} else if (t_m <= 6100000000.0) {
		tmp = 2.0 / (((pow(t_m, 3.0) / l) / l) * ((sin(k) * tan(k)) * (2.0 + ((k / t_m) / (t_m / k)))));
	} else {
		tmp = 2.0 / ((2.0 * k) * pow((t_m * cbrt((k / pow(l, 2.0)))), 3.0));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.35e-138) {
		tmp = 2.0 / ((2.0 * Math.pow(k, 2.0)) * (Math.pow(t_m, 1.5) * ((Math.pow(t_m, 1.5) / l) / l)));
	} else if (t_m <= 6100000000.0) {
		tmp = 2.0 / (((Math.pow(t_m, 3.0) / l) / l) * ((Math.sin(k) * Math.tan(k)) * (2.0 + ((k / t_m) / (t_m / k)))));
	} else {
		tmp = 2.0 / ((2.0 * k) * Math.pow((t_m * Math.cbrt((k / Math.pow(l, 2.0)))), 3.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.35e-138)
		tmp = Float64(2.0 / Float64(Float64(2.0 * (k ^ 2.0)) * Float64((t_m ^ 1.5) * Float64(Float64((t_m ^ 1.5) / l) / l))));
	elseif (t_m <= 6100000000.0)
		tmp = Float64(2.0 / Float64(Float64(Float64((t_m ^ 3.0) / l) / l) * Float64(Float64(sin(k) * tan(k)) * Float64(2.0 + Float64(Float64(k / t_m) / Float64(t_m / k))))));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * (Float64(t_m * cbrt(Float64(k / (l ^ 2.0)))) ^ 3.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.35e-138], N[(2.0 / N[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[t$95$m, 1.5], $MachinePrecision] * N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6100000000.0], N[(2.0 / N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(N[(k / t$95$m), $MachinePrecision] / N[(t$95$m / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[Power[N[(t$95$m * N[Power[N[(k / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.3500000000000001e-138

   1. Initial program 49.1%

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

   3. Simplified 48.5%

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

   5. Taylor expanded in k around 0 52.1%

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

      1. sqr-pow 10.7%

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

      2. *-un-lft-identity 10.7%

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

      3. times-frac 11.4%

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

      4. metadata-eval 11.4%

         \[\leadsto \frac{2}{\frac{\frac{{t}^{\color{blue}{1.5}}}{1} \cdot \frac{{t}^{\left(\frac{3}{2}\right)}}{\ell}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)} \]

      5. metadata-eval 11.4%

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

   7. Applied egg-rr 11.4%

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

      1. /-rgt-identity 11.4%

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

      2. associate-/l* 13.4%

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

   9. Applied egg-rr 13.4%

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

   if 2.3500000000000001e-138 < t < 6.1e9

   1. Initial program 76.5%

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

   3. Simplified 80.8%

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

      1. unpow2 80.8%

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

      2. clear-num 80.8%

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

      3. un-div-inv 80.8%

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

   6. Applied egg-rr 80.8%

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

   if 6.1e9 < t

   1. Initial program 65.9%

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

   3. Simplified 65.9%

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

      1. add-cube-cbrt 65.8%

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

      2. pow3 65.8%

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

      3. *-commutative 65.8%

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

      4. cbrt-prod 65.9%

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

      5. cbrt-div 67.4%

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

      6. rem-cbrt-cube 80.0%

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

      7. cbrt-prod 93.5%

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

      8. pow2 93.5%

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

   6. Applied egg-rr 93.5%

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

      1. *-commutative 93.5%

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

   8. Simplified 93.5%

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

   9. Taylor expanded in k around 0 80.3%

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

   10. Taylor expanded in k around 0 72.6%

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

4. Final simplification 39.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.35 \cdot 10^{-138}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \left({t}^{1.5} \cdot \frac{\frac{{t}^{1.5}}{\ell}}{\ell}\right)}\\ \mathbf{elif}\;t \leq 6100000000:\\ \;\;\;\;\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left(t \cdot \sqrt[3]{\frac{k}{{\ell}^{2}}}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 63.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.8 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot {\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left(t\_m \cdot \sqrt[3]{\frac{k}{{\ell}^{2}}}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.8e-14)
    (/ 2.0 (* (* (sin k) (pow (/ (pow t_m 1.5) l) 2.0)) (* 2.0 k)))
    (/ 2.0 (* (* 2.0 k) (pow (* t_m (cbrt (/ k (pow l 2.0)))) 3.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.8e-14) {
		tmp = 2.0 / ((sin(k) * pow((pow(t_m, 1.5) / l), 2.0)) * (2.0 * k));
	} else {
		tmp = 2.0 / ((2.0 * k) * pow((t_m * cbrt((k / pow(l, 2.0)))), 3.0));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.8e-14) {
		tmp = 2.0 / ((Math.sin(k) * Math.pow((Math.pow(t_m, 1.5) / l), 2.0)) * (2.0 * k));
	} else {
		tmp = 2.0 / ((2.0 * k) * Math.pow((t_m * Math.cbrt((k / Math.pow(l, 2.0)))), 3.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.8e-14)
		tmp = Float64(2.0 / Float64(Float64(sin(k) * (Float64((t_m ^ 1.5) / l) ^ 2.0)) * Float64(2.0 * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * (Float64(t_m * cbrt(Float64(k / (l ^ 2.0)))) ^ 3.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1.8e-14], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[Power[N[(t$95$m * N[Power[N[(k / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.7999999999999999e-14

   1. Initial program 58.8%

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

   3. Simplified 58.8%

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

      1. add-sqr-sqrt 25.0%

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

      2. pow2 25.0%

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

      3. *-commutative 25.0%

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

      4. sqrt-prod 8.9%

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

      5. sqrt-div 8.9%

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

      6. sqrt-pow1 10.0%

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

      7. metadata-eval 10.0%

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

      8. sqrt-prod 5.8%

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

      9. add-sqr-sqrt 13.6%

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

   6. Applied egg-rr 13.6%

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

      1. *-commutative 13.6%

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

   8. Simplified 13.6%

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

      1. pow1 13.6%

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

      2. *-commutative 13.6%

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

      3. associate-+r+ 13.6%

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

      4. metadata-eval 13.6%

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

      5. *-commutative 13.6%

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

      6. unpow-prod-down 13.1%

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

      7. pow2 13.1%

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

      8. add-sqr-sqrt 38.9%

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

   10. Applied egg-rr 38.9%

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

       1. unpow1 38.9%

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

   12. Simplified 38.9%

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

   13. Taylor expanded in k around 0 36.1%

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

   if 1.7999999999999999e-14 < k

   1. Initial program 54.8%

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

   3. Simplified 54.8%

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

      1. add-cube-cbrt 54.8%

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

      2. pow3 54.8%

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

      3. *-commutative 54.8%

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

      4. cbrt-prod 54.7%

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

      5. cbrt-div 54.7%

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

      6. rem-cbrt-cube 63.1%

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

      7. cbrt-prod 73.7%

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

      8. pow2 73.7%

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

   6. Applied egg-rr 73.7%

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

      1. *-commutative 73.7%

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

   8. Simplified 73.7%

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

   9. Taylor expanded in k around 0 50.1%

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

   10. Taylor expanded in k around 0 57.7%

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

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.8 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left(t \cdot \sqrt[3]{\frac{k}{{\ell}^{2}}}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 62.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.35 \cdot 10^{-12}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot {\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{\left(\frac{t\_m}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.35e-12)
    (/ 2.0 (* (* (sin k) (pow (/ (pow t_m 1.5) l) 2.0)) (* 2.0 k)))
    (/ 2.0 (* (* 2.0 (pow k 2.0)) (/ (pow (/ t_m (cbrt l)) 3.0) l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.35e-12) {
		tmp = 2.0 / ((sin(k) * pow((pow(t_m, 1.5) / l), 2.0)) * (2.0 * k));
	} else {
		tmp = 2.0 / ((2.0 * pow(k, 2.0)) * (pow((t_m / cbrt(l)), 3.0) / l));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.35e-12) {
		tmp = 2.0 / ((Math.sin(k) * Math.pow((Math.pow(t_m, 1.5) / l), 2.0)) * (2.0 * k));
	} else {
		tmp = 2.0 / ((2.0 * Math.pow(k, 2.0)) * (Math.pow((t_m / Math.cbrt(l)), 3.0) / l));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.35e-12)
		tmp = Float64(2.0 / Float64(Float64(sin(k) * (Float64((t_m ^ 1.5) / l) ^ 2.0)) * Float64(2.0 * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * (k ^ 2.0)) * Float64((Float64(t_m / cbrt(l)) ^ 3.0) / l)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1.35e-12], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.3499999999999999e-12

   1. Initial program 58.7%

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

   3. Simplified 58.7%

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

      1. add-sqr-sqrt 24.7%

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

      2. pow2 24.7%

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

      3. *-commutative 24.7%

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

      4. sqrt-prod 8.9%

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

      5. sqrt-div 8.9%

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

      6. sqrt-pow1 9.9%

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

      7. metadata-eval 9.9%

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

      8. sqrt-prod 5.7%

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

      9. add-sqr-sqrt 13.4%

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

   6. Applied egg-rr 13.4%

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

      1. *-commutative 13.4%

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

   8. Simplified 13.4%

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

      1. pow1 13.4%

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

      2. *-commutative 13.4%

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

      3. associate-+r+ 13.4%

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

      4. metadata-eval 13.4%

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

      5. *-commutative 13.4%

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

      6. unpow-prod-down 12.9%

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

      7. pow2 12.9%

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

      8. add-sqr-sqrt 38.5%

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

   10. Applied egg-rr 38.5%

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

       1. unpow1 38.5%

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

   12. Simplified 38.5%

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

   13. Taylor expanded in k around 0 35.7%

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

   if 1.3499999999999999e-12 < k

   1. Initial program 54.9%

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

   3. Simplified 60.1%

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

   5. Taylor expanded in k around 0 56.1%

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

      1. add-cube-cbrt 60.0%

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

      2. pow3 60.0%

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

      3. cbrt-div 60.0%

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

      4. rem-cbrt-cube 65.1%

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

   7. Applied egg-rr 56.1%

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

4. Final simplification 40.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.35 \cdot 10^{-12}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 62.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.7 \cdot 10^{+27}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot \left({t\_m}^{3} \cdot \frac{{k}^{2}}{\ell}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot {\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(2 \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.7e+27)
    (/ 2.0 (/ (* 2.0 (* (pow t_m 3.0) (/ (pow k 2.0) l))) l))
    (/ 2.0 (* (* (sin k) (pow (/ (pow t_m 1.5) l) 2.0)) (* 2.0 k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.7e+27) {
		tmp = 2.0 / ((2.0 * (pow(t_m, 3.0) * (pow(k, 2.0) / l))) / l);
	} else {
		tmp = 2.0 / ((sin(k) * pow((pow(t_m, 1.5) / l), 2.0)) * (2.0 * k));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 3.7d+27) then
        tmp = 2.0d0 / ((2.0d0 * ((t_m ** 3.0d0) * ((k ** 2.0d0) / l))) / l)
    else
        tmp = 2.0d0 / ((sin(k) * (((t_m ** 1.5d0) / l) ** 2.0d0)) * (2.0d0 * k))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.7e+27) {
		tmp = 2.0 / ((2.0 * (Math.pow(t_m, 3.0) * (Math.pow(k, 2.0) / l))) / l);
	} else {
		tmp = 2.0 / ((Math.sin(k) * Math.pow((Math.pow(t_m, 1.5) / l), 2.0)) * (2.0 * k));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 3.7e+27:
		tmp = 2.0 / ((2.0 * (math.pow(t_m, 3.0) * (math.pow(k, 2.0) / l))) / l)
	else:
		tmp = 2.0 / ((math.sin(k) * math.pow((math.pow(t_m, 1.5) / l), 2.0)) * (2.0 * k))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 3.7e+27)
		tmp = Float64(2.0 / Float64(Float64(2.0 * Float64((t_m ^ 3.0) * Float64((k ^ 2.0) / l))) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(sin(k) * (Float64((t_m ^ 1.5) / l) ^ 2.0)) * Float64(2.0 * k)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 3.7e+27)
		tmp = 2.0 / ((2.0 * ((t_m ^ 3.0) * ((k ^ 2.0) / l))) / l);
	else
		tmp = 2.0 / ((sin(k) * (((t_m ^ 1.5) / l) ^ 2.0)) * (2.0 * k));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 3.7e+27], N[(2.0 / N[(N[(2.0 * N[(N[Power[t$95$m, 3.0], $MachinePrecision] * N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.70000000000000002e27

   1. Initial program 55.4%

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

   3. Simplified 55.9%

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

   5. Taylor expanded in k around 0 56.9%

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

      1. associate-*l/ 57.9%

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

   7. Applied egg-rr 57.9%

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

   8. Taylor expanded in t around 0 57.0%

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

      1. *-commutative 57.0%

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

      2. associate-/l* 57.9%

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

   10. Simplified 57.9%

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

   if 3.70000000000000002e27 < t

   1. Initial program 65.7%

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

   3. Simplified 65.7%

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

      1. add-sqr-sqrt 30.4%

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

      2. pow2 30.4%

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

      3. *-commutative 30.4%

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

      4. sqrt-prod 30.4%

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

      5. sqrt-div 30.4%

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

      6. sqrt-pow1 35.4%

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

      7. metadata-eval 35.4%

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

      8. sqrt-prod 16.9%

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

      9. add-sqr-sqrt 40.4%

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

   6. Applied egg-rr 40.4%

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

      1. *-commutative 40.4%

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

   8. Simplified 40.4%

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

      1. pow1 40.4%

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

      2. *-commutative 40.4%

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

      3. associate-+r+ 40.4%

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

      4. metadata-eval 40.4%

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

      5. *-commutative 40.4%

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

      6. unpow-prod-down 40.4%

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

      7. pow2 40.4%

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

      8. add-sqr-sqrt 82.4%

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

   10. Applied egg-rr 82.4%

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

       1. unpow1 82.4%

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

   12. Simplified 82.4%

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

   13. Taylor expanded in k around 0 71.4%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.7 \cdot 10^{+27}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot \left({t}^{3} \cdot \frac{{k}^{2}}{\ell}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(2 \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 57.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7 \cdot 10^{+21}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot \left({t\_m}^{3} \cdot \frac{{k}^{2}}{\ell}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \frac{k \cdot \left(\sin k \cdot {t\_m}^{3}\right)}{{\ell}^{2}}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 7e+21)
    (/ 2.0 (/ (* 2.0 (* (pow t_m 3.0) (/ (pow k 2.0) l))) l))
    (/ 2.0 (* 2.0 (/ (* k (* (sin k) (pow t_m 3.0))) (pow l 2.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 7e+21) {
		tmp = 2.0 / ((2.0 * (pow(t_m, 3.0) * (pow(k, 2.0) / l))) / l);
	} else {
		tmp = 2.0 / (2.0 * ((k * (sin(k) * pow(t_m, 3.0))) / pow(l, 2.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 7d+21) then
        tmp = 2.0d0 / ((2.0d0 * ((t_m ** 3.0d0) * ((k ** 2.0d0) / l))) / l)
    else
        tmp = 2.0d0 / (2.0d0 * ((k * (sin(k) * (t_m ** 3.0d0))) / (l ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 7e+21) {
		tmp = 2.0 / ((2.0 * (Math.pow(t_m, 3.0) * (Math.pow(k, 2.0) / l))) / l);
	} else {
		tmp = 2.0 / (2.0 * ((k * (Math.sin(k) * Math.pow(t_m, 3.0))) / Math.pow(l, 2.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 7e+21:
		tmp = 2.0 / ((2.0 * (math.pow(t_m, 3.0) * (math.pow(k, 2.0) / l))) / l)
	else:
		tmp = 2.0 / (2.0 * ((k * (math.sin(k) * math.pow(t_m, 3.0))) / math.pow(l, 2.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 7e+21)
		tmp = Float64(2.0 / Float64(Float64(2.0 * Float64((t_m ^ 3.0) * Float64((k ^ 2.0) / l))) / l));
	else
		tmp = Float64(2.0 / Float64(2.0 * Float64(Float64(k * Float64(sin(k) * (t_m ^ 3.0))) / (l ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 7e+21)
		tmp = 2.0 / ((2.0 * ((t_m ^ 3.0) * ((k ^ 2.0) / l))) / l);
	else
		tmp = 2.0 / (2.0 * ((k * (sin(k) * (t_m ^ 3.0))) / (l ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 7e+21], N[(2.0 / N[(N[(2.0 * N[(N[Power[t$95$m, 3.0], $MachinePrecision] * N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(2.0 * N[(N[(k * N[(N[Sin[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 7e21

   1. Initial program 55.4%

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

   3. Simplified 55.9%

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

   5. Taylor expanded in k around 0 56.9%

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

      1. associate-*l/ 57.9%

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

   7. Applied egg-rr 57.9%

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

   8. Taylor expanded in t around 0 57.0%

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

      1. *-commutative 57.0%

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

      2. associate-/l* 57.9%

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

   10. Simplified 57.9%

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

   if 7e21 < t

   1. Initial program 65.7%

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

   3. Simplified 65.7%

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

      1. add-cube-cbrt 65.6%

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

      2. pow3 65.6%

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

      3. *-commutative 65.6%

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

      4. cbrt-prod 65.7%

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

      5. cbrt-div 67.3%

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

      6. rem-cbrt-cube 80.5%

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

      7. cbrt-prod 94.6%

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

      8. pow2 94.6%

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

   6. Applied egg-rr 94.6%

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

      1. *-commutative 94.6%

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

   8. Simplified 94.6%

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

   9. Taylor expanded in k around 0 82.4%

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

   10. Taylor expanded in t around 0 65.6%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7 \cdot 10^{+21}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot \left({t}^{3} \cdot \frac{{k}^{2}}{\ell}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \frac{k \cdot \left(\sin k \cdot {t}^{3}\right)}{{\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 56.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.7 \cdot 10^{-169}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot \frac{{t\_m}^{3}}{\ell \cdot \ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(2 \cdot {k}^{2}\right) \cdot \left({t\_m}^{2} \cdot \frac{t\_m}{\ell}\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.7e-169)
    (/ 2.0 (* (* 2.0 k) (* (sin k) (/ (pow t_m 3.0) (* l l)))))
    (/ 2.0 (/ (* (* 2.0 (pow k 2.0)) (* (pow t_m 2.0) (/ t_m l))) l)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.7e-169) {
		tmp = 2.0 / ((2.0 * k) * (sin(k) * (pow(t_m, 3.0) / (l * l))));
	} else {
		tmp = 2.0 / (((2.0 * pow(k, 2.0)) * (pow(t_m, 2.0) * (t_m / l))) / l);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.7d-169) then
        tmp = 2.0d0 / ((2.0d0 * k) * (sin(k) * ((t_m ** 3.0d0) / (l * l))))
    else
        tmp = 2.0d0 / (((2.0d0 * (k ** 2.0d0)) * ((t_m ** 2.0d0) * (t_m / l))) / l)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.7e-169) {
		tmp = 2.0 / ((2.0 * k) * (Math.sin(k) * (Math.pow(t_m, 3.0) / (l * l))));
	} else {
		tmp = 2.0 / (((2.0 * Math.pow(k, 2.0)) * (Math.pow(t_m, 2.0) * (t_m / l))) / l);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 1.7e-169:
		tmp = 2.0 / ((2.0 * k) * (math.sin(k) * (math.pow(t_m, 3.0) / (l * l))))
	else:
		tmp = 2.0 / (((2.0 * math.pow(k, 2.0)) * (math.pow(t_m, 2.0) * (t_m / l))) / l)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.7e-169)
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * Float64(sin(k) * Float64((t_m ^ 3.0) / Float64(l * l)))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(2.0 * (k ^ 2.0)) * Float64((t_m ^ 2.0) * Float64(t_m / l))) / l));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 1.7e-169)
		tmp = 2.0 / ((2.0 * k) * (sin(k) * ((t_m ^ 3.0) / (l * l))));
	else
		tmp = 2.0 / (((2.0 * (k ^ 2.0)) * ((t_m ^ 2.0) * (t_m / l))) / l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1.7e-169], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[t$95$m, 2.0], $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.7e-169

   1. Initial program 60.4%

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

   3. Simplified 60.4%

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

   5. Taylor expanded in k around 0 56.8%

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

   if 1.7e-169 < k

   1. Initial program 52.9%

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

   3. Simplified 62.1%

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

   5. Taylor expanded in k around 0 65.0%

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

      1. associate-*l/ 66.1%

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

   7. Applied egg-rr 66.1%

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

      1. unpow3 66.1%

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

      2. *-un-lft-identity 66.1%

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

      3. times-frac 66.1%

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

      4. pow2 66.1%

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

   9. Applied egg-rr 66.1%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.7 \cdot 10^{-169}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(2 \cdot {k}^{2}\right) \cdot \left({t}^{2} \cdot \frac{t}{\ell}\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 57.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.4 \cdot 10^{+27}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot \left({t\_m}^{3} \cdot \frac{{k}^{2}}{\ell}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot \frac{{t\_m}^{3}}{\ell \cdot \ell}\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.4e+27)
    (/ 2.0 (/ (* 2.0 (* (pow t_m 3.0) (/ (pow k 2.0) l))) l))
    (/ 2.0 (* (* 2.0 k) (* (sin k) (/ (pow t_m 3.0) (* l l))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.4e+27) {
		tmp = 2.0 / ((2.0 * (pow(t_m, 3.0) * (pow(k, 2.0) / l))) / l);
	} else {
		tmp = 2.0 / ((2.0 * k) * (sin(k) * (pow(t_m, 3.0) / (l * l))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 3.4d+27) then
        tmp = 2.0d0 / ((2.0d0 * ((t_m ** 3.0d0) * ((k ** 2.0d0) / l))) / l)
    else
        tmp = 2.0d0 / ((2.0d0 * k) * (sin(k) * ((t_m ** 3.0d0) / (l * l))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.4e+27) {
		tmp = 2.0 / ((2.0 * (Math.pow(t_m, 3.0) * (Math.pow(k, 2.0) / l))) / l);
	} else {
		tmp = 2.0 / ((2.0 * k) * (Math.sin(k) * (Math.pow(t_m, 3.0) / (l * l))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 3.4e+27:
		tmp = 2.0 / ((2.0 * (math.pow(t_m, 3.0) * (math.pow(k, 2.0) / l))) / l)
	else:
		tmp = 2.0 / ((2.0 * k) * (math.sin(k) * (math.pow(t_m, 3.0) / (l * l))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 3.4e+27)
		tmp = Float64(2.0 / Float64(Float64(2.0 * Float64((t_m ^ 3.0) * Float64((k ^ 2.0) / l))) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * Float64(sin(k) * Float64((t_m ^ 3.0) / Float64(l * l)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 3.4e+27)
		tmp = 2.0 / ((2.0 * ((t_m ^ 3.0) * ((k ^ 2.0) / l))) / l);
	else
		tmp = 2.0 / ((2.0 * k) * (sin(k) * ((t_m ^ 3.0) / (l * l))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 3.4e+27], N[(2.0 / N[(N[(2.0 * N[(N[Power[t$95$m, 3.0], $MachinePrecision] * N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.4e27

   1. Initial program 55.4%

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

   3. Simplified 55.9%

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

   5. Taylor expanded in k around 0 56.9%

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

      1. associate-*l/ 57.9%

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

   7. Applied egg-rr 57.9%

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

   8. Taylor expanded in t around 0 57.0%

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

      1. *-commutative 57.0%

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

      2. associate-/l* 57.9%

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

   10. Simplified 57.9%

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

   if 3.4e27 < t

   1. Initial program 65.7%

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

   3. Simplified 65.7%

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

   5. Taylor expanded in k around 0 64.0%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.4 \cdot 10^{+27}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot \left({t}^{3} \cdot \frac{{k}^{2}}{\ell}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 55.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 / ((2.0d0 * ((t_m ** 3.0d0) * ((k ** 2.0d0) / l))) / l))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / ((2.0 * (Math.pow(t_m, 3.0) * (Math.pow(k, 2.0) / l))) / l));
}

Python

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

Julia

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

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 / ((2.0 * ((t_m ^ 3.0) * ((k ^ 2.0) / l))) / l));
end

Wolfram

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

Derivation

1. Initial program 57.8%

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

3. Simplified 57.0%

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

5. Taylor expanded in k around 0 57.0%

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

   1. associate-*l/ 57.8%

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

7. Applied egg-rr 57.8%

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

8. Taylor expanded in t around 0 57.2%

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

   1. *-commutative 57.2%

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

   2. associate-/l* 57.8%

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

10. Simplified 57.8%

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

Alternative 21: 55.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 / (((2.0d0 * (k ** 2.0d0)) / l) * ((t_m ** 3.0d0) / l)))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / (((2.0 * Math.pow(k, 2.0)) / l) * (Math.pow(t_m, 3.0) / l)));
}

Python

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

Julia

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

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 / (((2.0 * (k ^ 2.0)) / l) * ((t_m ^ 3.0) / l)));
end

Wolfram

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

Derivation

1. Initial program 57.8%

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

3. Simplified 57.0%

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

5. Taylor expanded in k around 0 57.0%

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

   1. associate-*l/ 57.8%

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

7. Applied egg-rr 57.8%

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

   1. associate-/l* 57.7%

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

9. Applied egg-rr 57.7%

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

10. Final simplification 57.7%

    \[\leadsto \frac{2}{\frac{2 \cdot {k}^{2}}{\ell} \cdot \frac{{t}^{3}}{\ell}} \]
11. Add Preprocessing

Alternative 22: 55.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 / (((t_m ** 3.0d0) / l) * ((k ** 2.0d0) * (2.0d0 / l))))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / ((Math.pow(t_m, 3.0) / l) * (Math.pow(k, 2.0) * (2.0 / l))));
}

Python

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

Julia

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

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 / (((t_m ^ 3.0) / l) * ((k ^ 2.0) * (2.0 / l))));
end

Wolfram

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

Derivation

1. Initial program 57.8%

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

3. Simplified 57.0%

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

5. Taylor expanded in k around 0 57.0%

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

   1. associate-*l/ 57.8%

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

7. Applied egg-rr 57.8%

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

   1. associate-/l* 57.7%

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

9. Applied egg-rr 57.7%

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

10. Taylor expanded in k around 0 57.7%

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

    1. associate-*r/ 57.7%

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

    2. *-commutative 57.7%

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

    3. associate-/l* 57.7%

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

12. Simplified 57.7%

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

Alternative 23: 55.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{\frac{{t\_m}^{3}}{\ell}}{\ell}} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ 2.0 (* (* 2.0 (pow k 2.0)) (/ (/ (pow t_m 3.0) l) l)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / ((2.0 * pow(k, 2.0)) * ((pow(t_m, 3.0) / l) / l)));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 / ((2.0d0 * (k ** 2.0d0)) * (((t_m ** 3.0d0) / l) / l)))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / ((2.0 * Math.pow(k, 2.0)) * ((Math.pow(t_m, 3.0) / l) / l)));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 / ((2.0 * math.pow(k, 2.0)) * ((math.pow(t_m, 3.0) / l) / l)))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 / Float64(Float64(2.0 * (k ^ 2.0)) * Float64(Float64((t_m ^ 3.0) / l) / l))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 / ((2.0 * (k ^ 2.0)) * (((t_m ^ 3.0) / l) / l)));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(2.0 / N[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.8%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

3. Simplified 57.0%

   \[\leadsto \color{blue}{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in k around 0 57.0%

   \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(2 \cdot {k}^{2}\right)}} \]

6. Final simplification 57.0%

   \[\leadsto \frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024090 
(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))))