Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.0% → 86.3%

Time: 19.7s

Alternatives: 24

Speedup: 24.8×

• 26.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.0% 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: 86.3% accurate, 0.5× speedup

Math

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

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 8e-118)
    (*
     (pow l 2.0)
     (/ (* 2.0 (cos k)) (* (* t_m (pow k 2.0)) (pow (sin k) 2.0))))
    (/
     2.0
     (pow
      (*
       (* (/ t_m (cbrt l)) (cbrt (/ 1.0 l)))
       (* (cbrt (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))) (cbrt (sin k))))
      3.0)))))

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if t < 7.99999999999999988e-118

   1. Initial program 48.3%

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

   3. Simplified 47.8%

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

      1. associate-*r* 53.7%

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

      2. *-un-lft-identity 53.7%

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

      3. times-frac 52.0%

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

      4. associate-/l/ 52.0%

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

   6. Applied egg-rr 52.0%

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

      1. /-rgt-identity 52.0%

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

      2. associate-*r/ 53.7%

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

      3. associate-*l/ 54.4%

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

      4. associate-*l* 49.6%

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

   8. Simplified 49.6%

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

      1. add-cube-cbrt 49.5%

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

      2. pow3 49.5%

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

      3. associate-/l* 47.8%

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

      4. associate-/l* 47.8%

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

   10. Applied egg-rr 47.8%

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

   11. Taylor expanded in t around 0 64.7%

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

       1. associate-/l* 64.7%

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

       2. rem-cube-cbrt 64.9%

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

       3. associate-*r* 64.9%

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

   13. Simplified 64.9%

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

   if 7.99999999999999988e-118 < t

   1. Initial program 62.1%

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

   3. Simplified 62.1%

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

      1. associate-*l* 59.5%

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

      2. associate-/r* 66.2%

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

      3. associate-+r+ 66.2%

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

      4. metadata-eval 66.2%

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

      5. associate-*l* 66.2%

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

      6. add-cube-cbrt 66.0%

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

      7. pow3 66.0%

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

   6. Applied egg-rr 83.0%

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

      1. *-commutative 83.0%

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

      2. cbrt-prod 91.6%

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

   8. Applied egg-rr 91.6%

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

      1. add-cbrt-cube 72.6%

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

      2. unpow3 72.6%

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

      3. unpow2 72.6%

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

      4. cbrt-prod 62.9%

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

      5. unpow2 62.9%

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

      6. cbrt-div 61.9%

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

      7. unpow2 61.9%

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

      8. associate-/l/ 68.5%

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

      9. div-inv 68.5%

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

      10. cbrt-prod 70.6%

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

      11. cbrt-div 72.7%

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

      12. unpow3 72.7%

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

      13. add-cbrt-cube 91.8%

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

   10. Applied egg-rr 91.8%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8 \cdot 10^{-118}:\\ \;\;\;\;{\ell}^{2} \cdot \frac{2 \cdot \cos k}{\left(t \cdot {k}^{2}\right) \cdot {\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(\frac{t}{\sqrt[3]{\ell}} \cdot \sqrt[3]{\frac{1}{\ell}}\right) \cdot \left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.4 \cdot 10^{-118}:\\ \;\;\;\;{\ell}^{2} \cdot \frac{2 \cdot \cos k}{\left(t\_m \cdot {k}^{2}\right) \cdot {\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)} \cdot \sqrt[3]{\sin k}\right) \cdot \frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 7.4e-118)
    (*
     (pow l 2.0)
     (/ (* 2.0 (cos k)) (* (* t_m (pow k 2.0)) (pow (sin k) 2.0))))
    (/
     2.0
     (pow
      (*
       (* (cbrt (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))) (cbrt (sin k)))
       (/ t_m (pow (cbrt l) 2.0)))
      3.0)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 7.4e-118) {
		tmp = pow(l, 2.0) * ((2.0 * cos(k)) / ((t_m * pow(k, 2.0)) * pow(sin(k), 2.0)));
	} else {
		tmp = 2.0 / pow(((cbrt((tan(k) * (2.0 + pow((k / t_m), 2.0)))) * cbrt(sin(k))) * (t_m / pow(cbrt(l), 2.0))), 3.0);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 7.4e-118) {
		tmp = Math.pow(l, 2.0) * ((2.0 * Math.cos(k)) / ((t_m * Math.pow(k, 2.0)) * Math.pow(Math.sin(k), 2.0)));
	} else {
		tmp = 2.0 / Math.pow(((Math.cbrt((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0)))) * Math.cbrt(Math.sin(k))) * (t_m / Math.pow(Math.cbrt(l), 2.0))), 3.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 7.4e-118)
		tmp = Float64((l ^ 2.0) * Float64(Float64(2.0 * cos(k)) / Float64(Float64(t_m * (k ^ 2.0)) * (sin(k) ^ 2.0))));
	else
		tmp = Float64(2.0 / (Float64(Float64(cbrt(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0)))) * cbrt(sin(k))) * Float64(t_m / (cbrt(l) ^ 2.0))) ^ 3.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 7.4e-118], N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[(2.0 * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(N[Power[N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 7.40000000000000029e-118

   1. Initial program 48.3%

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

   3. Simplified 47.8%

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

      1. associate-*r* 53.7%

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

      2. *-un-lft-identity 53.7%

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

      3. times-frac 52.0%

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

      4. associate-/l/ 52.0%

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

   6. Applied egg-rr 52.0%

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

      1. /-rgt-identity 52.0%

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

      2. associate-*r/ 53.7%

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

      3. associate-*l/ 54.4%

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

      4. associate-*l* 49.6%

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

   8. Simplified 49.6%

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

      1. add-cube-cbrt 49.5%

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

      2. pow3 49.5%

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

      3. associate-/l* 47.8%

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

      4. associate-/l* 47.8%

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

   10. Applied egg-rr 47.8%

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

   11. Taylor expanded in t around 0 64.7%

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

       1. associate-/l* 64.7%

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

       2. rem-cube-cbrt 64.9%

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

       3. associate-*r* 64.9%

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

   13. Simplified 64.9%

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

   if 7.40000000000000029e-118 < t

   1. Initial program 62.1%

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

   3. Simplified 62.1%

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

      1. associate-*l* 59.5%

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

      2. associate-/r* 66.2%

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

      3. associate-+r+ 66.2%

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

      4. metadata-eval 66.2%

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

      5. associate-*l* 66.2%

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

      6. add-cube-cbrt 66.0%

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

      7. pow3 66.0%

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

   6. Applied egg-rr 83.0%

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

      1. *-commutative 83.0%

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

      2. cbrt-prod 91.6%

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

   8. Applied egg-rr 91.6%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.4 \cdot 10^{-118}:\\ \;\;\;\;{\ell}^{2} \cdot \frac{2 \cdot \cos k}{\left(t \cdot {k}^{2}\right) \cdot {\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \sqrt[3]{\sin k}\right) \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m}{\sqrt[3]{\ell}} \cdot \sqrt[3]{\frac{1}{\ell}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 0.000192:\\ \;\;\;\;\frac{2}{{\left(t\_2 \cdot \left(\sqrt[3]{\sin k} \cdot \sqrt[3]{2 \cdot k}\right)\right)}^{3}}\\ \mathbf{elif}\;k \leq 3.5 \cdot 10^{+141}:\\ \;\;\;\;{\ell}^{2} \cdot \frac{2 \cdot \cos k}{\left(t\_m \cdot {k}^{2}\right) \cdot {\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t\_2 \cdot \sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right)}\right)}^{3}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (/ t_m (cbrt l)) (cbrt (/ 1.0 l)))))
   (*
    t_s
    (if (<= k 0.000192)
      (/ 2.0 (pow (* t_2 (* (cbrt (sin k)) (cbrt (* 2.0 k)))) 3.0))
      (if (<= k 3.5e+141)
        (*
         (pow l 2.0)
         (/ (* 2.0 (cos k)) (* (* t_m (pow k 2.0)) (pow (sin k) 2.0))))
        (/
         2.0
         (pow
          (* t_2 (cbrt (* (sin k) (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0))))))
          3.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = (t_m / cbrt(l)) * cbrt((1.0 / l));
	double tmp;
	if (k <= 0.000192) {
		tmp = 2.0 / pow((t_2 * (cbrt(sin(k)) * cbrt((2.0 * k)))), 3.0);
	} else if (k <= 3.5e+141) {
		tmp = pow(l, 2.0) * ((2.0 * cos(k)) / ((t_m * pow(k, 2.0)) * pow(sin(k), 2.0)));
	} else {
		tmp = 2.0 / pow((t_2 * cbrt((sin(k) * (tan(k) * (2.0 + pow((k / t_m), 2.0)))))), 3.0);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = (t_m / Math.cbrt(l)) * Math.cbrt((1.0 / l));
	double tmp;
	if (k <= 0.000192) {
		tmp = 2.0 / Math.pow((t_2 * (Math.cbrt(Math.sin(k)) * Math.cbrt((2.0 * k)))), 3.0);
	} else if (k <= 3.5e+141) {
		tmp = Math.pow(l, 2.0) * ((2.0 * Math.cos(k)) / ((t_m * Math.pow(k, 2.0)) * Math.pow(Math.sin(k), 2.0)));
	} else {
		tmp = 2.0 / Math.pow((t_2 * Math.cbrt((Math.sin(k) * (Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0)))))), 3.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(Float64(t_m / cbrt(l)) * cbrt(Float64(1.0 / l)))
	tmp = 0.0
	if (k <= 0.000192)
		tmp = Float64(2.0 / (Float64(t_2 * Float64(cbrt(sin(k)) * cbrt(Float64(2.0 * k)))) ^ 3.0));
	elseif (k <= 3.5e+141)
		tmp = Float64((l ^ 2.0) * Float64(Float64(2.0 * cos(k)) / Float64(Float64(t_m * (k ^ 2.0)) * (sin(k) ^ 2.0))));
	else
		tmp = Float64(2.0 / (Float64(t_2 * cbrt(Float64(sin(k) * Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0)))))) ^ 3.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[(1.0 / l), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 0.000192], N[(2.0 / N[Power[N[(t$95$2 * N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(2.0 * k), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 3.5e+141], N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[(2.0 * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(t$95$2 * N[Power[N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 1.92e-4

   1. Initial program 54.0%

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

   3. Simplified 54.0%

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

      1. associate-*l* 48.9%

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

      2. associate-/r* 55.4%

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

      3. associate-+r+ 55.4%

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

      4. metadata-eval 55.4%

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

      5. associate-*l* 55.5%

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

      6. add-cube-cbrt 55.4%

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

      7. pow3 55.4%

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

   6. Applied egg-rr 74.3%

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

      1. *-commutative 74.3%

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

      2. cbrt-prod 86.4%

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

   8. Applied egg-rr 86.4%

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

      1. add-cbrt-cube 64.7%

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

      2. unpow3 64.7%

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

      3. unpow2 64.7%

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

      4. cbrt-prod 55.3%

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

      5. unpow2 55.3%

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

      6. cbrt-div 53.8%

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

      7. unpow2 53.8%

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

      8. associate-/l/ 60.8%

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

      9. div-inv 60.8%

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

      10. cbrt-prod 62.8%

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

      11. cbrt-div 64.7%

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

      12. unpow3 64.7%

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

      13. add-cbrt-cube 86.4%

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

   10. Applied egg-rr 86.4%

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

   11. Taylor expanded in k around 0 77.4%

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

   if 1.92e-4 < k < 3.5e141

   1. Initial program 48.4%

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

   3. Simplified 48.5%

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

      1. associate-*r* 50.7%

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

      2. *-un-lft-identity 50.7%

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

      3. times-frac 50.7%

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

      4. associate-/l/ 50.6%

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

   6. Applied egg-rr 50.6%

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

      1. /-rgt-identity 50.6%

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

      2. associate-*r/ 50.7%

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

      3. associate-*l/ 50.7%

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

      4. associate-*l* 50.5%

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

   8. Simplified 50.5%

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

      1. add-cube-cbrt 50.5%

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

      2. pow3 50.5%

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

      3. associate-/l* 50.5%

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

      4. associate-/l* 50.5%

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

   10. Applied egg-rr 50.5%

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

   11. Taylor expanded in t around 0 85.7%

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

       1. associate-/l* 85.7%

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

       2. rem-cube-cbrt 86.9%

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

       3. associate-*r* 86.8%

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

   13. Simplified 86.8%

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

   if 3.5e141 < k

   1. Initial program 51.3%

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

   3. Simplified 51.3%

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

      1. associate-*l* 51.3%

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

      2. associate-/r* 54.2%

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

      3. associate-+r+ 54.2%

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

      4. metadata-eval 54.2%

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

      5. associate-*l* 54.2%

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

      6. add-cube-cbrt 54.2%

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

      7. pow3 54.2%

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

   6. Applied egg-rr 76.0%

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

      1. add-cbrt-cube 54.2%

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

      2. unpow3 54.2%

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

      3. unpow2 54.2%

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

      4. cbrt-prod 51.3%

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

      5. unpow2 51.3%

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

      6. cbrt-div 51.3%

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

      7. unpow2 51.3%

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

      8. associate-/l/ 54.2%

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

      9. div-inv 54.2%

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

      10. cbrt-prod 54.2%

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

      11. cbrt-div 54.2%

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

      12. unpow3 54.2%

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

      13. add-cbrt-cube 75.9%

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

   8. Applied egg-rr 75.9%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.000192:\\ \;\;\;\;\frac{2}{{\left(\left(\frac{t}{\sqrt[3]{\ell}} \cdot \sqrt[3]{\frac{1}{\ell}}\right) \cdot \left(\sqrt[3]{\sin k} \cdot \sqrt[3]{2 \cdot k}\right)\right)}^{3}}\\ \mathbf{elif}\;k \leq 3.5 \cdot 10^{+141}:\\ \;\;\;\;{\ell}^{2} \cdot \frac{2 \cdot \cos k}{\left(t \cdot {k}^{2}\right) \cdot {\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(\frac{t}{\sqrt[3]{\ell}} \cdot \sqrt[3]{\frac{1}{\ell}}\right) \cdot \sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 0.55:\\ \;\;\;\;\frac{2}{{\left(\left(\frac{t\_m}{\sqrt[3]{\ell}} \cdot \sqrt[3]{\frac{1}{\ell}}\right) \cdot \left(\sqrt[3]{\sin k} \cdot \sqrt[3]{2 \cdot k}\right)\right)}^{3}}\\ \mathbf{elif}\;k \leq 1.76 \cdot 10^{+140}:\\ \;\;\;\;{\ell}^{2} \cdot \frac{2 \cdot \cos k}{\left(t\_m \cdot {k}^{2}\right) \cdot {\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{\sqrt[3]{\ell \cdot 2}}{t\_m}}{\sqrt[3]{\sin k \cdot \tan k}} \cdot \sqrt[3]{\frac{\ell}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 0.55)
    (/
     2.0
     (pow
      (*
       (* (/ t_m (cbrt l)) (cbrt (/ 1.0 l)))
       (* (cbrt (sin k)) (cbrt (* 2.0 k))))
      3.0))
    (if (<= k 1.76e+140)
      (*
       (pow l 2.0)
       (/ (* 2.0 (cos k)) (* (* t_m (pow k 2.0)) (pow (sin k) 2.0))))
      (pow
       (*
        (/ (/ (cbrt (* l 2.0)) t_m) (cbrt (* (sin k) (tan k))))
        (cbrt (/ l (+ 2.0 (pow (/ k t_m) 2.0)))))
       3.0)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 0.55) {
		tmp = 2.0 / pow((((t_m / cbrt(l)) * cbrt((1.0 / l))) * (cbrt(sin(k)) * cbrt((2.0 * k)))), 3.0);
	} else if (k <= 1.76e+140) {
		tmp = pow(l, 2.0) * ((2.0 * cos(k)) / ((t_m * pow(k, 2.0)) * pow(sin(k), 2.0)));
	} else {
		tmp = pow((((cbrt((l * 2.0)) / t_m) / cbrt((sin(k) * tan(k)))) * cbrt((l / (2.0 + pow((k / t_m), 2.0))))), 3.0);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 0.55) {
		tmp = 2.0 / Math.pow((((t_m / Math.cbrt(l)) * Math.cbrt((1.0 / l))) * (Math.cbrt(Math.sin(k)) * Math.cbrt((2.0 * k)))), 3.0);
	} else if (k <= 1.76e+140) {
		tmp = Math.pow(l, 2.0) * ((2.0 * Math.cos(k)) / ((t_m * Math.pow(k, 2.0)) * Math.pow(Math.sin(k), 2.0)));
	} else {
		tmp = Math.pow((((Math.cbrt((l * 2.0)) / t_m) / Math.cbrt((Math.sin(k) * Math.tan(k)))) * Math.cbrt((l / (2.0 + Math.pow((k / t_m), 2.0))))), 3.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 0.55)
		tmp = Float64(2.0 / (Float64(Float64(Float64(t_m / cbrt(l)) * cbrt(Float64(1.0 / l))) * Float64(cbrt(sin(k)) * cbrt(Float64(2.0 * k)))) ^ 3.0));
	elseif (k <= 1.76e+140)
		tmp = Float64((l ^ 2.0) * Float64(Float64(2.0 * cos(k)) / Float64(Float64(t_m * (k ^ 2.0)) * (sin(k) ^ 2.0))));
	else
		tmp = Float64(Float64(Float64(cbrt(Float64(l * 2.0)) / t_m) / cbrt(Float64(sin(k) * tan(k)))) * cbrt(Float64(l / Float64(2.0 + (Float64(k / t_m) ^ 2.0))))) ^ 3.0;
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 0.55], N[(2.0 / N[Power[N[(N[(N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[(1.0 / l), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(2.0 * k), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.76e+140], N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[(2.0 * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[(N[Power[N[(l * 2.0), $MachinePrecision], 1/3], $MachinePrecision] / t$95$m), $MachinePrecision] / N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[(l / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 0.55000000000000004

   1. Initial program 54.0%

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

   3. Simplified 54.0%

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

      1. associate-*l* 48.9%

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

      2. associate-/r* 55.4%

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

      3. associate-+r+ 55.4%

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

      4. metadata-eval 55.4%

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

      5. associate-*l* 55.5%

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

      6. add-cube-cbrt 55.4%

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

      7. pow3 55.4%

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

   6. Applied egg-rr 74.3%

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

      1. *-commutative 74.3%

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

      2. cbrt-prod 86.4%

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

   8. Applied egg-rr 86.4%

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

      1. add-cbrt-cube 64.7%

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

      2. unpow3 64.7%

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

      3. unpow2 64.7%

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

      4. cbrt-prod 55.3%

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

      5. unpow2 55.3%

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

      6. cbrt-div 53.8%

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

      7. unpow2 53.8%

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

      8. associate-/l/ 60.8%

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

      9. div-inv 60.8%

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

      10. cbrt-prod 62.8%

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

      11. cbrt-div 64.7%

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

      12. unpow3 64.7%

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

      13. add-cbrt-cube 86.4%

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

   10. Applied egg-rr 86.4%

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

   11. Taylor expanded in k around 0 77.4%

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

   if 0.55000000000000004 < k < 1.76e140

   1. Initial program 48.4%

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

   3. Simplified 48.5%

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

      1. associate-*r* 50.7%

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

      2. *-un-lft-identity 50.7%

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

      3. times-frac 50.7%

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

      4. associate-/l/ 50.6%

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

   6. Applied egg-rr 50.6%

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

      1. /-rgt-identity 50.6%

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

      2. associate-*r/ 50.7%

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

      3. associate-*l/ 50.7%

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

      4. associate-*l* 50.5%

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

   8. Simplified 50.5%

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

      1. add-cube-cbrt 50.5%

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

      2. pow3 50.5%

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

      3. associate-/l* 50.5%

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

      4. associate-/l* 50.5%

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

   10. Applied egg-rr 50.5%

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

   11. Taylor expanded in t around 0 85.7%

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

       1. associate-/l* 85.7%

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

       2. rem-cube-cbrt 86.9%

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

       3. associate-*r* 86.8%

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

   13. Simplified 86.8%

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

   if 1.76e140 < k

   1. Initial program 51.3%

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

   3. Simplified 48.7%

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

      1. associate-*r* 51.6%

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

      2. *-un-lft-identity 51.6%

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

      3. times-frac 51.6%

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

      4. associate-/l/ 51.6%

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

   6. Applied egg-rr 51.6%

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

      1. /-rgt-identity 51.6%

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

      2. associate-*r/ 51.6%

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

      3. associate-*l/ 54.2%

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

      4. associate-*l* 54.2%

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

   8. Simplified 54.2%

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

      1. add-cube-cbrt 54.2%

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

      2. pow3 54.2%

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

      3. associate-/l* 54.2%

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

      4. associate-/l* 54.2%

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

   10. Applied egg-rr 54.2%

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

       1. cbrt-prod 54.2%

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

       2. associate-*r/ 54.2%

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

       3. cbrt-div 54.2%

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

       4. cbrt-prod 54.2%

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

       5. unpow3 54.2%

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

       6. add-cbrt-cube 75.9%

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

       7. *-commutative 75.9%

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

   12. Applied egg-rr 75.9%

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

       1. associate-/r* 75.9%

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

       2. *-commutative 75.9%

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

   14. Simplified 75.9%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.55:\\ \;\;\;\;\frac{2}{{\left(\left(\frac{t}{\sqrt[3]{\ell}} \cdot \sqrt[3]{\frac{1}{\ell}}\right) \cdot \left(\sqrt[3]{\sin k} \cdot \sqrt[3]{2 \cdot k}\right)\right)}^{3}}\\ \mathbf{elif}\;k \leq 1.76 \cdot 10^{+140}:\\ \;\;\;\;{\ell}^{2} \cdot \frac{2 \cdot \cos k}{\left(t \cdot {k}^{2}\right) \cdot {\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{\sqrt[3]{\ell \cdot 2}}{t}}{\sqrt[3]{\sin k \cdot \tan k}} \cdot \sqrt[3]{\frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1160:\\ \;\;\;\;\frac{2}{{\left(\left(\frac{t\_m}{\sqrt[3]{\ell}} \cdot \sqrt[3]{\frac{1}{\ell}}\right) \cdot \left(\sqrt[3]{\sin k} \cdot \sqrt[3]{2 \cdot k}\right)\right)}^{3}}\\ \mathbf{elif}\;k \leq 1.8 \cdot 10^{+142}:\\ \;\;\;\;{\ell}^{2} \cdot \frac{2 \cdot \cos k}{\left(t\_m \cdot {k}^{2}\right) \cdot {\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right)}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1160.0)
    (/
     2.0
     (pow
      (*
       (* (/ t_m (cbrt l)) (cbrt (/ 1.0 l)))
       (* (cbrt (sin k)) (cbrt (* 2.0 k))))
      3.0))
    (if (<= k 1.8e+142)
      (*
       (pow l 2.0)
       (/ (* 2.0 (cos k)) (* (* t_m (pow k 2.0)) (pow (sin k) 2.0))))
      (/
       2.0
       (pow
        (*
         (/ t_m (pow (cbrt l) 2.0))
         (cbrt (* (sin k) (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0))))))
        3.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1160.0) {
		tmp = 2.0 / pow((((t_m / cbrt(l)) * cbrt((1.0 / l))) * (cbrt(sin(k)) * cbrt((2.0 * k)))), 3.0);
	} else if (k <= 1.8e+142) {
		tmp = pow(l, 2.0) * ((2.0 * cos(k)) / ((t_m * pow(k, 2.0)) * pow(sin(k), 2.0)));
	} else {
		tmp = 2.0 / pow(((t_m / pow(cbrt(l), 2.0)) * cbrt((sin(k) * (tan(k) * (2.0 + pow((k / t_m), 2.0)))))), 3.0);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1160.0) {
		tmp = 2.0 / Math.pow((((t_m / Math.cbrt(l)) * Math.cbrt((1.0 / l))) * (Math.cbrt(Math.sin(k)) * Math.cbrt((2.0 * k)))), 3.0);
	} else if (k <= 1.8e+142) {
		tmp = Math.pow(l, 2.0) * ((2.0 * Math.cos(k)) / ((t_m * Math.pow(k, 2.0)) * Math.pow(Math.sin(k), 2.0)));
	} else {
		tmp = 2.0 / Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt((Math.sin(k) * (Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0)))))), 3.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1160.0)
		tmp = Float64(2.0 / (Float64(Float64(Float64(t_m / cbrt(l)) * cbrt(Float64(1.0 / l))) * Float64(cbrt(sin(k)) * cbrt(Float64(2.0 * k)))) ^ 3.0));
	elseif (k <= 1.8e+142)
		tmp = Float64((l ^ 2.0) * Float64(Float64(2.0 * cos(k)) / Float64(Float64(t_m * (k ^ 2.0)) * (sin(k) ^ 2.0))));
	else
		tmp = Float64(2.0 / (Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * cbrt(Float64(sin(k) * Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0)))))) ^ 3.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1160.0], N[(2.0 / N[Power[N[(N[(N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[(1.0 / l), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(2.0 * k), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.8e+142], N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[(2.0 * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1160

   1. Initial program 54.0%

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

   3. Simplified 54.0%

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

      1. associate-*l* 48.9%

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

      2. associate-/r* 55.4%

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

      3. associate-+r+ 55.4%

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

      4. metadata-eval 55.4%

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

      5. associate-*l* 55.5%

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

      6. add-cube-cbrt 55.4%

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

      7. pow3 55.4%

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

   6. Applied egg-rr 74.3%

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

      1. *-commutative 74.3%

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

      2. cbrt-prod 86.4%

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

   8. Applied egg-rr 86.4%

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

      1. add-cbrt-cube 64.7%

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

      2. unpow3 64.7%

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

      3. unpow2 64.7%

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

      4. cbrt-prod 55.3%

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

      5. unpow2 55.3%

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

      6. cbrt-div 53.8%

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

      7. unpow2 53.8%

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

      8. associate-/l/ 60.8%

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

      9. div-inv 60.8%

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

      10. cbrt-prod 62.8%

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

      11. cbrt-div 64.7%

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

      12. unpow3 64.7%

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

      13. add-cbrt-cube 86.4%

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

   10. Applied egg-rr 86.4%

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

   11. Taylor expanded in k around 0 77.4%

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

   if 1160 < k < 1.8000000000000001e142

   1. Initial program 48.4%

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

   3. Simplified 48.5%

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

      1. associate-*r* 50.7%

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

      2. *-un-lft-identity 50.7%

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

      3. times-frac 50.7%

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

      4. associate-/l/ 50.6%

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

   6. Applied egg-rr 50.6%

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

      1. /-rgt-identity 50.6%

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

      2. associate-*r/ 50.7%

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

      3. associate-*l/ 50.7%

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

      4. associate-*l* 50.5%

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

   8. Simplified 50.5%

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

      1. add-cube-cbrt 50.5%

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

      2. pow3 50.5%

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

      3. associate-/l* 50.5%

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

      4. associate-/l* 50.5%

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

   10. Applied egg-rr 50.5%

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

   11. Taylor expanded in t around 0 85.7%

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

       1. associate-/l* 85.7%

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

       2. rem-cube-cbrt 86.9%

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

       3. associate-*r* 86.8%

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

   13. Simplified 86.8%

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

   if 1.8000000000000001e142 < k

   1. Initial program 51.3%

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

   3. Simplified 51.3%

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

      1. associate-*l* 51.3%

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

      2. associate-/r* 54.2%

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

      3. associate-+r+ 54.2%

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

      4. metadata-eval 54.2%

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

      5. associate-*l* 54.2%

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

      6. add-cube-cbrt 54.2%

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

      7. pow3 54.2%

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

   6. Applied egg-rr 76.0%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1160:\\ \;\;\;\;\frac{2}{{\left(\left(\frac{t}{\sqrt[3]{\ell}} \cdot \sqrt[3]{\frac{1}{\ell}}\right) \cdot \left(\sqrt[3]{\sin k} \cdot \sqrt[3]{2 \cdot k}\right)\right)}^{3}}\\ \mathbf{elif}\;k \leq 1.8 \cdot 10^{+142}:\\ \;\;\;\;{\ell}^{2} \cdot \frac{2 \cdot \cos k}{\left(t \cdot {k}^{2}\right) \cdot {\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 0.0007:\\ \;\;\;\;\frac{2}{{\left(\left(\frac{t\_m}{\sqrt[3]{\ell}} \cdot \sqrt[3]{\frac{1}{\ell}}\right) \cdot \left(\sqrt[3]{\sin k} \cdot \sqrt[3]{2 \cdot k}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;{\ell}^{2} \cdot \frac{2 \cdot \cos k}{\left(t\_m \cdot {k}^{2}\right) \cdot {\sin k}^{2}}\\ \end{array} \end{array} \]

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 (<= k 0.0007)
    (/
     2.0
     (pow
      (*
       (* (/ t_m (cbrt l)) (cbrt (/ 1.0 l)))
       (* (cbrt (sin k)) (cbrt (* 2.0 k))))
      3.0))
    (*
     (pow l 2.0)
     (/ (* 2.0 (cos k)) (* (* t_m (pow k 2.0)) (pow (sin k) 2.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 0.0007) {
		tmp = 2.0 / pow((((t_m / cbrt(l)) * cbrt((1.0 / l))) * (cbrt(sin(k)) * cbrt((2.0 * k)))), 3.0);
	} else {
		tmp = pow(l, 2.0) * ((2.0 * cos(k)) / ((t_m * pow(k, 2.0)) * pow(sin(k), 2.0)));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 0.0007) {
		tmp = 2.0 / Math.pow((((t_m / Math.cbrt(l)) * Math.cbrt((1.0 / l))) * (Math.cbrt(Math.sin(k)) * Math.cbrt((2.0 * k)))), 3.0);
	} else {
		tmp = Math.pow(l, 2.0) * ((2.0 * Math.cos(k)) / ((t_m * Math.pow(k, 2.0)) * Math.pow(Math.sin(k), 2.0)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 0.0007)
		tmp = Float64(2.0 / (Float64(Float64(Float64(t_m / cbrt(l)) * cbrt(Float64(1.0 / l))) * Float64(cbrt(sin(k)) * cbrt(Float64(2.0 * k)))) ^ 3.0));
	else
		tmp = Float64((l ^ 2.0) * Float64(Float64(2.0 * cos(k)) / Float64(Float64(t_m * (k ^ 2.0)) * (sin(k) ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 0.0007], N[(2.0 / N[Power[N[(N[(N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[(1.0 / l), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(2.0 * k), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[(2.0 * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 6.99999999999999993e-4

   1. Initial program 54.0%

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

   3. Simplified 54.0%

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

      1. associate-*l* 48.9%

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

      2. associate-/r* 55.4%

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

      3. associate-+r+ 55.4%

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

      4. metadata-eval 55.4%

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

      5. associate-*l* 55.5%

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

      6. add-cube-cbrt 55.4%

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

      7. pow3 55.4%

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

   6. Applied egg-rr 74.3%

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

      1. *-commutative 74.3%

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

      2. cbrt-prod 86.4%

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

   8. Applied egg-rr 86.4%

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

      1. add-cbrt-cube 64.7%

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

      2. unpow3 64.7%

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

      3. unpow2 64.7%

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

      4. cbrt-prod 55.3%

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

      5. unpow2 55.3%

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

      6. cbrt-div 53.8%

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

      7. unpow2 53.8%

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

      8. associate-/l/ 60.8%

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

      9. div-inv 60.8%

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

      10. cbrt-prod 62.8%

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

      11. cbrt-div 64.7%

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

      12. unpow3 64.7%

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

      13. add-cbrt-cube 86.4%

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

   10. Applied egg-rr 86.4%

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

   11. Taylor expanded in k around 0 77.4%

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

   if 6.99999999999999993e-4 < k

   1. Initial program 49.9%

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

   3. Simplified 48.6%

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

      1. associate-*r* 51.2%

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

      2. *-un-lft-identity 51.2%

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

      3. times-frac 51.2%

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

      4. associate-/l/ 51.2%

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

   6. Applied egg-rr 51.2%

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

      1. /-rgt-identity 51.2%

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

      2. associate-*r/ 51.2%

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

      3. associate-*l/ 52.6%

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

      4. associate-*l* 52.5%

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

   8. Simplified 52.5%

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

      1. add-cube-cbrt 52.5%

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

      2. pow3 52.5%

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

      3. associate-/l* 52.5%

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

      4. associate-/l* 52.5%

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

   10. Applied egg-rr 52.5%

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

   11. Taylor expanded in t around 0 76.2%

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

       1. associate-/l* 76.2%

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

       2. rem-cube-cbrt 76.8%

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

       3. associate-*r* 76.8%

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

   13. Simplified 76.8%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.0007:\\ \;\;\;\;\frac{2}{{\left(\left(\frac{t}{\sqrt[3]{\ell}} \cdot \sqrt[3]{\frac{1}{\ell}}\right) \cdot \left(\sqrt[3]{\sin k} \cdot \sqrt[3]{2 \cdot k}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;{\ell}^{2} \cdot \frac{2 \cdot \cos k}{\left(t \cdot {k}^{2}\right) \cdot {\sin k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 + {\left(\frac{k}{t\_m}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.55 \cdot 10^{-55}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{k}^{2} \cdot \left(t\_m \cdot {\sin k}^{2}\right)}\\ \mathbf{elif}\;t\_m \leq 7.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{\ell}{t\_2} \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot {t\_m}^{3}\right)}\right)\\ \mathbf{elif}\;t\_m \leq 1.9 \cdot 10^{+191}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{k}}{{\left(t\_m \cdot \sqrt[3]{\sin k}\right)}^{3}}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)}^{3}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (+ 2.0 (pow (/ k t_m) 2.0))))
   (*
    t_s
    (if (<= t_m 1.55e-55)
      (* 2.0 (/ (* (cos k) (* l l)) (* (pow k 2.0) (* t_m (pow (sin k) 2.0)))))
      (if (<= t_m 7.5e+102)
        (* (/ l t_2) (* l (/ 2.0 (* (tan k) (* (sin k) (pow t_m 3.0))))))
        (if (<= t_m 1.9e+191)
          (/ (* (* l l) (/ (/ 2.0 k) (pow (* t_m (cbrt (sin k))) 3.0))) t_2)
          (/
           2.0
           (pow
            (* (/ t_m (pow (cbrt l) 2.0)) (cbrt (* 2.0 (pow k 2.0))))
            3.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = 2.0 + pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 1.55e-55) {
		tmp = 2.0 * ((cos(k) * (l * l)) / (pow(k, 2.0) * (t_m * pow(sin(k), 2.0))));
	} else if (t_m <= 7.5e+102) {
		tmp = (l / t_2) * (l * (2.0 / (tan(k) * (sin(k) * pow(t_m, 3.0)))));
	} else if (t_m <= 1.9e+191) {
		tmp = ((l * l) * ((2.0 / k) / pow((t_m * cbrt(sin(k))), 3.0))) / t_2;
	} else {
		tmp = 2.0 / pow(((t_m / pow(cbrt(l), 2.0)) * cbrt((2.0 * pow(k, 2.0)))), 3.0);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = 2.0 + Math.pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 1.55e-55) {
		tmp = 2.0 * ((Math.cos(k) * (l * l)) / (Math.pow(k, 2.0) * (t_m * Math.pow(Math.sin(k), 2.0))));
	} else if (t_m <= 7.5e+102) {
		tmp = (l / t_2) * (l * (2.0 / (Math.tan(k) * (Math.sin(k) * Math.pow(t_m, 3.0)))));
	} else if (t_m <= 1.9e+191) {
		tmp = ((l * l) * ((2.0 / k) / Math.pow((t_m * Math.cbrt(Math.sin(k))), 3.0))) / t_2;
	} else {
		tmp = 2.0 / Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt((2.0 * Math.pow(k, 2.0)))), 3.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(2.0 + (Float64(k / t_m) ^ 2.0))
	tmp = 0.0
	if (t_m <= 1.55e-55)
		tmp = Float64(2.0 * Float64(Float64(cos(k) * Float64(l * l)) / Float64((k ^ 2.0) * Float64(t_m * (sin(k) ^ 2.0)))));
	elseif (t_m <= 7.5e+102)
		tmp = Float64(Float64(l / t_2) * Float64(l * Float64(2.0 / Float64(tan(k) * Float64(sin(k) * (t_m ^ 3.0))))));
	elseif (t_m <= 1.9e+191)
		tmp = Float64(Float64(Float64(l * l) * Float64(Float64(2.0 / k) / (Float64(t_m * cbrt(sin(k))) ^ 3.0))) / t_2);
	else
		tmp = Float64(2.0 / (Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * cbrt(Float64(2.0 * (k ^ 2.0)))) ^ 3.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.55e-55], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 7.5e+102], N[(N[(l / t$95$2), $MachinePrecision] * N[(l * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.9e+191], N[(N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 / k), $MachinePrecision] / N[Power[N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(2.0 / N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 1.54999999999999998e-55

   1. Initial program 48.8%

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

   3. Simplified 48.8%

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

   5. Taylor expanded in t around 0 65.7%

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

      1. pow2 65.7%

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

   7. Applied egg-rr 65.7%

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

   if 1.54999999999999998e-55 < t < 7.5e102

   1. Initial program 71.0%

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

   3. Simplified 64.6%

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

      1. associate-*r* 72.7%

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

      2. *-un-lft-identity 72.7%

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

      3. times-frac 78.6%

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

      4. associate-/l/ 78.6%

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

   6. Applied egg-rr 78.6%

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

   if 7.5e102 < t < 1.8999999999999999e191

   1. Initial program 37.3%

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

   3. Simplified 37.3%

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

   5. Taylor expanded in k around 0 37.3%

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

      1. add-cube-cbrt 37.3%

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

      2. pow2 37.3%

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

      3. cbrt-div 37.3%

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

      4. cbrt-prod 37.3%

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

      5. unpow3 37.3%

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

      6. add-cbrt-cube 37.3%

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

      7. cbrt-div 37.3%

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

      8. cbrt-prod 37.3%

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

      9. unpow3 37.3%

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

      10. add-cbrt-cube 79.0%

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

   7. Applied egg-rr 79.0%

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

      1. unpow2 79.0%

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

      2. unpow3 79.1%

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

      3. cube-div 79.0%

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

      4. rem-cube-cbrt 79.0%

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

   9. Simplified 79.0%

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

   if 1.8999999999999999e191 < t

   1. Initial program 68.1%

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

   3. Simplified 68.1%

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

      1. associate-*l* 68.0%

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

      2. associate-/r* 75.8%

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

      3. associate-+r+ 75.8%

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

      4. metadata-eval 75.8%

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

      5. associate-*l* 75.8%

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

      6. add-cube-cbrt 75.8%

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

      7. pow3 75.8%

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

   6. Applied egg-rr 94.5%

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

   7. Taylor expanded in k around 0 90.9%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.55 \cdot 10^{-55}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot {t}^{3}\right)}\right)\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{+191}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{k}}{{\left(t \cdot \sqrt[3]{\sin k}\right)}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 + {\left(\frac{k}{t\_m}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.2 \cdot 10^{-56}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{k}^{2} \cdot \left(t\_m \cdot {\sin k}^{2}\right)}\\ \mathbf{elif}\;t\_m \leq 7.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{\ell}{t\_2} \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot {t\_m}^{3}\right)}\right)\\ \mathbf{elif}\;t\_m \leq 5.3 \cdot 10^{+191}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{k}}{{\left(t\_m \cdot \sqrt[3]{\sin k}\right)}^{3}}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t\_m \cdot \left(\sqrt[3]{2 \cdot {k}^{2}} \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)}^{3}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (+ 2.0 (pow (/ k t_m) 2.0))))
   (*
    t_s
    (if (<= t_m 5.2e-56)
      (* 2.0 (/ (* (cos k) (* l l)) (* (pow k 2.0) (* t_m (pow (sin k) 2.0)))))
      (if (<= t_m 7.5e+102)
        (* (/ l t_2) (* l (/ 2.0 (* (tan k) (* (sin k) (pow t_m 3.0))))))
        (if (<= t_m 5.3e+191)
          (/ (* (* l l) (/ (/ 2.0 k) (pow (* t_m (cbrt (sin k))) 3.0))) t_2)
          (/
           2.0
           (pow
            (* t_m (* (cbrt (* 2.0 (pow k 2.0))) (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 = 2.0 + pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 5.2e-56) {
		tmp = 2.0 * ((cos(k) * (l * l)) / (pow(k, 2.0) * (t_m * pow(sin(k), 2.0))));
	} else if (t_m <= 7.5e+102) {
		tmp = (l / t_2) * (l * (2.0 / (tan(k) * (sin(k) * pow(t_m, 3.0)))));
	} else if (t_m <= 5.3e+191) {
		tmp = ((l * l) * ((2.0 / k) / pow((t_m * cbrt(sin(k))), 3.0))) / t_2;
	} else {
		tmp = 2.0 / pow((t_m * (cbrt((2.0 * pow(k, 2.0))) * 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 = 2.0 + Math.pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 5.2e-56) {
		tmp = 2.0 * ((Math.cos(k) * (l * l)) / (Math.pow(k, 2.0) * (t_m * Math.pow(Math.sin(k), 2.0))));
	} else if (t_m <= 7.5e+102) {
		tmp = (l / t_2) * (l * (2.0 / (Math.tan(k) * (Math.sin(k) * Math.pow(t_m, 3.0)))));
	} else if (t_m <= 5.3e+191) {
		tmp = ((l * l) * ((2.0 / k) / Math.pow((t_m * Math.cbrt(Math.sin(k))), 3.0))) / t_2;
	} else {
		tmp = 2.0 / Math.pow((t_m * (Math.cbrt((2.0 * Math.pow(k, 2.0))) * 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(2.0 + (Float64(k / t_m) ^ 2.0))
	tmp = 0.0
	if (t_m <= 5.2e-56)
		tmp = Float64(2.0 * Float64(Float64(cos(k) * Float64(l * l)) / Float64((k ^ 2.0) * Float64(t_m * (sin(k) ^ 2.0)))));
	elseif (t_m <= 7.5e+102)
		tmp = Float64(Float64(l / t_2) * Float64(l * Float64(2.0 / Float64(tan(k) * Float64(sin(k) * (t_m ^ 3.0))))));
	elseif (t_m <= 5.3e+191)
		tmp = Float64(Float64(Float64(l * l) * Float64(Float64(2.0 / k) / (Float64(t_m * cbrt(sin(k))) ^ 3.0))) / t_2);
	else
		tmp = Float64(2.0 / (Float64(t_m * Float64(cbrt(Float64(2.0 * (k ^ 2.0))) * (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[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 5.2e-56], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 7.5e+102], N[(N[(l / t$95$2), $MachinePrecision] * N[(l * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.3e+191], N[(N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 / k), $MachinePrecision] / N[Power[N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(2.0 / N[Power[N[(t$95$m * N[(N[Power[N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 5.19999999999999994e-56

   1. Initial program 48.8%

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

   3. Simplified 48.8%

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

   5. Taylor expanded in t around 0 65.7%

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

      1. pow2 65.7%

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

   7. Applied egg-rr 65.7%

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

   if 5.19999999999999994e-56 < t < 7.5e102

   1. Initial program 71.0%

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

   3. Simplified 64.6%

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

      1. associate-*r* 72.7%

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

      2. *-un-lft-identity 72.7%

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

      3. times-frac 78.6%

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

      4. associate-/l/ 78.6%

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

   6. Applied egg-rr 78.6%

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

   if 7.5e102 < t < 5.30000000000000031e191

   1. Initial program 37.3%

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

   3. Simplified 37.3%

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

   5. Taylor expanded in k around 0 37.3%

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

      1. add-cube-cbrt 37.3%

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

      2. pow2 37.3%

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

      3. cbrt-div 37.3%

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

      4. cbrt-prod 37.3%

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

      5. unpow3 37.3%

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

      6. add-cbrt-cube 37.3%

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

      7. cbrt-div 37.3%

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

      8. cbrt-prod 37.3%

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

      9. unpow3 37.3%

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

      10. add-cbrt-cube 79.0%

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

   7. Applied egg-rr 79.0%

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

      1. unpow2 79.0%

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

      2. unpow3 79.1%

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

      3. cube-div 79.0%

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

      4. rem-cube-cbrt 79.0%

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

   9. Simplified 79.0%

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

   if 5.30000000000000031e191 < t

   1. Initial program 68.1%

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

   3. Simplified 68.1%

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

      1. associate-*l* 68.0%

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

      2. associate-/r* 75.8%

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

      3. associate-+r+ 75.8%

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

      4. metadata-eval 75.8%

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

      5. associate-*l* 75.8%

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

      6. add-cube-cbrt 75.8%

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

      7. pow3 75.8%

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

   6. Applied egg-rr 94.5%

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

   7. Taylor expanded in k around 0 90.9%

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

   9. Applied egg-rr 90.8%

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

       1. unpow1 90.8%

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

       2. associate-*l* 90.8%

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

   11. Simplified 90.8%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.2 \cdot 10^{-56}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot {t}^{3}\right)}\right)\\ \mathbf{elif}\;t \leq 5.3 \cdot 10^{+191}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{k}}{{\left(t \cdot \sqrt[3]{\sin k}\right)}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t \cdot \left(\sqrt[3]{2 \cdot {k}^{2}} \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1720:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\sqrt[3]{2 \cdot k} \cdot \sqrt[3]{k}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;{\ell}^{2} \cdot \frac{2 \cdot \cos k}{\left(t\_m \cdot {k}^{2}\right) \cdot {\sin k}^{2}}\\ \end{array} \end{array} \]

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 (<= k 1720.0)
    (/
     2.0
     (pow (* (/ t_m (pow (cbrt l) 2.0)) (* (cbrt (* 2.0 k)) (cbrt k))) 3.0))
    (*
     (pow l 2.0)
     (/ (* 2.0 (cos k)) (* (* t_m (pow k 2.0)) (pow (sin k) 2.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1720.0) {
		tmp = 2.0 / pow(((t_m / pow(cbrt(l), 2.0)) * (cbrt((2.0 * k)) * cbrt(k))), 3.0);
	} else {
		tmp = pow(l, 2.0) * ((2.0 * cos(k)) / ((t_m * pow(k, 2.0)) * pow(sin(k), 2.0)));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1720.0) {
		tmp = 2.0 / Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * (Math.cbrt((2.0 * k)) * Math.cbrt(k))), 3.0);
	} else {
		tmp = Math.pow(l, 2.0) * ((2.0 * Math.cos(k)) / ((t_m * Math.pow(k, 2.0)) * Math.pow(Math.sin(k), 2.0)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1720.0)
		tmp = Float64(2.0 / (Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * Float64(cbrt(Float64(2.0 * k)) * cbrt(k))) ^ 3.0));
	else
		tmp = Float64((l ^ 2.0) * Float64(Float64(2.0 * cos(k)) / Float64(Float64(t_m * (k ^ 2.0)) * (sin(k) ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1720.0], N[(2.0 / N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(2.0 * k), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[k, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[(2.0 * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1720

   1. Initial program 54.0%

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

   3. Simplified 54.0%

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

      1. associate-*l* 48.9%

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

      2. associate-/r* 55.4%

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

      3. associate-+r+ 55.4%

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

      4. metadata-eval 55.4%

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

      5. associate-*l* 55.5%

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

      6. add-cube-cbrt 55.4%

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

      7. pow3 55.4%

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

   6. Applied egg-rr 74.3%

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

   7. Taylor expanded in k around 0 67.5%

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

      1. pow-to-exp 22.1%

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

      2. *-commutative 22.1%

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

      3. *-commutative 22.1%

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

      4. pow-to-exp 67.5%

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

      5. pow2 67.5%

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

      6. associate-*r* 67.5%

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

      7. cbrt-prod 77.5%

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

   9. Applied egg-rr 77.5%

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

   if 1720 < k

   1. Initial program 49.9%

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

   3. Simplified 48.6%

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

      1. associate-*r* 51.2%

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

      2. *-un-lft-identity 51.2%

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

      3. times-frac 51.2%

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

      4. associate-/l/ 51.2%

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

   6. Applied egg-rr 51.2%

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

      1. /-rgt-identity 51.2%

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

      2. associate-*r/ 51.2%

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

      3. associate-*l/ 52.6%

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

      4. associate-*l* 52.5%

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

   8. Simplified 52.5%

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

      1. add-cube-cbrt 52.5%

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

      2. pow3 52.5%

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

      3. associate-/l* 52.5%

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

      4. associate-/l* 52.5%

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

   10. Applied egg-rr 52.5%

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

   11. Taylor expanded in t around 0 76.2%

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

       1. associate-/l* 76.2%

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

       2. rem-cube-cbrt 76.8%

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

       3. associate-*r* 76.8%

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

   13. Simplified 76.8%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1720:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\sqrt[3]{2 \cdot k} \cdot \sqrt[3]{k}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;{\ell}^{2} \cdot \frac{2 \cdot \cos k}{\left(t \cdot {k}^{2}\right) \cdot {\sin k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 0.0064:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\sqrt[3]{2 \cdot k} \cdot \sqrt[3]{k}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{\cos k}{\left(t\_m \cdot {k}^{2}\right) \cdot {\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 0.0064)
    (/
     2.0
     (pow (* (/ t_m (pow (cbrt l) 2.0)) (* (cbrt (* 2.0 k)) (cbrt k))) 3.0))
    (*
     2.0
     (* (pow l 2.0) (/ (cos k) (* (* t_m (pow k 2.0)) (pow (sin k) 2.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 0.0064) {
		tmp = 2.0 / pow(((t_m / pow(cbrt(l), 2.0)) * (cbrt((2.0 * k)) * cbrt(k))), 3.0);
	} else {
		tmp = 2.0 * (pow(l, 2.0) * (cos(k) / ((t_m * pow(k, 2.0)) * pow(sin(k), 2.0))));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 0.0064) {
		tmp = 2.0 / Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * (Math.cbrt((2.0 * k)) * Math.cbrt(k))), 3.0);
	} else {
		tmp = 2.0 * (Math.pow(l, 2.0) * (Math.cos(k) / ((t_m * Math.pow(k, 2.0)) * Math.pow(Math.sin(k), 2.0))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 0.0064)
		tmp = Float64(2.0 / (Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * Float64(cbrt(Float64(2.0 * k)) * cbrt(k))) ^ 3.0));
	else
		tmp = Float64(2.0 * Float64((l ^ 2.0) * Float64(cos(k) / Float64(Float64(t_m * (k ^ 2.0)) * (sin(k) ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 0.0064], N[(2.0 / N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(2.0 * k), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[k, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 0.00640000000000000031

   1. Initial program 54.0%

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

   3. Simplified 54.0%

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

      1. associate-*l* 48.9%

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

      2. associate-/r* 55.4%

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

      3. associate-+r+ 55.4%

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

      4. metadata-eval 55.4%

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

      5. associate-*l* 55.5%

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

      6. add-cube-cbrt 55.4%

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

      7. pow3 55.4%

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

   6. Applied egg-rr 74.3%

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

   7. Taylor expanded in k around 0 67.5%

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

      1. pow-to-exp 22.1%

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

      2. *-commutative 22.1%

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

      3. *-commutative 22.1%

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

      4. pow-to-exp 67.5%

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

      5. pow2 67.5%

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

      6. associate-*r* 67.5%

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

      7. cbrt-prod 77.5%

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

   9. Applied egg-rr 77.5%

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

   if 0.00640000000000000031 < k

   1. Initial program 49.9%

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

   3. Simplified 50.0%

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

      1. add-cube-cbrt 49.9%

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

      2. pow3 49.9%

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

      3. associate-/r* 52.5%

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

      4. *-commutative 52.5%

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

      5. cbrt-prod 52.5%

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

      6. associate-/r* 49.9%

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

      7. cbrt-div 49.8%

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

      8. rem-cbrt-cube 57.0%

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

      9. cbrt-prod 62.1%

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

      10. pow2 62.1%

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

   6. Applied egg-rr 62.1%

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

   7. Taylor expanded in k around inf 76.7%

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

      1. associate-/l* 76.8%

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

      2. associate-*r* 76.8%

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

   9. Simplified 76.8%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.0064:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\sqrt[3]{2 \cdot k} \cdot \sqrt[3]{k}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{\cos k}{\left(t \cdot {k}^{2}\right) \cdot {\sin k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1020:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\sqrt[3]{2 \cdot k} \cdot \sqrt[3]{k}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{k}^{2} \cdot \left(t\_m \cdot {\sin k}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1020.0)
    (/
     2.0
     (pow (* (/ t_m (pow (cbrt l) 2.0)) (* (cbrt (* 2.0 k)) (cbrt k))) 3.0))
    (*
     2.0
     (/ (* (cos k) (* l l)) (* (pow k 2.0) (* t_m (pow (sin k) 2.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1020.0) {
		tmp = 2.0 / pow(((t_m / pow(cbrt(l), 2.0)) * (cbrt((2.0 * k)) * cbrt(k))), 3.0);
	} else {
		tmp = 2.0 * ((cos(k) * (l * l)) / (pow(k, 2.0) * (t_m * pow(sin(k), 2.0))));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1020.0) {
		tmp = 2.0 / Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * (Math.cbrt((2.0 * k)) * Math.cbrt(k))), 3.0);
	} else {
		tmp = 2.0 * ((Math.cos(k) * (l * l)) / (Math.pow(k, 2.0) * (t_m * Math.pow(Math.sin(k), 2.0))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1020.0)
		tmp = Float64(2.0 / (Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * Float64(cbrt(Float64(2.0 * k)) * cbrt(k))) ^ 3.0));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) * Float64(l * l)) / Float64((k ^ 2.0) * Float64(t_m * (sin(k) ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1020.0], N[(2.0 / N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(2.0 * k), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[k, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1020

   1. Initial program 54.0%

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

   3. Simplified 54.0%

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

      1. associate-*l* 48.9%

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

      2. associate-/r* 55.4%

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

      3. associate-+r+ 55.4%

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

      4. metadata-eval 55.4%

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

      5. associate-*l* 55.5%

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

      6. add-cube-cbrt 55.4%

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

      7. pow3 55.4%

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

   6. Applied egg-rr 74.3%

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

   7. Taylor expanded in k around 0 67.5%

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

      1. pow-to-exp 22.1%

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

      2. *-commutative 22.1%

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

      3. *-commutative 22.1%

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

      4. pow-to-exp 67.5%

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

      5. pow2 67.5%

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

      6. associate-*r* 67.5%

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

      7. cbrt-prod 77.5%

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

   9. Applied egg-rr 77.5%

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

   if 1020 < k

   1. Initial program 49.9%

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

   3. Simplified 50.0%

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

   5. Taylor expanded in t around 0 76.7%

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

      1. pow2 76.7%

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

   7. Applied egg-rr 76.7%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1020:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\sqrt[3]{2 \cdot k} \cdot \sqrt[3]{k}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 71.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 0.012:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left(\sqrt[3]{\sin k} \cdot \frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{k}^{2} \cdot \left(t\_m \cdot {\sin k}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 0.012)
    (/
     2.0
     (* (* 2.0 k) (pow (* (cbrt (sin k)) (/ t_m (pow (cbrt l) 2.0))) 3.0)))
    (*
     2.0
     (/ (* (cos k) (* l l)) (* (pow k 2.0) (* t_m (pow (sin k) 2.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 0.012) {
		tmp = 2.0 / ((2.0 * k) * pow((cbrt(sin(k)) * (t_m / pow(cbrt(l), 2.0))), 3.0));
	} else {
		tmp = 2.0 * ((cos(k) * (l * l)) / (pow(k, 2.0) * (t_m * pow(sin(k), 2.0))));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 0.012) {
		tmp = 2.0 / ((2.0 * k) * Math.pow((Math.cbrt(Math.sin(k)) * (t_m / Math.pow(Math.cbrt(l), 2.0))), 3.0));
	} else {
		tmp = 2.0 * ((Math.cos(k) * (l * l)) / (Math.pow(k, 2.0) * (t_m * Math.pow(Math.sin(k), 2.0))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 0.012)
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * (Float64(cbrt(sin(k)) * Float64(t_m / (cbrt(l) ^ 2.0))) ^ 3.0)));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) * Float64(l * l)) / Float64((k ^ 2.0) * Float64(t_m * (sin(k) ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 0.012], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[Power[N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 0.012

   1. Initial program 54.0%

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

   3. Simplified 54.0%

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

      1. add-cube-cbrt 53.9%

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

      2. pow3 53.9%

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

      3. associate-/r* 60.9%

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

      4. *-commutative 60.9%

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

      5. cbrt-prod 60.9%

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

      6. associate-/r* 53.8%

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

      7. cbrt-div 54.8%

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

      8. rem-cbrt-cube 67.8%

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

      9. cbrt-prod 80.0%

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

      10. pow2 80.0%

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

   6. Applied egg-rr 80.0%

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

   7. Taylor expanded in k around 0 73.1%

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

   if 0.012 < k

   1. Initial program 49.9%

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

   3. Simplified 50.0%

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

   5. Taylor expanded in t around 0 76.7%

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

      1. pow2 76.7%

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

   7. Applied egg-rr 76.7%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.012:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 74.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 + {\left(\frac{k}{t\_m}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.25 \cdot 10^{-55}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{k}^{2} \cdot \left(t\_m \cdot {\sin k}^{2}\right)}\\ \mathbf{elif}\;t\_m \leq 7.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{\ell}{t\_2} \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot {t\_m}^{3}\right)}\right)\\ \mathbf{elif}\;t\_m \leq 2.3 \cdot 10^{+191}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{k}}{{\left(t\_m \cdot \sqrt[3]{\sin k}\right)}^{3}}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3} \cdot \left(2 \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (+ 2.0 (pow (/ k t_m) 2.0))))
   (*
    t_s
    (if (<= t_m 1.25e-55)
      (* 2.0 (/ (* (cos k) (* l l)) (* (pow k 2.0) (* t_m (pow (sin k) 2.0)))))
      (if (<= t_m 7.5e+102)
        (* (/ l t_2) (* l (/ 2.0 (* (tan k) (* (sin k) (pow t_m 3.0))))))
        (if (<= t_m 2.3e+191)
          (/ (* (* l l) (/ (/ 2.0 k) (pow (* t_m (cbrt (sin k))) 3.0))) t_2)
          (/
           2.0
           (* (pow (* t_m (pow (cbrt l) -2.0)) 3.0) (* 2.0 (* k k))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = 2.0 + pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 1.25e-55) {
		tmp = 2.0 * ((cos(k) * (l * l)) / (pow(k, 2.0) * (t_m * pow(sin(k), 2.0))));
	} else if (t_m <= 7.5e+102) {
		tmp = (l / t_2) * (l * (2.0 / (tan(k) * (sin(k) * pow(t_m, 3.0)))));
	} else if (t_m <= 2.3e+191) {
		tmp = ((l * l) * ((2.0 / k) / pow((t_m * cbrt(sin(k))), 3.0))) / t_2;
	} else {
		tmp = 2.0 / (pow((t_m * pow(cbrt(l), -2.0)), 3.0) * (2.0 * (k * k)));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = 2.0 + Math.pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 1.25e-55) {
		tmp = 2.0 * ((Math.cos(k) * (l * l)) / (Math.pow(k, 2.0) * (t_m * Math.pow(Math.sin(k), 2.0))));
	} else if (t_m <= 7.5e+102) {
		tmp = (l / t_2) * (l * (2.0 / (Math.tan(k) * (Math.sin(k) * Math.pow(t_m, 3.0)))));
	} else if (t_m <= 2.3e+191) {
		tmp = ((l * l) * ((2.0 / k) / Math.pow((t_m * Math.cbrt(Math.sin(k))), 3.0))) / t_2;
	} else {
		tmp = 2.0 / (Math.pow((t_m * Math.pow(Math.cbrt(l), -2.0)), 3.0) * (2.0 * (k * k)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(2.0 + (Float64(k / t_m) ^ 2.0))
	tmp = 0.0
	if (t_m <= 1.25e-55)
		tmp = Float64(2.0 * Float64(Float64(cos(k) * Float64(l * l)) / Float64((k ^ 2.0) * Float64(t_m * (sin(k) ^ 2.0)))));
	elseif (t_m <= 7.5e+102)
		tmp = Float64(Float64(l / t_2) * Float64(l * Float64(2.0 / Float64(tan(k) * Float64(sin(k) * (t_m ^ 3.0))))));
	elseif (t_m <= 2.3e+191)
		tmp = Float64(Float64(Float64(l * l) * Float64(Float64(2.0 / k) / (Float64(t_m * cbrt(sin(k))) ^ 3.0))) / t_2);
	else
		tmp = Float64(2.0 / Float64((Float64(t_m * (cbrt(l) ^ -2.0)) ^ 3.0) * Float64(2.0 * Float64(k * k))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.25e-55], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 7.5e+102], N[(N[(l / t$95$2), $MachinePrecision] * N[(l * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.3e+191], N[(N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 / k), $MachinePrecision] / N[Power[N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(2.0 / N[(N[Power[N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 1.25e-55

   1. Initial program 48.8%

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

   3. Simplified 48.8%

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

   5. Taylor expanded in t around 0 65.7%

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

      1. pow2 65.7%

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

   7. Applied egg-rr 65.7%

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

   if 1.25e-55 < t < 7.5e102

   1. Initial program 71.0%

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

   3. Simplified 64.6%

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

      1. associate-*r* 72.7%

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

      2. *-un-lft-identity 72.7%

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

      3. times-frac 78.6%

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

      4. associate-/l/ 78.6%

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

   6. Applied egg-rr 78.6%

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

   if 7.5e102 < t < 2.2999999999999999e191

   1. Initial program 37.3%

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

   3. Simplified 37.3%

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

   5. Taylor expanded in k around 0 37.3%

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

      1. add-cube-cbrt 37.3%

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

      2. pow2 37.3%

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

      3. cbrt-div 37.3%

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

      4. cbrt-prod 37.3%

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

      5. unpow3 37.3%

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

      6. add-cbrt-cube 37.3%

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

      7. cbrt-div 37.3%

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

      8. cbrt-prod 37.3%

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

      9. unpow3 37.3%

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

      10. add-cbrt-cube 79.0%

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

   7. Applied egg-rr 79.0%

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

      1. unpow2 79.0%

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

      2. unpow3 79.1%

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

      3. cube-div 79.0%

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

      4. rem-cube-cbrt 79.0%

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

   9. Simplified 79.0%

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

   if 2.2999999999999999e191 < t

   1. Initial program 68.1%

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

   3. Simplified 75.8%

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

   5. Taylor expanded in k around 0 75.8%

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

      1. unpow2 75.8%

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

   7. Applied egg-rr 75.8%

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

      1. associate-/r* 68.0%

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

      2. unpow3 68.0%

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

      3. times-frac 75.8%

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

      4. pow2 75.8%

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

   9. Applied egg-rr 75.8%

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

       1. frac-times 68.0%

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

       2. unpow2 68.0%

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

       3. unpow3 68.0%

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

       4. associate-/l/ 75.8%

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

       5. add-cube-cbrt 75.8%

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

       6. pow3 75.8%

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

       7. associate-/l/ 68.0%

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

       8. cbrt-div 68.0%

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

       9. unpow3 68.0%

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

       10. add-cbrt-cube 72.4%

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

       11. cbrt-unprod 87.3%

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

       12. unpow2 87.3%

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

       13. div-inv 87.4%

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

       14. pow-flip 87.4%

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

       15. metadata-eval 87.4%

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

   11. Applied egg-rr 87.4%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.25 \cdot 10^{-55}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot {t}^{3}\right)}\right)\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{+191}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{k}}{{\left(t \cdot \sqrt[3]{\sin k}\right)}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3} \cdot \left(2 \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 74.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 + {\left(\frac{k}{t\_m}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 10^{-42}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{k}^{2} \cdot \left(t\_m \cdot {\sin k}^{2}\right)}\\ \mathbf{elif}\;t\_m \leq 7.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell \cdot 2}{\tan k \cdot \left(\sin k \cdot {t\_m}^{3}\right)}}{t\_2}\\ \mathbf{elif}\;t\_m \leq 3.3 \cdot 10^{+191}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{k}}{{\left(t\_m \cdot \sqrt[3]{\sin k}\right)}^{3}}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3} \cdot \left(2 \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (+ 2.0 (pow (/ k t_m) 2.0))))
   (*
    t_s
    (if (<= t_m 1e-42)
      (* 2.0 (/ (* (cos k) (* l l)) (* (pow k 2.0) (* t_m (pow (sin k) 2.0)))))
      (if (<= t_m 7.5e+102)
        (/ (* l (/ (* l 2.0) (* (tan k) (* (sin k) (pow t_m 3.0))))) t_2)
        (if (<= t_m 3.3e+191)
          (/ (* (* l l) (/ (/ 2.0 k) (pow (* t_m (cbrt (sin k))) 3.0))) t_2)
          (/
           2.0
           (* (pow (* t_m (pow (cbrt l) -2.0)) 3.0) (* 2.0 (* k k))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = 2.0 + pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 1e-42) {
		tmp = 2.0 * ((cos(k) * (l * l)) / (pow(k, 2.0) * (t_m * pow(sin(k), 2.0))));
	} else if (t_m <= 7.5e+102) {
		tmp = (l * ((l * 2.0) / (tan(k) * (sin(k) * pow(t_m, 3.0))))) / t_2;
	} else if (t_m <= 3.3e+191) {
		tmp = ((l * l) * ((2.0 / k) / pow((t_m * cbrt(sin(k))), 3.0))) / t_2;
	} else {
		tmp = 2.0 / (pow((t_m * pow(cbrt(l), -2.0)), 3.0) * (2.0 * (k * k)));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = 2.0 + Math.pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 1e-42) {
		tmp = 2.0 * ((Math.cos(k) * (l * l)) / (Math.pow(k, 2.0) * (t_m * Math.pow(Math.sin(k), 2.0))));
	} else if (t_m <= 7.5e+102) {
		tmp = (l * ((l * 2.0) / (Math.tan(k) * (Math.sin(k) * Math.pow(t_m, 3.0))))) / t_2;
	} else if (t_m <= 3.3e+191) {
		tmp = ((l * l) * ((2.0 / k) / Math.pow((t_m * Math.cbrt(Math.sin(k))), 3.0))) / t_2;
	} else {
		tmp = 2.0 / (Math.pow((t_m * Math.pow(Math.cbrt(l), -2.0)), 3.0) * (2.0 * (k * k)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(2.0 + (Float64(k / t_m) ^ 2.0))
	tmp = 0.0
	if (t_m <= 1e-42)
		tmp = Float64(2.0 * Float64(Float64(cos(k) * Float64(l * l)) / Float64((k ^ 2.0) * Float64(t_m * (sin(k) ^ 2.0)))));
	elseif (t_m <= 7.5e+102)
		tmp = Float64(Float64(l * Float64(Float64(l * 2.0) / Float64(tan(k) * Float64(sin(k) * (t_m ^ 3.0))))) / t_2);
	elseif (t_m <= 3.3e+191)
		tmp = Float64(Float64(Float64(l * l) * Float64(Float64(2.0 / k) / (Float64(t_m * cbrt(sin(k))) ^ 3.0))) / t_2);
	else
		tmp = Float64(2.0 / Float64((Float64(t_m * (cbrt(l) ^ -2.0)) ^ 3.0) * Float64(2.0 * Float64(k * k))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1e-42], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 7.5e+102], N[(N[(l * N[(N[(l * 2.0), $MachinePrecision] / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$m, 3.3e+191], N[(N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 / k), $MachinePrecision] / N[Power[N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(2.0 / N[(N[Power[N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 1.00000000000000004e-42

   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 49.2%

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

   5. Taylor expanded in t around 0 66.1%

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

      1. pow2 66.1%

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

   7. Applied egg-rr 66.1%

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

   if 1.00000000000000004e-42 < t < 7.5e102

   1. Initial program 73.0%

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

   3. Simplified 69.2%

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

      1. associate-*r* 75.1%

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

      2. *-un-lft-identity 75.1%

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

      3. times-frac 78.4%

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

      4. associate-/l/ 78.3%

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

   6. Applied egg-rr 78.3%

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

      1. /-rgt-identity 78.3%

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

      2. associate-*r/ 75.0%

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

      3. associate-*l/ 76.6%

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

      4. associate-*l* 67.9%

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

   8. Simplified 67.9%

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

      1. associate-/l* 67.9%

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

   10. Applied egg-rr 67.9%

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

       1. associate-*r/ 67.9%

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

       2. *-commutative 67.9%

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

       3. associate-*r* 76.6%

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

   12. Simplified 76.6%

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

   if 7.5e102 < t < 3.2999999999999998e191

   1. Initial program 37.3%

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

   3. Simplified 37.3%

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

   5. Taylor expanded in k around 0 37.3%

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

      1. add-cube-cbrt 37.3%

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

      2. pow2 37.3%

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

      3. cbrt-div 37.3%

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

      4. cbrt-prod 37.3%

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

      5. unpow3 37.3%

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

      6. add-cbrt-cube 37.3%

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

      7. cbrt-div 37.3%

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

      8. cbrt-prod 37.3%

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

      9. unpow3 37.3%

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

      10. add-cbrt-cube 79.0%

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

   7. Applied egg-rr 79.0%

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

      1. unpow2 79.0%

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

      2. unpow3 79.1%

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

      3. cube-div 79.0%

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

      4. rem-cube-cbrt 79.0%

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

   9. Simplified 79.0%

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

   if 3.2999999999999998e191 < t

   1. Initial program 68.1%

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

   3. Simplified 75.8%

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

   5. Taylor expanded in k around 0 75.8%

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

      1. unpow2 75.8%

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

   7. Applied egg-rr 75.8%

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

      1. associate-/r* 68.0%

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

      2. unpow3 68.0%

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

      3. times-frac 75.8%

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

      4. pow2 75.8%

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

   9. Applied egg-rr 75.8%

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

       1. frac-times 68.0%

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

       2. unpow2 68.0%

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

       3. unpow3 68.0%

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

       4. associate-/l/ 75.8%

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

       5. add-cube-cbrt 75.8%

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

       6. pow3 75.8%

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

       7. associate-/l/ 68.0%

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

       8. cbrt-div 68.0%

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

       9. unpow3 68.0%

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

       10. add-cbrt-cube 72.4%

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

       11. cbrt-unprod 87.3%

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

       12. unpow2 87.3%

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

       13. div-inv 87.4%

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

       14. pow-flip 87.4%

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

       15. metadata-eval 87.4%

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

   11. Applied egg-rr 87.4%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 10^{-42}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell \cdot 2}{\tan k \cdot \left(\sin k \cdot {t}^{3}\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{+191}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{k}}{{\left(t \cdot \sqrt[3]{\sin k}\right)}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3} \cdot \left(2 \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 73.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.8 \cdot 10^{-41}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{k}^{2} \cdot \left(t\_m \cdot {\sin k}^{2}\right)}\\ \mathbf{elif}\;t\_m \leq 5.2 \cdot 10^{+110}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell \cdot 2}{\tan k \cdot \left(\sin k \cdot {t\_m}^{3}\right)}}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3} \cdot \left(2 \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4.8e-41)
    (* 2.0 (/ (* (cos k) (* l l)) (* (pow k 2.0) (* t_m (pow (sin k) 2.0)))))
    (if (<= t_m 5.2e+110)
      (/
       (* l (/ (* l 2.0) (* (tan k) (* (sin k) (pow t_m 3.0)))))
       (+ 2.0 (pow (/ k t_m) 2.0)))
      (/ 2.0 (* (pow (* t_m (pow (cbrt l) -2.0)) 3.0) (* 2.0 (* k k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.8e-41) {
		tmp = 2.0 * ((cos(k) * (l * l)) / (pow(k, 2.0) * (t_m * pow(sin(k), 2.0))));
	} else if (t_m <= 5.2e+110) {
		tmp = (l * ((l * 2.0) / (tan(k) * (sin(k) * pow(t_m, 3.0))))) / (2.0 + pow((k / t_m), 2.0));
	} else {
		tmp = 2.0 / (pow((t_m * pow(cbrt(l), -2.0)), 3.0) * (2.0 * (k * k)));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.8e-41) {
		tmp = 2.0 * ((Math.cos(k) * (l * l)) / (Math.pow(k, 2.0) * (t_m * Math.pow(Math.sin(k), 2.0))));
	} else if (t_m <= 5.2e+110) {
		tmp = (l * ((l * 2.0) / (Math.tan(k) * (Math.sin(k) * Math.pow(t_m, 3.0))))) / (2.0 + Math.pow((k / t_m), 2.0));
	} else {
		tmp = 2.0 / (Math.pow((t_m * Math.pow(Math.cbrt(l), -2.0)), 3.0) * (2.0 * (k * k)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 4.8e-41)
		tmp = Float64(2.0 * Float64(Float64(cos(k) * Float64(l * l)) / Float64((k ^ 2.0) * Float64(t_m * (sin(k) ^ 2.0)))));
	elseif (t_m <= 5.2e+110)
		tmp = Float64(Float64(l * Float64(Float64(l * 2.0) / Float64(tan(k) * Float64(sin(k) * (t_m ^ 3.0))))) / Float64(2.0 + (Float64(k / t_m) ^ 2.0)));
	else
		tmp = Float64(2.0 / Float64((Float64(t_m * (cbrt(l) ^ -2.0)) ^ 3.0) * Float64(2.0 * Float64(k * k))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 4.8e-41], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.2e+110], N[(N[(l * N[(N[(l * 2.0), $MachinePrecision] / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 4.80000000000000044e-41

   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 49.2%

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

   5. Taylor expanded in t around 0 66.1%

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

      1. pow2 66.1%

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

   7. Applied egg-rr 66.1%

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

   if 4.80000000000000044e-41 < t < 5.2e110

   1. Initial program 73.0%

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

   3. Simplified 69.2%

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

      1. associate-*r* 75.1%

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

      2. *-un-lft-identity 75.1%

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

      3. times-frac 78.4%

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

      4. associate-/l/ 78.3%

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

   6. Applied egg-rr 78.3%

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

      1. /-rgt-identity 78.3%

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

      2. associate-*r/ 75.0%

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

      3. associate-*l/ 76.6%

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

      4. associate-*l* 67.9%

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

   8. Simplified 67.9%

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

      1. associate-/l* 67.9%

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

   10. Applied egg-rr 67.9%

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

       1. associate-*r/ 67.9%

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

       2. *-commutative 67.9%

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

       3. associate-*r* 76.6%

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

   12. Simplified 76.6%

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

   if 5.2e110 < t

   1. Initial program 57.0%

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

   3. Simplified 61.6%

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

   5. Taylor expanded in k around 0 61.6%

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

      1. unpow2 61.6%

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

   7. Applied egg-rr 61.6%

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

      1. associate-/r* 56.5%

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

      2. unpow3 56.5%

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

      3. times-frac 71.9%

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

      4. pow2 71.9%

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

   9. Applied egg-rr 71.9%

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

       1. frac-times 56.5%

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

       2. unpow2 56.5%

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

       3. unpow3 56.5%

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

       4. associate-/l/ 61.6%

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

       5. add-cube-cbrt 61.6%

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

       6. pow3 61.6%

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

       7. associate-/l/ 56.5%

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

       8. cbrt-div 56.5%

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

       9. unpow3 56.5%

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

       10. add-cbrt-cube 69.6%

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

       11. cbrt-unprod 79.2%

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

       12. unpow2 79.2%

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

       13. div-inv 79.3%

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

       14. pow-flip 79.3%

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

       15. metadata-eval 79.3%

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

   11. Applied egg-rr 79.3%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.8 \cdot 10^{-41}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{+110}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell \cdot 2}{\tan k \cdot \left(\sin k \cdot {t}^{3}\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3} \cdot \left(2 \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 61.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.15 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{{\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3} \cdot \left(2 \cdot \left(k \cdot k\right)\right)}\\ \mathbf{elif}\;k \leq 1.9 \cdot 10^{+91}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \frac{{t\_m}^{3}}{\ell \cdot \ell}\right) \cdot \left(2 \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot \left(t\_m \cdot {\sin k}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.15e-14)
    (/ 2.0 (* (pow (* t_m (pow (cbrt l) -2.0)) 3.0) (* 2.0 (* k k))))
    (if (<= k 1.9e+91)
      (/ 2.0 (* (* (sin k) (/ (pow t_m 3.0) (* l l))) (* 2.0 (tan k))))
      (* 2.0 (/ (pow l 2.0) (* (pow k 2.0) (* t_m (pow (sin k) 2.0)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.15e-14) {
		tmp = 2.0 / (pow((t_m * pow(cbrt(l), -2.0)), 3.0) * (2.0 * (k * k)));
	} else if (k <= 1.9e+91) {
		tmp = 2.0 / ((sin(k) * (pow(t_m, 3.0) / (l * l))) * (2.0 * tan(k)));
	} else {
		tmp = 2.0 * (pow(l, 2.0) / (pow(k, 2.0) * (t_m * pow(sin(k), 2.0))));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.15e-14) {
		tmp = 2.0 / (Math.pow((t_m * Math.pow(Math.cbrt(l), -2.0)), 3.0) * (2.0 * (k * k)));
	} else if (k <= 1.9e+91) {
		tmp = 2.0 / ((Math.sin(k) * (Math.pow(t_m, 3.0) / (l * l))) * (2.0 * Math.tan(k)));
	} else {
		tmp = 2.0 * (Math.pow(l, 2.0) / (Math.pow(k, 2.0) * (t_m * Math.pow(Math.sin(k), 2.0))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.15e-14)
		tmp = Float64(2.0 / Float64((Float64(t_m * (cbrt(l) ^ -2.0)) ^ 3.0) * Float64(2.0 * Float64(k * k))));
	elseif (k <= 1.9e+91)
		tmp = Float64(2.0 / Float64(Float64(sin(k) * Float64((t_m ^ 3.0) / Float64(l * l))) * Float64(2.0 * tan(k))));
	else
		tmp = Float64(2.0 * Float64((l ^ 2.0) / Float64((k ^ 2.0) * Float64(t_m * (sin(k) ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1.15e-14], N[(2.0 / N[(N[Power[N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.9e+91], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.14999999999999999e-14

   1. Initial program 53.7%

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

   3. Simplified 55.2%

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

   5. Taylor expanded in k around 0 54.9%

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

      1. unpow2 54.9%

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

   7. Applied egg-rr 54.9%

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

      1. associate-/r* 49.2%

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

      2. unpow3 49.2%

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

      3. times-frac 59.8%

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

      4. pow2 59.8%

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

   9. Applied egg-rr 59.8%

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

       1. frac-times 49.2%

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

       2. unpow2 49.2%

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

       3. unpow3 49.2%

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

       4. associate-/l/ 54.9%

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

       5. add-cube-cbrt 54.8%

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

       6. pow3 54.8%

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

       7. associate-/l/ 49.2%

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

       8. cbrt-div 49.2%

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

       9. unpow3 49.2%

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

       10. add-cbrt-cube 57.5%

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

       11. cbrt-unprod 63.9%

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

       12. unpow2 63.9%

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

       13. div-inv 64.0%

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

       14. pow-flip 64.0%

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

       15. metadata-eval 64.0%

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

   11. Applied egg-rr 64.0%

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

   if 1.14999999999999999e-14 < k < 1.8999999999999999e91

   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 58.6%

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

   5. Taylor expanded in k around 0 54.8%

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

   if 1.8999999999999999e91 < k

   1. Initial program 48.2%

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

   3. Simplified 48.2%

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

   5. Taylor expanded in t around 0 76.2%

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

   6. Taylor expanded in k around 0 63.9%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.15 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{{\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3} \cdot \left(2 \cdot \left(k \cdot k\right)\right)}\\ \mathbf{elif}\;k \leq 1.9 \cdot 10^{+91}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(2 \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 64.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 7.8 \cdot 10^{-8}:\\ \;\;\;\;\frac{2}{{\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3} \cdot \left(2 \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{k}^{2} \cdot \left(t\_m \cdot {\sin k}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 7.8e-8)
    (/ 2.0 (* (pow (* t_m (pow (cbrt l) -2.0)) 3.0) (* 2.0 (* k k))))
    (*
     2.0
     (/ (* (cos k) (* l l)) (* (pow k 2.0) (* t_m (pow (sin k) 2.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 7.8e-8) {
		tmp = 2.0 / (pow((t_m * pow(cbrt(l), -2.0)), 3.0) * (2.0 * (k * k)));
	} else {
		tmp = 2.0 * ((cos(k) * (l * l)) / (pow(k, 2.0) * (t_m * pow(sin(k), 2.0))));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 7.8e-8) {
		tmp = 2.0 / (Math.pow((t_m * Math.pow(Math.cbrt(l), -2.0)), 3.0) * (2.0 * (k * k)));
	} else {
		tmp = 2.0 * ((Math.cos(k) * (l * l)) / (Math.pow(k, 2.0) * (t_m * Math.pow(Math.sin(k), 2.0))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 7.8e-8)
		tmp = Float64(2.0 / Float64((Float64(t_m * (cbrt(l) ^ -2.0)) ^ 3.0) * Float64(2.0 * Float64(k * k))));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) * Float64(l * l)) / Float64((k ^ 2.0) * Float64(t_m * (sin(k) ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 7.8e-8], N[(2.0 / N[(N[Power[N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 7.7999999999999997e-8

   1. Initial program 54.0%

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

   3. Simplified 55.5%

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

   5. Taylor expanded in k around 0 55.1%

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

      1. unpow2 55.1%

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

   7. Applied egg-rr 55.1%

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

      1. associate-/r* 49.5%

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

      2. unpow3 49.5%

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

      3. times-frac 60.0%

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

      4. pow2 60.0%

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

   9. Applied egg-rr 60.0%

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

       1. frac-times 49.5%

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

       2. unpow2 49.5%

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

       3. unpow3 49.5%

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

       4. associate-/l/ 55.1%

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

       5. add-cube-cbrt 55.0%

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

       6. pow3 55.0%

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

       7. associate-/l/ 49.5%

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

       8. cbrt-div 49.5%

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

       9. unpow3 49.5%

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

       10. add-cbrt-cube 57.8%

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

       11. cbrt-unprod 64.1%

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

       12. unpow2 64.1%

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

       13. div-inv 64.2%

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

       14. pow-flip 64.2%

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

       15. metadata-eval 64.2%

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

   11. Applied egg-rr 64.2%

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

   if 7.7999999999999997e-8 < k

   1. Initial program 49.9%

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

   3. Simplified 50.0%

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

   5. Taylor expanded in t around 0 76.7%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. pow2 76.7%

         \[\leadsto 2 \cdot \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   7. Applied egg-rr 76.7%

      \[\leadsto 2 \cdot \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.8 \cdot 10^{-8}:\\ \;\;\;\;\frac{2}{{\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3} \cdot \left(2 \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 60.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 8.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{{\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3} \cdot \left(2 \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{t\_m \cdot {k}^{4}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 8.8e-5)
    (/ 2.0 (* (pow (* t_m (pow (cbrt l) -2.0)) 3.0) (* 2.0 (* k k))))
    (* 2.0 (/ (* (pow l 2.0) (cos k)) (* t_m (pow k 4.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 8.8e-5) {
		tmp = 2.0 / (pow((t_m * pow(cbrt(l), -2.0)), 3.0) * (2.0 * (k * k)));
	} else {
		tmp = 2.0 * ((pow(l, 2.0) * cos(k)) / (t_m * pow(k, 4.0)));
	}
	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 <= 8.8e-5) {
		tmp = 2.0 / (Math.pow((t_m * Math.pow(Math.cbrt(l), -2.0)), 3.0) * (2.0 * (k * k)));
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) * Math.cos(k)) / (t_m * Math.pow(k, 4.0)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 8.8e-5)
		tmp = Float64(2.0 / Float64((Float64(t_m * (cbrt(l) ^ -2.0)) ^ 3.0) * Float64(2.0 * Float64(k * k))));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k)) / Float64(t_m * (k ^ 4.0))));
	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, 8.8e-5], N[(2.0 / N[(N[Power[N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 8.7999999999999998e-5

   1. Initial program 54.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 55.5%

      \[\leadsto \color{blue}{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 55.1%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(2 \cdot {k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. unpow2 55.1%

         \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   7. Applied egg-rr 55.1%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]
   8. Step-by-step derivation

      1. associate-/r* 49.5%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

      2. unpow3 49.5%

         \[\leadsto \frac{2}{\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

      3. times-frac 60.0%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

      4. pow2 60.0%

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{2}}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

   9. Applied egg-rr 60.0%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]
   10. Step-by-step derivation

       1. frac-times 49.5%

          \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{2} \cdot t}{\ell \cdot \ell}} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

       2. unpow2 49.5%

          \[\leadsto \frac{2}{\frac{\color{blue}{\left(t \cdot t\right)} \cdot t}{\ell \cdot \ell} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

       3. unpow3 49.5%

          \[\leadsto \frac{2}{\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

       4. associate-/l/ 55.1%

          \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

       5. add-cube-cbrt 55.0%

          \[\leadsto \frac{2}{\color{blue}{\left(\left(\sqrt[3]{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sqrt[3]{\frac{\frac{{t}^{3}}{\ell}}{\ell}}\right) \cdot \sqrt[3]{\frac{\frac{{t}^{3}}{\ell}}{\ell}}\right)} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

       6. pow3 55.0%

          \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt[3]{\frac{\frac{{t}^{3}}{\ell}}{\ell}}\right)}^{3}} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

       7. associate-/l/ 49.5%

          \[\leadsto \frac{2}{{\left(\sqrt[3]{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}}\right)}^{3} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

       8. cbrt-div 49.5%

          \[\leadsto \frac{2}{{\color{blue}{\left(\frac{\sqrt[3]{{t}^{3}}}{\sqrt[3]{\ell \cdot \ell}}\right)}}^{3} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

       9. unpow3 49.5%

          \[\leadsto \frac{2}{{\left(\frac{\sqrt[3]{\color{blue}{\left(t \cdot t\right) \cdot t}}}{\sqrt[3]{\ell \cdot \ell}}\right)}^{3} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

       10. add-cbrt-cube 57.8%

           \[\leadsto \frac{2}{{\left(\frac{\color{blue}{t}}{\sqrt[3]{\ell \cdot \ell}}\right)}^{3} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

       11. cbrt-unprod 64.1%

           \[\leadsto \frac{2}{{\left(\frac{t}{\color{blue}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\right)}^{3} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

       12. unpow2 64.1%

           \[\leadsto \frac{2}{{\left(\frac{t}{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\right)}^{3} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

       13. div-inv 64.2%

           \[\leadsto \frac{2}{{\color{blue}{\left(t \cdot \frac{1}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}}^{3} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

       14. pow-flip 64.2%

           \[\leadsto \frac{2}{{\left(t \cdot \color{blue}{{\left(\sqrt[3]{\ell}\right)}^{\left(-2\right)}}\right)}^{3} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

       15. metadata-eval 64.2%

           \[\leadsto \frac{2}{{\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{\color{blue}{-2}}\right)}^{3} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

   11. Applied egg-rr 64.2%

       \[\leadsto \frac{2}{\color{blue}{{\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

   if 8.7999999999999998e-5 < k

   1. Initial program 49.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 50.0%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 76.7%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   6. Taylor expanded in k around 0 56.0%

      \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{4} \cdot t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{{\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3} \cdot \left(2 \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{t \cdot {k}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 59.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 51000:\\ \;\;\;\;\frac{2}{\frac{{t\_m}^{2}}{\ell} \cdot \left(\left(2 \cdot {k}^{2}\right) \cdot \frac{t\_m}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{t\_m \cdot {k}^{4}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 51000.0)
    (/ 2.0 (* (/ (pow t_m 2.0) l) (* (* 2.0 (pow k 2.0)) (/ t_m l))))
    (* 2.0 (/ (* (pow l 2.0) (cos k)) (* t_m (pow k 4.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 51000.0) {
		tmp = 2.0 / ((pow(t_m, 2.0) / l) * ((2.0 * pow(k, 2.0)) * (t_m / l)));
	} else {
		tmp = 2.0 * ((pow(l, 2.0) * cos(k)) / (t_m * pow(k, 4.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 51000.0d0) then
        tmp = 2.0d0 / (((t_m ** 2.0d0) / l) * ((2.0d0 * (k ** 2.0d0)) * (t_m / l)))
    else
        tmp = 2.0d0 * (((l ** 2.0d0) * cos(k)) / (t_m * (k ** 4.0d0)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 51000.0) {
		tmp = 2.0 / ((Math.pow(t_m, 2.0) / l) * ((2.0 * Math.pow(k, 2.0)) * (t_m / l)));
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) * Math.cos(k)) / (t_m * Math.pow(k, 4.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 51000.0:
		tmp = 2.0 / ((math.pow(t_m, 2.0) / l) * ((2.0 * math.pow(k, 2.0)) * (t_m / l)))
	else:
		tmp = 2.0 * ((math.pow(l, 2.0) * math.cos(k)) / (t_m * math.pow(k, 4.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 51000.0)
		tmp = Float64(2.0 / Float64(Float64((t_m ^ 2.0) / l) * Float64(Float64(2.0 * (k ^ 2.0)) * Float64(t_m / l))));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k)) / Float64(t_m * (k ^ 4.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 51000.0)
		tmp = 2.0 / (((t_m ^ 2.0) / l) * ((2.0 * (k ^ 2.0)) * (t_m / l)));
	else
		tmp = 2.0 * (((l ^ 2.0) * cos(k)) / (t_m * (k ^ 4.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 51000.0], N[(2.0 / N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 51000

   1. Initial program 54.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 54.0%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l* 48.9%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)\right)}} \]

      2. associate-/r* 55.4%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)\right)} \]

      3. associate-+r+ 55.4%

         \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}\right)\right)} \]

      4. metadata-eval 55.4%

         \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      5. associate-*l* 55.5%

         \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

      6. add-cube-cbrt 55.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\sqrt[3]{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \sqrt[3]{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right) \cdot \sqrt[3]{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}} \]

      7. pow3 55.4%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt[3]{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{3}}} \]

   6. Applied egg-rr 74.3%

      \[\leadsto \frac{2}{\color{blue}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{3}}} \]

   7. Taylor expanded in k around 0 67.5%

      \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\color{blue}{2 \cdot {k}^{2}}}\right)}^{3}} \]
   8. Step-by-step derivation

      1. unpow-prod-down 64.2%

         \[\leadsto \frac{2}{\color{blue}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot {\left(\sqrt[3]{2 \cdot {k}^{2}}\right)}^{3}}} \]

      2. pow1/3 63.7%

         \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot {\color{blue}{\left({\left(2 \cdot {k}^{2}\right)}^{0.3333333333333333}\right)}}^{3}} \]

      3. pow2 63.7%

         \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot {\left({\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right)}^{0.3333333333333333}\right)}^{3}} \]

      4. pow-pow 64.1%

         \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \color{blue}{{\left(2 \cdot \left(k \cdot k\right)\right)}^{\left(0.3333333333333333 \cdot 3\right)}}} \]

      5. metadata-eval 64.1%

         \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot {\left(2 \cdot \left(k \cdot k\right)\right)}^{\color{blue}{1}}} \]

      6. pow1 64.1%

         \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \color{blue}{\left(2 \cdot \left(k \cdot k\right)\right)}} \]

   9. Applied egg-rr 61.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{2}}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(2 \cdot {k}^{2}\right)\right)}} \]

   if 51000 < k

   1. Initial program 49.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 50.0%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 76.7%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   6. Taylor expanded in k around 0 56.0%

      \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{4} \cdot t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 51000:\\ \;\;\;\;\frac{2}{\frac{{t}^{2}}{\ell} \cdot \left(\left(2 \cdot {k}^{2}\right) \cdot \frac{t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{t \cdot {k}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 58.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 7.8 \cdot 10^{+162}:\\ \;\;\;\;\frac{2}{\frac{{t\_m}^{2}}{\ell} \cdot \left(\left(2 \cdot {k}^{2}\right) \cdot \frac{t\_m}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot {\ell}^{2}}{t\_m \cdot {k}^{4}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 7.8e+162)
    (/ 2.0 (* (/ (pow t_m 2.0) l) (* (* 2.0 (pow k 2.0)) (/ t_m l))))
    (/ (* 2.0 (pow l 2.0)) (* t_m (pow k 4.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 7.8e+162) {
		tmp = 2.0 / ((pow(t_m, 2.0) / l) * ((2.0 * pow(k, 2.0)) * (t_m / l)));
	} else {
		tmp = (2.0 * pow(l, 2.0)) / (t_m * pow(k, 4.0));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 7.8d+162) then
        tmp = 2.0d0 / (((t_m ** 2.0d0) / l) * ((2.0d0 * (k ** 2.0d0)) * (t_m / l)))
    else
        tmp = (2.0d0 * (l ** 2.0d0)) / (t_m * (k ** 4.0d0))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 7.8e+162) {
		tmp = 2.0 / ((Math.pow(t_m, 2.0) / l) * ((2.0 * Math.pow(k, 2.0)) * (t_m / l)));
	} else {
		tmp = (2.0 * Math.pow(l, 2.0)) / (t_m * Math.pow(k, 4.0));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 7.8e+162:
		tmp = 2.0 / ((math.pow(t_m, 2.0) / l) * ((2.0 * math.pow(k, 2.0)) * (t_m / l)))
	else:
		tmp = (2.0 * math.pow(l, 2.0)) / (t_m * math.pow(k, 4.0))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 7.8e+162)
		tmp = Float64(2.0 / Float64(Float64((t_m ^ 2.0) / l) * Float64(Float64(2.0 * (k ^ 2.0)) * Float64(t_m / l))));
	else
		tmp = Float64(Float64(2.0 * (l ^ 2.0)) / Float64(t_m * (k ^ 4.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 7.8e+162)
		tmp = 2.0 / (((t_m ^ 2.0) / l) * ((2.0 * (k ^ 2.0)) * (t_m / l)));
	else
		tmp = (2.0 * (l ^ 2.0)) / (t_m * (k ^ 4.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 7.8e+162], N[(2.0 / N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 7.80000000000000079e162

   1. Initial program 53.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 53.8%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l* 49.6%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)\right)}} \]

      2. associate-/r* 55.3%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)\right)} \]

      3. associate-+r+ 55.3%

         \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}\right)\right)} \]

      4. metadata-eval 55.3%

         \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      5. associate-*l* 55.3%

         \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

      6. add-cube-cbrt 55.2%

         \[\leadsto \frac{2}{\color{blue}{\left(\sqrt[3]{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \sqrt[3]{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right) \cdot \sqrt[3]{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}} \]

      7. pow3 55.2%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt[3]{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{3}}} \]

   6. Applied egg-rr 72.0%

      \[\leadsto \frac{2}{\color{blue}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{3}}} \]

   7. Taylor expanded in k around 0 64.2%

      \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\color{blue}{2 \cdot {k}^{2}}}\right)}^{3}} \]
   8. Step-by-step derivation

      1. unpow-prod-down 61.5%

         \[\leadsto \frac{2}{\color{blue}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot {\left(\sqrt[3]{2 \cdot {k}^{2}}\right)}^{3}}} \]

      2. pow1/3 61.1%

         \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot {\color{blue}{\left({\left(2 \cdot {k}^{2}\right)}^{0.3333333333333333}\right)}}^{3}} \]

      3. pow2 61.1%

         \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot {\left({\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right)}^{0.3333333333333333}\right)}^{3}} \]

      4. pow-pow 61.4%

         \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \color{blue}{{\left(2 \cdot \left(k \cdot k\right)\right)}^{\left(0.3333333333333333 \cdot 3\right)}}} \]

      5. metadata-eval 61.4%

         \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot {\left(2 \cdot \left(k \cdot k\right)\right)}^{\color{blue}{1}}} \]

      6. pow1 61.4%

         \[\leadsto \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \color{blue}{\left(2 \cdot \left(k \cdot k\right)\right)}} \]

   9. Applied egg-rr 59.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{2}}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(2 \cdot {k}^{2}\right)\right)}} \]

   if 7.80000000000000079e162 < k

   1. Initial program 46.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 46.8%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 66.0%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   6. Taylor expanded in k around 0 66.0%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   7. Step-by-step derivation

      1. associate-*r/ 66.0%

         \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

      2. *-commutative 66.0%

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{t \cdot {k}^{4}}} \]

   8. Simplified 66.0%

      \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{t \cdot {k}^{4}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.8 \cdot 10^{+162}:\\ \;\;\;\;\frac{2}{\frac{{t}^{2}}{\ell} \cdot \left(\left(2 \cdot {k}^{2}\right) \cdot \frac{t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot {\ell}^{2}}{t \cdot {k}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 58.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 7.8 \cdot 10^{+162}:\\ \;\;\;\;\frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot {\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot {\ell}^{2}}{t\_m \cdot {k}^{4}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 7.8e+162)
    (/ 2.0 (* (* 2.0 (* k k)) (pow (/ (pow t_m 1.5) l) 2.0)))
    (/ (* 2.0 (pow l 2.0)) (* t_m (pow k 4.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 7.8e+162) {
		tmp = 2.0 / ((2.0 * (k * k)) * pow((pow(t_m, 1.5) / l), 2.0));
	} else {
		tmp = (2.0 * pow(l, 2.0)) / (t_m * pow(k, 4.0));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 7.8d+162) then
        tmp = 2.0d0 / ((2.0d0 * (k * k)) * (((t_m ** 1.5d0) / l) ** 2.0d0))
    else
        tmp = (2.0d0 * (l ** 2.0d0)) / (t_m * (k ** 4.0d0))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 7.8e+162) {
		tmp = 2.0 / ((2.0 * (k * k)) * Math.pow((Math.pow(t_m, 1.5) / l), 2.0));
	} else {
		tmp = (2.0 * Math.pow(l, 2.0)) / (t_m * Math.pow(k, 4.0));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 7.8e+162:
		tmp = 2.0 / ((2.0 * (k * k)) * math.pow((math.pow(t_m, 1.5) / l), 2.0))
	else:
		tmp = (2.0 * math.pow(l, 2.0)) / (t_m * math.pow(k, 4.0))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 7.8e+162)
		tmp = Float64(2.0 / Float64(Float64(2.0 * Float64(k * k)) * (Float64((t_m ^ 1.5) / l) ^ 2.0)));
	else
		tmp = Float64(Float64(2.0 * (l ^ 2.0)) / Float64(t_m * (k ^ 4.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 7.8e+162)
		tmp = 2.0 / ((2.0 * (k * k)) * (((t_m ^ 1.5) / l) ^ 2.0));
	else
		tmp = (2.0 * (l ^ 2.0)) / (t_m * (k ^ 4.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 7.8e+162], N[(2.0 / N[(N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 7.80000000000000079e162

   1. Initial program 53.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 55.3%

      \[\leadsto \color{blue}{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 53.9%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(2 \cdot {k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. unpow2 53.9%

         \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   7. Applied egg-rr 53.9%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 31.9%

         \[\leadsto \frac{2}{\color{blue}{\left(\sqrt{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sqrt{\frac{\frac{{t}^{3}}{\ell}}{\ell}}\right)} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

      2. pow2 31.9%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{\frac{\frac{{t}^{3}}{\ell}}{\ell}}\right)}^{2}} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

      3. associate-/r* 27.6%

         \[\leadsto \frac{2}{{\left(\sqrt{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}}\right)}^{2} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

      4. sqrt-div 26.7%

         \[\leadsto \frac{2}{{\color{blue}{\left(\frac{\sqrt{{t}^{3}}}{\sqrt{\ell \cdot \ell}}\right)}}^{2} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

      5. sqrt-pow1 30.2%

         \[\leadsto \frac{2}{{\left(\frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

      6. metadata-eval 30.2%

         \[\leadsto \frac{2}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

      7. sqrt-prod 14.1%

         \[\leadsto \frac{2}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

      8. add-sqr-sqrt 34.5%

         \[\leadsto \frac{2}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

   9. Applied egg-rr 34.5%

      \[\leadsto \frac{2}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

   if 7.80000000000000079e162 < k

   1. Initial program 46.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 46.8%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 66.0%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   6. Taylor expanded in k around 0 66.0%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   7. Step-by-step derivation

      1. associate-*r/ 66.0%

         \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

      2. *-commutative 66.0%

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{t \cdot {k}^{4}}} \]

   8. Simplified 66.0%

      \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{t \cdot {k}^{4}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.8 \cdot 10^{+162}:\\ \;\;\;\;\frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot {\ell}^{2}}{t \cdot {k}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 57.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 8 \cdot 10^{+162}:\\ \;\;\;\;\frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{{t\_m}^{2}}{\ell} \cdot \frac{1}{\frac{\ell}{t\_m}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot {\ell}^{2}}{t\_m \cdot {k}^{4}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 8e+162)
    (/ 2.0 (* (* 2.0 (* k k)) (* (/ (pow t_m 2.0) l) (/ 1.0 (/ l t_m)))))
    (/ (* 2.0 (pow l 2.0)) (* t_m (pow k 4.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 8e+162) {
		tmp = 2.0 / ((2.0 * (k * k)) * ((pow(t_m, 2.0) / l) * (1.0 / (l / t_m))));
	} else {
		tmp = (2.0 * pow(l, 2.0)) / (t_m * pow(k, 4.0));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 8d+162) then
        tmp = 2.0d0 / ((2.0d0 * (k * k)) * (((t_m ** 2.0d0) / l) * (1.0d0 / (l / t_m))))
    else
        tmp = (2.0d0 * (l ** 2.0d0)) / (t_m * (k ** 4.0d0))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 8e+162) {
		tmp = 2.0 / ((2.0 * (k * k)) * ((Math.pow(t_m, 2.0) / l) * (1.0 / (l / t_m))));
	} else {
		tmp = (2.0 * Math.pow(l, 2.0)) / (t_m * Math.pow(k, 4.0));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 8e+162:
		tmp = 2.0 / ((2.0 * (k * k)) * ((math.pow(t_m, 2.0) / l) * (1.0 / (l / t_m))))
	else:
		tmp = (2.0 * math.pow(l, 2.0)) / (t_m * math.pow(k, 4.0))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 8e+162)
		tmp = Float64(2.0 / Float64(Float64(2.0 * Float64(k * k)) * Float64(Float64((t_m ^ 2.0) / l) * Float64(1.0 / Float64(l / t_m)))));
	else
		tmp = Float64(Float64(2.0 * (l ^ 2.0)) / Float64(t_m * (k ^ 4.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 8e+162)
		tmp = 2.0 / ((2.0 * (k * k)) * (((t_m ^ 2.0) / l) * (1.0 / (l / t_m))));
	else
		tmp = (2.0 * (l ^ 2.0)) / (t_m * (k ^ 4.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 8e+162], N[(2.0 / N[(N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(1.0 / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 7.9999999999999995e162

   1. Initial program 53.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 55.3%

      \[\leadsto \color{blue}{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 53.9%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(2 \cdot {k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. unpow2 53.9%

         \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   7. Applied egg-rr 53.9%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]
   8. Step-by-step derivation

      1. associate-/r* 49.0%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

      2. unpow3 48.9%

         \[\leadsto \frac{2}{\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

      3. times-frac 58.0%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

      4. pow2 58.0%

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{2}}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

   9. Applied egg-rr 58.0%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]
   10. Step-by-step derivation

       1. clear-num 58.0%

          \[\leadsto \frac{2}{\left(\frac{{t}^{2}}{\ell} \cdot \color{blue}{\frac{1}{\frac{\ell}{t}}}\right) \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

       2. inv-pow 58.0%

          \[\leadsto \frac{2}{\left(\frac{{t}^{2}}{\ell} \cdot \color{blue}{{\left(\frac{\ell}{t}\right)}^{-1}}\right) \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

   11. Applied egg-rr 58.0%

       \[\leadsto \frac{2}{\left(\frac{{t}^{2}}{\ell} \cdot \color{blue}{{\left(\frac{\ell}{t}\right)}^{-1}}\right) \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]
   12. Step-by-step derivation

       1. unpow-1 58.0%

          \[\leadsto \frac{2}{\left(\frac{{t}^{2}}{\ell} \cdot \color{blue}{\frac{1}{\frac{\ell}{t}}}\right) \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

   13. Simplified 58.0%

       \[\leadsto \frac{2}{\left(\frac{{t}^{2}}{\ell} \cdot \color{blue}{\frac{1}{\frac{\ell}{t}}}\right) \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

   if 7.9999999999999995e162 < k

   1. Initial program 46.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 46.8%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 66.0%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   6. Taylor expanded in k around 0 66.0%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   7. Step-by-step derivation

      1. associate-*r/ 66.0%

         \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

      2. *-commutative 66.0%

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{t \cdot {k}^{4}}} \]

   8. Simplified 66.0%

      \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{t \cdot {k}^{4}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8 \cdot 10^{+162}:\\ \;\;\;\;\frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{1}{\frac{\ell}{t}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot {\ell}^{2}}{t \cdot {k}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 56.5% accurate, 3.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{{t\_m}^{2}}{\ell} \cdot \frac{1}{\frac{\ell}{t\_m}}\right)} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ 2.0 (* (* 2.0 (* k k)) (* (/ (pow t_m 2.0) l) (/ 1.0 (/ l t_m)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / ((2.0 * (k * k)) * ((pow(t_m, 2.0) / l) * (1.0 / (l / t_m)))));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 / ((2.0d0 * (k * k)) * (((t_m ** 2.0d0) / l) * (1.0d0 / (l / t_m)))))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / ((2.0 * (k * k)) * ((Math.pow(t_m, 2.0) / l) * (1.0 / (l / t_m)))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 / ((2.0 * (k * k)) * ((math.pow(t_m, 2.0) / l) * (1.0 / (l / t_m)))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 / Float64(Float64(2.0 * Float64(k * k)) * Float64(Float64((t_m ^ 2.0) / l) * Float64(1.0 / Float64(l / t_m))))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 / ((2.0 * (k * k)) * (((t_m ^ 2.0) / l) * (1.0 / (l / t_m)))));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(2.0 / N[(N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(1.0 / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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 54.6%

   \[\leadsto \color{blue}{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in k around 0 53.5%

   \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(2 \cdot {k}^{2}\right)}} \]
6. Step-by-step derivation

   1. unpow2 53.5%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

7. Applied egg-rr 53.5%

   \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]
8. Step-by-step derivation

   1. associate-/r* 48.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

   2. unpow3 48.7%

      \[\leadsto \frac{2}{\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

   3. times-frac 57.8%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

   4. pow2 57.8%

      \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{2}}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

9. Applied egg-rr 57.8%

   \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]
10. Step-by-step derivation

    1. clear-num 57.8%

       \[\leadsto \frac{2}{\left(\frac{{t}^{2}}{\ell} \cdot \color{blue}{\frac{1}{\frac{\ell}{t}}}\right) \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

    2. inv-pow 57.8%

       \[\leadsto \frac{2}{\left(\frac{{t}^{2}}{\ell} \cdot \color{blue}{{\left(\frac{\ell}{t}\right)}^{-1}}\right) \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

11. Applied egg-rr 57.8%

    \[\leadsto \frac{2}{\left(\frac{{t}^{2}}{\ell} \cdot \color{blue}{{\left(\frac{\ell}{t}\right)}^{-1}}\right) \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]
12. Step-by-step derivation

    1. unpow-1 57.8%

       \[\leadsto \frac{2}{\left(\frac{{t}^{2}}{\ell} \cdot \color{blue}{\frac{1}{\frac{\ell}{t}}}\right) \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

13. Simplified 57.8%

    \[\leadsto \frac{2}{\left(\frac{{t}^{2}}{\ell} \cdot \color{blue}{\frac{1}{\frac{\ell}{t}}}\right) \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

14. Final simplification 57.8%

    \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{1}{\frac{\ell}{t}}\right)} \]
15. Add Preprocessing

Alternative 24: 56.5% accurate, 24.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{t\_m}{\ell} \cdot \frac{t\_m \cdot t\_m}{\ell}\right)} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ 2.0 (* (* 2.0 (* k k)) (* (/ t_m l) (/ (* t_m t_m) l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / ((2.0 * (k * k)) * ((t_m / l) * ((t_m * t_m) / l))));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 / ((2.0d0 * (k * k)) * ((t_m / l) * ((t_m * t_m) / l))))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / ((2.0 * (k * k)) * ((t_m / l) * ((t_m * t_m) / l))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 / ((2.0 * (k * k)) * ((t_m / l) * ((t_m * t_m) / l))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 / Float64(Float64(2.0 * Float64(k * k)) * Float64(Float64(t_m / l) * Float64(Float64(t_m * t_m) / l)))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 / ((2.0 * (k * k)) * ((t_m / l) * ((t_m * t_m) / l))));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(2.0 / N[(N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 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 54.6%

   \[\leadsto \color{blue}{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in k around 0 53.5%

   \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(2 \cdot {k}^{2}\right)}} \]
6. Step-by-step derivation

   1. unpow2 53.5%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

7. Applied egg-rr 53.5%

   \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]
8. Step-by-step derivation

   1. associate-/r* 48.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

   2. unpow3 48.7%

      \[\leadsto \frac{2}{\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

   3. times-frac 57.8%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

   4. pow2 57.8%

      \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{2}}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

9. Applied egg-rr 57.8%

   \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]
10. Step-by-step derivation

    1. unpow2 57.8%

       \[\leadsto \frac{2}{\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

11. Applied egg-rr 57.8%

    \[\leadsto \frac{2}{\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

12. Final simplification 57.8%

    \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right)} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024191 
(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))))