Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.8% → 87.0%

Time: 19.2s

Alternatives: 25

Speedup: 1.9×

• 28.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 87.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.65 \cdot 10^{-107}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{t\_m \cdot {k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(t\_m \cdot \left(\sqrt[3]{\sin k} \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right) \cdot \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.65e-107)
    (*
     2.0
     (* (/ (cos k) (* t_m (pow k 2.0))) (/ (pow l 2.0) (pow (sin k) 2.0))))
    (/
     2.0
     (pow
      (*
       (* t_m (* (cbrt (sin k)) (pow (cbrt l) -2.0)))
       (cbrt (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))))
      3.0)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.65e-107) {
		tmp = 2.0 * ((cos(k) / (t_m * pow(k, 2.0))) * (pow(l, 2.0) / pow(sin(k), 2.0)));
	} else {
		tmp = 2.0 / pow(((t_m * (cbrt(sin(k)) * pow(cbrt(l), -2.0))) * cbrt((tan(k) * (2.0 + pow((k / t_m), 2.0))))), 3.0);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.65e-107) {
		tmp = 2.0 * ((Math.cos(k) / (t_m * Math.pow(k, 2.0))) * (Math.pow(l, 2.0) / Math.pow(Math.sin(k), 2.0)));
	} else {
		tmp = 2.0 / Math.pow(((t_m * (Math.cbrt(Math.sin(k)) * Math.pow(Math.cbrt(l), -2.0))) * Math.cbrt((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0))))), 3.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.65e-107)
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(t_m * (k ^ 2.0))) * Float64((l ^ 2.0) / (sin(k) ^ 2.0))));
	else
		tmp = Float64(2.0 / (Float64(Float64(t_m * Float64(cbrt(sin(k)) * (cbrt(l) ^ -2.0))) * cbrt(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))))) ^ 3.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.65e-107], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(t$95$m * N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.65000000000000002e-107

   1. Initial program 53.8%

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

      1. associate-/r* 53.8%

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

      2. +-commutative 53.8%

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

      3. unpow2 53.8%

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

      4. sqr-neg 53.8%

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

      5. distribute-frac-neg2 53.8%

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

      6. distribute-frac-neg2 53.8%

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

      7. unpow2 53.8%

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

      8. +-commutative 53.8%

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

      9. associate-*l* 47.0%

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

      10. associate-*l/ 46.4%

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

      11. associate-/r/ 46.4%

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

      12. +-commutative 46.4%

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

      13. associate-+r+ 46.4%

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

   3. Simplified 46.4%

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

      1. unpow2 46.4%

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

   6. Applied egg-rr 46.4%

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

   7. Taylor expanded in t around 0 65.1%

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

      1. *-commutative 65.1%

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

      2. associate-*r* 65.1%

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

      3. times-frac 66.2%

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

   9. Simplified 66.2%

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

   if 1.65000000000000002e-107 < t

   1. Initial program 73.2%

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

   3. Simplified 73.2%

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

      1. add-cube-cbrt 73.2%

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

      2. pow3 73.2%

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

      3. *-commutative 73.2%

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

      4. cbrt-prod 73.0%

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

      5. cbrt-div 73.0%

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

      6. rem-cbrt-cube 79.0%

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

      7. cbrt-prod 88.6%

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

      8. pow2 88.6%

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

   6. Applied egg-rr 88.6%

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

      1. pow1 88.6%

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

      2. div-inv 88.5%

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

      3. pow-flip 88.6%

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

      4. metadata-eval 88.6%

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

   8. Applied egg-rr 88.6%

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

      1. unpow1 88.6%

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

      2. associate-*r* 88.6%

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

      3. *-commutative 88.6%

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

   10. Simplified 88.6%

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

       1. add-cube-cbrt 88.6%

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

       2. pow3 88.6%

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

   12. Applied egg-rr 95.9%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.65 \cdot 10^{-107}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{t \cdot {k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(t \cdot \left(\sqrt[3]{\sin k} \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right) \cdot \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 68.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\frac{k}{t\_m}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(1 + \left(t\_2 + 1\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{{t\_m}^{3}}{\ell \cdot \ell}\right)\right)} \leq 2 \cdot 10^{+257}:\\ \;\;\;\;\frac{2}{\frac{1}{\ell} \cdot \left(\left(\tan k \cdot \left(2 + t\_2\right)\right) \cdot \left(\sin k \cdot \frac{{t\_m}^{3}}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left({k}^{2} \cdot \frac{t\_m}{{\ell}^{2}}\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (/ k t_m) 2.0)))
   (*
    t_s
    (if (<=
         (/
          2.0
          (*
           (+ 1.0 (+ t_2 1.0))
           (* (tan k) (* (sin k) (/ (pow t_m 3.0) (* l l))))))
         2e+257)
      (/
       2.0
       (*
        (/ 1.0 l)
        (* (* (tan k) (+ 2.0 t_2)) (* (sin k) (/ (pow t_m 3.0) l)))))
      (/ 2.0 (* (pow k 2.0) (* (pow k 2.0) (/ t_m (pow l 2.0)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow((k / t_m), 2.0);
	double tmp;
	if ((2.0 / ((1.0 + (t_2 + 1.0)) * (tan(k) * (sin(k) * (pow(t_m, 3.0) / (l * l)))))) <= 2e+257) {
		tmp = 2.0 / ((1.0 / l) * ((tan(k) * (2.0 + t_2)) * (sin(k) * (pow(t_m, 3.0) / l))));
	} else {
		tmp = 2.0 / (pow(k, 2.0) * (pow(k, 2.0) * (t_m / pow(l, 2.0))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (k / t_m) ** 2.0d0
    if ((2.0d0 / ((1.0d0 + (t_2 + 1.0d0)) * (tan(k) * (sin(k) * ((t_m ** 3.0d0) / (l * l)))))) <= 2d+257) then
        tmp = 2.0d0 / ((1.0d0 / l) * ((tan(k) * (2.0d0 + t_2)) * (sin(k) * ((t_m ** 3.0d0) / l))))
    else
        tmp = 2.0d0 / ((k ** 2.0d0) * ((k ** 2.0d0) * (t_m / (l ** 2.0d0))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.pow((k / t_m), 2.0);
	double tmp;
	if ((2.0 / ((1.0 + (t_2 + 1.0)) * (Math.tan(k) * (Math.sin(k) * (Math.pow(t_m, 3.0) / (l * l)))))) <= 2e+257) {
		tmp = 2.0 / ((1.0 / l) * ((Math.tan(k) * (2.0 + t_2)) * (Math.sin(k) * (Math.pow(t_m, 3.0) / l))));
	} else {
		tmp = 2.0 / (Math.pow(k, 2.0) * (Math.pow(k, 2.0) * (t_m / Math.pow(l, 2.0))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = math.pow((k / t_m), 2.0)
	tmp = 0
	if (2.0 / ((1.0 + (t_2 + 1.0)) * (math.tan(k) * (math.sin(k) * (math.pow(t_m, 3.0) / (l * l)))))) <= 2e+257:
		tmp = 2.0 / ((1.0 / l) * ((math.tan(k) * (2.0 + t_2)) * (math.sin(k) * (math.pow(t_m, 3.0) / l))))
	else:
		tmp = 2.0 / (math.pow(k, 2.0) * (math.pow(k, 2.0) * (t_m / math.pow(l, 2.0))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(k / t_m) ^ 2.0
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(1.0 + Float64(t_2 + 1.0)) * Float64(tan(k) * Float64(sin(k) * Float64((t_m ^ 3.0) / Float64(l * l)))))) <= 2e+257)
		tmp = Float64(2.0 / Float64(Float64(1.0 / l) * Float64(Float64(tan(k) * Float64(2.0 + t_2)) * Float64(sin(k) * Float64((t_m ^ 3.0) / l)))));
	else
		tmp = Float64(2.0 / Float64((k ^ 2.0) * Float64((k ^ 2.0) * Float64(t_m / (l ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = (k / t_m) ^ 2.0;
	tmp = 0.0;
	if ((2.0 / ((1.0 + (t_2 + 1.0)) * (tan(k) * (sin(k) * ((t_m ^ 3.0) / (l * l)))))) <= 2e+257)
		tmp = 2.0 / ((1.0 / l) * ((tan(k) * (2.0 + t_2)) * (sin(k) * ((t_m ^ 3.0) / l))));
	else
		tmp = 2.0 / ((k ^ 2.0) * ((k ^ 2.0) * (t_m / (l ^ 2.0))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(2.0 / N[(N[(1.0 + N[(t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+257], N[(2.0 / N[(N[(1.0 / l), $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 2.00000000000000006e257

   1. Initial program 84.7%

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

   3. Simplified 84.7%

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

      1. associate-*l* 72.5%

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

      2. associate-/r* 75.0%

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

      3. associate-+r+ 75.0%

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

      4. metadata-eval 75.0%

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

      5. associate-*l* 75.0%

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

      6. associate-*l/ 75.5%

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

      7. clear-num 75.5%

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

      8. associate-*l* 75.4%

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

   6. Applied egg-rr 75.4%

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

      1. associate-/r/ 75.4%

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

      2. associate-*r* 88.5%

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

   8. Simplified 88.5%

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

   if 2.00000000000000006e257 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 23.6%

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

   3. Simplified 23.6%

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

      1. add-cube-cbrt 23.6%

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

      2. pow3 23.6%

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

      3. *-commutative 23.6%

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

      4. cbrt-prod 23.6%

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

      5. cbrt-div 23.6%

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

      6. rem-cbrt-cube 36.3%

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

      7. cbrt-prod 47.2%

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

      8. pow2 47.2%

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

   6. Applied egg-rr 47.2%

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

   7. Taylor expanded in k around inf 61.1%

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

      1. associate-/l* 59.5%

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

      2. *-commutative 59.5%

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

      3. *-commutative 59.5%

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

      4. times-frac 60.1%

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

   9. Simplified 60.1%

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

   10. Taylor expanded in k around 0 47.1%

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

       1. associate-/l* 47.5%

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

   12. Simplified 47.5%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right)} \leq 2 \cdot 10^{+257}:\\ \;\;\;\;\frac{2}{\frac{1}{\ell} \cdot \left(\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3 \cdot 10^{-115}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)} \cdot \left(t\_m \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{k}\right)\right)\right)}^{3}}\\ \mathbf{elif}\;k \leq 2.4 \cdot 10^{-26}:\\ \;\;\;\;{\left(\frac{\sqrt{2}}{\frac{{t\_m}^{1.5}}{\ell} \cdot \left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t\_m}\right)\right) \cdot \sqrt{\sin k \cdot \tan k}\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{t\_m \cdot {k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right)\\ \end{array} \end{array} \]

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 3e-115)
    (/
     2.0
     (pow
      (*
       (cbrt (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0))))
       (* t_m (* (pow (cbrt l) -2.0) (cbrt k))))
      3.0))
    (if (<= k 2.4e-26)
      (pow
       (/
        (sqrt 2.0)
        (*
         (/ (pow t_m 1.5) l)
         (* (hypot 1.0 (hypot 1.0 (/ k t_m))) (sqrt (* (sin k) (tan k))))))
       2.0)
      (*
       2.0
       (*
        (/ (cos k) (* t_m (pow k 2.0)))
        (/ (pow l 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 <= 3e-115) {
		tmp = 2.0 / pow((cbrt((tan(k) * (2.0 + pow((k / t_m), 2.0)))) * (t_m * (pow(cbrt(l), -2.0) * cbrt(k)))), 3.0);
	} else if (k <= 2.4e-26) {
		tmp = pow((sqrt(2.0) / ((pow(t_m, 1.5) / l) * (hypot(1.0, hypot(1.0, (k / t_m))) * sqrt((sin(k) * tan(k)))))), 2.0);
	} else {
		tmp = 2.0 * ((cos(k) / (t_m * pow(k, 2.0))) * (pow(l, 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 <= 3e-115) {
		tmp = 2.0 / Math.pow((Math.cbrt((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0)))) * (t_m * (Math.pow(Math.cbrt(l), -2.0) * Math.cbrt(k)))), 3.0);
	} else if (k <= 2.4e-26) {
		tmp = Math.pow((Math.sqrt(2.0) / ((Math.pow(t_m, 1.5) / l) * (Math.hypot(1.0, Math.hypot(1.0, (k / t_m))) * Math.sqrt((Math.sin(k) * Math.tan(k)))))), 2.0);
	} else {
		tmp = 2.0 * ((Math.cos(k) / (t_m * Math.pow(k, 2.0))) * (Math.pow(l, 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 <= 3e-115)
		tmp = Float64(2.0 / (Float64(cbrt(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0)))) * Float64(t_m * Float64((cbrt(l) ^ -2.0) * cbrt(k)))) ^ 3.0));
	elseif (k <= 2.4e-26)
		tmp = Float64(sqrt(2.0) / Float64(Float64((t_m ^ 1.5) / l) * Float64(hypot(1.0, hypot(1.0, Float64(k / t_m))) * sqrt(Float64(sin(k) * tan(k)))))) ^ 2.0;
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(t_m * (k ^ 2.0))) * Float64((l ^ 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, 3e-115], N[(2.0 / N[Power[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[(t$95$m * N[(N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * N[Power[k, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.4e-26], N[Power[N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision] * N[Sqrt[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 3.0000000000000002e-115

   1. Initial program 64.6%

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

   3. Simplified 64.6%

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

      1. add-cube-cbrt 64.6%

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

      2. pow3 64.6%

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

      3. *-commutative 64.6%

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

      4. cbrt-prod 64.5%

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

      5. cbrt-div 64.5%

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

      6. rem-cbrt-cube 71.2%

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

      7. cbrt-prod 79.1%

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

      8. pow2 79.1%

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

   6. Applied egg-rr 79.1%

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

      1. pow1 79.1%

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

      2. div-inv 79.1%

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

      3. pow-flip 79.1%

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

      4. metadata-eval 79.1%

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

   8. Applied egg-rr 79.1%

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

      1. unpow1 79.1%

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

      2. associate-*r* 79.2%

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

      3. *-commutative 79.2%

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

   10. Simplified 79.2%

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

       1. add-cube-cbrt 79.2%

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

       2. pow3 79.2%

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

   12. Applied egg-rr 87.6%

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

   13. Taylor expanded in k around 0 82.6%

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

   if 3.0000000000000002e-115 < k < 2.4000000000000001e-26

   1. Initial program 56.5%

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

   3. Simplified 56.5%

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

   6. Applied egg-rr 44.2%

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

      1. unpow2 44.2%

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

   8. Simplified 44.2%

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

   if 2.4000000000000001e-26 < k

   1. Initial program 51.6%

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

      1. associate-/r* 51.6%

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

      2. +-commutative 51.6%

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

      3. unpow2 51.6%

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

      4. sqr-neg 51.6%

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

      5. distribute-frac-neg2 51.6%

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

      6. distribute-frac-neg2 51.6%

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

      7. unpow2 51.6%

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

      8. +-commutative 51.6%

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

      9. associate-*l* 51.6%

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

      10. associate-*l/ 51.6%

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

      11. associate-/r/ 51.6%

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

      12. +-commutative 51.6%

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

      13. associate-+r+ 51.6%

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

   3. Simplified 51.6%

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

      1. unpow2 51.6%

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

   6. Applied egg-rr 51.6%

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

   7. Taylor expanded in t around 0 70.9%

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

      1. *-commutative 70.9%

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

      2. associate-*r* 70.9%

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

      3. times-frac 72.1%

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

   9. Simplified 72.1%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3 \cdot 10^{-115}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{k}\right)\right)\right)}^{3}}\\ \mathbf{elif}\;k \leq 2.4 \cdot 10^{-26}:\\ \;\;\;\;{\left(\frac{\sqrt{2}}{\frac{{t}^{1.5}}{\ell} \cdot \left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right) \cdot \sqrt{\sin k \cdot \tan k}\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{t \cdot {k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.6 \cdot 10^{-107}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{t\_m \cdot {k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t\_m \cdot \left(\left(\sqrt[3]{\sin k} \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)}\right)\right)}^{3}}\\ \end{array} \end{array} \]

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 (<= t_m 3.6e-107)
    (*
     2.0
     (* (/ (cos k) (* t_m (pow k 2.0))) (/ (pow l 2.0) (pow (sin k) 2.0))))
    (/
     2.0
     (pow
      (*
       t_m
       (*
        (* (cbrt (sin k)) (pow (cbrt l) -2.0))
        (cbrt (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0))))))
      3.0)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.6e-107) {
		tmp = 2.0 * ((cos(k) / (t_m * pow(k, 2.0))) * (pow(l, 2.0) / pow(sin(k), 2.0)));
	} else {
		tmp = 2.0 / pow((t_m * ((cbrt(sin(k)) * pow(cbrt(l), -2.0)) * cbrt((tan(k) * (2.0 + pow((k / t_m), 2.0)))))), 3.0);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.6e-107) {
		tmp = 2.0 * ((Math.cos(k) / (t_m * Math.pow(k, 2.0))) * (Math.pow(l, 2.0) / Math.pow(Math.sin(k), 2.0)));
	} else {
		tmp = 2.0 / Math.pow((t_m * ((Math.cbrt(Math.sin(k)) * Math.pow(Math.cbrt(l), -2.0)) * Math.cbrt((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0)))))), 3.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 3.6e-107)
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(t_m * (k ^ 2.0))) * Float64((l ^ 2.0) / (sin(k) ^ 2.0))));
	else
		tmp = Float64(2.0 / (Float64(t_m * Float64(Float64(cbrt(sin(k)) * (cbrt(l) ^ -2.0)) * cbrt(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0)))))) ^ 3.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 3.6e-107], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(t$95$m * N[(N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.59999999999999976e-107

   1. Initial program 53.8%

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

      1. associate-/r* 53.8%

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

      2. +-commutative 53.8%

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

      3. unpow2 53.8%

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

      4. sqr-neg 53.8%

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

      5. distribute-frac-neg2 53.8%

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

      6. distribute-frac-neg2 53.8%

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

      7. unpow2 53.8%

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

      8. +-commutative 53.8%

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

      9. associate-*l* 47.0%

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

      10. associate-*l/ 46.4%

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

      11. associate-/r/ 46.4%

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

      12. +-commutative 46.4%

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

      13. associate-+r+ 46.4%

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

   3. Simplified 46.4%

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

      1. unpow2 46.4%

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

   6. Applied egg-rr 46.4%

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

   7. Taylor expanded in t around 0 65.1%

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

      1. *-commutative 65.1%

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

      2. associate-*r* 65.1%

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

      3. times-frac 66.2%

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

   9. Simplified 66.2%

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

   if 3.59999999999999976e-107 < t

   1. Initial program 73.2%

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

   3. Simplified 73.2%

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

      1. add-cube-cbrt 73.2%

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

      2. pow3 73.2%

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

      3. *-commutative 73.2%

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

      4. cbrt-prod 73.0%

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

      5. cbrt-div 73.0%

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

      6. rem-cbrt-cube 79.0%

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

      7. cbrt-prod 88.6%

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

      8. pow2 88.6%

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

   6. Applied egg-rr 88.6%

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

      1. pow1 88.6%

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

      2. div-inv 88.5%

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

      3. pow-flip 88.6%

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

      4. metadata-eval 88.6%

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

   8. Applied egg-rr 88.6%

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

      1. unpow1 88.6%

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

      2. associate-*r* 88.6%

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

      3. *-commutative 88.6%

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

   10. Simplified 88.6%

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

       1. add-cube-cbrt 88.6%

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

       2. pow3 88.6%

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

   12. Applied egg-rr 95.9%

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

       1. associate-*l* 95.3%

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

   14. Simplified 95.3%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.6 \cdot 10^{-107}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{t \cdot {k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t \cdot \left(\left(\sqrt[3]{\sin k} \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t\_m}\right)\right)}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.35 \cdot 10^{-102}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{t\_m \cdot {k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right)\\ \mathbf{elif}\;t\_m \leq 1.2 \cdot 10^{+26}:\\ \;\;\;\;t\_2 \cdot \left(\frac{2}{\left(\sin k \cdot \tan k\right) \cdot {t\_m}^{3}} \cdot t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(t\_m \cdot \left(\sqrt[3]{\sin k} \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right) \cdot \left(\sqrt[3]{k} \cdot \sqrt[3]{2}\right)\right)}^{3}}\\ \end{array} \end{array} \end{array} \]

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 (/ l (hypot 1.0 (hypot 1.0 (/ k t_m))))))
   (*
    t_s
    (if (<= t_m 1.35e-102)
      (*
       2.0
       (* (/ (cos k) (* t_m (pow k 2.0))) (/ (pow l 2.0) (pow (sin k) 2.0))))
      (if (<= t_m 1.2e+26)
        (* t_2 (* (/ 2.0 (* (* (sin k) (tan k)) (pow t_m 3.0))) t_2))
        (/
         2.0
         (pow
          (*
           (* t_m (* (cbrt (sin k)) (pow (cbrt l) -2.0)))
           (* (cbrt k) (cbrt 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 = l / hypot(1.0, hypot(1.0, (k / t_m)));
	double tmp;
	if (t_m <= 1.35e-102) {
		tmp = 2.0 * ((cos(k) / (t_m * pow(k, 2.0))) * (pow(l, 2.0) / pow(sin(k), 2.0)));
	} else if (t_m <= 1.2e+26) {
		tmp = t_2 * ((2.0 / ((sin(k) * tan(k)) * pow(t_m, 3.0))) * t_2);
	} else {
		tmp = 2.0 / pow(((t_m * (cbrt(sin(k)) * pow(cbrt(l), -2.0))) * (cbrt(k) * cbrt(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 = l / Math.hypot(1.0, Math.hypot(1.0, (k / t_m)));
	double tmp;
	if (t_m <= 1.35e-102) {
		tmp = 2.0 * ((Math.cos(k) / (t_m * Math.pow(k, 2.0))) * (Math.pow(l, 2.0) / Math.pow(Math.sin(k), 2.0)));
	} else if (t_m <= 1.2e+26) {
		tmp = t_2 * ((2.0 / ((Math.sin(k) * Math.tan(k)) * Math.pow(t_m, 3.0))) * t_2);
	} else {
		tmp = 2.0 / Math.pow(((t_m * (Math.cbrt(Math.sin(k)) * Math.pow(Math.cbrt(l), -2.0))) * (Math.cbrt(k) * Math.cbrt(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(l / hypot(1.0, hypot(1.0, Float64(k / t_m))))
	tmp = 0.0
	if (t_m <= 1.35e-102)
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(t_m * (k ^ 2.0))) * Float64((l ^ 2.0) / (sin(k) ^ 2.0))));
	elseif (t_m <= 1.2e+26)
		tmp = Float64(t_2 * Float64(Float64(2.0 / Float64(Float64(sin(k) * tan(k)) * (t_m ^ 3.0))) * t_2));
	else
		tmp = Float64(2.0 / (Float64(Float64(t_m * Float64(cbrt(sin(k)) * (cbrt(l) ^ -2.0))) * Float64(cbrt(k) * cbrt(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[(l / N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.35e-102], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.2e+26], N[(t$95$2 * N[(N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(t$95$m * N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[k, 1/3], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.35e-102

   1. Initial program 54.6%

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

      1. associate-/r* 54.6%

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

      2. +-commutative 54.6%

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

      3. unpow2 54.6%

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

      4. sqr-neg 54.6%

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

      5. distribute-frac-neg2 54.6%

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

      6. distribute-frac-neg2 54.6%

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

      7. unpow2 54.6%

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

      8. +-commutative 54.6%

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

      9. associate-*l* 47.9%

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

      10. associate-*l/ 47.4%

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

      11. associate-/r/ 46.2%

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

      12. +-commutative 46.2%

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

      13. associate-+r+ 46.2%

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

   3. Simplified 46.2%

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

      1. unpow2 46.2%

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

   6. Applied egg-rr 46.2%

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

   7. Taylor expanded in t around 0 65.7%

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

      1. *-commutative 65.7%

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

      2. associate-*r* 65.7%

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

      3. times-frac 66.8%

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

   9. Simplified 66.8%

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

   if 1.35e-102 < t < 1.20000000000000002e26

   1. Initial program 84.5%

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

      1. associate-/r* 84.6%

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

      2. +-commutative 84.6%

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

      3. unpow2 84.6%

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

      4. sqr-neg 84.6%

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

      5. distribute-frac-neg2 84.6%

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

      6. distribute-frac-neg2 84.6%

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

      7. unpow2 84.6%

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

      8. +-commutative 84.6%

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

      9. associate-*l* 84.6%

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

      10. associate-*l/ 84.6%

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

      11. associate-/r/ 84.6%

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

      12. +-commutative 84.6%

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

      13. associate-+r+ 84.6%

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

   3. Simplified 84.6%

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

      1. associate-*r* 88.3%

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

      2. add-sqr-sqrt 88.4%

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

      3. times-frac 88.6%

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

   6. Applied egg-rr 92.4%

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

      1. associate-/l* 96.1%

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

      2. associate-*l* 96.1%

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

   8. Simplified 96.1%

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

   if 1.20000000000000002e26 < t

   1. Initial program 66.7%

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

   3. Simplified 66.8%

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

      1. add-cube-cbrt 66.7%

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

      2. pow3 66.7%

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

      3. *-commutative 66.7%

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

      4. cbrt-prod 66.7%

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

      5. cbrt-div 66.7%

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

      6. rem-cbrt-cube 75.5%

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

      7. cbrt-prod 86.8%

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

      8. pow2 86.8%

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

   6. Applied egg-rr 86.8%

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

      1. pow1 86.8%

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

      2. div-inv 86.8%

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

      3. pow-flip 86.8%

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

      4. metadata-eval 86.8%

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

   8. Applied egg-rr 86.8%

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

      1. unpow1 86.8%

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

      2. associate-*r* 86.9%

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

      3. *-commutative 86.9%

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

   10. Simplified 86.9%

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

       1. add-cube-cbrt 86.9%

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

       2. pow3 86.9%

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

   12. Applied egg-rr 97.8%

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

   13. Taylor expanded in k around 0 88.4%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.35 \cdot 10^{-102}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{t \cdot {k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right)\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{+26}:\\ \;\;\;\;\frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \left(\frac{2}{\left(\sin k \cdot \tan k\right) \cdot {t}^{3}} \cdot \frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(t \cdot \left(\sqrt[3]{\sin k} \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right) \cdot \left(\sqrt[3]{k} \cdot \sqrt[3]{2}\right)\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.7 \cdot 10^{-105}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{t\_m \cdot {k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(t\_m \cdot \sqrt[3]{\sin k}\right)\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t\_m}\right)}^{2} + 1\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.7e-105)
    (*
     2.0
     (* (/ (cos k) (* t_m (pow k 2.0))) (/ (pow l 2.0) (pow (sin k) 2.0))))
    (/
     2.0
     (*
      (pow (* (pow (cbrt l) -2.0) (* t_m (cbrt (sin k)))) 3.0)
      (* (tan k) (+ 1.0 (+ (pow (/ k t_m) 2.0) 1.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.7e-105) {
		tmp = 2.0 * ((cos(k) / (t_m * pow(k, 2.0))) * (pow(l, 2.0) / pow(sin(k), 2.0)));
	} else {
		tmp = 2.0 / (pow((pow(cbrt(l), -2.0) * (t_m * cbrt(sin(k)))), 3.0) * (tan(k) * (1.0 + (pow((k / t_m), 2.0) + 1.0))));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.7e-105) {
		tmp = 2.0 * ((Math.cos(k) / (t_m * Math.pow(k, 2.0))) * (Math.pow(l, 2.0) / Math.pow(Math.sin(k), 2.0)));
	} else {
		tmp = 2.0 / (Math.pow((Math.pow(Math.cbrt(l), -2.0) * (t_m * Math.cbrt(Math.sin(k)))), 3.0) * (Math.tan(k) * (1.0 + (Math.pow((k / t_m), 2.0) + 1.0))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 3.7e-105)
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(t_m * (k ^ 2.0))) * Float64((l ^ 2.0) / (sin(k) ^ 2.0))));
	else
		tmp = Float64(2.0 / Float64((Float64((cbrt(l) ^ -2.0) * Float64(t_m * cbrt(sin(k)))) ^ 3.0) * Float64(tan(k) * Float64(1.0 + Float64((Float64(k / t_m) ^ 2.0) + 1.0)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 3.7e-105], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.70000000000000008e-105

   1. Initial program 54.3%

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

      1. associate-/r* 54.3%

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

      2. +-commutative 54.3%

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

      3. unpow2 54.3%

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

      4. sqr-neg 54.3%

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

      5. distribute-frac-neg2 54.3%

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

      6. distribute-frac-neg2 54.3%

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

      7. unpow2 54.3%

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

      8. +-commutative 54.3%

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

      9. associate-*l* 47.6%

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

      10. associate-*l/ 47.1%

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

      11. associate-/r/ 46.5%

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

      12. +-commutative 46.5%

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

      13. associate-+r+ 46.5%

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

   3. Simplified 46.5%

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

      1. unpow2 46.5%

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

   6. Applied egg-rr 46.5%

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

   7. Taylor expanded in t around 0 65.5%

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

      1. *-commutative 65.5%

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

      2. associate-*r* 65.5%

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

      3. times-frac 66.6%

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

   9. Simplified 66.6%

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

   if 3.70000000000000008e-105 < t

   1. Initial program 72.6%

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

   3. Simplified 72.6%

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

      1. add-cube-cbrt 72.5%

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

      2. pow3 72.6%

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

      3. *-commutative 72.6%

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

      4. cbrt-prod 72.4%

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

      5. cbrt-div 72.4%

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

      6. rem-cbrt-cube 78.5%

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

      7. cbrt-prod 88.3%

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

      8. pow2 88.3%

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

   6. Applied egg-rr 88.3%

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

      1. pow1 88.3%

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

      2. div-inv 88.2%

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

      3. pow-flip 88.3%

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

      4. metadata-eval 88.3%

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

   8. Applied egg-rr 88.3%

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

      1. unpow1 88.3%

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

      2. associate-*r* 88.3%

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

      3. *-commutative 88.3%

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

   10. Simplified 88.3%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.7 \cdot 10^{-105}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{t \cdot {k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.6 \cdot 10^{-106}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{t\_m \cdot {k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right)\\ \mathbf{elif}\;t\_m \leq 3 \cdot 10^{+165}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right) \cdot \left(\tan k \cdot \left({\left(\frac{{t\_m}^{1.5}}{\sqrt{\ell}}\right)}^{2} \cdot \frac{\sin k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(t\_m \cdot \left(\sqrt[3]{\sin k} \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right) \cdot \left(\sqrt[3]{k} \cdot \sqrt[3]{2}\right)\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4.6e-106)
    (*
     2.0
     (* (/ (cos k) (* t_m (pow k 2.0))) (/ (pow l 2.0) (pow (sin k) 2.0))))
    (if (<= t_m 3e+165)
      (/
       2.0
       (*
        (+ 2.0 (pow (/ k t_m) 2.0))
        (* (tan k) (* (pow (/ (pow t_m 1.5) (sqrt l)) 2.0) (/ (sin k) l)))))
      (/
       2.0
       (pow
        (*
         (* t_m (* (cbrt (sin k)) (pow (cbrt l) -2.0)))
         (* (cbrt k) (cbrt 2.0)))
        3.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.6e-106) {
		tmp = 2.0 * ((cos(k) / (t_m * pow(k, 2.0))) * (pow(l, 2.0) / pow(sin(k), 2.0)));
	} else if (t_m <= 3e+165) {
		tmp = 2.0 / ((2.0 + pow((k / t_m), 2.0)) * (tan(k) * (pow((pow(t_m, 1.5) / sqrt(l)), 2.0) * (sin(k) / l))));
	} else {
		tmp = 2.0 / pow(((t_m * (cbrt(sin(k)) * pow(cbrt(l), -2.0))) * (cbrt(k) * cbrt(2.0))), 3.0);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.6e-106) {
		tmp = 2.0 * ((Math.cos(k) / (t_m * Math.pow(k, 2.0))) * (Math.pow(l, 2.0) / Math.pow(Math.sin(k), 2.0)));
	} else if (t_m <= 3e+165) {
		tmp = 2.0 / ((2.0 + Math.pow((k / t_m), 2.0)) * (Math.tan(k) * (Math.pow((Math.pow(t_m, 1.5) / Math.sqrt(l)), 2.0) * (Math.sin(k) / l))));
	} else {
		tmp = 2.0 / Math.pow(((t_m * (Math.cbrt(Math.sin(k)) * Math.pow(Math.cbrt(l), -2.0))) * (Math.cbrt(k) * Math.cbrt(2.0))), 3.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 4.6e-106)
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(t_m * (k ^ 2.0))) * Float64((l ^ 2.0) / (sin(k) ^ 2.0))));
	elseif (t_m <= 3e+165)
		tmp = Float64(2.0 / Float64(Float64(2.0 + (Float64(k / t_m) ^ 2.0)) * Float64(tan(k) * Float64((Float64((t_m ^ 1.5) / sqrt(l)) ^ 2.0) * Float64(sin(k) / l)))));
	else
		tmp = Float64(2.0 / (Float64(Float64(t_m * Float64(cbrt(sin(k)) * (cbrt(l) ^ -2.0))) * Float64(cbrt(k) * cbrt(2.0))) ^ 3.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 4.6e-106], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3e+165], N[(2.0 / N[(N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(t$95$m * N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[k, 1/3], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 4.6000000000000002e-106

   1. Initial program 54.3%

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

      1. associate-/r* 54.3%

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

      2. +-commutative 54.3%

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

      3. unpow2 54.3%

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

      4. sqr-neg 54.3%

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

      5. distribute-frac-neg2 54.3%

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

      6. distribute-frac-neg2 54.3%

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

      7. unpow2 54.3%

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

      8. +-commutative 54.3%

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

      9. associate-*l* 47.6%

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

      10. associate-*l/ 47.1%

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

      11. associate-/r/ 46.5%

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

      12. +-commutative 46.5%

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

      13. associate-+r+ 46.5%

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

   3. Simplified 46.5%

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

      1. unpow2 46.5%

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

   6. Applied egg-rr 46.5%

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

   7. Taylor expanded in t around 0 65.5%

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

      1. *-commutative 65.5%

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

      2. associate-*r* 65.5%

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

      3. times-frac 66.6%

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

   9. Simplified 66.6%

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

   if 4.6000000000000002e-106 < t < 2.9999999999999999e165

   1. Initial program 77.7%

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

   3. Simplified 77.8%

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

      1. associate-/r* 82.1%

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

      2. add-sqr-sqrt 40.3%

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

      3. *-un-lft-identity 40.3%

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

      4. times-frac 40.3%

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

      5. sqrt-div 40.3%

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

      6. sqrt-pow1 40.3%

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

      7. metadata-eval 40.3%

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

      8. sqrt-div 40.3%

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

      9. sqrt-pow1 44.3%

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

      10. metadata-eval 44.3%

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

   6. Applied egg-rr 44.3%

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

      1. /-rgt-identity 44.3%

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

      2. associate-*r/ 44.3%

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

      3. unpow2 44.3%

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

   8. Simplified 44.3%

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

      1. pow1 44.3%

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

      2. associate-*l* 44.3%

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

      3. associate-+r+ 44.3%

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

      4. metadata-eval 44.3%

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

   10. Applied egg-rr 44.3%

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

       1. unpow1 44.3%

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

       2. associate-*r* 44.3%

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

       3. *-commutative 44.3%

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

       4. associate-*r* 44.3%

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

       5. associate-*r/ 44.3%

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

       6. *-commutative 44.3%

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

       7. associate-/l* 44.4%

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

   12. Simplified 44.4%

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

   if 2.9999999999999999e165 < t

   1. Initial program 65.9%

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

   3. Simplified 65.9%

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

      1. add-cube-cbrt 65.9%

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

      2. pow3 65.9%

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

      3. *-commutative 65.9%

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

      4. cbrt-prod 65.9%

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

      5. cbrt-div 65.9%

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

      6. rem-cbrt-cube 71.4%

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

      7. cbrt-prod 84.2%

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

      8. pow2 84.2%

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

   6. Applied egg-rr 84.2%

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

      1. pow1 84.2%

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

      2. div-inv 84.1%

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

      3. pow-flip 84.1%

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

      4. metadata-eval 84.1%

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

   8. Applied egg-rr 84.1%

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

      1. unpow1 84.1%

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

      2. associate-*r* 84.2%

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

      3. *-commutative 84.2%

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

   10. Simplified 84.2%

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

       1. add-cube-cbrt 84.1%

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

       2. pow3 84.1%

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

   12. Applied egg-rr 98.6%

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

   13. Taylor expanded in k around 0 93.8%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.6 \cdot 10^{-106}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{t \cdot {k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right)\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+165}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \left({\left(\frac{{t}^{1.5}}{\sqrt{\ell}}\right)}^{2} \cdot \frac{\sin k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(t \cdot \left(\sqrt[3]{\sin k} \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right) \cdot \left(\sqrt[3]{k} \cdot \sqrt[3]{2}\right)\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.2 \cdot 10^{-105}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{t\_m \cdot {k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right) \cdot {\left(t\_m \cdot \left(\sqrt[3]{\sin k} \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.2e-105)
    (*
     2.0
     (* (/ (cos k) (* t_m (pow k 2.0))) (/ (pow l 2.0) (pow (sin k) 2.0))))
    (/
     2.0
     (*
      (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))
      (pow (* t_m (* (cbrt (sin k)) (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 <= 1.2e-105) {
		tmp = 2.0 * ((cos(k) / (t_m * pow(k, 2.0))) * (pow(l, 2.0) / pow(sin(k), 2.0)));
	} else {
		tmp = 2.0 / ((tan(k) * (2.0 + pow((k / t_m), 2.0))) * pow((t_m * (cbrt(sin(k)) * 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 <= 1.2e-105) {
		tmp = 2.0 * ((Math.cos(k) / (t_m * Math.pow(k, 2.0))) * (Math.pow(l, 2.0) / Math.pow(Math.sin(k), 2.0)));
	} else {
		tmp = 2.0 / ((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0))) * Math.pow((t_m * (Math.cbrt(Math.sin(k)) * 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 <= 1.2e-105)
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(t_m * (k ^ 2.0))) * Float64((l ^ 2.0) / (sin(k) ^ 2.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * (Float64(t_m * Float64(cbrt(sin(k)) * (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, 1.2e-105], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(t$95$m * N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.20000000000000007e-105

   1. Initial program 54.3%

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

      1. associate-/r* 54.3%

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

      2. +-commutative 54.3%

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

      3. unpow2 54.3%

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

      4. sqr-neg 54.3%

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

      5. distribute-frac-neg2 54.3%

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

      6. distribute-frac-neg2 54.3%

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

      7. unpow2 54.3%

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

      8. +-commutative 54.3%

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

      9. associate-*l* 47.6%

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

      10. associate-*l/ 47.1%

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

      11. associate-/r/ 46.5%

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

      12. +-commutative 46.5%

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

      13. associate-+r+ 46.5%

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

   3. Simplified 46.5%

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

      1. unpow2 46.5%

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

   6. Applied egg-rr 46.5%

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

   7. Taylor expanded in t around 0 65.5%

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

      1. *-commutative 65.5%

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

      2. associate-*r* 65.5%

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

      3. times-frac 66.6%

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

   9. Simplified 66.6%

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

   if 1.20000000000000007e-105 < t

   1. Initial program 72.6%

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

   3. Simplified 72.6%

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

      1. add-cube-cbrt 72.5%

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

      2. pow3 72.6%

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

      3. *-commutative 72.6%

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

      4. cbrt-prod 72.4%

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

      5. cbrt-div 72.4%

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

      6. rem-cbrt-cube 78.5%

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

      7. cbrt-prod 88.3%

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

      8. pow2 88.3%

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

   6. Applied egg-rr 88.3%

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

      1. pow1 88.3%

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

      2. div-inv 88.2%

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

      3. pow-flip 88.3%

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

      4. metadata-eval 88.3%

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

   8. Applied egg-rr 88.3%

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

      1. unpow1 88.3%

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

      2. associate-*r* 88.3%

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

      3. *-commutative 88.3%

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

   10. Simplified 88.3%

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

       1. add-cube-cbrt 88.3%

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

       2. pow3 88.3%

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

   12. Applied egg-rr 95.8%

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

       1. *-commutative 95.8%

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

       2. cube-prod 88.3%

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

       3. rem-cube-cbrt 88.4%

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

   14. Simplified 88.4%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.2 \cdot 10^{-105}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{t \cdot {k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(t \cdot \left(\sqrt[3]{\sin k} \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9 \cdot 10^{-106}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{t\_m \cdot {k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t\_m}\right)}^{2} + 1\right)\right)\right) \cdot \left(\sin k \cdot {\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2}\right)}\\ \end{array} \end{array} \]

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 (<= t_m 9e-106)
    (*
     2.0
     (* (/ (cos k) (* t_m (pow k 2.0))) (/ (pow l 2.0) (pow (sin k) 2.0))))
    (/
     2.0
     (*
      (* (tan k) (+ 1.0 (+ (pow (/ k t_m) 2.0) 1.0)))
      (* (sin k) (pow (/ (pow t_m 1.5) l) 2.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 9e-106) {
		tmp = 2.0 * ((cos(k) / (t_m * pow(k, 2.0))) * (pow(l, 2.0) / pow(sin(k), 2.0)));
	} else {
		tmp = 2.0 / ((tan(k) * (1.0 + (pow((k / t_m), 2.0) + 1.0))) * (sin(k) * pow((pow(t_m, 1.5) / l), 2.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 9d-106) then
        tmp = 2.0d0 * ((cos(k) / (t_m * (k ** 2.0d0))) * ((l ** 2.0d0) / (sin(k) ** 2.0d0)))
    else
        tmp = 2.0d0 / ((tan(k) * (1.0d0 + (((k / t_m) ** 2.0d0) + 1.0d0))) * (sin(k) * (((t_m ** 1.5d0) / l) ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 9e-106) {
		tmp = 2.0 * ((Math.cos(k) / (t_m * Math.pow(k, 2.0))) * (Math.pow(l, 2.0) / Math.pow(Math.sin(k), 2.0)));
	} else {
		tmp = 2.0 / ((Math.tan(k) * (1.0 + (Math.pow((k / t_m), 2.0) + 1.0))) * (Math.sin(k) * Math.pow((Math.pow(t_m, 1.5) / l), 2.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 9e-106:
		tmp = 2.0 * ((math.cos(k) / (t_m * math.pow(k, 2.0))) * (math.pow(l, 2.0) / math.pow(math.sin(k), 2.0)))
	else:
		tmp = 2.0 / ((math.tan(k) * (1.0 + (math.pow((k / t_m), 2.0) + 1.0))) * (math.sin(k) * math.pow((math.pow(t_m, 1.5) / l), 2.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 9e-106)
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(t_m * (k ^ 2.0))) * Float64((l ^ 2.0) / (sin(k) ^ 2.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(1.0 + Float64((Float64(k / t_m) ^ 2.0) + 1.0))) * Float64(sin(k) * (Float64((t_m ^ 1.5) / l) ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 9e-106)
		tmp = 2.0 * ((cos(k) / (t_m * (k ^ 2.0))) * ((l ^ 2.0) / (sin(k) ^ 2.0)));
	else
		tmp = 2.0 / ((tan(k) * (1.0 + (((k / t_m) ^ 2.0) + 1.0))) * (sin(k) * (((t_m ^ 1.5) / l) ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 9e-106], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 8.99999999999999911e-106

   1. Initial program 54.3%

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

      1. associate-/r* 54.3%

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

      2. +-commutative 54.3%

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

      3. unpow2 54.3%

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

      4. sqr-neg 54.3%

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

      5. distribute-frac-neg2 54.3%

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

      6. distribute-frac-neg2 54.3%

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

      7. unpow2 54.3%

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

      8. +-commutative 54.3%

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

      9. associate-*l* 47.6%

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

      10. associate-*l/ 47.1%

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

      11. associate-/r/ 46.5%

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

      12. +-commutative 46.5%

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

      13. associate-+r+ 46.5%

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

   3. Simplified 46.5%

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

      1. unpow2 46.5%

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

   6. Applied egg-rr 46.5%

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

   7. Taylor expanded in t around 0 65.5%

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

      1. *-commutative 65.5%

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

      2. associate-*r* 65.5%

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

      3. times-frac 66.6%

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

   9. Simplified 66.6%

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

   if 8.99999999999999911e-106 < t

   1. Initial program 72.6%

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

   3. Simplified 72.6%

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

      1. add-sqr-sqrt 72.6%

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

      2. pow2 72.6%

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

      3. sqrt-div 72.6%

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

      4. sqrt-pow1 77.4%

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

      5. metadata-eval 77.4%

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

      6. sqrt-prod 46.3%

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

      7. add-sqr-sqrt 86.6%

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

   6. Applied egg-rr 86.6%

      \[\leadsto \frac{2}{\left(\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9 \cdot 10^{-106}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{t \cdot {k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right) \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.35 \cdot 10^{-105}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{t\_m \cdot {k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t\_m}\right)}^{2} + 1\right)\right)\right) \cdot \left(\sin k \cdot \left(\frac{{t\_m}^{2}}{\ell} \cdot \frac{t\_m}{\ell}\right)\right)}\\ \end{array} \end{array} \]

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 1.35e-105)
    (*
     2.0
     (* (/ (cos k) (* t_m (pow k 2.0))) (/ (pow l 2.0) (pow (sin k) 2.0))))
    (/
     2.0
     (*
      (* (tan k) (+ 1.0 (+ (pow (/ k t_m) 2.0) 1.0)))
      (* (sin k) (* (/ (pow t_m 2.0) l) (/ t_m l))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.35e-105) {
		tmp = 2.0 * ((cos(k) / (t_m * pow(k, 2.0))) * (pow(l, 2.0) / pow(sin(k), 2.0)));
	} else {
		tmp = 2.0 / ((tan(k) * (1.0 + (pow((k / t_m), 2.0) + 1.0))) * (sin(k) * ((pow(t_m, 2.0) / l) * (t_m / l))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.35d-105) then
        tmp = 2.0d0 * ((cos(k) / (t_m * (k ** 2.0d0))) * ((l ** 2.0d0) / (sin(k) ** 2.0d0)))
    else
        tmp = 2.0d0 / ((tan(k) * (1.0d0 + (((k / t_m) ** 2.0d0) + 1.0d0))) * (sin(k) * (((t_m ** 2.0d0) / l) * (t_m / l))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.35e-105) {
		tmp = 2.0 * ((Math.cos(k) / (t_m * Math.pow(k, 2.0))) * (Math.pow(l, 2.0) / Math.pow(Math.sin(k), 2.0)));
	} else {
		tmp = 2.0 / ((Math.tan(k) * (1.0 + (Math.pow((k / t_m), 2.0) + 1.0))) * (Math.sin(k) * ((Math.pow(t_m, 2.0) / l) * (t_m / l))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.35e-105:
		tmp = 2.0 * ((math.cos(k) / (t_m * math.pow(k, 2.0))) * (math.pow(l, 2.0) / math.pow(math.sin(k), 2.0)))
	else:
		tmp = 2.0 / ((math.tan(k) * (1.0 + (math.pow((k / t_m), 2.0) + 1.0))) * (math.sin(k) * ((math.pow(t_m, 2.0) / l) * (t_m / l))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.35e-105)
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(t_m * (k ^ 2.0))) * Float64((l ^ 2.0) / (sin(k) ^ 2.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(1.0 + Float64((Float64(k / t_m) ^ 2.0) + 1.0))) * Float64(sin(k) * Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.35e-105)
		tmp = 2.0 * ((cos(k) / (t_m * (k ^ 2.0))) * ((l ^ 2.0) / (sin(k) ^ 2.0)));
	else
		tmp = 2.0 / ((tan(k) * (1.0 + (((k / t_m) ^ 2.0) + 1.0))) * (sin(k) * (((t_m ^ 2.0) / l) * (t_m / l))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.35e-105], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.34999999999999996e-105

   1. Initial program 54.3%

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

      1. associate-/r* 54.3%

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

      2. +-commutative 54.3%

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

      3. unpow2 54.3%

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

      4. sqr-neg 54.3%

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

      5. distribute-frac-neg2 54.3%

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

      6. distribute-frac-neg2 54.3%

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

      7. unpow2 54.3%

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

      8. +-commutative 54.3%

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

      9. associate-*l* 47.6%

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

      10. associate-*l/ 47.1%

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

      11. associate-/r/ 46.5%

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

      12. +-commutative 46.5%

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

      13. associate-+r+ 46.5%

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

   3. Simplified 46.5%

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

      1. unpow2 46.5%

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

   6. Applied egg-rr 46.5%

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

   7. Taylor expanded in t around 0 65.5%

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

      1. *-commutative 65.5%

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

      2. associate-*r* 65.5%

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

      3. times-frac 66.6%

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

   9. Simplified 66.6%

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

   if 1.34999999999999996e-105 < t

   1. Initial program 72.6%

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

   3. Simplified 72.6%

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

      1. unpow3 72.6%

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

      2. times-frac 81.0%

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

      3. pow2 81.0%

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

   6. Applied egg-rr 81.0%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.35 \cdot 10^{-105}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{t \cdot {k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right) \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 76.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.1 \cdot 10^{-107}:\\ \;\;\;\;{\left(\left(\frac{\ell}{k} \cdot \frac{\sqrt{2}}{\sin k}\right) \cdot \sqrt{\frac{\cos k}{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t\_m}\right)}^{2} + 1\right)\right)\right) \cdot \left(\sin k \cdot \left(\frac{{t\_m}^{2}}{\ell} \cdot \frac{t\_m}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.1e-107)
    (pow (* (* (/ l k) (/ (sqrt 2.0) (sin k))) (sqrt (/ (cos k) t_m))) 2.0)
    (/
     2.0
     (*
      (* (tan k) (+ 1.0 (+ (pow (/ k t_m) 2.0) 1.0)))
      (* (sin k) (* (/ (pow t_m 2.0) l) (/ t_m l))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.1e-107) {
		tmp = pow((((l / k) * (sqrt(2.0) / sin(k))) * sqrt((cos(k) / t_m))), 2.0);
	} else {
		tmp = 2.0 / ((tan(k) * (1.0 + (pow((k / t_m), 2.0) + 1.0))) * (sin(k) * ((pow(t_m, 2.0) / l) * (t_m / l))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2.1d-107) then
        tmp = (((l / k) * (sqrt(2.0d0) / sin(k))) * sqrt((cos(k) / t_m))) ** 2.0d0
    else
        tmp = 2.0d0 / ((tan(k) * (1.0d0 + (((k / t_m) ** 2.0d0) + 1.0d0))) * (sin(k) * (((t_m ** 2.0d0) / l) * (t_m / l))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.1e-107) {
		tmp = Math.pow((((l / k) * (Math.sqrt(2.0) / Math.sin(k))) * Math.sqrt((Math.cos(k) / t_m))), 2.0);
	} else {
		tmp = 2.0 / ((Math.tan(k) * (1.0 + (Math.pow((k / t_m), 2.0) + 1.0))) * (Math.sin(k) * ((Math.pow(t_m, 2.0) / l) * (t_m / l))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2.1e-107:
		tmp = math.pow((((l / k) * (math.sqrt(2.0) / math.sin(k))) * math.sqrt((math.cos(k) / t_m))), 2.0)
	else:
		tmp = 2.0 / ((math.tan(k) * (1.0 + (math.pow((k / t_m), 2.0) + 1.0))) * (math.sin(k) * ((math.pow(t_m, 2.0) / l) * (t_m / l))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.1e-107)
		tmp = Float64(Float64(Float64(l / k) * Float64(sqrt(2.0) / sin(k))) * sqrt(Float64(cos(k) / t_m))) ^ 2.0;
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(1.0 + Float64((Float64(k / t_m) ^ 2.0) + 1.0))) * Float64(sin(k) * Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2.1e-107)
		tmp = (((l / k) * (sqrt(2.0) / sin(k))) * sqrt((cos(k) / t_m))) ^ 2.0;
	else
		tmp = 2.0 / ((tan(k) * (1.0 + (((k / t_m) ^ 2.0) + 1.0))) * (sin(k) * (((t_m ^ 2.0) / l) * (t_m / l))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.1e-107], N[Power[N[(N[(N[(l / k), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.0999999999999999e-107

   1. Initial program 53.8%

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

      1. associate-/r* 53.8%

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

      2. +-commutative 53.8%

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

      3. unpow2 53.8%

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

      4. sqr-neg 53.8%

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

      5. distribute-frac-neg2 53.8%

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

      6. distribute-frac-neg2 53.8%

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

      7. unpow2 53.8%

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

      8. +-commutative 53.8%

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

      9. associate-*l* 47.0%

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

      10. associate-*l/ 46.4%

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

      11. associate-/r/ 46.4%

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

      12. +-commutative 46.4%

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

      13. associate-+r+ 46.4%

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

   3. Simplified 46.4%

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

      1. add-sqr-sqrt 30.4%

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

      2. pow2 30.4%

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

   6. Applied egg-rr 35.5%

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

   7. Taylor expanded in t around 0 40.6%

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

      1. times-frac 40.6%

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

   9. Simplified 40.6%

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

   if 2.0999999999999999e-107 < t

   1. Initial program 73.2%

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

   3. Simplified 73.2%

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

      1. unpow3 73.2%

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

      2. times-frac 81.5%

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

      3. pow2 81.5%

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

   6. Applied egg-rr 81.5%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.1 \cdot 10^{-107}:\\ \;\;\;\;{\left(\left(\frac{\ell}{k} \cdot \frac{\sqrt{2}}{\sin k}\right) \cdot \sqrt{\frac{\cos k}{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right) \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 66.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 + {\left(\frac{k}{t\_m}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.5 \cdot 10^{-106}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left({k}^{2} \cdot \frac{t\_m}{{\ell}^{2}}\right)}\\ \mathbf{elif}\;t\_m \leq 2.3 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{\frac{\frac{{t\_m}^{3}}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot t\_2\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t\_2} \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(k \cdot {t\_m}^{3}\right)}\right)\\ \end{array} \end{array} \end{array} \]

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 6.5e-106)
      (/ 2.0 (* (pow k 2.0) (* (pow k 2.0) (/ t_m (pow l 2.0)))))
      (if (<= t_m 2.3e-14)
        (/ 2.0 (/ (* (/ (pow t_m 3.0) l) (* (sin k) (* (tan k) t_2))) l))
        (* (/ l t_2) (* l (/ 2.0 (* (tan k) (* k (pow t_m 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 <= 6.5e-106) {
		tmp = 2.0 / (pow(k, 2.0) * (pow(k, 2.0) * (t_m / pow(l, 2.0))));
	} else if (t_m <= 2.3e-14) {
		tmp = 2.0 / (((pow(t_m, 3.0) / l) * (sin(k) * (tan(k) * t_2))) / l);
	} else {
		tmp = (l / t_2) * (l * (2.0 / (tan(k) * (k * pow(t_m, 3.0)))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = 2.0d0 + ((k / t_m) ** 2.0d0)
    if (t_m <= 6.5d-106) then
        tmp = 2.0d0 / ((k ** 2.0d0) * ((k ** 2.0d0) * (t_m / (l ** 2.0d0))))
    else if (t_m <= 2.3d-14) then
        tmp = 2.0d0 / ((((t_m ** 3.0d0) / l) * (sin(k) * (tan(k) * t_2))) / l)
    else
        tmp = (l / t_2) * (l * (2.0d0 / (tan(k) * (k * (t_m ** 3.0d0)))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = 2.0 + Math.pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 6.5e-106) {
		tmp = 2.0 / (Math.pow(k, 2.0) * (Math.pow(k, 2.0) * (t_m / Math.pow(l, 2.0))));
	} else if (t_m <= 2.3e-14) {
		tmp = 2.0 / (((Math.pow(t_m, 3.0) / l) * (Math.sin(k) * (Math.tan(k) * t_2))) / l);
	} else {
		tmp = (l / t_2) * (l * (2.0 / (Math.tan(k) * (k * Math.pow(t_m, 3.0)))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = 2.0 + math.pow((k / t_m), 2.0)
	tmp = 0
	if t_m <= 6.5e-106:
		tmp = 2.0 / (math.pow(k, 2.0) * (math.pow(k, 2.0) * (t_m / math.pow(l, 2.0))))
	elif t_m <= 2.3e-14:
		tmp = 2.0 / (((math.pow(t_m, 3.0) / l) * (math.sin(k) * (math.tan(k) * t_2))) / l)
	else:
		tmp = (l / t_2) * (l * (2.0 / (math.tan(k) * (k * math.pow(t_m, 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 <= 6.5e-106)
		tmp = Float64(2.0 / Float64((k ^ 2.0) * Float64((k ^ 2.0) * Float64(t_m / (l ^ 2.0)))));
	elseif (t_m <= 2.3e-14)
		tmp = Float64(2.0 / Float64(Float64(Float64((t_m ^ 3.0) / l) * Float64(sin(k) * Float64(tan(k) * t_2))) / l));
	else
		tmp = Float64(Float64(l / t_2) * Float64(l * Float64(2.0 / Float64(tan(k) * Float64(k * (t_m ^ 3.0))))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = 2.0 + ((k / t_m) ^ 2.0);
	tmp = 0.0;
	if (t_m <= 6.5e-106)
		tmp = 2.0 / ((k ^ 2.0) * ((k ^ 2.0) * (t_m / (l ^ 2.0))));
	elseif (t_m <= 2.3e-14)
		tmp = 2.0 / ((((t_m ^ 3.0) / l) * (sin(k) * (tan(k) * t_2))) / l);
	else
		tmp = (l / t_2) * (l * (2.0 / (tan(k) * (k * (t_m ^ 3.0)))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 6.5e-106], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.3e-14], N[(2.0 / N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[(l / t$95$2), $MachinePrecision] * N[(l * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(k * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 6.4999999999999997e-106

   1. Initial program 54.3%

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

   3. Simplified 54.3%

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

      1. add-cube-cbrt 54.3%

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

      2. pow3 54.3%

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

      3. *-commutative 54.3%

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

      4. cbrt-prod 54.3%

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

      5. cbrt-div 54.2%

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

      6. rem-cbrt-cube 62.6%

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

      7. cbrt-prod 66.9%

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

      8. pow2 66.9%

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

   6. Applied egg-rr 66.9%

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

   7. Taylor expanded in k around inf 65.4%

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

      1. associate-/l* 64.5%

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

      2. *-commutative 64.5%

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

      3. *-commutative 64.5%

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

      4. times-frac 65.9%

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

   9. Simplified 65.9%

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

   10. Taylor expanded in k around 0 56.6%

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

       1. associate-/l* 57.9%

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

   12. Simplified 57.9%

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

   if 6.4999999999999997e-106 < t < 2.29999999999999998e-14

   1. Initial program 86.4%

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

   3. Simplified 86.5%

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

      1. associate-*l* 86.4%

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

      2. associate-/r* 95.2%

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

      3. associate-+r+ 95.2%

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

      4. metadata-eval 95.2%

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

      5. associate-*l* 95.2%

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

      6. associate-*l/ 95.5%

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

      7. associate-*l* 95.2%

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

   6. Applied egg-rr 95.2%

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

   if 2.29999999999999998e-14 < t

   1. Initial program 67.7%

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

      1. associate-/r* 67.8%

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

      2. +-commutative 67.8%

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

      3. unpow2 67.8%

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

      4. sqr-neg 67.8%

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

      5. distribute-frac-neg2 67.8%

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

      6. distribute-frac-neg2 67.8%

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

      7. unpow2 67.8%

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

      8. +-commutative 67.8%

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

      9. associate-*l* 55.9%

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

      10. associate-*l/ 56.0%

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

      11. associate-/r/ 56.0%

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

      12. +-commutative 56.0%

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

      13. associate-+r+ 56.0%

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

   3. Simplified 56.0%

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

      1. associate-*r* 58.0%

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

      2. *-un-lft-identity 58.0%

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

      3. times-frac 58.0%

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

      4. associate-*r* 69.9%

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

   6. Applied egg-rr 69.9%

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

   7. Taylor expanded in k around 0 65.4%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.5 \cdot 10^{-106}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right)}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(k \cdot {t}^{3}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 71.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.6 \cdot 10^{-106}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left({k}^{2} \cdot \frac{t\_m}{{\ell}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t\_m}\right)}^{2} + 1\right)\right)\right) \cdot \left(\sin k \cdot \left(\frac{{t\_m}^{2}}{\ell} \cdot \frac{t\_m}{\ell}\right)\right)}\\ \end{array} \end{array} \]

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 7.6e-106)
    (/ 2.0 (* (pow k 2.0) (* (pow k 2.0) (/ t_m (pow l 2.0)))))
    (/
     2.0
     (*
      (* (tan k) (+ 1.0 (+ (pow (/ k t_m) 2.0) 1.0)))
      (* (sin k) (* (/ (pow t_m 2.0) l) (/ t_m l))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 7.6e-106) {
		tmp = 2.0 / (pow(k, 2.0) * (pow(k, 2.0) * (t_m / pow(l, 2.0))));
	} else {
		tmp = 2.0 / ((tan(k) * (1.0 + (pow((k / t_m), 2.0) + 1.0))) * (sin(k) * ((pow(t_m, 2.0) / l) * (t_m / l))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 7.6d-106) then
        tmp = 2.0d0 / ((k ** 2.0d0) * ((k ** 2.0d0) * (t_m / (l ** 2.0d0))))
    else
        tmp = 2.0d0 / ((tan(k) * (1.0d0 + (((k / t_m) ** 2.0d0) + 1.0d0))) * (sin(k) * (((t_m ** 2.0d0) / l) * (t_m / l))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 7.6e-106) {
		tmp = 2.0 / (Math.pow(k, 2.0) * (Math.pow(k, 2.0) * (t_m / Math.pow(l, 2.0))));
	} else {
		tmp = 2.0 / ((Math.tan(k) * (1.0 + (Math.pow((k / t_m), 2.0) + 1.0))) * (Math.sin(k) * ((Math.pow(t_m, 2.0) / l) * (t_m / l))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 7.6e-106:
		tmp = 2.0 / (math.pow(k, 2.0) * (math.pow(k, 2.0) * (t_m / math.pow(l, 2.0))))
	else:
		tmp = 2.0 / ((math.tan(k) * (1.0 + (math.pow((k / t_m), 2.0) + 1.0))) * (math.sin(k) * ((math.pow(t_m, 2.0) / l) * (t_m / l))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 7.6e-106)
		tmp = Float64(2.0 / Float64((k ^ 2.0) * Float64((k ^ 2.0) * Float64(t_m / (l ^ 2.0)))));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(1.0 + Float64((Float64(k / t_m) ^ 2.0) + 1.0))) * Float64(sin(k) * Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 7.6e-106)
		tmp = 2.0 / ((k ^ 2.0) * ((k ^ 2.0) * (t_m / (l ^ 2.0))));
	else
		tmp = 2.0 / ((tan(k) * (1.0 + (((k / t_m) ^ 2.0) + 1.0))) * (sin(k) * (((t_m ^ 2.0) / l) * (t_m / l))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 7.6e-106], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 7.5999999999999999e-106

   1. Initial program 54.3%

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

   3. Simplified 54.3%

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

      1. add-cube-cbrt 54.3%

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

      2. pow3 54.3%

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

      3. *-commutative 54.3%

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

      4. cbrt-prod 54.3%

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

      5. cbrt-div 54.2%

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

      6. rem-cbrt-cube 62.6%

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

      7. cbrt-prod 66.9%

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

      8. pow2 66.9%

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

   6. Applied egg-rr 66.9%

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

   7. Taylor expanded in k around inf 65.4%

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

      1. associate-/l* 64.5%

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

      2. *-commutative 64.5%

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

      3. *-commutative 64.5%

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

      4. times-frac 65.9%

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

   9. Simplified 65.9%

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

   10. Taylor expanded in k around 0 56.6%

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

       1. associate-/l* 57.9%

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

   12. Simplified 57.9%

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

   if 7.5999999999999999e-106 < t

   1. Initial program 72.6%

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

   3. Simplified 72.6%

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

      1. unpow3 72.6%

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

      2. times-frac 81.0%

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

      3. pow2 81.0%

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

   6. Applied egg-rr 81.0%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.6 \cdot 10^{-106}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right) \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 69.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.7 \cdot 10^{-101}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left({k}^{2} \cdot \frac{t\_m}{{\ell}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot {t\_m}^{3}\right)}\right) \cdot \frac{\ell}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.7e-101)
    (/ 2.0 (* (pow k 2.0) (* (pow k 2.0) (/ t_m (pow l 2.0)))))
    (*
     (* l (/ 2.0 (* (tan k) (* (sin k) (pow t_m 3.0)))))
     (/ l (+ 2.0 (pow (/ k t_m) 2.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.7e-101) {
		tmp = 2.0 / (pow(k, 2.0) * (pow(k, 2.0) * (t_m / pow(l, 2.0))));
	} else {
		tmp = (l * (2.0 / (tan(k) * (sin(k) * pow(t_m, 3.0))))) * (l / (2.0 + pow((k / t_m), 2.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.7d-101) then
        tmp = 2.0d0 / ((k ** 2.0d0) * ((k ** 2.0d0) * (t_m / (l ** 2.0d0))))
    else
        tmp = (l * (2.0d0 / (tan(k) * (sin(k) * (t_m ** 3.0d0))))) * (l / (2.0d0 + ((k / t_m) ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.7e-101) {
		tmp = 2.0 / (Math.pow(k, 2.0) * (Math.pow(k, 2.0) * (t_m / Math.pow(l, 2.0))));
	} else {
		tmp = (l * (2.0 / (Math.tan(k) * (Math.sin(k) * Math.pow(t_m, 3.0))))) * (l / (2.0 + Math.pow((k / t_m), 2.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.7e-101:
		tmp = 2.0 / (math.pow(k, 2.0) * (math.pow(k, 2.0) * (t_m / math.pow(l, 2.0))))
	else:
		tmp = (l * (2.0 / (math.tan(k) * (math.sin(k) * math.pow(t_m, 3.0))))) * (l / (2.0 + math.pow((k / t_m), 2.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.7e-101)
		tmp = Float64(2.0 / Float64((k ^ 2.0) * Float64((k ^ 2.0) * Float64(t_m / (l ^ 2.0)))));
	else
		tmp = Float64(Float64(l * Float64(2.0 / Float64(tan(k) * Float64(sin(k) * (t_m ^ 3.0))))) * Float64(l / Float64(2.0 + (Float64(k / t_m) ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.7e-101)
		tmp = 2.0 / ((k ^ 2.0) * ((k ^ 2.0) * (t_m / (l ^ 2.0))));
	else
		tmp = (l * (2.0 / (tan(k) * (sin(k) * (t_m ^ 3.0))))) * (l / (2.0 + ((k / t_m) ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.7e-101], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.69999999999999995e-101

   1. Initial program 54.6%

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

   3. Simplified 54.6%

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

      1. add-cube-cbrt 54.5%

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

      2. pow3 54.5%

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

      3. *-commutative 54.5%

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

      4. cbrt-prod 54.5%

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

      5. cbrt-div 54.5%

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

      6. rem-cbrt-cube 62.8%

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

      7. cbrt-prod 67.1%

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

      8. pow2 67.1%

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

   6. Applied egg-rr 67.1%

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

   7. Taylor expanded in k around inf 65.6%

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

      1. associate-/l* 64.7%

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

      2. *-commutative 64.7%

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

      3. *-commutative 64.7%

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

      4. times-frac 66.1%

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

   9. Simplified 66.1%

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

   10. Taylor expanded in k around 0 56.9%

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

       1. associate-/l* 58.1%

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

   12. Simplified 58.1%

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

   if 1.69999999999999995e-101 < t

   1. Initial program 72.2%

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

      1. associate-/r* 72.3%

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

      2. +-commutative 72.3%

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

      3. unpow2 72.3%

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

      4. sqr-neg 72.3%

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

      5. distribute-frac-neg2 72.3%

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

      6. distribute-frac-neg2 72.3%

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

      7. unpow2 72.3%

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

      8. +-commutative 72.3%

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

      9. associate-*l* 63.4%

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

      10. associate-*l/ 63.4%

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

      11. associate-/r/ 63.4%

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

      12. +-commutative 63.4%

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

      13. associate-+r+ 63.4%

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

   3. Simplified 63.4%

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

      1. associate-*r* 66.1%

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

      2. *-un-lft-identity 66.1%

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

      3. times-frac 66.2%

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

      4. associate-*r* 75.0%

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

   6. Applied egg-rr 75.0%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.7 \cdot 10^{-101}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot {t}^{3}\right)}\right) \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 66.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2 \cdot 10^{-105}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left({k}^{2} \cdot \frac{t\_m}{{\ell}^{2}}\right)}\\ \mathbf{elif}\;t\_m \leq 7.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{2}{\frac{\frac{{t\_m}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{k}{t\_m} \cdot \frac{k}{t\_m}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{2 + {\left(\frac{k}{t\_m}\right)}^{2}} \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(k \cdot {t\_m}^{3}\right)}\right)\\ \end{array} \end{array} \]

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 2e-105)
    (/ 2.0 (* (pow k 2.0) (* (pow k 2.0) (/ t_m (pow l 2.0)))))
    (if (<= t_m 7.2e-8)
      (/
       2.0
       (*
        (/ (/ (pow t_m 3.0) l) l)
        (* (* (sin k) (tan k)) (+ 2.0 (* (/ k t_m) (/ k t_m))))))
      (*
       (/ l (+ 2.0 (pow (/ k t_m) 2.0)))
       (* l (/ 2.0 (* (tan k) (* k (pow t_m 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 <= 2e-105) {
		tmp = 2.0 / (pow(k, 2.0) * (pow(k, 2.0) * (t_m / pow(l, 2.0))));
	} else if (t_m <= 7.2e-8) {
		tmp = 2.0 / (((pow(t_m, 3.0) / l) / l) * ((sin(k) * tan(k)) * (2.0 + ((k / t_m) * (k / t_m)))));
	} else {
		tmp = (l / (2.0 + pow((k / t_m), 2.0))) * (l * (2.0 / (tan(k) * (k * pow(t_m, 3.0)))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2d-105) then
        tmp = 2.0d0 / ((k ** 2.0d0) * ((k ** 2.0d0) * (t_m / (l ** 2.0d0))))
    else if (t_m <= 7.2d-8) then
        tmp = 2.0d0 / ((((t_m ** 3.0d0) / l) / l) * ((sin(k) * tan(k)) * (2.0d0 + ((k / t_m) * (k / t_m)))))
    else
        tmp = (l / (2.0d0 + ((k / t_m) ** 2.0d0))) * (l * (2.0d0 / (tan(k) * (k * (t_m ** 3.0d0)))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2e-105) {
		tmp = 2.0 / (Math.pow(k, 2.0) * (Math.pow(k, 2.0) * (t_m / Math.pow(l, 2.0))));
	} else if (t_m <= 7.2e-8) {
		tmp = 2.0 / (((Math.pow(t_m, 3.0) / l) / l) * ((Math.sin(k) * Math.tan(k)) * (2.0 + ((k / t_m) * (k / t_m)))));
	} else {
		tmp = (l / (2.0 + Math.pow((k / t_m), 2.0))) * (l * (2.0 / (Math.tan(k) * (k * Math.pow(t_m, 3.0)))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2e-105:
		tmp = 2.0 / (math.pow(k, 2.0) * (math.pow(k, 2.0) * (t_m / math.pow(l, 2.0))))
	elif t_m <= 7.2e-8:
		tmp = 2.0 / (((math.pow(t_m, 3.0) / l) / l) * ((math.sin(k) * math.tan(k)) * (2.0 + ((k / t_m) * (k / t_m)))))
	else:
		tmp = (l / (2.0 + math.pow((k / t_m), 2.0))) * (l * (2.0 / (math.tan(k) * (k * math.pow(t_m, 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 <= 2e-105)
		tmp = Float64(2.0 / Float64((k ^ 2.0) * Float64((k ^ 2.0) * Float64(t_m / (l ^ 2.0)))));
	elseif (t_m <= 7.2e-8)
		tmp = Float64(2.0 / Float64(Float64(Float64((t_m ^ 3.0) / l) / l) * Float64(Float64(sin(k) * tan(k)) * Float64(2.0 + Float64(Float64(k / t_m) * Float64(k / t_m))))));
	else
		tmp = Float64(Float64(l / Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * Float64(l * Float64(2.0 / Float64(tan(k) * Float64(k * (t_m ^ 3.0))))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2e-105)
		tmp = 2.0 / ((k ^ 2.0) * ((k ^ 2.0) * (t_m / (l ^ 2.0))));
	elseif (t_m <= 7.2e-8)
		tmp = 2.0 / ((((t_m ^ 3.0) / l) / l) * ((sin(k) * tan(k)) * (2.0 + ((k / t_m) * (k / t_m)))));
	else
		tmp = (l / (2.0 + ((k / t_m) ^ 2.0))) * (l * (2.0 / (tan(k) * (k * (t_m ^ 3.0)))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2e-105], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 7.2e-8], N[(2.0 / N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(k * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.99999999999999993e-105

   1. Initial program 54.3%

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

   3. Simplified 54.3%

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

      1. add-cube-cbrt 54.3%

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

      2. pow3 54.3%

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

      3. *-commutative 54.3%

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

      4. cbrt-prod 54.3%

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

      5. cbrt-div 54.2%

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

      6. rem-cbrt-cube 62.6%

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

      7. cbrt-prod 66.9%

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

      8. pow2 66.9%

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

   6. Applied egg-rr 66.9%

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

   7. Taylor expanded in k around inf 65.4%

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

      1. associate-/l* 64.5%

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

      2. *-commutative 64.5%

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

      3. *-commutative 64.5%

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

      4. times-frac 65.9%

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

   9. Simplified 65.9%

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

   10. Taylor expanded in k around 0 56.6%

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

       1. associate-/l* 57.9%

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

   12. Simplified 57.9%

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

   if 1.99999999999999993e-105 < t < 7.19999999999999962e-8

   1. Initial program 86.9%

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

   3. Simplified 95.3%

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

      1. unpow2 82.6%

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

   6. Applied egg-rr 95.3%

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

   if 7.19999999999999962e-8 < t

   1. Initial program 67.2%

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

      1. associate-/r* 67.3%

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

      2. +-commutative 67.3%

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

      3. unpow2 67.3%

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

      4. sqr-neg 67.3%

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

      5. distribute-frac-neg2 67.3%

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

      6. distribute-frac-neg2 67.3%

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

      7. unpow2 67.3%

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

      8. +-commutative 67.3%

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

      9. associate-*l* 55.2%

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

      10. associate-*l/ 55.3%

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

      11. associate-/r/ 55.3%

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

      12. +-commutative 55.3%

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

      13. associate-+r+ 55.3%

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

   3. Simplified 55.3%

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

      1. associate-*r* 57.3%

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

      2. *-un-lft-identity 57.3%

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

      3. times-frac 57.4%

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

      4. associate-*r* 69.4%

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

   6. Applied egg-rr 69.4%

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

   7. Taylor expanded in k around 0 64.9%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2 \cdot 10^{-105}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right)}\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{k}{t} \cdot \frac{k}{t}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(k \cdot {t}^{3}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 64.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.16 \cdot 10^{-107}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left({k}^{2} \cdot \frac{t\_m}{{\ell}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{2 + {\left(\frac{k}{t\_m}\right)}^{2}} \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(k \cdot {t\_m}^{3}\right)}\right)\\ \end{array} \end{array} \]

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 1.16e-107)
    (/ 2.0 (* (pow k 2.0) (* (pow k 2.0) (/ t_m (pow l 2.0)))))
    (*
     (/ l (+ 2.0 (pow (/ k t_m) 2.0)))
     (* l (/ 2.0 (* (tan k) (* k (pow t_m 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 <= 1.16e-107) {
		tmp = 2.0 / (pow(k, 2.0) * (pow(k, 2.0) * (t_m / pow(l, 2.0))));
	} else {
		tmp = (l / (2.0 + pow((k / t_m), 2.0))) * (l * (2.0 / (tan(k) * (k * pow(t_m, 3.0)))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.16d-107) then
        tmp = 2.0d0 / ((k ** 2.0d0) * ((k ** 2.0d0) * (t_m / (l ** 2.0d0))))
    else
        tmp = (l / (2.0d0 + ((k / t_m) ** 2.0d0))) * (l * (2.0d0 / (tan(k) * (k * (t_m ** 3.0d0)))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.16e-107) {
		tmp = 2.0 / (Math.pow(k, 2.0) * (Math.pow(k, 2.0) * (t_m / Math.pow(l, 2.0))));
	} else {
		tmp = (l / (2.0 + Math.pow((k / t_m), 2.0))) * (l * (2.0 / (Math.tan(k) * (k * Math.pow(t_m, 3.0)))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.16e-107:
		tmp = 2.0 / (math.pow(k, 2.0) * (math.pow(k, 2.0) * (t_m / math.pow(l, 2.0))))
	else:
		tmp = (l / (2.0 + math.pow((k / t_m), 2.0))) * (l * (2.0 / (math.tan(k) * (k * math.pow(t_m, 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 <= 1.16e-107)
		tmp = Float64(2.0 / Float64((k ^ 2.0) * Float64((k ^ 2.0) * Float64(t_m / (l ^ 2.0)))));
	else
		tmp = Float64(Float64(l / Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * Float64(l * Float64(2.0 / Float64(tan(k) * Float64(k * (t_m ^ 3.0))))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.16e-107)
		tmp = 2.0 / ((k ^ 2.0) * ((k ^ 2.0) * (t_m / (l ^ 2.0))));
	else
		tmp = (l / (2.0 + ((k / t_m) ^ 2.0))) * (l * (2.0 / (tan(k) * (k * (t_m ^ 3.0)))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.16e-107], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(k * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.16e-107

   1. Initial program 54.1%

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

   3. Simplified 54.1%

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

      1. add-cube-cbrt 54.0%

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

      2. pow3 54.0%

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

      3. *-commutative 54.0%

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

      4. cbrt-prod 54.0%

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

      5. cbrt-div 54.0%

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

      6. rem-cbrt-cube 62.6%

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

      7. cbrt-prod 66.9%

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

      8. pow2 66.9%

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

   6. Applied egg-rr 66.9%

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

   7. Taylor expanded in k around inf 65.4%

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

      1. associate-/l* 64.5%

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

      2. *-commutative 64.5%

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

      3. *-commutative 64.5%

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

      4. times-frac 65.9%

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

   9. Simplified 65.9%

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

   10. Taylor expanded in k around 0 56.4%

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

       1. associate-/l* 57.7%

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

   12. Simplified 57.7%

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

   if 1.16e-107 < t

   1. Initial program 72.3%

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

      1. associate-/r* 72.4%

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

      2. +-commutative 72.4%

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

      3. unpow2 72.4%

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

      4. sqr-neg 72.4%

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

      5. distribute-frac-neg2 72.4%

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

      6. distribute-frac-neg2 72.4%

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

      7. unpow2 72.4%

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

      8. +-commutative 72.4%

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

      9. associate-*l* 63.9%

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

      10. associate-*l/ 64.0%

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

      11. associate-/r/ 61.7%

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

      12. +-commutative 61.7%

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

      13. associate-+r+ 61.7%

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

   3. Simplified 61.7%

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

      1. associate-*r* 64.2%

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

      2. *-un-lft-identity 64.2%

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

      3. times-frac 64.3%

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

      4. associate-*r* 72.7%

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

   6. Applied egg-rr 72.7%

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

   7. Taylor expanded in k around 0 68.7%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.16 \cdot 10^{-107}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(k \cdot {t}^{3}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 40.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.6 \cdot 10^{-105}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left({k}^{2} \cdot \frac{t\_m}{{\ell}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t\_m \cdot {\ell}^{-0.6666666666666666}\right)}^{3} \cdot \left(2 \cdot {k}^{2}\right)}\\ \end{array} \end{array} \]

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 (<= t_m 1.6e-105)
    (/ 2.0 (* (pow k 2.0) (* (pow k 2.0) (/ t_m (pow l 2.0)))))
    (/
     2.0
     (* (pow (* t_m (pow l -0.6666666666666666)) 3.0) (* 2.0 (pow k 2.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.6e-105) {
		tmp = 2.0 / (pow(k, 2.0) * (pow(k, 2.0) * (t_m / pow(l, 2.0))));
	} else {
		tmp = 2.0 / (pow((t_m * pow(l, -0.6666666666666666)), 3.0) * (2.0 * pow(k, 2.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.6d-105) then
        tmp = 2.0d0 / ((k ** 2.0d0) * ((k ** 2.0d0) * (t_m / (l ** 2.0d0))))
    else
        tmp = 2.0d0 / (((t_m * (l ** (-0.6666666666666666d0))) ** 3.0d0) * (2.0d0 * (k ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.6e-105) {
		tmp = 2.0 / (Math.pow(k, 2.0) * (Math.pow(k, 2.0) * (t_m / Math.pow(l, 2.0))));
	} else {
		tmp = 2.0 / (Math.pow((t_m * Math.pow(l, -0.6666666666666666)), 3.0) * (2.0 * Math.pow(k, 2.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.6e-105:
		tmp = 2.0 / (math.pow(k, 2.0) * (math.pow(k, 2.0) * (t_m / math.pow(l, 2.0))))
	else:
		tmp = 2.0 / (math.pow((t_m * math.pow(l, -0.6666666666666666)), 3.0) * (2.0 * math.pow(k, 2.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.6e-105)
		tmp = Float64(2.0 / Float64((k ^ 2.0) * Float64((k ^ 2.0) * Float64(t_m / (l ^ 2.0)))));
	else
		tmp = Float64(2.0 / Float64((Float64(t_m * (l ^ -0.6666666666666666)) ^ 3.0) * Float64(2.0 * (k ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.6e-105)
		tmp = 2.0 / ((k ^ 2.0) * ((k ^ 2.0) * (t_m / (l ^ 2.0))));
	else
		tmp = 2.0 / (((t_m * (l ^ -0.6666666666666666)) ^ 3.0) * (2.0 * (k ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.6e-105], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(t$95$m * N[Power[l, -0.6666666666666666], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.59999999999999991e-105

   1. Initial program 54.3%

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

   3. Simplified 54.3%

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

      1. add-cube-cbrt 54.3%

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

      2. pow3 54.3%

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

      3. *-commutative 54.3%

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

      4. cbrt-prod 54.3%

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

      5. cbrt-div 54.2%

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

      6. rem-cbrt-cube 62.6%

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

      7. cbrt-prod 66.9%

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

      8. pow2 66.9%

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

   6. Applied egg-rr 66.9%

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

   7. Taylor expanded in k around inf 65.4%

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

      1. associate-/l* 64.5%

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

      2. *-commutative 64.5%

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

      3. *-commutative 64.5%

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

      4. times-frac 65.9%

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

   9. Simplified 65.9%

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

   10. Taylor expanded in k around 0 56.6%

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

       1. associate-/l* 57.9%

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

   12. Simplified 57.9%

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

   if 1.59999999999999991e-105 < t

   1. Initial program 72.6%

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

   3. Simplified 67.6%

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

   5. Taylor expanded in k around 0 61.0%

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

      1. add-cube-cbrt 61.0%

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

      2. pow3 61.0%

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

      3. associate-/l/ 57.3%

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

      4. cbrt-div 57.2%

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

      5. unpow3 57.2%

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

      6. add-cbrt-cube 61.1%

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

      7. cbrt-unprod 65.6%

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

      8. unpow2 65.6%

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

      9. div-inv 65.6%

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

      10. unpow-prod-down 57.2%

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

      11. pow-flip 57.3%

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

      12. metadata-eval 57.3%

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

   7. Applied egg-rr 57.3%

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

      1. cube-prod 65.6%

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

   9. Simplified 65.6%

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

       1. add-exp-log 65.3%

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

       2. log-pow 32.3%

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

   11. Applied egg-rr 32.3%

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

       1. *-commutative 32.3%

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

       2. pow-to-exp 65.6%

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

       3. pow1/3 32.2%

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

       4. pow-pow 32.2%

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

       5. metadata-eval 32.2%

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

   13. Applied egg-rr 32.2%

       \[\leadsto \frac{2}{{\left(t \cdot \color{blue}{{\ell}^{-0.6666666666666666}}\right)}^{3} \cdot \left(2 \cdot {k}^{2}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 61.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4 \cdot 10^{-105}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left({k}^{2} \cdot \frac{t\_m}{{\ell}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{t\_m \cdot \frac{{t\_m}^{2}}{\ell}}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4e-105)
    (/ 2.0 (* (pow k 2.0) (* (pow k 2.0) (/ t_m (pow l 2.0)))))
    (/ 2.0 (* (* 2.0 (pow k 2.0)) (/ (* t_m (/ (pow t_m 2.0) l)) l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4e-105) {
		tmp = 2.0 / (pow(k, 2.0) * (pow(k, 2.0) * (t_m / pow(l, 2.0))));
	} else {
		tmp = 2.0 / ((2.0 * pow(k, 2.0)) * ((t_m * (pow(t_m, 2.0) / l)) / l));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 4d-105) then
        tmp = 2.0d0 / ((k ** 2.0d0) * ((k ** 2.0d0) * (t_m / (l ** 2.0d0))))
    else
        tmp = 2.0d0 / ((2.0d0 * (k ** 2.0d0)) * ((t_m * ((t_m ** 2.0d0) / l)) / l))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4e-105) {
		tmp = 2.0 / (Math.pow(k, 2.0) * (Math.pow(k, 2.0) * (t_m / Math.pow(l, 2.0))));
	} else {
		tmp = 2.0 / ((2.0 * Math.pow(k, 2.0)) * ((t_m * (Math.pow(t_m, 2.0) / l)) / l));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 4e-105:
		tmp = 2.0 / (math.pow(k, 2.0) * (math.pow(k, 2.0) * (t_m / math.pow(l, 2.0))))
	else:
		tmp = 2.0 / ((2.0 * math.pow(k, 2.0)) * ((t_m * (math.pow(t_m, 2.0) / l)) / l))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 4e-105)
		tmp = Float64(2.0 / Float64((k ^ 2.0) * Float64((k ^ 2.0) * Float64(t_m / (l ^ 2.0)))));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * (k ^ 2.0)) * Float64(Float64(t_m * Float64((t_m ^ 2.0) / l)) / l)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 4e-105)
		tmp = 2.0 / ((k ^ 2.0) * ((k ^ 2.0) * (t_m / (l ^ 2.0))));
	else
		tmp = 2.0 / ((2.0 * (k ^ 2.0)) * ((t_m * ((t_m ^ 2.0) / l)) / l));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 4e-105], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.99999999999999986e-105

   1. Initial program 54.3%

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

   3. Simplified 54.3%

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

      1. add-cube-cbrt 54.3%

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

      2. pow3 54.3%

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

      3. *-commutative 54.3%

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

      4. cbrt-prod 54.3%

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

      5. cbrt-div 54.2%

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

      6. rem-cbrt-cube 62.6%

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

      7. cbrt-prod 66.9%

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

      8. pow2 66.9%

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

   6. Applied egg-rr 66.9%

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

   7. Taylor expanded in k around inf 65.4%

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

      1. associate-/l* 64.5%

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

      2. *-commutative 64.5%

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

      3. *-commutative 64.5%

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

      4. times-frac 65.9%

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

   9. Simplified 65.9%

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

   10. Taylor expanded in k around 0 56.6%

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

       1. associate-/l* 57.9%

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

   12. Simplified 57.9%

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

   if 3.99999999999999986e-105 < t

   1. Initial program 72.6%

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

   3. Simplified 67.6%

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

   5. Taylor expanded in k around 0 61.0%

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

      1. cube-mult 61.0%

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

      2. *-un-lft-identity 61.0%

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

      3. times-frac 64.5%

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

      4. pow2 64.5%

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

   7. Applied egg-rr 64.5%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4 \cdot 10^{-105}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{t \cdot \frac{{t}^{2}}{\ell}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 60.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3 \cdot 10^{-105}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{{\ell}^{2}} \cdot {k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{t\_m \cdot \frac{{t\_m}^{2}}{\ell}}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3e-105)
    (/ 2.0 (* (/ t_m (pow l 2.0)) (pow k 4.0)))
    (/ 2.0 (* (* 2.0 (pow k 2.0)) (/ (* t_m (/ (pow t_m 2.0) l)) l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3e-105) {
		tmp = 2.0 / ((t_m / pow(l, 2.0)) * pow(k, 4.0));
	} else {
		tmp = 2.0 / ((2.0 * pow(k, 2.0)) * ((t_m * (pow(t_m, 2.0) / l)) / l));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 3d-105) then
        tmp = 2.0d0 / ((t_m / (l ** 2.0d0)) * (k ** 4.0d0))
    else
        tmp = 2.0d0 / ((2.0d0 * (k ** 2.0d0)) * ((t_m * ((t_m ** 2.0d0) / l)) / l))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3e-105) {
		tmp = 2.0 / ((t_m / Math.pow(l, 2.0)) * Math.pow(k, 4.0));
	} else {
		tmp = 2.0 / ((2.0 * Math.pow(k, 2.0)) * ((t_m * (Math.pow(t_m, 2.0) / l)) / l));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 3e-105:
		tmp = 2.0 / ((t_m / math.pow(l, 2.0)) * math.pow(k, 4.0))
	else:
		tmp = 2.0 / ((2.0 * math.pow(k, 2.0)) * ((t_m * (math.pow(t_m, 2.0) / l)) / l))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 3e-105)
		tmp = Float64(2.0 / Float64(Float64(t_m / (l ^ 2.0)) * (k ^ 4.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * (k ^ 2.0)) * Float64(Float64(t_m * Float64((t_m ^ 2.0) / l)) / l)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 3e-105)
		tmp = 2.0 / ((t_m / (l ^ 2.0)) * (k ^ 4.0));
	else
		tmp = 2.0 / ((2.0 * (k ^ 2.0)) * ((t_m * ((t_m ^ 2.0) / l)) / l));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 3e-105], N[(2.0 / N[(N[(t$95$m / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.0000000000000001e-105

   1. Initial program 54.3%

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

   3. Simplified 54.3%

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

      1. add-cube-cbrt 54.3%

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

      2. pow3 54.3%

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

      3. *-commutative 54.3%

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

      4. cbrt-prod 54.3%

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

      5. cbrt-div 54.2%

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

      6. rem-cbrt-cube 62.6%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{\color{blue}{t}}{\sqrt[3]{\ell \cdot \ell}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      7. cbrt-prod 66.9%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{\color{blue}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      8. pow2 66.9%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   6. Applied egg-rr 66.9%

      \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   7. Taylor expanded in k around inf 65.4%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   8. Step-by-step derivation

      1. associate-/l* 64.5%

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      2. *-commutative 64.5%

         \[\leadsto \frac{2}{{k}^{2} \cdot \frac{\color{blue}{{\sin k}^{2} \cdot t}}{{\ell}^{2} \cdot \cos k}} \]

      3. *-commutative 64.5%

         \[\leadsto \frac{2}{{k}^{2} \cdot \frac{{\sin k}^{2} \cdot t}{\color{blue}{\cos k \cdot {\ell}^{2}}}} \]

      4. times-frac 65.9%

         \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\left(\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t}{{\ell}^{2}}\right)}} \]

   9. Simplified 65.9%

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t}{{\ell}^{2}}\right)}} \]

   10. Taylor expanded in k around 0 54.0%

       \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
   11. Step-by-step derivation

       1. associate-/l* 55.6%

          \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}} \]

   12. Simplified 55.6%

       \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}} \]

   if 3.0000000000000001e-105 < t

   1. Initial program 72.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 67.6%

      \[\leadsto \color{blue}{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 61.0%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(2 \cdot {k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. cube-mult 61.0%

         \[\leadsto \frac{2}{\frac{\frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)} \]

      2. *-un-lft-identity 61.0%

         \[\leadsto \frac{2}{\frac{\frac{t \cdot \left(t \cdot t\right)}{\color{blue}{1 \cdot \ell}}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)} \]

      3. times-frac 64.5%

         \[\leadsto \frac{2}{\frac{\color{blue}{\frac{t}{1} \cdot \frac{t \cdot t}{\ell}}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)} \]

      4. pow2 64.5%

         \[\leadsto \frac{2}{\frac{\frac{t}{1} \cdot \frac{\color{blue}{{t}^{2}}}{\ell}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)} \]

   7. Applied egg-rr 64.5%

      \[\leadsto \frac{2}{\frac{\color{blue}{\frac{t}{1} \cdot \frac{{t}^{2}}{\ell}}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3 \cdot 10^{-105}:\\ \;\;\;\;\frac{2}{\frac{t}{{\ell}^{2}} \cdot {k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{t \cdot \frac{{t}^{2}}{\ell}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 59.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.5 \cdot 10^{-105}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{{\ell}^{2}} \cdot {k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{{t\_m}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.5e-105)
    (/ 2.0 (* (/ t_m (pow l 2.0)) (pow k 4.0)))
    (/ 2.0 (/ (* (/ (pow t_m 3.0) l) (* 2.0 (pow k 2.0))) l)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.5e-105) {
		tmp = 2.0 / ((t_m / pow(l, 2.0)) * pow(k, 4.0));
	} else {
		tmp = 2.0 / (((pow(t_m, 3.0) / l) * (2.0 * pow(k, 2.0))) / l);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2.5d-105) then
        tmp = 2.0d0 / ((t_m / (l ** 2.0d0)) * (k ** 4.0d0))
    else
        tmp = 2.0d0 / ((((t_m ** 3.0d0) / l) * (2.0d0 * (k ** 2.0d0))) / l)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.5e-105) {
		tmp = 2.0 / ((t_m / Math.pow(l, 2.0)) * Math.pow(k, 4.0));
	} else {
		tmp = 2.0 / (((Math.pow(t_m, 3.0) / l) * (2.0 * Math.pow(k, 2.0))) / l);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2.5e-105:
		tmp = 2.0 / ((t_m / math.pow(l, 2.0)) * math.pow(k, 4.0))
	else:
		tmp = 2.0 / (((math.pow(t_m, 3.0) / l) * (2.0 * math.pow(k, 2.0))) / l)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.5e-105)
		tmp = Float64(2.0 / Float64(Float64(t_m / (l ^ 2.0)) * (k ^ 4.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((t_m ^ 3.0) / l) * Float64(2.0 * (k ^ 2.0))) / l));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2.5e-105)
		tmp = 2.0 / ((t_m / (l ^ 2.0)) * (k ^ 4.0));
	else
		tmp = 2.0 / ((((t_m ^ 3.0) / l) * (2.0 * (k ^ 2.0))) / l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.5e-105], N[(2.0 / N[(N[(t$95$m / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.49999999999999982e-105

   1. Initial program 54.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 54.3%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-cube-cbrt 54.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}\right)} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      2. pow3 54.3%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}\right)}^{3}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      3. *-commutative 54.3%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\color{blue}{\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      4. cbrt-prod 54.3%

         \[\leadsto \frac{2}{{\color{blue}{\left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right)}}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      5. cbrt-div 54.2%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \color{blue}{\frac{\sqrt[3]{{t}^{3}}}{\sqrt[3]{\ell \cdot \ell}}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      6. rem-cbrt-cube 62.6%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{\color{blue}{t}}{\sqrt[3]{\ell \cdot \ell}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      7. cbrt-prod 66.9%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{\color{blue}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      8. pow2 66.9%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   6. Applied egg-rr 66.9%

      \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   7. Taylor expanded in k around inf 65.4%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   8. Step-by-step derivation

      1. associate-/l* 64.5%

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      2. *-commutative 64.5%

         \[\leadsto \frac{2}{{k}^{2} \cdot \frac{\color{blue}{{\sin k}^{2} \cdot t}}{{\ell}^{2} \cdot \cos k}} \]

      3. *-commutative 64.5%

         \[\leadsto \frac{2}{{k}^{2} \cdot \frac{{\sin k}^{2} \cdot t}{\color{blue}{\cos k \cdot {\ell}^{2}}}} \]

      4. times-frac 65.9%

         \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\left(\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t}{{\ell}^{2}}\right)}} \]

   9. Simplified 65.9%

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t}{{\ell}^{2}}\right)}} \]

   10. Taylor expanded in k around 0 54.0%

       \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
   11. Step-by-step derivation

       1. associate-/l* 55.6%

          \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}} \]

   12. Simplified 55.6%

       \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}} \]

   if 2.49999999999999982e-105 < t

   1. Initial program 72.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 67.6%

      \[\leadsto \color{blue}{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 61.0%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(2 \cdot {k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. associate-*l/ 61.1%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}} \]

   7. Applied egg-rr 61.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.5 \cdot 10^{-105}:\\ \;\;\;\;\frac{2}{\frac{t}{{\ell}^{2}} \cdot {k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 59.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.5 \cdot 10^{-105}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{{\ell}^{2}} \cdot {k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{{t\_m}^{3}}{\ell}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}\\ \end{array} \end{array} \]

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 (<= t_m 3.5e-105)
    (/ 2.0 (* (/ t_m (pow l 2.0)) (pow k 4.0)))
    (/ 2.0 (* (/ (/ (pow t_m 3.0) l) l) (* 2.0 (pow k 2.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.5e-105) {
		tmp = 2.0 / ((t_m / pow(l, 2.0)) * pow(k, 4.0));
	} else {
		tmp = 2.0 / (((pow(t_m, 3.0) / l) / l) * (2.0 * pow(k, 2.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 3.5d-105) then
        tmp = 2.0d0 / ((t_m / (l ** 2.0d0)) * (k ** 4.0d0))
    else
        tmp = 2.0d0 / ((((t_m ** 3.0d0) / l) / l) * (2.0d0 * (k ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.5e-105) {
		tmp = 2.0 / ((t_m / Math.pow(l, 2.0)) * Math.pow(k, 4.0));
	} else {
		tmp = 2.0 / (((Math.pow(t_m, 3.0) / l) / l) * (2.0 * Math.pow(k, 2.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 3.5e-105:
		tmp = 2.0 / ((t_m / math.pow(l, 2.0)) * math.pow(k, 4.0))
	else:
		tmp = 2.0 / (((math.pow(t_m, 3.0) / l) / l) * (2.0 * math.pow(k, 2.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 3.5e-105)
		tmp = Float64(2.0 / Float64(Float64(t_m / (l ^ 2.0)) * (k ^ 4.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((t_m ^ 3.0) / l) / l) * Float64(2.0 * (k ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 3.5e-105)
		tmp = 2.0 / ((t_m / (l ^ 2.0)) * (k ^ 4.0));
	else
		tmp = 2.0 / ((((t_m ^ 3.0) / l) / l) * (2.0 * (k ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 3.5e-105], N[(2.0 / N[(N[(t$95$m / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.5e-105

   1. Initial program 54.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 54.3%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-cube-cbrt 54.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}\right)} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      2. pow3 54.3%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}\right)}^{3}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      3. *-commutative 54.3%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\color{blue}{\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      4. cbrt-prod 54.3%

         \[\leadsto \frac{2}{{\color{blue}{\left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right)}}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      5. cbrt-div 54.2%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \color{blue}{\frac{\sqrt[3]{{t}^{3}}}{\sqrt[3]{\ell \cdot \ell}}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      6. rem-cbrt-cube 62.6%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{\color{blue}{t}}{\sqrt[3]{\ell \cdot \ell}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      7. cbrt-prod 66.9%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{\color{blue}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      8. pow2 66.9%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   6. Applied egg-rr 66.9%

      \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   7. Taylor expanded in k around inf 65.4%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   8. Step-by-step derivation

      1. associate-/l* 64.5%

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      2. *-commutative 64.5%

         \[\leadsto \frac{2}{{k}^{2} \cdot \frac{\color{blue}{{\sin k}^{2} \cdot t}}{{\ell}^{2} \cdot \cos k}} \]

      3. *-commutative 64.5%

         \[\leadsto \frac{2}{{k}^{2} \cdot \frac{{\sin k}^{2} \cdot t}{\color{blue}{\cos k \cdot {\ell}^{2}}}} \]

      4. times-frac 65.9%

         \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\left(\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t}{{\ell}^{2}}\right)}} \]

   9. Simplified 65.9%

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t}{{\ell}^{2}}\right)}} \]

   10. Taylor expanded in k around 0 54.0%

       \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
   11. Step-by-step derivation

       1. associate-/l* 55.6%

          \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}} \]

   12. Simplified 55.6%

       \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}} \]

   if 3.5e-105 < t

   1. Initial program 72.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 67.6%

      \[\leadsto \color{blue}{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 61.0%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(2 \cdot {k}^{2}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.5 \cdot 10^{-105}:\\ \;\;\;\;\frac{2}{\frac{t}{{\ell}^{2}} \cdot {k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 59.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6 \cdot 10^{-116}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{{\ell}^{2}} \cdot {k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{2}{\frac{{t\_m}^{3} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 6e-116)
    (/ 2.0 (* (/ t_m (pow l 2.0)) (pow k 4.0)))
    (* l (/ 2.0 (/ (* (pow t_m 3.0) (* 2.0 (pow k 2.0))) l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 6e-116) {
		tmp = 2.0 / ((t_m / pow(l, 2.0)) * pow(k, 4.0));
	} else {
		tmp = l * (2.0 / ((pow(t_m, 3.0) * (2.0 * pow(k, 2.0))) / l));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 6d-116) then
        tmp = 2.0d0 / ((t_m / (l ** 2.0d0)) * (k ** 4.0d0))
    else
        tmp = l * (2.0d0 / (((t_m ** 3.0d0) * (2.0d0 * (k ** 2.0d0))) / l))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 6e-116) {
		tmp = 2.0 / ((t_m / Math.pow(l, 2.0)) * Math.pow(k, 4.0));
	} else {
		tmp = l * (2.0 / ((Math.pow(t_m, 3.0) * (2.0 * Math.pow(k, 2.0))) / l));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 6e-116:
		tmp = 2.0 / ((t_m / math.pow(l, 2.0)) * math.pow(k, 4.0))
	else:
		tmp = l * (2.0 / ((math.pow(t_m, 3.0) * (2.0 * math.pow(k, 2.0))) / l))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 6e-116)
		tmp = Float64(2.0 / Float64(Float64(t_m / (l ^ 2.0)) * (k ^ 4.0)));
	else
		tmp = Float64(l * Float64(2.0 / Float64(Float64((t_m ^ 3.0) * Float64(2.0 * (k ^ 2.0))) / l)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 6e-116)
		tmp = 2.0 / ((t_m / (l ^ 2.0)) * (k ^ 4.0));
	else
		tmp = l * (2.0 / (((t_m ^ 3.0) * (2.0 * (k ^ 2.0))) / l));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 6e-116], N[(2.0 / N[(N[(t$95$m / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(2.0 / N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] * N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 6.00000000000000053e-116

   1. Initial program 54.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 54.5%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-cube-cbrt 54.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}\right)} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      2. pow3 54.4%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}\right)}^{3}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      3. *-commutative 54.4%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\color{blue}{\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      4. cbrt-prod 54.4%

         \[\leadsto \frac{2}{{\color{blue}{\left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right)}}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      5. cbrt-div 54.4%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \color{blue}{\frac{\sqrt[3]{{t}^{3}}}{\sqrt[3]{\ell \cdot \ell}}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      6. rem-cbrt-cube 63.1%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{\color{blue}{t}}{\sqrt[3]{\ell \cdot \ell}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      7. cbrt-prod 67.6%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{\color{blue}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      8. pow2 67.6%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   6. Applied egg-rr 67.6%

      \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   7. Taylor expanded in k around inf 64.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   8. Step-by-step derivation

      1. associate-/l* 63.9%

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      2. *-commutative 63.9%

         \[\leadsto \frac{2}{{k}^{2} \cdot \frac{\color{blue}{{\sin k}^{2} \cdot t}}{{\ell}^{2} \cdot \cos k}} \]

      3. *-commutative 63.9%

         \[\leadsto \frac{2}{{k}^{2} \cdot \frac{{\sin k}^{2} \cdot t}{\color{blue}{\cos k \cdot {\ell}^{2}}}} \]

      4. times-frac 65.3%

         \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\left(\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t}{{\ell}^{2}}\right)}} \]

   9. Simplified 65.3%

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t}{{\ell}^{2}}\right)}} \]

   10. Taylor expanded in k around 0 54.1%

       \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
   11. Step-by-step derivation

       1. associate-/l* 55.8%

          \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}} \]

   12. Simplified 55.8%

       \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}} \]

   if 6.00000000000000053e-116 < t

   1. Initial program 71.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 66.4%

      \[\leadsto \color{blue}{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 60.3%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(2 \cdot {k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. associate-*l/ 60.4%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}} \]

   7. Applied egg-rr 60.4%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}} \]
   8. Step-by-step derivation

      1. associate-/r/ 60.4%

         \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)} \cdot \ell} \]

      2. associate-*l/ 59.4%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}} \cdot \ell \]

   9. Applied egg-rr 59.4%

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}} \cdot \ell} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6 \cdot 10^{-116}:\\ \;\;\;\;\frac{2}{\frac{t}{{\ell}^{2}} \cdot {k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{2}{\frac{{t}^{3} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 53.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{\frac{t\_m}{{\ell}^{2}} \cdot {k}^{4}} \end{array} \]

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 (* (/ t_m (pow l 2.0)) (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) {
	return t_s * (2.0 / ((t_m / pow(l, 2.0)) * pow(k, 4.0)));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 / ((t_m / (l ** 2.0d0)) * (k ** 4.0d0)))
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 / ((t_m / Math.pow(l, 2.0)) * Math.pow(k, 4.0)));
}

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 / ((t_m / math.pow(l, 2.0)) * math.pow(k, 4.0)))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 / Float64(Float64(t_m / (l ^ 2.0)) * (k ^ 4.0))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 / ((t_m / (l ^ 2.0)) * (k ^ 4.0)));
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[(t$95$m / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.4%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

3. Simplified 60.4%

   \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-cube-cbrt 60.3%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}\right)} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   2. pow3 60.3%

      \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}\right)}^{3}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   3. *-commutative 60.3%

      \[\leadsto \frac{2}{{\left(\sqrt[3]{\color{blue}{\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   4. cbrt-prod 60.3%

      \[\leadsto \frac{2}{{\color{blue}{\left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right)}}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   5. cbrt-div 60.3%

      \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \color{blue}{\frac{\sqrt[3]{{t}^{3}}}{\sqrt[3]{\ell \cdot \ell}}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   6. rem-cbrt-cube 67.9%

      \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{\color{blue}{t}}{\sqrt[3]{\ell \cdot \ell}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   7. cbrt-prod 74.0%

      \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{\color{blue}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   8. pow2 74.0%

      \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

6. Applied egg-rr 74.0%

   \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

7. Taylor expanded in k around inf 62.8%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
8. Step-by-step derivation

   1. associate-/l* 62.6%

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

   2. *-commutative 62.6%

      \[\leadsto \frac{2}{{k}^{2} \cdot \frac{\color{blue}{{\sin k}^{2} \cdot t}}{{\ell}^{2} \cdot \cos k}} \]

   3. *-commutative 62.6%

      \[\leadsto \frac{2}{{k}^{2} \cdot \frac{{\sin k}^{2} \cdot t}{\color{blue}{\cos k \cdot {\ell}^{2}}}} \]

   4. times-frac 64.2%

      \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\left(\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t}{{\ell}^{2}}\right)}} \]

9. Simplified 64.2%

   \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t}{{\ell}^{2}}\right)}} \]

10. Taylor expanded in k around 0 53.2%

    \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
11. Step-by-step derivation

    1. associate-/l* 53.8%

       \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}} \]

12. Simplified 53.8%

    \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}} \]

13. Final simplification 53.8%

    \[\leadsto \frac{2}{\frac{t}{{\ell}^{2}} \cdot {k}^{4}} \]
14. Add Preprocessing

Alternative 24: 53.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{{k}^{4} \cdot \left(t\_m \cdot {\ell}^{-2}\right)} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ 2.0 (* (pow k 4.0) (* t_m (pow l -2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / (pow(k, 4.0) * (t_m * pow(l, -2.0))));
}

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 / ((k ** 4.0d0) * (t_m * (l ** (-2.0d0)))))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / (Math.pow(k, 4.0) * (t_m * Math.pow(l, -2.0))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 / (math.pow(k, 4.0) * (t_m * math.pow(l, -2.0))))

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((k ^ 4.0) * Float64(t_m * (l ^ -2.0)))))
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 / ((k ^ 4.0) * (t_m * (l ^ -2.0))));
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[Power[k, 4.0], $MachinePrecision] * N[(t$95$m * N[Power[l, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.4%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

3. Simplified 60.4%

   \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-cube-cbrt 60.3%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}\right)} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   2. pow3 60.3%

      \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}\right)}^{3}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   3. *-commutative 60.3%

      \[\leadsto \frac{2}{{\left(\sqrt[3]{\color{blue}{\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   4. cbrt-prod 60.3%

      \[\leadsto \frac{2}{{\color{blue}{\left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right)}}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   5. cbrt-div 60.3%

      \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \color{blue}{\frac{\sqrt[3]{{t}^{3}}}{\sqrt[3]{\ell \cdot \ell}}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   6. rem-cbrt-cube 67.9%

      \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{\color{blue}{t}}{\sqrt[3]{\ell \cdot \ell}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   7. cbrt-prod 74.0%

      \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{\color{blue}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   8. pow2 74.0%

      \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

6. Applied egg-rr 74.0%

   \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

7. Taylor expanded in k around inf 62.8%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
8. Step-by-step derivation

   1. associate-/l* 62.6%

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

   2. *-commutative 62.6%

      \[\leadsto \frac{2}{{k}^{2} \cdot \frac{\color{blue}{{\sin k}^{2} \cdot t}}{{\ell}^{2} \cdot \cos k}} \]

   3. *-commutative 62.6%

      \[\leadsto \frac{2}{{k}^{2} \cdot \frac{{\sin k}^{2} \cdot t}{\color{blue}{\cos k \cdot {\ell}^{2}}}} \]

   4. times-frac 64.2%

      \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\left(\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t}{{\ell}^{2}}\right)}} \]

9. Simplified 64.2%

   \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t}{{\ell}^{2}}\right)}} \]

10. Taylor expanded in k around 0 53.2%

    \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
11. Step-by-step derivation

    1. associate-/l* 53.8%

       \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}} \]

12. Simplified 53.8%

    \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}} \]
13. Step-by-step derivation

    1. div-inv 53.4%

       \[\leadsto \frac{2}{{k}^{4} \cdot \color{blue}{\left(t \cdot \frac{1}{{\ell}^{2}}\right)}} \]

    2. pow-flip 53.4%

       \[\leadsto \frac{2}{{k}^{4} \cdot \left(t \cdot \color{blue}{{\ell}^{\left(-2\right)}}\right)} \]

    3. metadata-eval 53.4%

       \[\leadsto \frac{2}{{k}^{4} \cdot \left(t \cdot {\ell}^{\color{blue}{-2}}\right)} \]

14. Applied egg-rr 53.4%

    \[\leadsto \frac{2}{{k}^{4} \cdot \color{blue}{\left(t \cdot {\ell}^{-2}\right)}} \]
15. Add Preprocessing

Alternative 25: 51.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left({\ell}^{2} \cdot \frac{2}{t\_m \cdot {k}^{4}}\right) \end{array} \]

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 (* (pow l 2.0) (/ 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) {
	return t_s * (pow(l, 2.0) * (2.0 / (t_m * pow(k, 4.0))));
}

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 * ((l ** 2.0d0) * (2.0d0 / (t_m * (k ** 4.0d0))))
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 * (Math.pow(l, 2.0) * (2.0 / (t_m * Math.pow(k, 4.0))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (math.pow(l, 2.0) * (2.0 / (t_m * math.pow(k, 4.0))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64((l ^ 2.0) * Float64(2.0 / Float64(t_m * (k ^ 4.0)))))
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 * ((l ^ 2.0) * (2.0 / (t_m * (k ^ 4.0))));
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[(N[Power[l, 2.0], $MachinePrecision] * N[(2.0 / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.4%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

3. Simplified 60.4%

   \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-cube-cbrt 60.3%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}\right)} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   2. pow3 60.3%

      \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}\right)}^{3}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   3. *-commutative 60.3%

      \[\leadsto \frac{2}{{\left(\sqrt[3]{\color{blue}{\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   4. cbrt-prod 60.3%

      \[\leadsto \frac{2}{{\color{blue}{\left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right)}}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   5. cbrt-div 60.3%

      \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \color{blue}{\frac{\sqrt[3]{{t}^{3}}}{\sqrt[3]{\ell \cdot \ell}}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   6. rem-cbrt-cube 67.9%

      \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{\color{blue}{t}}{\sqrt[3]{\ell \cdot \ell}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   7. cbrt-prod 74.0%

      \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{\color{blue}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   8. pow2 74.0%

      \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

6. Applied egg-rr 74.0%

   \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

7. Taylor expanded in k around inf 62.8%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
8. Step-by-step derivation

   1. associate-/l* 62.6%

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

   2. *-commutative 62.6%

      \[\leadsto \frac{2}{{k}^{2} \cdot \frac{\color{blue}{{\sin k}^{2} \cdot t}}{{\ell}^{2} \cdot \cos k}} \]

   3. *-commutative 62.6%

      \[\leadsto \frac{2}{{k}^{2} \cdot \frac{{\sin k}^{2} \cdot t}{\color{blue}{\cos k \cdot {\ell}^{2}}}} \]

   4. times-frac 64.2%

      \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\left(\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t}{{\ell}^{2}}\right)}} \]

9. Simplified 64.2%

   \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t}{{\ell}^{2}}\right)}} \]

10. Taylor expanded in k around 0 53.2%

    \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
11. Step-by-step derivation

    1. associate-/l* 53.8%

       \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}} \]

12. Simplified 53.8%

    \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}} \]
13. Step-by-step derivation

    1. div-inv 53.8%

       \[\leadsto \color{blue}{2 \cdot \frac{1}{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}} \]

    2. pow2 53.8%

       \[\leadsto 2 \cdot \frac{1}{{k}^{4} \cdot \frac{t}{\color{blue}{\ell \cdot \ell}}} \]

    3. associate-*r/ 53.2%

       \[\leadsto 2 \cdot \frac{1}{\color{blue}{\frac{{k}^{4} \cdot t}{\ell \cdot \ell}}} \]

    4. pow2 53.2%

       \[\leadsto 2 \cdot \frac{1}{\frac{{k}^{4} \cdot t}{\color{blue}{{\ell}^{2}}}} \]

14. Applied egg-rr 53.2%

    \[\leadsto \color{blue}{2 \cdot \frac{1}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
15. Step-by-step derivation

    1. associate-*r/ 53.2%

       \[\leadsto \color{blue}{\frac{2 \cdot 1}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]

    2. metadata-eval 53.2%

       \[\leadsto \frac{\color{blue}{2}}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}} \]

    3. associate-/r/ 53.2%

       \[\leadsto \color{blue}{\frac{2}{{k}^{4} \cdot t} \cdot {\ell}^{2}} \]

    4. *-commutative 53.2%

       \[\leadsto \frac{2}{\color{blue}{t \cdot {k}^{4}}} \cdot {\ell}^{2} \]

16. Simplified 53.2%

    \[\leadsto \color{blue}{\frac{2}{t \cdot {k}^{4}} \cdot {\ell}^{2}} \]

17. Final simplification 53.2%

    \[\leadsto {\ell}^{2} \cdot \frac{2}{t \cdot {k}^{4}} \]
18. Add Preprocessing

Reproduce

herbie shell --seed 2024103 
(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))))