Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.5% → 82.5%

Time: 18.4s

Alternatives: 26

Speedup: 24.8×

• 26.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.5% 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: 82.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\\ t_3 := \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t\_m}\right)\right)\\ t_4 := \frac{\ell}{t\_3}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.45 \cdot 10^{-185}:\\ \;\;\;\;2 \cdot \left({k}^{-2} \cdot \left({\ell}^{2} \cdot \left(\cos k \cdot \frac{{\sin k}^{-2}}{t\_m}\right)\right)\right)\\ \mathbf{elif}\;t\_m \leq 2.2 \cdot 10^{-83}:\\ \;\;\;\;\frac{2}{{\left(t\_2 \cdot \sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right)}\right)}^{3}}\\ \mathbf{elif}\;t\_m \leq 2.7 \cdot 10^{+82}:\\ \;\;\;\;t\_4 \cdot \left(\frac{2}{{t\_m}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot t\_4\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t\_2 \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot {t\_3}^{2}\right)}\\ \end{array} \end{array} \end{array} \]

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 (/ t_m (pow (cbrt l) 2.0)))
        (t_3 (hypot 1.0 (hypot 1.0 (/ k t_m))))
        (t_4 (/ l t_3)))
   (*
    t_s
    (if (<= t_m 2.45e-185)
      (*
       2.0
       (* (pow k -2.0) (* (pow l 2.0) (* (cos k) (/ (pow (sin k) -2.0) t_m)))))
      (if (<= t_m 2.2e-83)
        (/
         2.0
         (pow
          (* t_2 (cbrt (* (sin k) (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0))))))
          3.0))
        (if (<= t_m 2.7e+82)
          (* t_4 (* (/ 2.0 (* (pow t_m 3.0) (* (sin k) (tan k)))) t_4))
          (/
           2.0
           (*
            (pow (* t_2 (cbrt (sin k))) 3.0)
            (* (tan k) (pow t_3 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 = t_m / pow(cbrt(l), 2.0);
	double t_3 = hypot(1.0, hypot(1.0, (k / t_m)));
	double t_4 = l / t_3;
	double tmp;
	if (t_m <= 2.45e-185) {
		tmp = 2.0 * (pow(k, -2.0) * (pow(l, 2.0) * (cos(k) * (pow(sin(k), -2.0) / t_m))));
	} else if (t_m <= 2.2e-83) {
		tmp = 2.0 / pow((t_2 * cbrt((sin(k) * (tan(k) * (2.0 + pow((k / t_m), 2.0)))))), 3.0);
	} else if (t_m <= 2.7e+82) {
		tmp = t_4 * ((2.0 / (pow(t_m, 3.0) * (sin(k) * tan(k)))) * t_4);
	} else {
		tmp = 2.0 / (pow((t_2 * cbrt(sin(k))), 3.0) * (tan(k) * pow(t_3, 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 t_2 = t_m / Math.pow(Math.cbrt(l), 2.0);
	double t_3 = Math.hypot(1.0, Math.hypot(1.0, (k / t_m)));
	double t_4 = l / t_3;
	double tmp;
	if (t_m <= 2.45e-185) {
		tmp = 2.0 * (Math.pow(k, -2.0) * (Math.pow(l, 2.0) * (Math.cos(k) * (Math.pow(Math.sin(k), -2.0) / t_m))));
	} else if (t_m <= 2.2e-83) {
		tmp = 2.0 / Math.pow((t_2 * Math.cbrt((Math.sin(k) * (Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0)))))), 3.0);
	} else if (t_m <= 2.7e+82) {
		tmp = t_4 * ((2.0 / (Math.pow(t_m, 3.0) * (Math.sin(k) * Math.tan(k)))) * t_4);
	} else {
		tmp = 2.0 / (Math.pow((t_2 * Math.cbrt(Math.sin(k))), 3.0) * (Math.tan(k) * Math.pow(t_3, 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(t_m / (cbrt(l) ^ 2.0))
	t_3 = hypot(1.0, hypot(1.0, Float64(k / t_m)))
	t_4 = Float64(l / t_3)
	tmp = 0.0
	if (t_m <= 2.45e-185)
		tmp = Float64(2.0 * Float64((k ^ -2.0) * Float64((l ^ 2.0) * Float64(cos(k) * Float64((sin(k) ^ -2.0) / t_m)))));
	elseif (t_m <= 2.2e-83)
		tmp = Float64(2.0 / (Float64(t_2 * cbrt(Float64(sin(k) * Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0)))))) ^ 3.0));
	elseif (t_m <= 2.7e+82)
		tmp = Float64(t_4 * Float64(Float64(2.0 / Float64((t_m ^ 3.0) * Float64(sin(k) * tan(k)))) * t_4));
	else
		tmp = Float64(2.0 / Float64((Float64(t_2 * cbrt(sin(k))) ^ 3.0) * Float64(tan(k) * (t_3 ^ 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_] := Block[{t$95$2 = N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$4 = N[(l / t$95$3), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.45e-185], N[(2.0 * N[(N[Power[k, -2.0], $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], -2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.2e-83], N[(2.0 / N[Power[N[(t$95$2 * N[Power[N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.7e+82], N[(t$95$4 * N[(N[(2.0 / N[(N[Power[t$95$m, 3.0], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(t$95$2 * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Power[t$95$3, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < 2.4500000000000001e-185

   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. Taylor expanded in t around 0 68.0%

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

      1. associate-*r/ 68.0%

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

      2. times-frac 66.8%

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

      3. times-frac 68.8%

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

   7. Simplified 68.8%

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

      1. pow1 68.8%

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

      2. div-inv 68.8%

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

      3. pow-flip 69.6%

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

      4. metadata-eval 69.6%

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

      5. div-inv 69.5%

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

      6. pow-flip 69.5%

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

      7. metadata-eval 69.5%

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

   9. Applied egg-rr 69.5%

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

       1. unpow1 69.5%

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

       2. associate-*l* 69.5%

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

       3. associate-*l/ 68.2%

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

       4. associate-/l* 67.6%

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

       5. associate-/l* 67.6%

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

   11. Simplified 67.6%

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

   if 2.4500000000000001e-185 < t < 2.20000000000000008e-83

   1. Initial program 36.7%

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

   3. Simplified 36.7%

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

      1. associate-*l* 36.8%

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

      2. associate-/r* 45.7%

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

      3. associate-+r+ 45.7%

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

      4. metadata-eval 45.7%

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

      5. associate-*l* 45.7%

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

      6. add-cube-cbrt 45.5%

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

      7. pow3 45.6%

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

   6. Applied egg-rr 73.0%

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

   if 2.20000000000000008e-83 < t < 2.6999999999999999e82

   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 54.8%

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

      1. associate-*r* 57.3%

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

      2. add-sqr-sqrt 57.2%

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

      3. times-frac 62.8%

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

   6. Applied egg-rr 73.7%

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

      1. associate-/l* 81.5%

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

      2. associate-*l* 81.7%

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

   8. Simplified 81.7%

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

   if 2.6999999999999999e82 < t

   1. Initial program 52.9%

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

   3. Simplified 52.9%

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

      1. add-cube-cbrt 52.9%

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

      2. pow3 52.9%

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

      3. associate-/r* 65.8%

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

      4. *-commutative 65.8%

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

      5. cbrt-prod 65.8%

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

      6. associate-/r* 52.9%

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

      7. cbrt-div 56.7%

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

      8. rem-cbrt-cube 67.3%

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

      9. cbrt-prod 96.3%

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

      10. pow2 96.3%

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

   6. Applied egg-rr 96.3%

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

      1. add-sqr-sqrt 56.4%

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

      2. pow2 56.4%

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

      3. associate-+r+ 56.4%

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

      4. metadata-eval 56.4%

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

      5. sqrt-prod 56.5%

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

      6. metadata-eval 56.5%

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

      7. associate-+r+ 56.5%

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

      8. add-sqr-sqrt 56.5%

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

      9. hypot-1-def 56.5%

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

      10. unpow2 56.5%

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

      11. hypot-1-def 56.5%

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

   8. Applied egg-rr 56.5%

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

      1. unpow2 56.5%

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

      2. swap-sqr 56.5%

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

      3. rem-square-sqrt 96.3%

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

      4. unpow2 96.3%

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

   10. Simplified 96.3%

       \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \color{blue}{\left(\tan k \cdot {\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.45 \cdot 10^{-185}:\\ \;\;\;\;2 \cdot \left({k}^{-2} \cdot \left({\ell}^{2} \cdot \left(\cos k \cdot \frac{{\sin k}^{-2}}{t}\right)\right)\right)\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{-83}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{3}}\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{+82}:\\ \;\;\;\;\frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \left(\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot {\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\\ t_3 := \frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t\_m}\right)\right)}\\ t_4 := \frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.8 \cdot 10^{-185}:\\ \;\;\;\;2 \cdot \left({k}^{-2} \cdot \left({\ell}^{2} \cdot \left(\cos k \cdot \frac{{\sin k}^{-2}}{t\_m}\right)\right)\right)\\ \mathbf{elif}\;t\_m \leq 2.95 \cdot 10^{-83}:\\ \;\;\;\;\frac{2}{{\left(t\_4 \cdot \sqrt[3]{\sin k \cdot t\_2}\right)}^{3}}\\ \mathbf{elif}\;t\_m \leq 2.05 \cdot 10^{+82}:\\ \;\;\;\;t\_3 \cdot \left(\frac{2}{{t\_m}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot t\_3\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_2 \cdot {\left(t\_4 \cdot \sqrt[3]{\sin k}\right)}^{3}}\\ \end{array} \end{array} \end{array} \]

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 (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0))))
        (t_3 (/ l (hypot 1.0 (hypot 1.0 (/ k t_m)))))
        (t_4 (/ t_m (pow (cbrt l) 2.0))))
   (*
    t_s
    (if (<= t_m 5.8e-185)
      (*
       2.0
       (* (pow k -2.0) (* (pow l 2.0) (* (cos k) (/ (pow (sin k) -2.0) t_m)))))
      (if (<= t_m 2.95e-83)
        (/ 2.0 (pow (* t_4 (cbrt (* (sin k) t_2))) 3.0))
        (if (<= t_m 2.05e+82)
          (* t_3 (* (/ 2.0 (* (pow t_m 3.0) (* (sin k) (tan k)))) t_3))
          (/ 2.0 (* t_2 (pow (* t_4 (cbrt (sin k))) 3.0)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = tan(k) * (2.0 + pow((k / t_m), 2.0));
	double t_3 = l / hypot(1.0, hypot(1.0, (k / t_m)));
	double t_4 = t_m / pow(cbrt(l), 2.0);
	double tmp;
	if (t_m <= 5.8e-185) {
		tmp = 2.0 * (pow(k, -2.0) * (pow(l, 2.0) * (cos(k) * (pow(sin(k), -2.0) / t_m))));
	} else if (t_m <= 2.95e-83) {
		tmp = 2.0 / pow((t_4 * cbrt((sin(k) * t_2))), 3.0);
	} else if (t_m <= 2.05e+82) {
		tmp = t_3 * ((2.0 / (pow(t_m, 3.0) * (sin(k) * tan(k)))) * t_3);
	} else {
		tmp = 2.0 / (t_2 * pow((t_4 * cbrt(sin(k))), 3.0));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0));
	double t_3 = l / Math.hypot(1.0, Math.hypot(1.0, (k / t_m)));
	double t_4 = t_m / Math.pow(Math.cbrt(l), 2.0);
	double tmp;
	if (t_m <= 5.8e-185) {
		tmp = 2.0 * (Math.pow(k, -2.0) * (Math.pow(l, 2.0) * (Math.cos(k) * (Math.pow(Math.sin(k), -2.0) / t_m))));
	} else if (t_m <= 2.95e-83) {
		tmp = 2.0 / Math.pow((t_4 * Math.cbrt((Math.sin(k) * t_2))), 3.0);
	} else if (t_m <= 2.05e+82) {
		tmp = t_3 * ((2.0 / (Math.pow(t_m, 3.0) * (Math.sin(k) * Math.tan(k)))) * t_3);
	} else {
		tmp = 2.0 / (t_2 * Math.pow((t_4 * Math.cbrt(Math.sin(k))), 3.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0)))
	t_3 = Float64(l / hypot(1.0, hypot(1.0, Float64(k / t_m))))
	t_4 = Float64(t_m / (cbrt(l) ^ 2.0))
	tmp = 0.0
	if (t_m <= 5.8e-185)
		tmp = Float64(2.0 * Float64((k ^ -2.0) * Float64((l ^ 2.0) * Float64(cos(k) * Float64((sin(k) ^ -2.0) / t_m)))));
	elseif (t_m <= 2.95e-83)
		tmp = Float64(2.0 / (Float64(t_4 * cbrt(Float64(sin(k) * t_2))) ^ 3.0));
	elseif (t_m <= 2.05e+82)
		tmp = Float64(t_3 * Float64(Float64(2.0 / Float64((t_m ^ 3.0) * Float64(sin(k) * tan(k)))) * t_3));
	else
		tmp = Float64(2.0 / Float64(t_2 * (Float64(t_4 * cbrt(sin(k))) ^ 3.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(l / N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 5.8e-185], N[(2.0 * N[(N[Power[k, -2.0], $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], -2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.95e-83], N[(2.0 / N[Power[N[(t$95$4 * N[Power[N[(N[Sin[k], $MachinePrecision] * t$95$2), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.05e+82], N[(t$95$3 * N[(N[(2.0 / N[(N[Power[t$95$m, 3.0], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$2 * N[Power[N[(t$95$4 * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < 5.79999999999999989e-185

   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. Taylor expanded in t around 0 68.0%

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

      1. associate-*r/ 68.0%

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

      2. times-frac 66.8%

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

      3. times-frac 68.8%

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

   7. Simplified 68.8%

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

      1. pow1 68.8%

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

      2. div-inv 68.8%

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

      3. pow-flip 69.6%

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

      4. metadata-eval 69.6%

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

      5. div-inv 69.5%

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

      6. pow-flip 69.5%

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

      7. metadata-eval 69.5%

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

   9. Applied egg-rr 69.5%

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

       1. unpow1 69.5%

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

       2. associate-*l* 69.5%

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

       3. associate-*l/ 68.2%

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

       4. associate-/l* 67.6%

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

       5. associate-/l* 67.6%

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

   11. Simplified 67.6%

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

   if 5.79999999999999989e-185 < t < 2.9499999999999998e-83

   1. Initial program 36.7%

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

   3. Simplified 36.7%

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

      1. associate-*l* 36.8%

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

      2. associate-/r* 45.7%

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

      3. associate-+r+ 45.7%

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

      4. metadata-eval 45.7%

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

      5. associate-*l* 45.7%

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

      6. add-cube-cbrt 45.5%

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

      7. pow3 45.6%

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

   6. Applied egg-rr 73.0%

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

   if 2.9499999999999998e-83 < t < 2.04999999999999998e82

   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 54.8%

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

      1. associate-*r* 57.3%

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

      2. add-sqr-sqrt 57.2%

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

      3. times-frac 62.8%

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

   6. Applied egg-rr 73.7%

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

      1. associate-/l* 81.5%

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

      2. associate-*l* 81.7%

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

   8. Simplified 81.7%

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

   if 2.04999999999999998e82 < t

   1. Initial program 52.9%

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

   3. Simplified 52.9%

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

      1. add-cube-cbrt 52.9%

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

      2. pow3 52.9%

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

      3. associate-/r* 65.8%

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

      4. *-commutative 65.8%

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

      5. cbrt-prod 65.8%

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

      6. associate-/r* 52.9%

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

      7. cbrt-div 56.7%

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

      8. rem-cbrt-cube 67.3%

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

      9. cbrt-prod 96.3%

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

      10. pow2 96.3%

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

   6. Applied egg-rr 96.3%

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

      1. pow1 96.3%

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

      2. associate-+r+ 96.3%

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

      3. metadata-eval 96.3%

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

   8. Applied egg-rr 96.3%

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

      1. unpow1 96.3%

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

   10. Simplified 96.3%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.8 \cdot 10^{-185}:\\ \;\;\;\;2 \cdot \left({k}^{-2} \cdot \left({\ell}^{2} \cdot \left(\cos k \cdot \frac{{\sin k}^{-2}}{t}\right)\right)\right)\\ \mathbf{elif}\;t \leq 2.95 \cdot 10^{-83}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{3}}\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{+82}:\\ \;\;\;\;\frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \left(\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\\ t_3 := \frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.4 \cdot 10^{-185}:\\ \;\;\;\;2 \cdot \left({k}^{-2} \cdot \left({\ell}^{2} \cdot \left(\cos k \cdot \frac{{\sin k}^{-2}}{t\_m}\right)\right)\right)\\ \mathbf{elif}\;t\_m \leq 5.4 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{{\left(t\_3 \cdot \sqrt[3]{\sin k \cdot t\_2}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_2 \cdot {\left(t\_3 \cdot \sqrt[3]{\sin k}\right)}^{3}}\\ \end{array} \end{array} \end{array} \]

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 (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0))))
        (t_3 (/ t_m (pow (cbrt l) 2.0))))
   (*
    t_s
    (if (<= t_m 5.4e-185)
      (*
       2.0
       (* (pow k -2.0) (* (pow l 2.0) (* (cos k) (/ (pow (sin k) -2.0) t_m)))))
      (if (<= t_m 5.4e-18)
        (/ 2.0 (pow (* t_3 (cbrt (* (sin k) t_2))) 3.0))
        (/ 2.0 (* t_2 (pow (* t_3 (cbrt (sin k))) 3.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = tan(k) * (2.0 + pow((k / t_m), 2.0));
	double t_3 = t_m / pow(cbrt(l), 2.0);
	double tmp;
	if (t_m <= 5.4e-185) {
		tmp = 2.0 * (pow(k, -2.0) * (pow(l, 2.0) * (cos(k) * (pow(sin(k), -2.0) / t_m))));
	} else if (t_m <= 5.4e-18) {
		tmp = 2.0 / pow((t_3 * cbrt((sin(k) * t_2))), 3.0);
	} else {
		tmp = 2.0 / (t_2 * pow((t_3 * cbrt(sin(k))), 3.0));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0));
	double t_3 = t_m / Math.pow(Math.cbrt(l), 2.0);
	double tmp;
	if (t_m <= 5.4e-185) {
		tmp = 2.0 * (Math.pow(k, -2.0) * (Math.pow(l, 2.0) * (Math.cos(k) * (Math.pow(Math.sin(k), -2.0) / t_m))));
	} else if (t_m <= 5.4e-18) {
		tmp = 2.0 / Math.pow((t_3 * Math.cbrt((Math.sin(k) * t_2))), 3.0);
	} else {
		tmp = 2.0 / (t_2 * Math.pow((t_3 * Math.cbrt(Math.sin(k))), 3.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0)))
	t_3 = Float64(t_m / (cbrt(l) ^ 2.0))
	tmp = 0.0
	if (t_m <= 5.4e-185)
		tmp = Float64(2.0 * Float64((k ^ -2.0) * Float64((l ^ 2.0) * Float64(cos(k) * Float64((sin(k) ^ -2.0) / t_m)))));
	elseif (t_m <= 5.4e-18)
		tmp = Float64(2.0 / (Float64(t_3 * cbrt(Float64(sin(k) * t_2))) ^ 3.0));
	else
		tmp = Float64(2.0 / Float64(t_2 * (Float64(t_3 * cbrt(sin(k))) ^ 3.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 5.4e-185], N[(2.0 * N[(N[Power[k, -2.0], $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], -2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.4e-18], N[(2.0 / N[Power[N[(t$95$3 * N[Power[N[(N[Sin[k], $MachinePrecision] * t$95$2), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$2 * N[Power[N[(t$95$3 * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 5.39999999999999976e-185

   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. Taylor expanded in t around 0 68.0%

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

      1. associate-*r/ 68.0%

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

      2. times-frac 66.8%

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

      3. times-frac 68.8%

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

   7. Simplified 68.8%

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

      1. pow1 68.8%

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

      2. div-inv 68.8%

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

      3. pow-flip 69.6%

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

      4. metadata-eval 69.6%

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

      5. div-inv 69.5%

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

      6. pow-flip 69.5%

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

      7. metadata-eval 69.5%

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

   9. Applied egg-rr 69.5%

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

       1. unpow1 69.5%

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

       2. associate-*l* 69.5%

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

       3. associate-*l/ 68.2%

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

       4. associate-/l* 67.6%

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

       5. associate-/l* 67.6%

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

   11. Simplified 67.6%

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

   if 5.39999999999999976e-185 < t < 5.39999999999999977e-18

   1. Initial program 36.3%

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

   3. Simplified 36.3%

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

      1. associate-*l* 33.6%

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

      2. associate-/r* 42.1%

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

      3. associate-+r+ 42.1%

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

      4. metadata-eval 42.1%

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

      5. associate-*l* 42.1%

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

      6. add-cube-cbrt 41.9%

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

      7. pow3 41.9%

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

   6. Applied egg-rr 67.9%

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

   if 5.39999999999999977e-18 < t

   1. Initial program 62.5%

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

   3. Simplified 62.5%

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

      1. add-cube-cbrt 62.4%

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

      2. pow3 62.5%

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

      3. associate-/r* 71.4%

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

      4. *-commutative 71.4%

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

      5. cbrt-prod 71.4%

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

      6. associate-/r* 62.3%

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

      7. cbrt-div 64.9%

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

      8. rem-cbrt-cube 72.0%

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

      9. cbrt-prod 91.7%

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

      10. pow2 91.7%

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

   6. Applied egg-rr 91.7%

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

      1. pow1 91.7%

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

      2. associate-+r+ 91.7%

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

      3. metadata-eval 91.7%

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

   8. Applied egg-rr 91.7%

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

      1. unpow1 91.7%

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

   10. Simplified 91.7%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.4 \cdot 10^{-185}:\\ \;\;\;\;2 \cdot \left({k}^{-2} \cdot \left({\ell}^{2} \cdot \left(\cos k \cdot \frac{{\sin k}^{-2}}{t}\right)\right)\right)\\ \mathbf{elif}\;t \leq 5.4 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.8 \cdot 10^{-185}:\\ \;\;\;\;2 \cdot \left({k}^{-2} \cdot \left({\ell}^{2} \cdot \left(\cos k \cdot \frac{{\sin k}^{-2}}{t\_m}\right)\right)\right)\\ \mathbf{elif}\;t\_m \leq 5.3 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{{\left(t\_m \cdot \left(\sqrt[3]{\sin k \cdot t\_2} \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_2 \cdot {\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3}}\\ \end{array} \end{array} \end{array} \]

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 (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))))
   (*
    t_s
    (if (<= t_m 2.8e-185)
      (*
       2.0
       (* (pow k -2.0) (* (pow l 2.0) (* (cos k) (/ (pow (sin k) -2.0) t_m)))))
      (if (<= t_m 5.3e-18)
        (/
         2.0
         (pow (* t_m (* (cbrt (* (sin k) t_2)) (pow (cbrt l) -2.0))) 3.0))
        (/
         2.0
         (* t_2 (pow (* (/ t_m (pow (cbrt l) 2.0)) (cbrt (sin k))) 3.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = tan(k) * (2.0 + pow((k / t_m), 2.0));
	double tmp;
	if (t_m <= 2.8e-185) {
		tmp = 2.0 * (pow(k, -2.0) * (pow(l, 2.0) * (cos(k) * (pow(sin(k), -2.0) / t_m))));
	} else if (t_m <= 5.3e-18) {
		tmp = 2.0 / pow((t_m * (cbrt((sin(k) * t_2)) * pow(cbrt(l), -2.0))), 3.0);
	} else {
		tmp = 2.0 / (t_2 * pow(((t_m / pow(cbrt(l), 2.0)) * cbrt(sin(k))), 3.0));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0));
	double tmp;
	if (t_m <= 2.8e-185) {
		tmp = 2.0 * (Math.pow(k, -2.0) * (Math.pow(l, 2.0) * (Math.cos(k) * (Math.pow(Math.sin(k), -2.0) / t_m))));
	} else if (t_m <= 5.3e-18) {
		tmp = 2.0 / Math.pow((t_m * (Math.cbrt((Math.sin(k) * t_2)) * Math.pow(Math.cbrt(l), -2.0))), 3.0);
	} else {
		tmp = 2.0 / (t_2 * Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt(Math.sin(k))), 3.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0)))
	tmp = 0.0
	if (t_m <= 2.8e-185)
		tmp = Float64(2.0 * Float64((k ^ -2.0) * Float64((l ^ 2.0) * Float64(cos(k) * Float64((sin(k) ^ -2.0) / t_m)))));
	elseif (t_m <= 5.3e-18)
		tmp = Float64(2.0 / (Float64(t_m * Float64(cbrt(Float64(sin(k) * t_2)) * (cbrt(l) ^ -2.0))) ^ 3.0));
	else
		tmp = Float64(2.0 / Float64(t_2 * (Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * cbrt(sin(k))) ^ 3.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.8e-185], N[(2.0 * N[(N[Power[k, -2.0], $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], -2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.3e-18], N[(2.0 / N[Power[N[(t$95$m * N[(N[Power[N[(N[Sin[k], $MachinePrecision] * t$95$2), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$2 * N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 2.79999999999999991e-185

   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. Taylor expanded in t around 0 68.0%

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

      1. associate-*r/ 68.0%

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

      2. times-frac 66.8%

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

      3. times-frac 68.8%

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

   7. Simplified 68.8%

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

      1. pow1 68.8%

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

      2. div-inv 68.8%

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

      3. pow-flip 69.6%

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

      4. metadata-eval 69.6%

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

      5. div-inv 69.5%

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

      6. pow-flip 69.5%

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

      7. metadata-eval 69.5%

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

   9. Applied egg-rr 69.5%

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

       1. unpow1 69.5%

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

       2. associate-*l* 69.5%

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

       3. associate-*l/ 68.2%

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

       4. associate-/l* 67.6%

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

       5. associate-/l* 67.6%

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

   11. Simplified 67.6%

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

   if 2.79999999999999991e-185 < t < 5.3000000000000003e-18

   1. Initial program 36.3%

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

   3. Simplified 42.1%

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

      1. add-cube-cbrt 41.8%

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

      2. *-un-lft-identity 41.8%

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

      3. times-frac 41.8%

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

      4. pow2 41.8%

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

      5. cbrt-div 41.8%

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

      6. rem-cbrt-cube 42.0%

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

      7. cbrt-div 42.0%

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

      8. rem-cbrt-cube 56.9%

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

   6. Applied egg-rr 56.9%

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

      1. add-cube-cbrt 56.7%

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

      2. pow3 56.7%

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

   8. Applied egg-rr 68.0%

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

      1. associate-*l* 67.9%

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

   10. Simplified 67.9%

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

   if 5.3000000000000003e-18 < t

   1. Initial program 62.5%

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

   3. Simplified 62.5%

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

      1. add-cube-cbrt 62.4%

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

      2. pow3 62.5%

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

      3. associate-/r* 71.4%

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

      4. *-commutative 71.4%

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

      5. cbrt-prod 71.4%

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

      6. associate-/r* 62.3%

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

      7. cbrt-div 64.9%

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

      8. rem-cbrt-cube 72.0%

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

      9. cbrt-prod 91.7%

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

      10. pow2 91.7%

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

   6. Applied egg-rr 91.7%

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

      1. pow1 91.7%

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

      2. associate-+r+ 91.7%

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

      3. metadata-eval 91.7%

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

   8. Applied egg-rr 91.7%

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

      1. unpow1 91.7%

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

   10. Simplified 91.7%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.8 \cdot 10^{-185}:\\ \;\;\;\;2 \cdot \left({k}^{-2} \cdot \left({\ell}^{2} \cdot \left(\cos k \cdot \frac{{\sin k}^{-2}}{t}\right)\right)\right)\\ \mathbf{elif}\;t \leq 5.3 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{{\left(t \cdot \left(\sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sin k \cdot \tan k\\ t_3 := \frac{t\_m}{\sqrt[3]{\ell}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 5.2 \cdot 10^{-159}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \mathbf{elif}\;k \leq 0.25:\\ \;\;\;\;\frac{2}{{\left(\frac{{t\_m}^{1.5}}{\ell} \cdot \left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t\_m}\right)\right) \cdot \sqrt{t\_2}\right)\right)}^{2}}\\ \mathbf{elif}\;k \leq 5 \cdot 10^{+123}:\\ \;\;\;\;\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left(t\_m \cdot {k}^{2}\right) \cdot {\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left({t\_3}^{2} \cdot \frac{t\_3}{\ell}\right) \cdot \left(\left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right) \cdot t\_2\right)}\\ \end{array} \end{array} \end{array} \]

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 (* (sin k) (tan k))) (t_3 (/ t_m (cbrt l))))
   (*
    t_s
    (if (<= k 5.2e-159)
      (/
       2.0
       (* (pow (* (/ t_m (pow (cbrt l) 2.0)) (cbrt (sin k))) 3.0) (* 2.0 k)))
      (if (<= k 0.25)
        (/
         2.0
         (pow
          (*
           (/ (pow t_m 1.5) l)
           (* (hypot 1.0 (hypot 1.0 (/ k t_m))) (sqrt t_2)))
          2.0))
        (if (<= k 5e+123)
          (/
           (* 2.0 (* (pow l 2.0) (cos k)))
           (* (* t_m (pow k 2.0)) (pow (sin k) 2.0)))
          (/
           2.0
           (*
            (* (pow t_3 2.0) (/ t_3 l))
            (* (+ 2.0 (pow (/ k t_m) 2.0)) t_2)))))))))

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 = sin(k) * tan(k);
	double t_3 = t_m / cbrt(l);
	double tmp;
	if (k <= 5.2e-159) {
		tmp = 2.0 / (pow(((t_m / pow(cbrt(l), 2.0)) * cbrt(sin(k))), 3.0) * (2.0 * k));
	} else if (k <= 0.25) {
		tmp = 2.0 / pow(((pow(t_m, 1.5) / l) * (hypot(1.0, hypot(1.0, (k / t_m))) * sqrt(t_2))), 2.0);
	} else if (k <= 5e+123) {
		tmp = (2.0 * (pow(l, 2.0) * cos(k))) / ((t_m * pow(k, 2.0)) * pow(sin(k), 2.0));
	} else {
		tmp = 2.0 / ((pow(t_3, 2.0) * (t_3 / l)) * ((2.0 + pow((k / t_m), 2.0)) * t_2));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.sin(k) * Math.tan(k);
	double t_3 = t_m / Math.cbrt(l);
	double tmp;
	if (k <= 5.2e-159) {
		tmp = 2.0 / (Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt(Math.sin(k))), 3.0) * (2.0 * k));
	} else if (k <= 0.25) {
		tmp = 2.0 / Math.pow(((Math.pow(t_m, 1.5) / l) * (Math.hypot(1.0, Math.hypot(1.0, (k / t_m))) * Math.sqrt(t_2))), 2.0);
	} else if (k <= 5e+123) {
		tmp = (2.0 * (Math.pow(l, 2.0) * Math.cos(k))) / ((t_m * Math.pow(k, 2.0)) * Math.pow(Math.sin(k), 2.0));
	} else {
		tmp = 2.0 / ((Math.pow(t_3, 2.0) * (t_3 / l)) * ((2.0 + Math.pow((k / t_m), 2.0)) * t_2));
	}
	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(sin(k) * tan(k))
	t_3 = Float64(t_m / cbrt(l))
	tmp = 0.0
	if (k <= 5.2e-159)
		tmp = Float64(2.0 / Float64((Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * cbrt(sin(k))) ^ 3.0) * Float64(2.0 * k)));
	elseif (k <= 0.25)
		tmp = Float64(2.0 / (Float64(Float64((t_m ^ 1.5) / l) * Float64(hypot(1.0, hypot(1.0, Float64(k / t_m))) * sqrt(t_2))) ^ 2.0));
	elseif (k <= 5e+123)
		tmp = Float64(Float64(2.0 * Float64((l ^ 2.0) * cos(k))) / Float64(Float64(t_m * (k ^ 2.0)) * (sin(k) ^ 2.0)));
	else
		tmp = Float64(2.0 / Float64(Float64((t_3 ^ 2.0) * Float64(t_3 / l)) * Float64(Float64(2.0 + (Float64(k / t_m) ^ 2.0)) * t_2)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 5.2e-159], N[(2.0 / N[(N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.25], N[(2.0 / N[Power[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[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 5e+123], N[(N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[t$95$3, 2.0], $MachinePrecision] * N[(t$95$3 / l), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if k < 5.1999999999999997e-159

   1. Initial program 60.7%

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

   3. Simplified 60.7%

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

      1. add-cube-cbrt 60.7%

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

      2. pow3 60.7%

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

      3. associate-/r* 67.8%

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

      4. *-commutative 67.8%

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

      5. cbrt-prod 67.7%

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

      6. associate-/r* 60.6%

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

      7. cbrt-div 63.0%

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

      8. rem-cbrt-cube 72.4%

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

      9. cbrt-prod 81.8%

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

      10. pow2 81.8%

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

   6. Applied egg-rr 81.8%

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

   7. Taylor expanded in k around 0 71.3%

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

   if 5.1999999999999997e-159 < k < 0.25

   1. Initial program 61.9%

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

   3. Simplified 61.9%

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

      1. associate-*l* 61.8%

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

      2. associate-/r* 63.5%

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

      3. associate-+r+ 63.5%

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

      4. metadata-eval 63.5%

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

      5. associate-*l* 63.5%

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

      6. add-sqr-sqrt 38.8%

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

      7. pow2 38.8%

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

   6. Applied egg-rr 50.0%

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

   if 0.25 < k < 4.99999999999999974e123

   1. Initial program 40.0%

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

   3. Simplified 40.0%

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

   5. Taylor expanded in t around 0 79.1%

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

      1. associate-*r/ 79.1%

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

      2. associate-*r* 79.1%

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

   7. Simplified 79.1%

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

   if 4.99999999999999974e123 < k

   1. Initial program 34.1%

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

   3. Simplified 34.6%

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

      1. add-cube-cbrt 34.6%

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

      2. *-un-lft-identity 34.6%

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

      3. times-frac 34.6%

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

      4. pow2 34.6%

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

      5. cbrt-div 34.6%

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

      6. rem-cbrt-cube 34.6%

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

      7. cbrt-div 34.6%

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

      8. rem-cbrt-cube 43.8%

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

   6. Applied egg-rr 43.8%

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

      1. /-rgt-identity 43.8%

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

      2. *-un-lft-identity 43.8%

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

   8. Applied egg-rr 43.8%

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

      1. *-lft-identity 43.8%

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

   10. Simplified 43.8%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.2 \cdot 10^{-159}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \mathbf{elif}\;k \leq 0.25:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right) \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}\\ \mathbf{elif}\;k \leq 5 \cdot 10^{+123}:\\ \;\;\;\;\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left(t \cdot {k}^{2}\right) \cdot {\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left({\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right) \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.75 \cdot 10^{-126}:\\ \;\;\;\;2 \cdot \left({k}^{-2} \cdot \left({\ell}^{2} \cdot \left(\cos k \cdot \frac{{\sin k}^{-2}}{t\_m}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right) \cdot {\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.75e-126)
    (*
     2.0
     (* (pow k -2.0) (* (pow l 2.0) (* (cos k) (/ (pow (sin k) -2.0) t_m)))))
    (/
     2.0
     (*
      (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))
      (pow (* (/ t_m (pow (cbrt l) 2.0)) (cbrt (sin k))) 3.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.75e-126) {
		tmp = 2.0 * (pow(k, -2.0) * (pow(l, 2.0) * (cos(k) * (pow(sin(k), -2.0) / t_m))));
	} else {
		tmp = 2.0 / ((tan(k) * (2.0 + pow((k / t_m), 2.0))) * pow(((t_m / pow(cbrt(l), 2.0)) * cbrt(sin(k))), 3.0));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.75e-126) {
		tmp = 2.0 * (Math.pow(k, -2.0) * (Math.pow(l, 2.0) * (Math.cos(k) * (Math.pow(Math.sin(k), -2.0) / t_m))));
	} else {
		tmp = 2.0 / ((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0))) * Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt(Math.sin(k))), 3.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.75e-126)
		tmp = Float64(2.0 * Float64((k ^ -2.0) * Float64((l ^ 2.0) * Float64(cos(k) * Float64((sin(k) ^ -2.0) / t_m)))));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * (Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * cbrt(sin(k))) ^ 3.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.75e-126], N[(2.0 * N[(N[Power[k, -2.0], $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], -2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $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[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.74999999999999993e-126

   1. Initial program 52.9%

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

   3. Simplified 52.9%

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

   5. Taylor expanded in t around 0 66.7%

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

      1. associate-*r/ 66.7%

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

      2. times-frac 66.2%

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

      3. times-frac 68.0%

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

   7. Simplified 68.0%

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

      1. pow1 68.0%

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

      2. div-inv 68.0%

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

      3. pow-flip 68.7%

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

      4. metadata-eval 68.7%

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

      5. div-inv 68.7%

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

      6. pow-flip 68.7%

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

      7. metadata-eval 68.7%

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

   9. Applied egg-rr 68.7%

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

       1. unpow1 68.7%

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

       2. associate-*l* 68.7%

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

       3. associate-*l/ 67.5%

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

       4. associate-/l* 66.9%

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

       5. associate-/l* 66.9%

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

   11. Simplified 66.9%

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

   if 2.74999999999999993e-126 < t

   1. Initial program 56.8%

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

   3. Simplified 56.7%

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

      1. add-cube-cbrt 56.7%

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

      2. pow3 56.7%

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

      3. associate-/r* 66.5%

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

      4. *-commutative 66.5%

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

      5. cbrt-prod 66.5%

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

      6. associate-/r* 56.6%

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

      7. cbrt-div 58.6%

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

      8. rem-cbrt-cube 65.2%

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

      9. cbrt-prod 84.3%

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

      10. pow2 84.3%

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

   6. Applied egg-rr 84.3%

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

      1. pow1 84.3%

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

      2. associate-+r+ 84.3%

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

      3. metadata-eval 84.3%

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

   8. Applied egg-rr 84.3%

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

      1. unpow1 84.3%

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

   10. Simplified 84.3%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.75 \cdot 10^{-126}:\\ \;\;\;\;2 \cdot \left({k}^{-2} \cdot \left({\ell}^{2} \cdot \left(\cos k \cdot \frac{{\sin k}^{-2}}{t}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m}{\sqrt[3]{\ell}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 7.4 \cdot 10^{-28}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \mathbf{elif}\;k \leq 1.62 \cdot 10^{+124}:\\ \;\;\;\;\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left(t\_m \cdot {k}^{2}\right) \cdot {\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left({t\_2}^{2} \cdot \frac{t\_2}{\ell}\right) \cdot \left(\left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ t_m (cbrt l))))
   (*
    t_s
    (if (<= k 7.4e-28)
      (/
       2.0
       (* (pow (* (/ t_m (pow (cbrt l) 2.0)) (cbrt (sin k))) 3.0) (* 2.0 k)))
      (if (<= k 1.62e+124)
        (/
         (* 2.0 (* (pow l 2.0) (cos k)))
         (* (* t_m (pow k 2.0)) (pow (sin k) 2.0)))
        (/
         2.0
         (*
          (* (pow t_2 2.0) (/ t_2 l))
          (* (+ 2.0 (pow (/ k t_m) 2.0)) (* (sin k) (tan k))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = t_m / cbrt(l);
	double tmp;
	if (k <= 7.4e-28) {
		tmp = 2.0 / (pow(((t_m / pow(cbrt(l), 2.0)) * cbrt(sin(k))), 3.0) * (2.0 * k));
	} else if (k <= 1.62e+124) {
		tmp = (2.0 * (pow(l, 2.0) * cos(k))) / ((t_m * pow(k, 2.0)) * pow(sin(k), 2.0));
	} else {
		tmp = 2.0 / ((pow(t_2, 2.0) * (t_2 / l)) * ((2.0 + pow((k / t_m), 2.0)) * (sin(k) * tan(k))));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = t_m / Math.cbrt(l);
	double tmp;
	if (k <= 7.4e-28) {
		tmp = 2.0 / (Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt(Math.sin(k))), 3.0) * (2.0 * k));
	} else if (k <= 1.62e+124) {
		tmp = (2.0 * (Math.pow(l, 2.0) * Math.cos(k))) / ((t_m * Math.pow(k, 2.0)) * Math.pow(Math.sin(k), 2.0));
	} else {
		tmp = 2.0 / ((Math.pow(t_2, 2.0) * (t_2 / l)) * ((2.0 + Math.pow((k / t_m), 2.0)) * (Math.sin(k) * Math.tan(k))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(t_m / cbrt(l))
	tmp = 0.0
	if (k <= 7.4e-28)
		tmp = Float64(2.0 / Float64((Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * cbrt(sin(k))) ^ 3.0) * Float64(2.0 * k)));
	elseif (k <= 1.62e+124)
		tmp = Float64(Float64(2.0 * Float64((l ^ 2.0) * cos(k))) / Float64(Float64(t_m * (k ^ 2.0)) * (sin(k) ^ 2.0)));
	else
		tmp = Float64(2.0 / Float64(Float64((t_2 ^ 2.0) * Float64(t_2 / l)) * Float64(Float64(2.0 + (Float64(k / t_m) ^ 2.0)) * Float64(sin(k) * tan(k)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 7.4e-28], N[(2.0 / N[(N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.62e+124], N[(N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[t$95$2, 2.0], $MachinePrecision] * N[(t$95$2 / l), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 7.40000000000000039e-28

   1. Initial program 62.4%

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

   3. Simplified 62.4%

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

      1. add-cube-cbrt 62.3%

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

      2. pow3 62.3%

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

      3. associate-/r* 68.5%

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

      4. *-commutative 68.5%

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

      5. cbrt-prod 68.4%

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

      6. associate-/r* 62.2%

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

      7. cbrt-div 64.3%

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

      8. rem-cbrt-cube 72.9%

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

      9. cbrt-prod 82.9%

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

      10. pow2 82.9%

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

   6. Applied egg-rr 82.9%

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

   7. Taylor expanded in k around 0 73.5%

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

   if 7.40000000000000039e-28 < k < 1.62e124

   1. Initial program 35.9%

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

   3. Simplified 35.9%

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

   5. Taylor expanded in t around 0 77.0%

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

      1. associate-*r/ 77.0%

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

      2. associate-*r* 77.0%

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

   7. Simplified 77.0%

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

   if 1.62e124 < k

   1. Initial program 34.1%

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

   3. Simplified 34.6%

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

      1. add-cube-cbrt 34.6%

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

      2. *-un-lft-identity 34.6%

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

      3. times-frac 34.6%

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

      4. pow2 34.6%

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

      5. cbrt-div 34.6%

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

      6. rem-cbrt-cube 34.6%

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

      7. cbrt-div 34.6%

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

      8. rem-cbrt-cube 43.8%

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

   6. Applied egg-rr 43.8%

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

      1. /-rgt-identity 43.8%

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

      2. *-un-lft-identity 43.8%

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

   8. Applied egg-rr 43.8%

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

      1. *-lft-identity 43.8%

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

   10. Simplified 43.8%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.4 \cdot 10^{-28}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \mathbf{elif}\;k \leq 1.62 \cdot 10^{+124}:\\ \;\;\;\;\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left(t \cdot {k}^{2}\right) \cdot {\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left({\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2} \cdot \frac{\frac{t}{\sqrt[3]{\ell}}}{\ell}\right) \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 7.4 \cdot 10^{-28}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \mathbf{elif}\;k \leq 1.6 \cdot 10^{+124}:\\ \;\;\;\;\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left(t\_m \cdot {k}^{2}\right) \cdot {\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right)\right) \cdot {\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 7.4e-28)
    (/
     2.0
     (* (pow (* (/ t_m (pow (cbrt l) 2.0)) (cbrt (sin k))) 3.0) (* 2.0 k)))
    (if (<= k 1.6e+124)
      (/
       (* 2.0 (* (pow l 2.0) (cos k)))
       (* (* t_m (pow k 2.0)) (pow (sin k) 2.0)))
      (/
       2.0
       (*
        (* (sin k) (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0))))
        (pow (* t_m (pow (cbrt l) -2.0)) 3.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 7.4e-28) {
		tmp = 2.0 / (pow(((t_m / pow(cbrt(l), 2.0)) * cbrt(sin(k))), 3.0) * (2.0 * k));
	} else if (k <= 1.6e+124) {
		tmp = (2.0 * (pow(l, 2.0) * cos(k))) / ((t_m * pow(k, 2.0)) * pow(sin(k), 2.0));
	} else {
		tmp = 2.0 / ((sin(k) * (tan(k) * (2.0 + pow((k / t_m), 2.0)))) * pow((t_m * pow(cbrt(l), -2.0)), 3.0));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 7.4e-28) {
		tmp = 2.0 / (Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt(Math.sin(k))), 3.0) * (2.0 * k));
	} else if (k <= 1.6e+124) {
		tmp = (2.0 * (Math.pow(l, 2.0) * Math.cos(k))) / ((t_m * Math.pow(k, 2.0)) * Math.pow(Math.sin(k), 2.0));
	} else {
		tmp = 2.0 / ((Math.sin(k) * (Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0)))) * Math.pow((t_m * Math.pow(Math.cbrt(l), -2.0)), 3.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 7.4e-28)
		tmp = Float64(2.0 / Float64((Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * cbrt(sin(k))) ^ 3.0) * Float64(2.0 * k)));
	elseif (k <= 1.6e+124)
		tmp = Float64(Float64(2.0 * Float64((l ^ 2.0) * cos(k))) / Float64(Float64(t_m * (k ^ 2.0)) * (sin(k) ^ 2.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(sin(k) * Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0)))) * (Float64(t_m * (cbrt(l) ^ -2.0)) ^ 3.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 7.4e-28], N[(2.0 / N[(N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.6e+124], N[(N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 7.40000000000000039e-28

   1. Initial program 62.4%

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

   3. Simplified 62.4%

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

      1. add-cube-cbrt 62.3%

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

      2. pow3 62.3%

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

      3. associate-/r* 68.5%

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

      4. *-commutative 68.5%

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

      5. cbrt-prod 68.4%

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

      6. associate-/r* 62.2%

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

      7. cbrt-div 64.3%

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

      8. rem-cbrt-cube 72.9%

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

      9. cbrt-prod 82.9%

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

      10. pow2 82.9%

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

   6. Applied egg-rr 82.9%

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

   7. Taylor expanded in k around 0 73.5%

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

   if 7.40000000000000039e-28 < k < 1.59999999999999996e124

   1. Initial program 35.9%

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

   3. Simplified 35.9%

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

   5. Taylor expanded in t around 0 77.0%

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

      1. associate-*r/ 77.0%

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

      2. associate-*r* 77.0%

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

   7. Simplified 77.0%

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

   if 1.59999999999999996e124 < k

   1. Initial program 34.1%

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

   3. Simplified 34.6%

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

      1. add-cube-cbrt 34.6%

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

      2. *-un-lft-identity 34.6%

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

      3. times-frac 34.6%

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

      4. pow2 34.6%

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

      5. cbrt-div 34.6%

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

      6. rem-cbrt-cube 34.6%

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

      7. cbrt-div 34.6%

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

      8. rem-cbrt-cube 43.8%

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

   6. Applied egg-rr 43.8%

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

      1. add-cube-cbrt 43.8%

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

      2. pow3 43.8%

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

   8. Applied egg-rr 52.4%

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

      1. *-commutative 52.4%

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

      2. cube-prod 43.8%

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

      3. rem-cube-cbrt 43.7%

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

      4. *-commutative 43.7%

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

   10. Simplified 43.7%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.4 \cdot 10^{-28}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \mathbf{elif}\;k \leq 1.6 \cdot 10^{+124}:\\ \;\;\;\;\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left(t \cdot {k}^{2}\right) \cdot {\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right) \cdot {\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 + {\left(\frac{k}{t\_m}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9.2 \cdot 10^{-99}:\\ \;\;\;\;\frac{2}{k \cdot k} \cdot \left(\frac{{\ell}^{2}}{t\_m} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \mathbf{elif}\;t\_m \leq 9.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{2}{\frac{{t\_m}^{3}}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \frac{t\_2}{\ell}\right)\right)}\\ \mathbf{elif}\;t\_m \leq 6 \cdot 10^{+103}:\\ \;\;\;\;\left(\ell \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot {t\_m}^{3}\right)}\right) \cdot \frac{\ell}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (+ 2.0 (pow (/ k t_m) 2.0))))
   (*
    t_s
    (if (<= t_m 9.2e-99)
      (* (/ 2.0 (* k k)) (* (/ (pow l 2.0) t_m) (/ (cos k) (pow (sin k) 2.0))))
      (if (<= t_m 9.5e+19)
        (/ 2.0 (* (/ (pow t_m 3.0) l) (* (sin k) (* (tan k) (/ t_2 l)))))
        (if (<= t_m 6e+103)
          (* (* l (/ 2.0 (* (tan k) (* (sin k) (pow t_m 3.0))))) (/ l t_2))
          (/
           2.0
           (*
            (pow (* (/ t_m (pow (cbrt l) 2.0)) (cbrt (sin k))) 3.0)
            (* 2.0 k)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = 2.0 + pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 9.2e-99) {
		tmp = (2.0 / (k * k)) * ((pow(l, 2.0) / t_m) * (cos(k) / pow(sin(k), 2.0)));
	} else if (t_m <= 9.5e+19) {
		tmp = 2.0 / ((pow(t_m, 3.0) / l) * (sin(k) * (tan(k) * (t_2 / l))));
	} else if (t_m <= 6e+103) {
		tmp = (l * (2.0 / (tan(k) * (sin(k) * pow(t_m, 3.0))))) * (l / t_2);
	} else {
		tmp = 2.0 / (pow(((t_m / pow(cbrt(l), 2.0)) * cbrt(sin(k))), 3.0) * (2.0 * k));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = 2.0 + Math.pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 9.2e-99) {
		tmp = (2.0 / (k * k)) * ((Math.pow(l, 2.0) / t_m) * (Math.cos(k) / Math.pow(Math.sin(k), 2.0)));
	} else if (t_m <= 9.5e+19) {
		tmp = 2.0 / ((Math.pow(t_m, 3.0) / l) * (Math.sin(k) * (Math.tan(k) * (t_2 / l))));
	} else if (t_m <= 6e+103) {
		tmp = (l * (2.0 / (Math.tan(k) * (Math.sin(k) * Math.pow(t_m, 3.0))))) * (l / t_2);
	} else {
		tmp = 2.0 / (Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt(Math.sin(k))), 3.0) * (2.0 * k));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(2.0 + (Float64(k / t_m) ^ 2.0))
	tmp = 0.0
	if (t_m <= 9.2e-99)
		tmp = Float64(Float64(2.0 / Float64(k * k)) * Float64(Float64((l ^ 2.0) / t_m) * Float64(cos(k) / (sin(k) ^ 2.0))));
	elseif (t_m <= 9.5e+19)
		tmp = Float64(2.0 / Float64(Float64((t_m ^ 3.0) / l) * Float64(sin(k) * Float64(tan(k) * Float64(t_2 / l)))));
	elseif (t_m <= 6e+103)
		tmp = Float64(Float64(l * Float64(2.0 / Float64(tan(k) * Float64(sin(k) * (t_m ^ 3.0))))) * Float64(l / t_2));
	else
		tmp = Float64(2.0 / Float64((Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * cbrt(sin(k))) ^ 3.0) * Float64(2.0 * k)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 9.2e-99], N[(N[(2.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[l, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 9.5e+19], N[(2.0 / N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(t$95$2 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6e+103], 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 / t$95$2), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 9.1999999999999994e-99

   1. Initial program 51.1%

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

   3. Simplified 51.1%

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

   5. Taylor expanded in t around 0 65.6%

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

      1. associate-*r/ 65.6%

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

      2. times-frac 65.1%

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

      3. times-frac 66.9%

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

   7. Simplified 66.9%

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

      1. unpow2 66.9%

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

   9. Applied egg-rr 66.9%

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

   if 9.1999999999999994e-99 < t < 9.5e19

   1. Initial program 64.7%

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

   3. Simplified 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. associate-*l* 49.9%

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

      2. associate-/r* 54.2%

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

      3. associate-+r+ 54.2%

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

      4. metadata-eval 54.2%

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

      5. associate-*l* 54.2%

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

      6. associate-*l/ 54.5%

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

      7. associate-*l* 54.5%

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

   6. Applied egg-rr 54.5%

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

      1. associate-/l* 66.5%

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

      2. associate-*r* 66.4%

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

   8. Simplified 66.4%

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

      1. *-un-lft-identity 66.4%

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

      2. associate-/l* 66.4%

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

   10. Applied egg-rr 66.4%

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

       1. *-lft-identity 66.4%

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

       2. associate-*l* 78.6%

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

   12. Simplified 78.6%

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

   if 9.5e19 < t < 6e103

   1. Initial program 61.5%

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

   3. Simplified 70.3%

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

      1. associate-*r* 78.8%

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

      2. *-un-lft-identity 78.8%

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

      3. times-frac 79.0%

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

      4. associate-/l/ 79.0%

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

   6. Applied egg-rr 79.0%

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

   if 6e103 < t

   1. Initial program 57.6%

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

   3. Simplified 57.6%

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

      1. add-cube-cbrt 57.6%

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

      2. pow3 57.6%

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

      3. associate-/r* 67.9%

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

      4. *-commutative 67.9%

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

      5. cbrt-prod 67.9%

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

      6. associate-/r* 57.6%

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

      7. cbrt-div 57.6%

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

      8. rem-cbrt-cube 69.6%

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

      9. cbrt-prod 96.3%

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

      10. pow2 96.3%

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

   6. Applied egg-rr 96.3%

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

   7. Taylor expanded in k around 0 86.0%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9.2 \cdot 10^{-99}:\\ \;\;\;\;\frac{2}{k \cdot k} \cdot \left(\frac{{\ell}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{2}{\frac{{t}^{3}}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \frac{2 + {\left(\frac{k}{t}\right)}^{2}}{\ell}\right)\right)}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+103}:\\ \;\;\;\;\left(\ell \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot {t}^{3}\right)}\right) \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6.6 \cdot 10^{-28}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left(t\_m \cdot {k}^{2}\right) \cdot {\sin k}^{2}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 6.6e-28)
    (/
     2.0
     (* (pow (* (/ t_m (pow (cbrt l) 2.0)) (cbrt (sin k))) 3.0) (* 2.0 k)))
    (/
     (* 2.0 (* (pow l 2.0) (cos k)))
     (* (* t_m (pow k 2.0)) (pow (sin k) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 6.6e-28) {
		tmp = 2.0 / (pow(((t_m / pow(cbrt(l), 2.0)) * cbrt(sin(k))), 3.0) * (2.0 * k));
	} else {
		tmp = (2.0 * (pow(l, 2.0) * cos(k))) / ((t_m * pow(k, 2.0)) * pow(sin(k), 2.0));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 6.6e-28) {
		tmp = 2.0 / (Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt(Math.sin(k))), 3.0) * (2.0 * k));
	} else {
		tmp = (2.0 * (Math.pow(l, 2.0) * Math.cos(k))) / ((t_m * Math.pow(k, 2.0)) * Math.pow(Math.sin(k), 2.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 6.6e-28)
		tmp = Float64(2.0 / Float64((Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * cbrt(sin(k))) ^ 3.0) * Float64(2.0 * k)));
	else
		tmp = Float64(Float64(2.0 * Float64((l ^ 2.0) * cos(k))) / Float64(Float64(t_m * (k ^ 2.0)) * (sin(k) ^ 2.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 6.6e-28], N[(2.0 / N[(N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 6.6000000000000003e-28

   1. Initial program 62.4%

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

   3. Simplified 62.4%

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

      1. add-cube-cbrt 62.3%

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

      2. pow3 62.3%

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

      3. associate-/r* 68.5%

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

      4. *-commutative 68.5%

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

      5. cbrt-prod 68.4%

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

      6. associate-/r* 62.2%

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

      7. cbrt-div 64.3%

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

      8. rem-cbrt-cube 72.9%

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

      9. cbrt-prod 82.9%

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

      10. pow2 82.9%

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

   6. Applied egg-rr 82.9%

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

   7. Taylor expanded in k around 0 73.5%

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

   if 6.6000000000000003e-28 < k

   1. Initial program 34.9%

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

   3. Simplified 34.9%

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

   5. Taylor expanded in t around 0 59.5%

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

      1. associate-*r/ 59.5%

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

      2. associate-*r* 59.5%

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

   7. Simplified 59.5%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.6 \cdot 10^{-28}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left(t \cdot {k}^{2}\right) \cdot {\sin k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 70.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 5.5 \cdot 10^{-28}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{\cos k \cdot \frac{{\sin k}^{-2}}{t\_m}}{{k}^{2}}\right)\\ \end{array} \end{array} \]

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 5.5e-28)
    (/
     2.0
     (* (pow (* (/ t_m (pow (cbrt l) 2.0)) (cbrt (sin k))) 3.0) (* 2.0 k)))
    (*
     2.0
     (* (pow l 2.0) (/ (* (cos k) (/ (pow (sin k) -2.0) t_m)) (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 (k <= 5.5e-28) {
		tmp = 2.0 / (pow(((t_m / pow(cbrt(l), 2.0)) * cbrt(sin(k))), 3.0) * (2.0 * k));
	} else {
		tmp = 2.0 * (pow(l, 2.0) * ((cos(k) * (pow(sin(k), -2.0) / t_m)) / pow(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 <= 5.5e-28) {
		tmp = 2.0 / (Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt(Math.sin(k))), 3.0) * (2.0 * k));
	} else {
		tmp = 2.0 * (Math.pow(l, 2.0) * ((Math.cos(k) * (Math.pow(Math.sin(k), -2.0) / t_m)) / 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 (k <= 5.5e-28)
		tmp = Float64(2.0 / Float64((Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * cbrt(sin(k))) ^ 3.0) * Float64(2.0 * k)));
	else
		tmp = Float64(2.0 * Float64((l ^ 2.0) * Float64(Float64(cos(k) * Float64((sin(k) ^ -2.0) / t_m)) / (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, 5.5e-28], N[(2.0 / N[(N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], -2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 5.49999999999999967e-28

   1. Initial program 62.4%

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

   3. Simplified 62.4%

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

      1. add-cube-cbrt 62.3%

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

      2. pow3 62.3%

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

      3. associate-/r* 68.5%

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

      4. *-commutative 68.5%

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

      5. cbrt-prod 68.4%

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

      6. associate-/r* 62.2%

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

      7. cbrt-div 64.3%

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

      8. rem-cbrt-cube 72.9%

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

      9. cbrt-prod 82.9%

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

      10. pow2 82.9%

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

   6. Applied egg-rr 82.9%

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

   7. Taylor expanded in k around 0 73.5%

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

   if 5.49999999999999967e-28 < k

   1. Initial program 34.9%

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

   3. Simplified 34.9%

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

   5. Taylor expanded in t around 0 59.5%

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

      1. associate-*r/ 59.5%

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

      2. times-frac 57.1%

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

      3. times-frac 57.2%

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

   7. Simplified 57.2%

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

      1. associate-*l/ 57.2%

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

      2. div-inv 57.2%

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

      3. pow-flip 57.2%

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

      4. metadata-eval 57.2%

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

   9. Applied egg-rr 57.2%

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

       1. associate-/l* 57.2%

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

       2. associate-*l/ 57.1%

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

       3. associate-/l* 57.2%

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

       4. associate-/l* 59.4%

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

       5. associate-/l* 59.5%

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

   11. Simplified 59.5%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.5 \cdot 10^{-28}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{\cos k \cdot \frac{{\sin k}^{-2}}{t}}{{k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 69.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 4 \cdot 10^{-28}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({k}^{-2} \cdot \left({\ell}^{2} \cdot \left(\cos k \cdot \frac{{\sin k}^{-2}}{t\_m}\right)\right)\right)\\ \end{array} \end{array} \]

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 (<= k 4e-28)
    (/
     2.0
     (* (pow (* (/ t_m (pow (cbrt l) 2.0)) (cbrt (sin k))) 3.0) (* 2.0 k)))
    (*
     2.0
     (*
      (pow k -2.0)
      (* (pow l 2.0) (* (cos k) (/ (pow (sin k) -2.0) t_m))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 4e-28) {
		tmp = 2.0 / (pow(((t_m / pow(cbrt(l), 2.0)) * cbrt(sin(k))), 3.0) * (2.0 * k));
	} else {
		tmp = 2.0 * (pow(k, -2.0) * (pow(l, 2.0) * (cos(k) * (pow(sin(k), -2.0) / t_m))));
	}
	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 <= 4e-28) {
		tmp = 2.0 / (Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt(Math.sin(k))), 3.0) * (2.0 * k));
	} else {
		tmp = 2.0 * (Math.pow(k, -2.0) * (Math.pow(l, 2.0) * (Math.cos(k) * (Math.pow(Math.sin(k), -2.0) / t_m))));
	}
	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 <= 4e-28)
		tmp = Float64(2.0 / Float64((Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * cbrt(sin(k))) ^ 3.0) * Float64(2.0 * k)));
	else
		tmp = Float64(2.0 * Float64((k ^ -2.0) * Float64((l ^ 2.0) * Float64(cos(k) * Float64((sin(k) ^ -2.0) / t_m)))));
	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, 4e-28], N[(2.0 / N[(N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[k, -2.0], $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], -2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.99999999999999988e-28

   1. Initial program 62.4%

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

   3. Simplified 62.4%

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

      1. add-cube-cbrt 62.3%

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

      2. pow3 62.3%

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

      3. associate-/r* 68.5%

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

      4. *-commutative 68.5%

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

      5. cbrt-prod 68.4%

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

      6. associate-/r* 62.2%

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

      7. cbrt-div 64.3%

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

      8. rem-cbrt-cube 72.9%

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

      9. cbrt-prod 82.9%

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

      10. pow2 82.9%

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

   6. Applied egg-rr 82.9%

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

   7. Taylor expanded in k around 0 73.5%

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

   if 3.99999999999999988e-28 < k

   1. Initial program 34.9%

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

   3. Simplified 34.9%

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

   5. Taylor expanded in t around 0 59.5%

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

      1. associate-*r/ 59.5%

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

      2. times-frac 57.1%

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

      3. times-frac 57.2%

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

   7. Simplified 57.2%

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

      1. pow1 57.2%

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

      2. div-inv 57.2%

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

      3. pow-flip 58.7%

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

      4. metadata-eval 58.7%

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

      5. div-inv 58.6%

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

      6. pow-flip 58.6%

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

      7. metadata-eval 58.6%

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

   9. Applied egg-rr 58.6%

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

       1. unpow1 58.6%

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

       2. associate-*l* 58.6%

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

       3. associate-*l/ 58.6%

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

       4. associate-/l* 58.7%

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

       5. associate-/l* 58.7%

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

   11. Simplified 58.7%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4 \cdot 10^{-28}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({k}^{-2} \cdot \left({\ell}^{2} \cdot \left(\cos k \cdot \frac{{\sin k}^{-2}}{t}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 63.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.5 \cdot 10^{-125}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t\_m \cdot {k}^{2}}\\ \mathbf{elif}\;t\_m \leq 8.5 \cdot 10^{+23}:\\ \;\;\;\;\frac{2}{\frac{{t\_m}^{3}}{\ell} \cdot \frac{2 \cdot {k}^{2}}{\ell}}\\ \mathbf{elif}\;t\_m \leq 2.22 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{\frac{2}{k}}{k \cdot {t\_m}^{3}} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left({\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{1.5}\right)}^{2} \cdot \left(2 \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.5e-125)
    (* (/ 2.0 (pow k 2.0)) (/ (pow l 2.0) (* t_m (pow k 2.0))))
    (if (<= t_m 8.5e+23)
      (/ 2.0 (* (/ (pow t_m 3.0) l) (/ (* 2.0 (pow k 2.0)) l)))
      (if (<= t_m 2.22e+102)
        (/
         (* (/ (/ 2.0 k) (* k (pow t_m 3.0))) (* l l))
         (+ 2.0 (pow (/ k t_m) 2.0)))
        (/
         2.0
         (*
          (pow (pow (* t_m (pow (cbrt l) -2.0)) 1.5) 2.0)
          (* 2.0 (* k k)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.5e-125) {
		tmp = (2.0 / pow(k, 2.0)) * (pow(l, 2.0) / (t_m * pow(k, 2.0)));
	} else if (t_m <= 8.5e+23) {
		tmp = 2.0 / ((pow(t_m, 3.0) / l) * ((2.0 * pow(k, 2.0)) / l));
	} else if (t_m <= 2.22e+102) {
		tmp = (((2.0 / k) / (k * pow(t_m, 3.0))) * (l * l)) / (2.0 + pow((k / t_m), 2.0));
	} else {
		tmp = 2.0 / (pow(pow((t_m * pow(cbrt(l), -2.0)), 1.5), 2.0) * (2.0 * (k * k)));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.5e-125) {
		tmp = (2.0 / Math.pow(k, 2.0)) * (Math.pow(l, 2.0) / (t_m * Math.pow(k, 2.0)));
	} else if (t_m <= 8.5e+23) {
		tmp = 2.0 / ((Math.pow(t_m, 3.0) / l) * ((2.0 * Math.pow(k, 2.0)) / l));
	} else if (t_m <= 2.22e+102) {
		tmp = (((2.0 / k) / (k * Math.pow(t_m, 3.0))) * (l * l)) / (2.0 + Math.pow((k / t_m), 2.0));
	} else {
		tmp = 2.0 / (Math.pow(Math.pow((t_m * Math.pow(Math.cbrt(l), -2.0)), 1.5), 2.0) * (2.0 * (k * k)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 3.5e-125)
		tmp = Float64(Float64(2.0 / (k ^ 2.0)) * Float64((l ^ 2.0) / Float64(t_m * (k ^ 2.0))));
	elseif (t_m <= 8.5e+23)
		tmp = Float64(2.0 / Float64(Float64((t_m ^ 3.0) / l) * Float64(Float64(2.0 * (k ^ 2.0)) / l)));
	elseif (t_m <= 2.22e+102)
		tmp = Float64(Float64(Float64(Float64(2.0 / k) / Float64(k * (t_m ^ 3.0))) * Float64(l * l)) / Float64(2.0 + (Float64(k / t_m) ^ 2.0)));
	else
		tmp = Float64(2.0 / Float64(((Float64(t_m * (cbrt(l) ^ -2.0)) ^ 1.5) ^ 2.0) * Float64(2.0 * Float64(k * k))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 3.5e-125], N[(N[(2.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 8.5e+23], N[(2.0 / N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.22e+102], N[(N[(N[(N[(2.0 / k), $MachinePrecision] / N[(k * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[Power[N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 2.0], $MachinePrecision] * N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if t < 3.49999999999999998e-125

   1. Initial program 52.6%

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

   3. Simplified 52.6%

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

   5. Taylor expanded in t around 0 66.9%

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

      1. associate-*r/ 66.9%

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

      2. times-frac 66.4%

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

      3. times-frac 68.2%

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

   7. Simplified 68.2%

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

   8. Taylor expanded in k around 0 60.8%

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

   if 3.49999999999999998e-125 < t < 8.5000000000000001e23

   1. Initial program 58.0%

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

   3. Simplified 57.9%

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

      1. associate-*l* 44.3%

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

      2. associate-/r* 53.5%

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

      3. associate-+r+ 53.5%

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

      4. metadata-eval 53.5%

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

      5. associate-*l* 53.5%

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

      6. associate-*l/ 53.7%

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

      7. associate-*l* 53.7%

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

   6. Applied egg-rr 53.7%

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

      1. associate-/l* 60.0%

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

      2. associate-*r* 60.0%

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

   8. Simplified 60.0%

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

   9. Taylor expanded in k around 0 45.4%

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

   if 8.5000000000000001e23 < t < 2.22000000000000009e102

   1. Initial program 58.7%

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

   3. Simplified 73.8%

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

   5. Taylor expanded in k around 0 68.2%

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

   6. Taylor expanded in k around 0 73.4%

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

   if 2.22000000000000009e102 < t

   1. Initial program 56.3%

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

   3. Simplified 54.7%

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

   5. Taylor expanded in k around 0 54.7%

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

      1. unpow2 41.9%

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

   7. Applied egg-rr 54.7%

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

      1. associate-/r* 44.7%

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

      2. unpow3 44.7%

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

      3. times-frac 57.2%

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

      4. pow2 57.2%

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

   9. Applied egg-rr 57.2%

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

       1. add-cube-cbrt 57.2%

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

       2. pow3 57.2%

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

       3. frac-times 44.7%

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

       4. unpow2 44.7%

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

       5. unpow3 44.7%

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

       6. cbrt-div 44.7%

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

       7. unpow3 44.7%

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

       8. add-cbrt-cube 49.9%

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

       9. cbrt-prod 64.3%

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

       10. unpow2 64.3%

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

       11. pow-to-exp 28.2%

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

       12. *-commutative 28.2%

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

       13. sqr-pow 28.2%

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

       14. pow2 28.2%

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

   11. Applied egg-rr 64.4%

       \[\leadsto \frac{2}{\color{blue}{{\left({\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{1.5}\right)}^{2}} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.5 \cdot 10^{-125}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t \cdot {k}^{2}}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{+23}:\\ \;\;\;\;\frac{2}{\frac{{t}^{3}}{\ell} \cdot \frac{2 \cdot {k}^{2}}{\ell}}\\ \mathbf{elif}\;t \leq 2.22 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{\frac{2}{k}}{k \cdot {t}^{3}} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left({\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{1.5}\right)}^{2} \cdot \left(2 \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 73.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 + {\left(\frac{k}{t\_m}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5 \cdot 10^{-100}:\\ \;\;\;\;\frac{2}{k \cdot k} \cdot \left(\frac{{\ell}^{2}}{t\_m} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \mathbf{elif}\;t\_m \leq 9.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{2}{\frac{{t\_m}^{3}}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \frac{t\_2}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot {t\_m}^{3}\right)}\right) \cdot \frac{\ell}{t\_2}\\ \end{array} \end{array} \end{array} \]

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 5e-100)
      (* (/ 2.0 (* k k)) (* (/ (pow l 2.0) t_m) (/ (cos k) (pow (sin k) 2.0))))
      (if (<= t_m 9.5e+19)
        (/ 2.0 (* (/ (pow t_m 3.0) l) (* (sin k) (* (tan k) (/ t_2 l)))))
        (* (* l (/ 2.0 (* (tan k) (* (sin k) (pow t_m 3.0))))) (/ l t_2)))))))

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 <= 5e-100) {
		tmp = (2.0 / (k * k)) * ((pow(l, 2.0) / t_m) * (cos(k) / pow(sin(k), 2.0)));
	} else if (t_m <= 9.5e+19) {
		tmp = 2.0 / ((pow(t_m, 3.0) / l) * (sin(k) * (tan(k) * (t_2 / l))));
	} else {
		tmp = (l * (2.0 / (tan(k) * (sin(k) * pow(t_m, 3.0))))) * (l / t_2);
	}
	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 <= 5d-100) then
        tmp = (2.0d0 / (k * k)) * (((l ** 2.0d0) / t_m) * (cos(k) / (sin(k) ** 2.0d0)))
    else if (t_m <= 9.5d+19) then
        tmp = 2.0d0 / (((t_m ** 3.0d0) / l) * (sin(k) * (tan(k) * (t_2 / l))))
    else
        tmp = (l * (2.0d0 / (tan(k) * (sin(k) * (t_m ** 3.0d0))))) * (l / t_2)
    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 <= 5e-100) {
		tmp = (2.0 / (k * k)) * ((Math.pow(l, 2.0) / t_m) * (Math.cos(k) / Math.pow(Math.sin(k), 2.0)));
	} else if (t_m <= 9.5e+19) {
		tmp = 2.0 / ((Math.pow(t_m, 3.0) / l) * (Math.sin(k) * (Math.tan(k) * (t_2 / l))));
	} else {
		tmp = (l * (2.0 / (Math.tan(k) * (Math.sin(k) * Math.pow(t_m, 3.0))))) * (l / t_2);
	}
	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 <= 5e-100:
		tmp = (2.0 / (k * k)) * ((math.pow(l, 2.0) / t_m) * (math.cos(k) / math.pow(math.sin(k), 2.0)))
	elif t_m <= 9.5e+19:
		tmp = 2.0 / ((math.pow(t_m, 3.0) / l) * (math.sin(k) * (math.tan(k) * (t_2 / l))))
	else:
		tmp = (l * (2.0 / (math.tan(k) * (math.sin(k) * math.pow(t_m, 3.0))))) * (l / t_2)
	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 <= 5e-100)
		tmp = Float64(Float64(2.0 / Float64(k * k)) * Float64(Float64((l ^ 2.0) / t_m) * Float64(cos(k) / (sin(k) ^ 2.0))));
	elseif (t_m <= 9.5e+19)
		tmp = Float64(2.0 / Float64(Float64((t_m ^ 3.0) / l) * Float64(sin(k) * Float64(tan(k) * Float64(t_2 / l)))));
	else
		tmp = Float64(Float64(l * Float64(2.0 / Float64(tan(k) * Float64(sin(k) * (t_m ^ 3.0))))) * Float64(l / t_2));
	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 <= 5e-100)
		tmp = (2.0 / (k * k)) * (((l ^ 2.0) / t_m) * (cos(k) / (sin(k) ^ 2.0)));
	elseif (t_m <= 9.5e+19)
		tmp = 2.0 / (((t_m ^ 3.0) / l) * (sin(k) * (tan(k) * (t_2 / l))));
	else
		tmp = (l * (2.0 / (tan(k) * (sin(k) * (t_m ^ 3.0))))) * (l / t_2);
	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, 5e-100], N[(N[(2.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[l, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 9.5e+19], N[(2.0 / N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(t$95$2 / l), $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 / t$95$2), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 5.0000000000000001e-100

   1. Initial program 51.1%

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

   3. Simplified 51.1%

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

   5. Taylor expanded in t around 0 65.6%

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

      1. associate-*r/ 65.6%

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

      2. times-frac 65.1%

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

      3. times-frac 66.9%

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

   7. Simplified 66.9%

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

      1. unpow2 66.9%

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

   9. Applied egg-rr 66.9%

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

   if 5.0000000000000001e-100 < t < 9.5e19

   1. Initial program 64.7%

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

   3. Simplified 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. associate-*l* 49.9%

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

      2. associate-/r* 54.2%

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

      3. associate-+r+ 54.2%

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

      4. metadata-eval 54.2%

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

      5. associate-*l* 54.2%

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

      6. associate-*l/ 54.5%

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

      7. associate-*l* 54.5%

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

   6. Applied egg-rr 54.5%

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

      1. associate-/l* 66.5%

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

      2. associate-*r* 66.4%

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

   8. Simplified 66.4%

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

      1. *-un-lft-identity 66.4%

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

      2. associate-/l* 66.4%

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

   10. Applied egg-rr 66.4%

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

       1. *-lft-identity 66.4%

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

       2. associate-*l* 78.6%

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

   12. Simplified 78.6%

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

   if 9.5e19 < t

   1. Initial program 59.0%

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

   3. Simplified 62.1%

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

      1. associate-*r* 71.7%

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

      2. *-un-lft-identity 71.7%

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

      3. times-frac 71.7%

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

      4. associate-/l/ 71.8%

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

   6. Applied egg-rr 71.8%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5 \cdot 10^{-100}:\\ \;\;\;\;\frac{2}{k \cdot k} \cdot \left(\frac{{\ell}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{2}{\frac{{t}^{3}}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \frac{2 + {\left(\frac{k}{t}\right)}^{2}}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot {t}^{3}\right)}\right) \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 71.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2 \cdot 10^{-40}:\\ \;\;\;\;\frac{2}{k \cdot k} \cdot \left(\frac{{\ell}^{2}}{t\_m} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \mathbf{elif}\;t\_m \leq 6 \cdot 10^{+103}:\\ \;\;\;\;\ell \cdot \frac{\frac{2}{{t\_m}^{3}} \cdot \frac{\ell}{\sin k \cdot \tan k}}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left({\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{1.5}\right)}^{2} \cdot \left(2 \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2e-40)
    (* (/ 2.0 (* k k)) (* (/ (pow l 2.0) t_m) (/ (cos k) (pow (sin k) 2.0))))
    (if (<= t_m 6e+103)
      (*
       l
       (/
        (* (/ 2.0 (pow t_m 3.0)) (/ l (* (sin k) (tan k))))
        (+ 2.0 (pow (/ k t_m) 2.0))))
      (/
       2.0
       (* (pow (pow (* t_m (pow (cbrt l) -2.0)) 1.5) 2.0) (* 2.0 (* k k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2e-40) {
		tmp = (2.0 / (k * k)) * ((pow(l, 2.0) / t_m) * (cos(k) / pow(sin(k), 2.0)));
	} else if (t_m <= 6e+103) {
		tmp = l * (((2.0 / pow(t_m, 3.0)) * (l / (sin(k) * tan(k)))) / (2.0 + pow((k / t_m), 2.0)));
	} else {
		tmp = 2.0 / (pow(pow((t_m * pow(cbrt(l), -2.0)), 1.5), 2.0) * (2.0 * (k * k)));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2e-40) {
		tmp = (2.0 / (k * k)) * ((Math.pow(l, 2.0) / t_m) * (Math.cos(k) / Math.pow(Math.sin(k), 2.0)));
	} else if (t_m <= 6e+103) {
		tmp = l * (((2.0 / Math.pow(t_m, 3.0)) * (l / (Math.sin(k) * Math.tan(k)))) / (2.0 + Math.pow((k / t_m), 2.0)));
	} else {
		tmp = 2.0 / (Math.pow(Math.pow((t_m * Math.pow(Math.cbrt(l), -2.0)), 1.5), 2.0) * (2.0 * (k * k)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2e-40)
		tmp = Float64(Float64(2.0 / Float64(k * k)) * Float64(Float64((l ^ 2.0) / t_m) * Float64(cos(k) / (sin(k) ^ 2.0))));
	elseif (t_m <= 6e+103)
		tmp = Float64(l * Float64(Float64(Float64(2.0 / (t_m ^ 3.0)) * Float64(l / Float64(sin(k) * tan(k)))) / Float64(2.0 + (Float64(k / t_m) ^ 2.0))));
	else
		tmp = Float64(2.0 / Float64(((Float64(t_m * (cbrt(l) ^ -2.0)) ^ 1.5) ^ 2.0) * Float64(2.0 * Float64(k * k))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2e-40], N[(N[(2.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[l, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6e+103], N[(l * N[(N[(N[(2.0 / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] * N[(l / N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[Power[N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 2.0], $MachinePrecision] * N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.9999999999999999e-40

   1. Initial program 50.8%

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

   3. Simplified 50.8%

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

   5. Taylor expanded in t around 0 64.4%

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

      1. associate-*r/ 64.4%

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

      2. times-frac 63.9%

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

      3. times-frac 65.6%

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

   7. Simplified 65.6%

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

      1. unpow2 65.6%

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

   9. Applied egg-rr 65.6%

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

   if 1.9999999999999999e-40 < t < 6e103

   1. Initial program 69.6%

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

   3. Simplified 65.4%

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

      1. associate-*r* 73.9%

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

      2. *-un-lft-identity 73.9%

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

      3. times-frac 74.1%

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

      4. associate-/l/ 74.1%

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

   6. Applied egg-rr 74.1%

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

      1. times-frac 74.0%

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

      2. *-commutative 74.0%

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

      3. times-frac 72.7%

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

      4. associate-*l/ 72.7%

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

      5. associate-*l* 67.0%

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

      6. times-frac 67.1%

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

      7. /-rgt-identity 67.1%

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

   8. Simplified 67.1%

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

   if 6e103 < t

   1. Initial program 57.6%

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

   3. Simplified 55.9%

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

   5. Taylor expanded in k around 0 55.9%

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

      1. unpow2 42.9%

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

   7. Applied egg-rr 55.9%

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

      1. associate-/r* 45.8%

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

      2. unpow3 45.8%

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

      3. times-frac 58.4%

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

      4. pow2 58.4%

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

   9. Applied egg-rr 58.4%

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

       1. add-cube-cbrt 58.4%

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

       2. pow3 58.4%

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

       3. frac-times 45.8%

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

       4. unpow2 45.8%

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

       5. unpow3 45.8%

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

       6. cbrt-div 45.8%

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

       7. unpow3 45.8%

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

       8. add-cbrt-cube 51.1%

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

       9. cbrt-prod 65.7%

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

       10. unpow2 65.7%

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

       11. pow-to-exp 28.8%

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

       12. *-commutative 28.8%

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

       13. sqr-pow 28.8%

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

       14. pow2 28.8%

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

   11. Applied egg-rr 65.8%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2 \cdot 10^{-40}:\\ \;\;\;\;\frac{2}{k \cdot k} \cdot \left(\frac{{\ell}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+103}:\\ \;\;\;\;\ell \cdot \frac{\frac{2}{{t}^{3}} \cdot \frac{\ell}{\sin k \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left({\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{1.5}\right)}^{2} \cdot \left(2 \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 59.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 8 \cdot 10^{-140}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right) \cdot \frac{k \cdot {t\_m}^{3}}{{\ell}^{2}}}\\ \mathbf{elif}\;k \leq 4.8 \cdot 10^{-28}:\\ \;\;\;\;\frac{2}{{\left({\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{1.5}\right)}^{2} \cdot \left(2 \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot k} \cdot \left(\frac{{\ell}^{2}}{t\_m} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \end{array} \end{array} \]

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 8e-140)
    (/
     2.0
     (*
      (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))
      (/ (* k (pow t_m 3.0)) (pow l 2.0))))
    (if (<= k 4.8e-28)
      (/
       2.0
       (* (pow (pow (* t_m (pow (cbrt l) -2.0)) 1.5) 2.0) (* 2.0 (* k k))))
      (*
       (/ 2.0 (* k k))
       (* (/ (pow l 2.0) t_m) (/ (cos k) (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 <= 8e-140) {
		tmp = 2.0 / ((tan(k) * (2.0 + pow((k / t_m), 2.0))) * ((k * pow(t_m, 3.0)) / pow(l, 2.0)));
	} else if (k <= 4.8e-28) {
		tmp = 2.0 / (pow(pow((t_m * pow(cbrt(l), -2.0)), 1.5), 2.0) * (2.0 * (k * k)));
	} else {
		tmp = (2.0 / (k * k)) * ((pow(l, 2.0) / t_m) * (cos(k) / 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 <= 8e-140) {
		tmp = 2.0 / ((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0))) * ((k * Math.pow(t_m, 3.0)) / Math.pow(l, 2.0)));
	} else if (k <= 4.8e-28) {
		tmp = 2.0 / (Math.pow(Math.pow((t_m * Math.pow(Math.cbrt(l), -2.0)), 1.5), 2.0) * (2.0 * (k * k)));
	} else {
		tmp = (2.0 / (k * k)) * ((Math.pow(l, 2.0) / t_m) * (Math.cos(k) / 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 <= 8e-140)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * Float64(Float64(k * (t_m ^ 3.0)) / (l ^ 2.0))));
	elseif (k <= 4.8e-28)
		tmp = Float64(2.0 / Float64(((Float64(t_m * (cbrt(l) ^ -2.0)) ^ 1.5) ^ 2.0) * Float64(2.0 * Float64(k * k))));
	else
		tmp = Float64(Float64(2.0 / Float64(k * k)) * Float64(Float64((l ^ 2.0) / t_m) * Float64(cos(k) / (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, 8e-140], 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[(N[(k * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 4.8e-28], N[(2.0 / N[(N[Power[N[Power[N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 2.0], $MachinePrecision] * N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[l, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 7.9999999999999999e-140

   1. Initial program 61.3%

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

   3. Simplified 61.2%

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

      1. add-cube-cbrt 61.2%

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

      2. pow3 61.2%

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

      3. associate-/r* 68.2%

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

      4. *-commutative 68.2%

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

      5. cbrt-prod 68.1%

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

      6. associate-/r* 61.1%

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

      7. cbrt-div 63.5%

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

      8. rem-cbrt-cube 72.8%

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

      9. cbrt-prod 82.0%

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

      10. pow2 82.0%

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

   6. Applied egg-rr 82.0%

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

      1. pow1 82.0%

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

      2. associate-+r+ 82.0%

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

      3. metadata-eval 82.0%

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

   8. Applied egg-rr 82.0%

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

      1. unpow1 82.0%

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

   10. Simplified 82.0%

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

   11. Taylor expanded in k around 0 60.1%

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

   if 7.9999999999999999e-140 < k < 4.8000000000000004e-28

   1. Initial program 69.4%

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

   3. Simplified 70.4%

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

   5. Taylor expanded in k around 0 70.5%

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

      1. unpow2 48.3%

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

   7. Applied egg-rr 70.5%

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

      1. associate-/r* 69.3%

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

      2. unpow3 69.3%

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

      3. times-frac 77.9%

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

      4. pow2 77.9%

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

   9. Applied egg-rr 77.9%

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

       1. add-cube-cbrt 77.9%

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

       2. pow3 77.9%

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

       3. frac-times 69.2%

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

       4. unpow2 69.2%

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

       5. unpow3 69.2%

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

       6. cbrt-div 69.2%

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

       7. unpow3 69.2%

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

       8. add-cbrt-cube 73.8%

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

       9. cbrt-prod 81.0%

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

       10. unpow2 81.0%

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

       11. pow-to-exp 34.6%

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

       12. *-commutative 34.6%

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

       13. sqr-pow 26.9%

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

       14. pow2 26.9%

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

   11. Applied egg-rr 53.7%

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

   if 4.8000000000000004e-28 < k

   1. Initial program 34.9%

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

   3. Simplified 34.9%

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

   5. Taylor expanded in t around 0 59.5%

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

      1. associate-*r/ 59.5%

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

      2. times-frac 57.1%

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

      3. times-frac 57.2%

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

   7. Simplified 57.2%

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

      1. unpow2 57.2%

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

   9. Applied egg-rr 57.2%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8 \cdot 10^{-140}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \frac{k \cdot {t}^{3}}{{\ell}^{2}}}\\ \mathbf{elif}\;k \leq 4.8 \cdot 10^{-28}:\\ \;\;\;\;\frac{2}{{\left({\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{1.5}\right)}^{2} \cdot \left(2 \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot k} \cdot \left(\frac{{\ell}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 68.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.3 \cdot 10^{+24}:\\ \;\;\;\;\frac{2}{k \cdot k} \cdot \left(\frac{{\ell}^{2}}{t\_m} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \mathbf{elif}\;t\_m \leq 1.55 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{\frac{2}{k}}{k \cdot {t\_m}^{3}} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left({\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{1.5}\right)}^{2} \cdot \left(2 \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.3e+24)
    (* (/ 2.0 (* k k)) (* (/ (pow l 2.0) t_m) (/ (cos k) (pow (sin k) 2.0))))
    (if (<= t_m 1.55e+102)
      (/
       (* (/ (/ 2.0 k) (* k (pow t_m 3.0))) (* l l))
       (+ 2.0 (pow (/ k t_m) 2.0)))
      (/
       2.0
       (* (pow (pow (* t_m (pow (cbrt l) -2.0)) 1.5) 2.0) (* 2.0 (* k k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.3e+24) {
		tmp = (2.0 / (k * k)) * ((pow(l, 2.0) / t_m) * (cos(k) / pow(sin(k), 2.0)));
	} else if (t_m <= 1.55e+102) {
		tmp = (((2.0 / k) / (k * pow(t_m, 3.0))) * (l * l)) / (2.0 + pow((k / t_m), 2.0));
	} else {
		tmp = 2.0 / (pow(pow((t_m * pow(cbrt(l), -2.0)), 1.5), 2.0) * (2.0 * (k * k)));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.3e+24) {
		tmp = (2.0 / (k * k)) * ((Math.pow(l, 2.0) / t_m) * (Math.cos(k) / Math.pow(Math.sin(k), 2.0)));
	} else if (t_m <= 1.55e+102) {
		tmp = (((2.0 / k) / (k * Math.pow(t_m, 3.0))) * (l * l)) / (2.0 + Math.pow((k / t_m), 2.0));
	} else {
		tmp = 2.0 / (Math.pow(Math.pow((t_m * Math.pow(Math.cbrt(l), -2.0)), 1.5), 2.0) * (2.0 * (k * k)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.3e+24)
		tmp = Float64(Float64(2.0 / Float64(k * k)) * Float64(Float64((l ^ 2.0) / t_m) * Float64(cos(k) / (sin(k) ^ 2.0))));
	elseif (t_m <= 1.55e+102)
		tmp = Float64(Float64(Float64(Float64(2.0 / k) / Float64(k * (t_m ^ 3.0))) * Float64(l * l)) / Float64(2.0 + (Float64(k / t_m) ^ 2.0)));
	else
		tmp = Float64(2.0 / Float64(((Float64(t_m * (cbrt(l) ^ -2.0)) ^ 1.5) ^ 2.0) * Float64(2.0 * Float64(k * k))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.3e+24], N[(N[(2.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[l, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.55e+102], N[(N[(N[(N[(2.0 / k), $MachinePrecision] / N[(k * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[Power[N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 2.0], $MachinePrecision] * N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.2999999999999999e24

   1. Initial program 53.5%

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

   3. Simplified 53.5%

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

   5. Taylor expanded in t around 0 63.9%

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

      1. associate-*r/ 63.9%

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

      2. times-frac 63.5%

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

      3. times-frac 65.1%

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

   7. Simplified 65.1%

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

      1. unpow2 65.1%

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

   9. Applied egg-rr 65.1%

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

   if 2.2999999999999999e24 < t < 1.54999999999999993e102

   1. Initial program 58.7%

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

   3. Simplified 73.8%

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

   5. Taylor expanded in k around 0 68.2%

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

   6. Taylor expanded in k around 0 73.4%

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

   if 1.54999999999999993e102 < t

   1. Initial program 56.3%

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

   3. Simplified 54.7%

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

   5. Taylor expanded in k around 0 54.7%

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

      1. unpow2 41.9%

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

   7. Applied egg-rr 54.7%

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

      1. associate-/r* 44.7%

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

      2. unpow3 44.7%

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

      3. times-frac 57.2%

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

      4. pow2 57.2%

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

   9. Applied egg-rr 57.2%

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

       1. add-cube-cbrt 57.2%

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

       2. pow3 57.2%

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

       3. frac-times 44.7%

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

       4. unpow2 44.7%

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

       5. unpow3 44.7%

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

       6. cbrt-div 44.7%

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

       7. unpow3 44.7%

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

       8. add-cbrt-cube 49.9%

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

       9. cbrt-prod 64.3%

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

       10. unpow2 64.3%

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

       11. pow-to-exp 28.2%

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

       12. *-commutative 28.2%

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

       13. sqr-pow 28.2%

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

       14. pow2 28.2%

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

   11. Applied egg-rr 64.4%

       \[\leadsto \frac{2}{\color{blue}{{\left({\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{1.5}\right)}^{2}} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 71.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.1 \cdot 10^{-99}:\\ \;\;\;\;\frac{2}{k \cdot k} \cdot \left(\frac{{\ell}^{2}}{t\_m} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{t\_m}^{3}}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \frac{2 + {\left(\frac{k}{t\_m}\right)}^{2}}{\ell}\right)\right)}\\ \end{array} \end{array} \]

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 6.1e-99)
    (* (/ 2.0 (* k k)) (* (/ (pow l 2.0) t_m) (/ (cos k) (pow (sin k) 2.0))))
    (/
     2.0
     (*
      (/ (pow t_m 3.0) l)
      (* (sin k) (* (tan k) (/ (+ 2.0 (pow (/ k t_m) 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 <= 6.1e-99) {
		tmp = (2.0 / (k * k)) * ((pow(l, 2.0) / t_m) * (cos(k) / pow(sin(k), 2.0)));
	} else {
		tmp = 2.0 / ((pow(t_m, 3.0) / l) * (sin(k) * (tan(k) * ((2.0 + pow((k / t_m), 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 <= 6.1d-99) then
        tmp = (2.0d0 / (k * k)) * (((l ** 2.0d0) / t_m) * (cos(k) / (sin(k) ** 2.0d0)))
    else
        tmp = 2.0d0 / (((t_m ** 3.0d0) / l) * (sin(k) * (tan(k) * ((2.0d0 + ((k / t_m) ** 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 <= 6.1e-99) {
		tmp = (2.0 / (k * k)) * ((Math.pow(l, 2.0) / t_m) * (Math.cos(k) / Math.pow(Math.sin(k), 2.0)));
	} else {
		tmp = 2.0 / ((Math.pow(t_m, 3.0) / l) * (Math.sin(k) * (Math.tan(k) * ((2.0 + Math.pow((k / t_m), 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 <= 6.1e-99:
		tmp = (2.0 / (k * k)) * ((math.pow(l, 2.0) / t_m) * (math.cos(k) / math.pow(math.sin(k), 2.0)))
	else:
		tmp = 2.0 / ((math.pow(t_m, 3.0) / l) * (math.sin(k) * (math.tan(k) * ((2.0 + math.pow((k / t_m), 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 <= 6.1e-99)
		tmp = Float64(Float64(2.0 / Float64(k * k)) * Float64(Float64((l ^ 2.0) / t_m) * Float64(cos(k) / (sin(k) ^ 2.0))));
	else
		tmp = Float64(2.0 / Float64(Float64((t_m ^ 3.0) / l) * Float64(sin(k) * Float64(tan(k) * Float64(Float64(2.0 + (Float64(k / t_m) ^ 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 <= 6.1e-99)
		tmp = (2.0 / (k * k)) * (((l ^ 2.0) / t_m) * (cos(k) / (sin(k) ^ 2.0)));
	else
		tmp = 2.0 / (((t_m ^ 3.0) / l) * (sin(k) * (tan(k) * ((2.0 + ((k / t_m) ^ 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, 6.1e-99], N[(N[(2.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[l, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 6.1000000000000003e-99

   1. Initial program 51.1%

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

   3. Simplified 51.1%

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

   5. Taylor expanded in t around 0 65.6%

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

      1. associate-*r/ 65.6%

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

      2. times-frac 65.1%

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

      3. times-frac 66.9%

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

   7. Simplified 66.9%

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

      1. unpow2 66.9%

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

   9. Applied egg-rr 66.9%

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

   if 6.1000000000000003e-99 < t

   1. Initial program 60.5%

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

   3. Simplified 60.4%

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

      1. associate-*l* 49.9%

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

      2. associate-/r* 58.1%

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

      3. associate-+r+ 58.1%

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

      4. metadata-eval 58.1%

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

      5. associate-*l* 58.1%

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

      6. associate-*l/ 57.3%

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

      7. associate-*l* 57.3%

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

   6. Applied egg-rr 57.3%

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

      1. associate-/l* 61.4%

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

      2. associate-*r* 61.4%

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

   8. Simplified 61.4%

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

      1. *-un-lft-identity 61.4%

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

      2. associate-/l* 61.3%

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

   10. Applied egg-rr 61.3%

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

       1. *-lft-identity 61.3%

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

       2. associate-*l* 69.0%

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

   12. Simplified 69.0%

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

Alternative 19: 63.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.5 \cdot 10^{-125}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t\_m \cdot {k}^{2}}\\ \mathbf{elif}\;t\_m \leq 3.6 \cdot 10^{+24}:\\ \;\;\;\;\frac{2}{\frac{{t\_m}^{3}}{\ell} \cdot \frac{2 \cdot {k}^{2}}{\ell}}\\ \mathbf{elif}\;t\_m \leq 2.22 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{\frac{2}{k}}{k \cdot {t\_m}^{3}} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3} \cdot \left(2 \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.5e-125)
    (* (/ 2.0 (pow k 2.0)) (/ (pow l 2.0) (* t_m (pow k 2.0))))
    (if (<= t_m 3.6e+24)
      (/ 2.0 (* (/ (pow t_m 3.0) l) (/ (* 2.0 (pow k 2.0)) l)))
      (if (<= t_m 2.22e+102)
        (/
         (* (/ (/ 2.0 k) (* k (pow t_m 3.0))) (* l l))
         (+ 2.0 (pow (/ k t_m) 2.0)))
        (/ 2.0 (* (pow (* t_m (pow (cbrt l) -2.0)) 3.0) (* 2.0 (* k k)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.5e-125) {
		tmp = (2.0 / pow(k, 2.0)) * (pow(l, 2.0) / (t_m * pow(k, 2.0)));
	} else if (t_m <= 3.6e+24) {
		tmp = 2.0 / ((pow(t_m, 3.0) / l) * ((2.0 * pow(k, 2.0)) / l));
	} else if (t_m <= 2.22e+102) {
		tmp = (((2.0 / k) / (k * pow(t_m, 3.0))) * (l * l)) / (2.0 + pow((k / t_m), 2.0));
	} else {
		tmp = 2.0 / (pow((t_m * pow(cbrt(l), -2.0)), 3.0) * (2.0 * (k * k)));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.5e-125) {
		tmp = (2.0 / Math.pow(k, 2.0)) * (Math.pow(l, 2.0) / (t_m * Math.pow(k, 2.0)));
	} else if (t_m <= 3.6e+24) {
		tmp = 2.0 / ((Math.pow(t_m, 3.0) / l) * ((2.0 * Math.pow(k, 2.0)) / l));
	} else if (t_m <= 2.22e+102) {
		tmp = (((2.0 / k) / (k * Math.pow(t_m, 3.0))) * (l * l)) / (2.0 + Math.pow((k / t_m), 2.0));
	} else {
		tmp = 2.0 / (Math.pow((t_m * Math.pow(Math.cbrt(l), -2.0)), 3.0) * (2.0 * (k * k)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 3.5e-125)
		tmp = Float64(Float64(2.0 / (k ^ 2.0)) * Float64((l ^ 2.0) / Float64(t_m * (k ^ 2.0))));
	elseif (t_m <= 3.6e+24)
		tmp = Float64(2.0 / Float64(Float64((t_m ^ 3.0) / l) * Float64(Float64(2.0 * (k ^ 2.0)) / l)));
	elseif (t_m <= 2.22e+102)
		tmp = Float64(Float64(Float64(Float64(2.0 / k) / Float64(k * (t_m ^ 3.0))) * Float64(l * l)) / Float64(2.0 + (Float64(k / t_m) ^ 2.0)));
	else
		tmp = Float64(2.0 / Float64((Float64(t_m * (cbrt(l) ^ -2.0)) ^ 3.0) * Float64(2.0 * Float64(k * k))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 3.5e-125], N[(N[(2.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.6e+24], N[(2.0 / N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.22e+102], N[(N[(N[(N[(2.0 / k), $MachinePrecision] / N[(k * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if t < 3.49999999999999998e-125

   1. Initial program 52.6%

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

   3. Simplified 52.6%

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

   5. Taylor expanded in t around 0 66.9%

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

      1. associate-*r/ 66.9%

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

      2. times-frac 66.4%

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

      3. times-frac 68.2%

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

   7. Simplified 68.2%

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

   8. Taylor expanded in k around 0 60.8%

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

   if 3.49999999999999998e-125 < t < 3.59999999999999983e24

   1. Initial program 58.0%

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

   3. Simplified 57.9%

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

      1. associate-*l* 44.3%

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

      2. associate-/r* 53.5%

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

      3. associate-+r+ 53.5%

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

      4. metadata-eval 53.5%

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

      5. associate-*l* 53.5%

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

      6. associate-*l/ 53.7%

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

      7. associate-*l* 53.7%

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

   6. Applied egg-rr 53.7%

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

      1. associate-/l* 60.0%

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

      2. associate-*r* 60.0%

         \[\leadsto \frac{2}{\frac{{t}^{3}}{\ell} \cdot \frac{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}}{\ell}} \]

   8. Simplified 60.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \frac{\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{\ell}}} \]

   9. Taylor expanded in k around 0 45.4%

      \[\leadsto \frac{2}{\frac{{t}^{3}}{\ell} \cdot \frac{\color{blue}{2 \cdot {k}^{2}}}{\ell}} \]

   if 3.59999999999999983e24 < t < 2.22000000000000009e102

   1. Initial program 58.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 73.8%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 68.2%

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{k \cdot {t}^{3}}} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   6. Taylor expanded in k around 0 73.4%

      \[\leadsto \frac{\frac{\color{blue}{\frac{2}{k}}}{k \cdot {t}^{3}} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   if 2.22000000000000009e102 < t

   1. Initial program 56.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 54.7%

      \[\leadsto \color{blue}{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 54.7%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(2 \cdot {k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. unpow2 41.9%

         \[\leadsto \frac{2}{\color{blue}{k \cdot k}} \cdot \left(\frac{{\ell}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right) \]

   7. Applied egg-rr 54.7%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]
   8. Step-by-step derivation

      1. associate-/r* 44.7%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

      2. unpow3 44.7%

         \[\leadsto \frac{2}{\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

      3. times-frac 57.2%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

      4. pow2 57.2%

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{2}}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

   9. Applied egg-rr 57.2%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]
   10. Step-by-step derivation

       1. add-cube-cbrt 57.2%

          \[\leadsto \frac{2}{\color{blue}{\left(\left(\sqrt[3]{\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}} \cdot \sqrt[3]{\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}}\right) \cdot \sqrt[3]{\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}}\right)} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

       2. pow3 57.2%

          \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt[3]{\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}}\right)}^{3}} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

       3. frac-times 44.7%

          \[\leadsto \frac{2}{{\left(\sqrt[3]{\color{blue}{\frac{{t}^{2} \cdot t}{\ell \cdot \ell}}}\right)}^{3} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

       4. unpow2 44.7%

          \[\leadsto \frac{2}{{\left(\sqrt[3]{\frac{\color{blue}{\left(t \cdot t\right)} \cdot t}{\ell \cdot \ell}}\right)}^{3} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

       5. unpow3 44.7%

          \[\leadsto \frac{2}{{\left(\sqrt[3]{\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell}}\right)}^{3} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

       6. cbrt-div 44.7%

          \[\leadsto \frac{2}{{\color{blue}{\left(\frac{\sqrt[3]{{t}^{3}}}{\sqrt[3]{\ell \cdot \ell}}\right)}}^{3} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

       7. unpow3 44.7%

          \[\leadsto \frac{2}{{\left(\frac{\sqrt[3]{\color{blue}{\left(t \cdot t\right) \cdot t}}}{\sqrt[3]{\ell \cdot \ell}}\right)}^{3} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

       8. add-cbrt-cube 49.9%

          \[\leadsto \frac{2}{{\left(\frac{\color{blue}{t}}{\sqrt[3]{\ell \cdot \ell}}\right)}^{3} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

       9. cbrt-prod 64.3%

          \[\leadsto \frac{2}{{\left(\frac{t}{\color{blue}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\right)}^{3} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

       10. unpow2 64.3%

           \[\leadsto \frac{2}{{\left(\frac{t}{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\right)}^{3} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

       11. div-inv 64.4%

           \[\leadsto \frac{2}{{\color{blue}{\left(t \cdot \frac{1}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}}^{3} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

       12. pow-flip 64.4%

           \[\leadsto \frac{2}{{\left(t \cdot \color{blue}{{\left(\sqrt[3]{\ell}\right)}^{\left(-2\right)}}\right)}^{3} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

       13. metadata-eval 64.4%

           \[\leadsto \frac{2}{{\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{\color{blue}{-2}}\right)}^{3} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

   11. Applied egg-rr 64.4%

       \[\leadsto \frac{2}{\color{blue}{{\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.5 \cdot 10^{-125}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t \cdot {k}^{2}}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{+24}:\\ \;\;\;\;\frac{2}{\frac{{t}^{3}}{\ell} \cdot \frac{2 \cdot {k}^{2}}{\ell}}\\ \mathbf{elif}\;t \leq 2.22 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{\frac{2}{k}}{k \cdot {t}^{3}} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3} \cdot \left(2 \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 52.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {k}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.5 \cdot 10^{-125}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t\_m \cdot {k}^{2}}\\ \mathbf{elif}\;t\_m \leq 2.3 \cdot 10^{+24}:\\ \;\;\;\;\frac{2}{\frac{{t\_m}^{3}}{\ell} \cdot \frac{t\_2}{\ell}}\\ \mathbf{elif}\;t\_m \leq 2.22 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{\frac{2}{k}}{k \cdot {t\_m}^{3}} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_2 \cdot {\left(\frac{t\_m}{{\ell}^{0.6666666666666666}}\right)}^{3}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* 2.0 (pow k 2.0))))
   (*
    t_s
    (if (<= t_m 3.5e-125)
      (* (/ 2.0 (pow k 2.0)) (/ (pow l 2.0) (* t_m (pow k 2.0))))
      (if (<= t_m 2.3e+24)
        (/ 2.0 (* (/ (pow t_m 3.0) l) (/ t_2 l)))
        (if (<= t_m 2.22e+102)
          (/
           (* (/ (/ 2.0 k) (* k (pow t_m 3.0))) (* l l))
           (+ 2.0 (pow (/ k t_m) 2.0)))
          (/ 2.0 (* t_2 (pow (/ t_m (pow l 0.6666666666666666)) 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, 2.0);
	double tmp;
	if (t_m <= 3.5e-125) {
		tmp = (2.0 / pow(k, 2.0)) * (pow(l, 2.0) / (t_m * pow(k, 2.0)));
	} else if (t_m <= 2.3e+24) {
		tmp = 2.0 / ((pow(t_m, 3.0) / l) * (t_2 / l));
	} else if (t_m <= 2.22e+102) {
		tmp = (((2.0 / k) / (k * pow(t_m, 3.0))) * (l * l)) / (2.0 + pow((k / t_m), 2.0));
	} else {
		tmp = 2.0 / (t_2 * pow((t_m / pow(l, 0.6666666666666666)), 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 ** 2.0d0)
    if (t_m <= 3.5d-125) then
        tmp = (2.0d0 / (k ** 2.0d0)) * ((l ** 2.0d0) / (t_m * (k ** 2.0d0)))
    else if (t_m <= 2.3d+24) then
        tmp = 2.0d0 / (((t_m ** 3.0d0) / l) * (t_2 / l))
    else if (t_m <= 2.22d+102) then
        tmp = (((2.0d0 / k) / (k * (t_m ** 3.0d0))) * (l * l)) / (2.0d0 + ((k / t_m) ** 2.0d0))
    else
        tmp = 2.0d0 / (t_2 * ((t_m / (l ** 0.6666666666666666d0)) ** 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, 2.0);
	double tmp;
	if (t_m <= 3.5e-125) {
		tmp = (2.0 / Math.pow(k, 2.0)) * (Math.pow(l, 2.0) / (t_m * Math.pow(k, 2.0)));
	} else if (t_m <= 2.3e+24) {
		tmp = 2.0 / ((Math.pow(t_m, 3.0) / l) * (t_2 / l));
	} else if (t_m <= 2.22e+102) {
		tmp = (((2.0 / k) / (k * Math.pow(t_m, 3.0))) * (l * l)) / (2.0 + Math.pow((k / t_m), 2.0));
	} else {
		tmp = 2.0 / (t_2 * Math.pow((t_m / Math.pow(l, 0.6666666666666666)), 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, 2.0)
	tmp = 0
	if t_m <= 3.5e-125:
		tmp = (2.0 / math.pow(k, 2.0)) * (math.pow(l, 2.0) / (t_m * math.pow(k, 2.0)))
	elif t_m <= 2.3e+24:
		tmp = 2.0 / ((math.pow(t_m, 3.0) / l) * (t_2 / l))
	elif t_m <= 2.22e+102:
		tmp = (((2.0 / k) / (k * math.pow(t_m, 3.0))) * (l * l)) / (2.0 + math.pow((k / t_m), 2.0))
	else:
		tmp = 2.0 / (t_2 * math.pow((t_m / math.pow(l, 0.6666666666666666)), 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 * (k ^ 2.0))
	tmp = 0.0
	if (t_m <= 3.5e-125)
		tmp = Float64(Float64(2.0 / (k ^ 2.0)) * Float64((l ^ 2.0) / Float64(t_m * (k ^ 2.0))));
	elseif (t_m <= 2.3e+24)
		tmp = Float64(2.0 / Float64(Float64((t_m ^ 3.0) / l) * Float64(t_2 / l)));
	elseif (t_m <= 2.22e+102)
		tmp = Float64(Float64(Float64(Float64(2.0 / k) / Float64(k * (t_m ^ 3.0))) * Float64(l * l)) / Float64(2.0 + (Float64(k / t_m) ^ 2.0)));
	else
		tmp = Float64(2.0 / Float64(t_2 * (Float64(t_m / (l ^ 0.6666666666666666)) ^ 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 ^ 2.0);
	tmp = 0.0;
	if (t_m <= 3.5e-125)
		tmp = (2.0 / (k ^ 2.0)) * ((l ^ 2.0) / (t_m * (k ^ 2.0)));
	elseif (t_m <= 2.3e+24)
		tmp = 2.0 / (((t_m ^ 3.0) / l) * (t_2 / l));
	elseif (t_m <= 2.22e+102)
		tmp = (((2.0 / k) / (k * (t_m ^ 3.0))) * (l * l)) / (2.0 + ((k / t_m) ^ 2.0));
	else
		tmp = 2.0 / (t_2 * ((t_m / (l ^ 0.6666666666666666)) ^ 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[k, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 3.5e-125], N[(N[(2.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.3e+24], N[(2.0 / N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$2 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.22e+102], N[(N[(N[(N[(2.0 / k), $MachinePrecision] / N[(k * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$2 * N[Power[N[(t$95$m / N[Power[l, 0.6666666666666666], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 3.49999999999999998e-125

   1. Initial program 52.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 52.6%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 66.9%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. associate-*r/ 66.9%

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. times-frac 66.4%

         \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2} \cdot \cos k}{t \cdot {\sin k}^{2}}} \]

      3. times-frac 68.2%

         \[\leadsto \frac{2}{{k}^{2}} \cdot \color{blue}{\left(\frac{{\ell}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)} \]

   7. Simplified 68.2%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \left(\frac{{\ell}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)} \]

   8. Taylor expanded in k around 0 60.8%

      \[\leadsto \frac{2}{{k}^{2}} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]

   if 3.49999999999999998e-125 < t < 2.2999999999999999e24

   1. Initial program 58.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 57.9%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l* 44.3%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)\right)}} \]

      2. associate-/r* 53.5%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)\right)} \]

      3. associate-+r+ 53.5%

         \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}\right)\right)} \]

      4. metadata-eval 53.5%

         \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      5. associate-*l* 53.5%

         \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

      6. associate-*l/ 53.7%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}{\ell}}} \]

      7. associate-*l* 53.7%

         \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell} \cdot \color{blue}{\left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}}{\ell}} \]

   6. Applied egg-rr 53.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}{\ell}}} \]
   7. Step-by-step derivation

      1. associate-/l* 60.0%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}{\ell}}} \]

      2. associate-*r* 60.0%

         \[\leadsto \frac{2}{\frac{{t}^{3}}{\ell} \cdot \frac{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}}{\ell}} \]

   8. Simplified 60.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \frac{\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{\ell}}} \]

   9. Taylor expanded in k around 0 45.4%

      \[\leadsto \frac{2}{\frac{{t}^{3}}{\ell} \cdot \frac{\color{blue}{2 \cdot {k}^{2}}}{\ell}} \]

   if 2.2999999999999999e24 < t < 2.22000000000000009e102

   1. Initial program 58.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 73.8%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 68.2%

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{k \cdot {t}^{3}}} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   6. Taylor expanded in k around 0 73.4%

      \[\leadsto \frac{\frac{\color{blue}{\frac{2}{k}}}{k \cdot {t}^{3}} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   if 2.22000000000000009e102 < t

   1. Initial program 56.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 54.7%

      \[\leadsto \color{blue}{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 54.7%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(2 \cdot {k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. add-cube-cbrt 54.7%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\sqrt[3]{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sqrt[3]{\frac{\frac{{t}^{3}}{\ell}}{\ell}}\right) \cdot \sqrt[3]{\frac{\frac{{t}^{3}}{\ell}}{\ell}}\right)} \cdot \left(2 \cdot {k}^{2}\right)} \]

      2. pow3 54.7%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt[3]{\frac{\frac{{t}^{3}}{\ell}}{\ell}}\right)}^{3}} \cdot \left(2 \cdot {k}^{2}\right)} \]

      3. associate-/r* 44.7%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}}\right)}^{3} \cdot \left(2 \cdot {k}^{2}\right)} \]

      4. cbrt-div 44.7%

         \[\leadsto \frac{2}{{\color{blue}{\left(\frac{\sqrt[3]{{t}^{3}}}{\sqrt[3]{\ell \cdot \ell}}\right)}}^{3} \cdot \left(2 \cdot {k}^{2}\right)} \]

      5. rem-cbrt-cube 49.9%

         \[\leadsto \frac{2}{{\left(\frac{\color{blue}{t}}{\sqrt[3]{\ell \cdot \ell}}\right)}^{3} \cdot \left(2 \cdot {k}^{2}\right)} \]

      6. cbrt-prod 64.3%

         \[\leadsto \frac{2}{{\left(\frac{t}{\color{blue}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\right)}^{3} \cdot \left(2 \cdot {k}^{2}\right)} \]

      7. pow2 64.3%

         \[\leadsto \frac{2}{{\left(\frac{t}{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\right)}^{3} \cdot \left(2 \cdot {k}^{2}\right)} \]

   7. Applied egg-rr 64.3%

      \[\leadsto \frac{2}{\color{blue}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \cdot \left(2 \cdot {k}^{2}\right)} \]
   8. Step-by-step derivation

      1. add-exp-log 64.0%

         \[\leadsto \frac{2}{{\left(\frac{t}{\color{blue}{e^{\log \left({\left(\sqrt[3]{\ell}\right)}^{2}\right)}}}\right)}^{3} \cdot \left(2 \cdot {k}^{2}\right)} \]

      2. log-pow 28.2%

         \[\leadsto \frac{2}{{\left(\frac{t}{e^{\color{blue}{2 \cdot \log \left(\sqrt[3]{\ell}\right)}}}\right)}^{3} \cdot \left(2 \cdot {k}^{2}\right)} \]

   9. Applied egg-rr 28.2%

      \[\leadsto \frac{2}{{\left(\frac{t}{\color{blue}{e^{2 \cdot \log \left(\sqrt[3]{\ell}\right)}}}\right)}^{3} \cdot \left(2 \cdot {k}^{2}\right)} \]
   10. Step-by-step derivation

       1. *-commutative 28.2%

          \[\leadsto \frac{2}{{\left(\frac{t}{e^{\color{blue}{\log \left(\sqrt[3]{\ell}\right) \cdot 2}}}\right)}^{3} \cdot \left(2 \cdot {k}^{2}\right)} \]

       2. pow-to-exp 64.3%

          \[\leadsto \frac{2}{{\left(\frac{t}{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\right)}^{3} \cdot \left(2 \cdot {k}^{2}\right)} \]

       3. pow1/3 28.2%

          \[\leadsto \frac{2}{{\left(\frac{t}{{\color{blue}{\left({\ell}^{0.3333333333333333}\right)}}^{2}}\right)}^{3} \cdot \left(2 \cdot {k}^{2}\right)} \]

       4. pow-pow 28.2%

          \[\leadsto \frac{2}{{\left(\frac{t}{\color{blue}{{\ell}^{\left(0.3333333333333333 \cdot 2\right)}}}\right)}^{3} \cdot \left(2 \cdot {k}^{2}\right)} \]

       5. metadata-eval 28.2%

          \[\leadsto \frac{2}{{\left(\frac{t}{{\ell}^{\color{blue}{0.6666666666666666}}}\right)}^{3} \cdot \left(2 \cdot {k}^{2}\right)} \]

   11. Applied egg-rr 28.2%

       \[\leadsto \frac{2}{{\left(\frac{t}{\color{blue}{{\ell}^{0.6666666666666666}}}\right)}^{3} \cdot \left(2 \cdot {k}^{2}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.5 \cdot 10^{-125}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t \cdot {k}^{2}}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{+24}:\\ \;\;\;\;\frac{2}{\frac{{t}^{3}}{\ell} \cdot \frac{2 \cdot {k}^{2}}{\ell}}\\ \mathbf{elif}\;t \leq 2.22 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{\frac{2}{k}}{k \cdot {t}^{3}} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot {\left(\frac{t}{{\ell}^{0.6666666666666666}}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 61.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.5 \cdot 10^{-125}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t\_m \cdot {k}^{2}}\\ \mathbf{elif}\;t\_m \leq 1.3 \cdot 10^{+24}:\\ \;\;\;\;\frac{2}{\frac{{t\_m}^{3}}{\ell} \cdot \frac{2 \cdot {k}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{k}}{k \cdot {t\_m}^{3}} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}\\ \end{array} \end{array} \]

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 3.5e-125)
    (* (/ 2.0 (pow k 2.0)) (/ (pow l 2.0) (* t_m (pow k 2.0))))
    (if (<= t_m 1.3e+24)
      (/ 2.0 (* (/ (pow t_m 3.0) l) (/ (* 2.0 (pow k 2.0)) l)))
      (/
       (* (/ (/ 2.0 k) (* k (pow t_m 3.0))) (* l 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 <= 3.5e-125) {
		tmp = (2.0 / pow(k, 2.0)) * (pow(l, 2.0) / (t_m * pow(k, 2.0)));
	} else if (t_m <= 1.3e+24) {
		tmp = 2.0 / ((pow(t_m, 3.0) / l) * ((2.0 * pow(k, 2.0)) / l));
	} else {
		tmp = (((2.0 / k) / (k * pow(t_m, 3.0))) * (l * 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 <= 3.5d-125) then
        tmp = (2.0d0 / (k ** 2.0d0)) * ((l ** 2.0d0) / (t_m * (k ** 2.0d0)))
    else if (t_m <= 1.3d+24) then
        tmp = 2.0d0 / (((t_m ** 3.0d0) / l) * ((2.0d0 * (k ** 2.0d0)) / l))
    else
        tmp = (((2.0d0 / k) / (k * (t_m ** 3.0d0))) * (l * 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 <= 3.5e-125) {
		tmp = (2.0 / Math.pow(k, 2.0)) * (Math.pow(l, 2.0) / (t_m * Math.pow(k, 2.0)));
	} else if (t_m <= 1.3e+24) {
		tmp = 2.0 / ((Math.pow(t_m, 3.0) / l) * ((2.0 * Math.pow(k, 2.0)) / l));
	} else {
		tmp = (((2.0 / k) / (k * Math.pow(t_m, 3.0))) * (l * 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 <= 3.5e-125:
		tmp = (2.0 / math.pow(k, 2.0)) * (math.pow(l, 2.0) / (t_m * math.pow(k, 2.0)))
	elif t_m <= 1.3e+24:
		tmp = 2.0 / ((math.pow(t_m, 3.0) / l) * ((2.0 * math.pow(k, 2.0)) / l))
	else:
		tmp = (((2.0 / k) / (k * math.pow(t_m, 3.0))) * (l * 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 <= 3.5e-125)
		tmp = Float64(Float64(2.0 / (k ^ 2.0)) * Float64((l ^ 2.0) / Float64(t_m * (k ^ 2.0))));
	elseif (t_m <= 1.3e+24)
		tmp = Float64(2.0 / Float64(Float64((t_m ^ 3.0) / l) * Float64(Float64(2.0 * (k ^ 2.0)) / l)));
	else
		tmp = Float64(Float64(Float64(Float64(2.0 / k) / Float64(k * (t_m ^ 3.0))) * Float64(l * 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 <= 3.5e-125)
		tmp = (2.0 / (k ^ 2.0)) * ((l ^ 2.0) / (t_m * (k ^ 2.0)));
	elseif (t_m <= 1.3e+24)
		tmp = 2.0 / (((t_m ^ 3.0) / l) * ((2.0 * (k ^ 2.0)) / l));
	else
		tmp = (((2.0 / k) / (k * (t_m ^ 3.0))) * (l * 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, 3.5e-125], N[(N[(2.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.3e+24], N[(2.0 / N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(2.0 / k), $MachinePrecision] / N[(k * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 3.49999999999999998e-125

   1. Initial program 52.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 52.6%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 66.9%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. associate-*r/ 66.9%

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. times-frac 66.4%

         \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2} \cdot \cos k}{t \cdot {\sin k}^{2}}} \]

      3. times-frac 68.2%

         \[\leadsto \frac{2}{{k}^{2}} \cdot \color{blue}{\left(\frac{{\ell}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)} \]

   7. Simplified 68.2%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \left(\frac{{\ell}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)} \]

   8. Taylor expanded in k around 0 60.8%

      \[\leadsto \frac{2}{{k}^{2}} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]

   if 3.49999999999999998e-125 < t < 1.2999999999999999e24

   1. Initial program 58.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 57.9%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l* 44.3%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)\right)}} \]

      2. associate-/r* 53.5%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)\right)} \]

      3. associate-+r+ 53.5%

         \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}\right)\right)} \]

      4. metadata-eval 53.5%

         \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      5. associate-*l* 53.5%

         \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

      6. associate-*l/ 53.7%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}{\ell}}} \]

      7. associate-*l* 53.7%

         \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell} \cdot \color{blue}{\left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}}{\ell}} \]

   6. Applied egg-rr 53.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}{\ell}}} \]
   7. Step-by-step derivation

      1. associate-/l* 60.0%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}{\ell}}} \]

      2. associate-*r* 60.0%

         \[\leadsto \frac{2}{\frac{{t}^{3}}{\ell} \cdot \frac{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}}{\ell}} \]

   8. Simplified 60.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \frac{\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{\ell}}} \]

   9. Taylor expanded in k around 0 45.4%

      \[\leadsto \frac{2}{\frac{{t}^{3}}{\ell} \cdot \frac{\color{blue}{2 \cdot {k}^{2}}}{\ell}} \]

   if 1.2999999999999999e24 < t

   1. Initial program 57.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 61.6%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 59.9%

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{k \cdot {t}^{3}}} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   6. Taylor expanded in k around 0 61.5%

      \[\leadsto \frac{\frac{\color{blue}{\frac{2}{k}}}{k \cdot {t}^{3}} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.5 \cdot 10^{-125}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t \cdot {k}^{2}}\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{+24}:\\ \;\;\;\;\frac{2}{\frac{{t}^{3}}{\ell} \cdot \frac{2 \cdot {k}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{k}}{k \cdot {t}^{3}} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 61.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.5 \cdot 10^{-125}:\\ \;\;\;\;\frac{2}{t\_m} \cdot \frac{{\ell}^{2}}{{k}^{4}}\\ \mathbf{elif}\;t\_m \leq 9.2 \cdot 10^{+25}:\\ \;\;\;\;\frac{2}{\frac{{t\_m}^{3}}{\ell} \cdot \frac{2 \cdot {k}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{k}}{k \cdot {t\_m}^{3}} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}\\ \end{array} \end{array} \]

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 3.5e-125)
    (* (/ 2.0 t_m) (/ (pow l 2.0) (pow k 4.0)))
    (if (<= t_m 9.2e+25)
      (/ 2.0 (* (/ (pow t_m 3.0) l) (/ (* 2.0 (pow k 2.0)) l)))
      (/
       (* (/ (/ 2.0 k) (* k (pow t_m 3.0))) (* l 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 <= 3.5e-125) {
		tmp = (2.0 / t_m) * (pow(l, 2.0) / pow(k, 4.0));
	} else if (t_m <= 9.2e+25) {
		tmp = 2.0 / ((pow(t_m, 3.0) / l) * ((2.0 * pow(k, 2.0)) / l));
	} else {
		tmp = (((2.0 / k) / (k * pow(t_m, 3.0))) * (l * 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 <= 3.5d-125) then
        tmp = (2.0d0 / t_m) * ((l ** 2.0d0) / (k ** 4.0d0))
    else if (t_m <= 9.2d+25) then
        tmp = 2.0d0 / (((t_m ** 3.0d0) / l) * ((2.0d0 * (k ** 2.0d0)) / l))
    else
        tmp = (((2.0d0 / k) / (k * (t_m ** 3.0d0))) * (l * 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 <= 3.5e-125) {
		tmp = (2.0 / t_m) * (Math.pow(l, 2.0) / Math.pow(k, 4.0));
	} else if (t_m <= 9.2e+25) {
		tmp = 2.0 / ((Math.pow(t_m, 3.0) / l) * ((2.0 * Math.pow(k, 2.0)) / l));
	} else {
		tmp = (((2.0 / k) / (k * Math.pow(t_m, 3.0))) * (l * 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 <= 3.5e-125:
		tmp = (2.0 / t_m) * (math.pow(l, 2.0) / math.pow(k, 4.0))
	elif t_m <= 9.2e+25:
		tmp = 2.0 / ((math.pow(t_m, 3.0) / l) * ((2.0 * math.pow(k, 2.0)) / l))
	else:
		tmp = (((2.0 / k) / (k * math.pow(t_m, 3.0))) * (l * 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 <= 3.5e-125)
		tmp = Float64(Float64(2.0 / t_m) * Float64((l ^ 2.0) / (k ^ 4.0)));
	elseif (t_m <= 9.2e+25)
		tmp = Float64(2.0 / Float64(Float64((t_m ^ 3.0) / l) * Float64(Float64(2.0 * (k ^ 2.0)) / l)));
	else
		tmp = Float64(Float64(Float64(Float64(2.0 / k) / Float64(k * (t_m ^ 3.0))) * Float64(l * 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 <= 3.5e-125)
		tmp = (2.0 / t_m) * ((l ^ 2.0) / (k ^ 4.0));
	elseif (t_m <= 9.2e+25)
		tmp = 2.0 / (((t_m ^ 3.0) / l) * ((2.0 * (k ^ 2.0)) / l));
	else
		tmp = (((2.0 / k) / (k * (t_m ^ 3.0))) * (l * 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, 3.5e-125], N[(N[(2.0 / t$95$m), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 9.2e+25], N[(2.0 / N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(2.0 / k), $MachinePrecision] / N[(k * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 3.49999999999999998e-125

   1. Initial program 52.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 52.6%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 66.9%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. associate-*r/ 66.9%

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. times-frac 66.4%

         \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2} \cdot \cos k}{t \cdot {\sin k}^{2}}} \]

      3. times-frac 68.2%

         \[\leadsto \frac{2}{{k}^{2}} \cdot \color{blue}{\left(\frac{{\ell}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)} \]

   7. Simplified 68.2%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \left(\frac{{\ell}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)} \]

   8. Taylor expanded in k around 0 58.3%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   9. Step-by-step derivation

      1. associate-*r/ 58.3%

         \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

      2. *-commutative 58.3%

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{t \cdot {k}^{4}}} \]

      3. times-frac 59.5%

         \[\leadsto \color{blue}{\frac{2}{t} \cdot \frac{{\ell}^{2}}{{k}^{4}}} \]

   10. Simplified 59.5%

       \[\leadsto \color{blue}{\frac{2}{t} \cdot \frac{{\ell}^{2}}{{k}^{4}}} \]

   if 3.49999999999999998e-125 < t < 9.1999999999999992e25

   1. Initial program 58.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 57.9%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l* 44.3%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)\right)}} \]

      2. associate-/r* 53.5%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)\right)} \]

      3. associate-+r+ 53.5%

         \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}\right)\right)} \]

      4. metadata-eval 53.5%

         \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      5. associate-*l* 53.5%

         \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

      6. associate-*l/ 53.7%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}{\ell}}} \]

      7. associate-*l* 53.7%

         \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell} \cdot \color{blue}{\left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}}{\ell}} \]

   6. Applied egg-rr 53.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}{\ell}}} \]
   7. Step-by-step derivation

      1. associate-/l* 60.0%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}{\ell}}} \]

      2. associate-*r* 60.0%

         \[\leadsto \frac{2}{\frac{{t}^{3}}{\ell} \cdot \frac{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}}{\ell}} \]

   8. Simplified 60.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \frac{\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{\ell}}} \]

   9. Taylor expanded in k around 0 45.4%

      \[\leadsto \frac{2}{\frac{{t}^{3}}{\ell} \cdot \frac{\color{blue}{2 \cdot {k}^{2}}}{\ell}} \]

   if 9.1999999999999992e25 < t

   1. Initial program 57.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 61.6%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 59.9%

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{k \cdot {t}^{3}}} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   6. Taylor expanded in k around 0 61.5%

      \[\leadsto \frac{\frac{\color{blue}{\frac{2}{k}}}{k \cdot {t}^{3}} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 23: 61.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.15 \cdot 10^{-125}:\\ \;\;\;\;\frac{2}{t\_m} \cdot \frac{{\ell}^{2}}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot {\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2}}\\ \end{array} \end{array} \]

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 2.15e-125)
    (* (/ 2.0 t_m) (/ (pow l 2.0) (pow k 4.0)))
    (/ 2.0 (* (* 2.0 (* k 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 <= 2.15e-125) {
		tmp = (2.0 / t_m) * (pow(l, 2.0) / pow(k, 4.0));
	} else {
		tmp = 2.0 / ((2.0 * (k * 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 <= 2.15d-125) then
        tmp = (2.0d0 / t_m) * ((l ** 2.0d0) / (k ** 4.0d0))
    else
        tmp = 2.0d0 / ((2.0d0 * (k * 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 <= 2.15e-125) {
		tmp = (2.0 / t_m) * (Math.pow(l, 2.0) / Math.pow(k, 4.0));
	} else {
		tmp = 2.0 / ((2.0 * (k * 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 <= 2.15e-125:
		tmp = (2.0 / t_m) * (math.pow(l, 2.0) / math.pow(k, 4.0))
	else:
		tmp = 2.0 / ((2.0 * (k * 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 <= 2.15e-125)
		tmp = Float64(Float64(2.0 / t_m) * Float64((l ^ 2.0) / (k ^ 4.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * Float64(k * 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 <= 2.15e-125)
		tmp = (2.0 / t_m) * ((l ^ 2.0) / (k ^ 4.0));
	else
		tmp = 2.0 / ((2.0 * (k * 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, 2.15e-125], N[(N[(2.0 / t$95$m), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.1500000000000001e-125

   1. Initial program 52.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 52.6%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 66.9%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. associate-*r/ 66.9%

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. times-frac 66.4%

         \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2} \cdot \cos k}{t \cdot {\sin k}^{2}}} \]

      3. times-frac 68.2%

         \[\leadsto \frac{2}{{k}^{2}} \cdot \color{blue}{\left(\frac{{\ell}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)} \]

   7. Simplified 68.2%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \left(\frac{{\ell}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)} \]

   8. Taylor expanded in k around 0 58.3%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   9. Step-by-step derivation

      1. associate-*r/ 58.3%

         \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

      2. *-commutative 58.3%

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{t \cdot {k}^{4}}} \]

      3. times-frac 59.5%

         \[\leadsto \color{blue}{\frac{2}{t} \cdot \frac{{\ell}^{2}}{{k}^{4}}} \]

   10. Simplified 59.5%

       \[\leadsto \color{blue}{\frac{2}{t} \cdot \frac{{\ell}^{2}}{{k}^{4}}} \]

   if 2.1500000000000001e-125 < t

   1. Initial program 57.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 57.1%

      \[\leadsto \color{blue}{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 51.4%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(2 \cdot {k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. unpow2 44.2%

         \[\leadsto \frac{2}{\color{blue}{k \cdot k}} \cdot \left(\frac{{\ell}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right) \]

   7. Applied egg-rr 51.4%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 51.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\sqrt{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sqrt{\frac{\frac{{t}^{3}}{\ell}}{\ell}}\right)} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

      2. pow2 51.4%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{\frac{\frac{{t}^{3}}{\ell}}{\ell}}\right)}^{2}} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

      3. associate-/r* 45.4%

         \[\leadsto \frac{2}{{\left(\sqrt{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}}\right)}^{2} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

      4. sqrt-div 45.4%

         \[\leadsto \frac{2}{{\color{blue}{\left(\frac{\sqrt{{t}^{3}}}{\sqrt{\ell \cdot \ell}}\right)}}^{2} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

      5. sqrt-pow1 47.8%

         \[\leadsto \frac{2}{{\left(\frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

      6. metadata-eval 47.8%

         \[\leadsto \frac{2}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

      7. sqrt-prod 24.7%

         \[\leadsto \frac{2}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

      8. add-sqr-sqrt 55.0%

         \[\leadsto \frac{2}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

   9. Applied egg-rr 55.0%

      \[\leadsto \frac{2}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.15 \cdot 10^{-125}:\\ \;\;\;\;\frac{2}{t} \cdot \frac{{\ell}^{2}}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 60.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.5 \cdot 10^{-125}:\\ \;\;\;\;\frac{2}{t\_m} \cdot \frac{{\ell}^{2}}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{t\_m \cdot t\_m}{\ell} \cdot \frac{t\_m}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.5e-125)
    (* (/ 2.0 t_m) (/ (pow l 2.0) (pow k 4.0)))
    (/ 2.0 (* (* 2.0 (* k k)) (* (/ (* t_m t_m) 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 <= 3.5e-125) {
		tmp = (2.0 / t_m) * (pow(l, 2.0) / pow(k, 4.0));
	} else {
		tmp = 2.0 / ((2.0 * (k * k)) * (((t_m * t_m) / 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 <= 3.5d-125) then
        tmp = (2.0d0 / t_m) * ((l ** 2.0d0) / (k ** 4.0d0))
    else
        tmp = 2.0d0 / ((2.0d0 * (k * k)) * (((t_m * t_m) / 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 <= 3.5e-125) {
		tmp = (2.0 / t_m) * (Math.pow(l, 2.0) / Math.pow(k, 4.0));
	} else {
		tmp = 2.0 / ((2.0 * (k * k)) * (((t_m * t_m) / 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 <= 3.5e-125:
		tmp = (2.0 / t_m) * (math.pow(l, 2.0) / math.pow(k, 4.0))
	else:
		tmp = 2.0 / ((2.0 * (k * k)) * (((t_m * t_m) / 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 <= 3.5e-125)
		tmp = Float64(Float64(2.0 / t_m) * Float64((l ^ 2.0) / (k ^ 4.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * Float64(k * k)) * Float64(Float64(Float64(t_m * t_m) / 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 <= 3.5e-125)
		tmp = (2.0 / t_m) * ((l ^ 2.0) / (k ^ 4.0));
	else
		tmp = 2.0 / ((2.0 * (k * k)) * (((t_m * t_m) / 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, 3.5e-125], N[(N[(2.0 / t$95$m), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.49999999999999998e-125

   1. Initial program 52.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 52.6%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 66.9%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. associate-*r/ 66.9%

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. times-frac 66.4%

         \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2} \cdot \cos k}{t \cdot {\sin k}^{2}}} \]

      3. times-frac 68.2%

         \[\leadsto \frac{2}{{k}^{2}} \cdot \color{blue}{\left(\frac{{\ell}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)} \]

   7. Simplified 68.2%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \left(\frac{{\ell}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)} \]

   8. Taylor expanded in k around 0 58.3%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   9. Step-by-step derivation

      1. associate-*r/ 58.3%

         \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

      2. *-commutative 58.3%

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{t \cdot {k}^{4}}} \]

      3. times-frac 59.5%

         \[\leadsto \color{blue}{\frac{2}{t} \cdot \frac{{\ell}^{2}}{{k}^{4}}} \]

   10. Simplified 59.5%

       \[\leadsto \color{blue}{\frac{2}{t} \cdot \frac{{\ell}^{2}}{{k}^{4}}} \]

   if 3.49999999999999998e-125 < t

   1. Initial program 57.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 57.1%

      \[\leadsto \color{blue}{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 51.4%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(2 \cdot {k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. unpow2 44.2%

         \[\leadsto \frac{2}{\color{blue}{k \cdot k}} \cdot \left(\frac{{\ell}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right) \]

   7. Applied egg-rr 51.4%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]
   8. Step-by-step derivation

      1. associate-/r* 45.4%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

      2. unpow3 45.4%

         \[\leadsto \frac{2}{\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

      3. times-frac 53.6%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

      4. pow2 53.6%

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{2}}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

   9. Applied egg-rr 53.6%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 53.6%

          \[\leadsto \frac{2}{\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

   11. Applied egg-rr 53.6%

       \[\leadsto \frac{2}{\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.5 \cdot 10^{-125}:\\ \;\;\;\;\frac{2}{t} \cdot \frac{{\ell}^{2}}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 60.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.4 \cdot 10^{-125}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{t\_m \cdot {k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{t\_m \cdot t\_m}{\ell} \cdot \frac{t\_m}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.4e-125)
    (* 2.0 (/ (pow l 2.0) (* t_m (pow k 4.0))))
    (/ 2.0 (* (* 2.0 (* k k)) (* (/ (* t_m t_m) 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 <= 3.4e-125) {
		tmp = 2.0 * (pow(l, 2.0) / (t_m * pow(k, 4.0)));
	} else {
		tmp = 2.0 / ((2.0 * (k * k)) * (((t_m * t_m) / 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 <= 3.4d-125) then
        tmp = 2.0d0 * ((l ** 2.0d0) / (t_m * (k ** 4.0d0)))
    else
        tmp = 2.0d0 / ((2.0d0 * (k * k)) * (((t_m * t_m) / 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 <= 3.4e-125) {
		tmp = 2.0 * (Math.pow(l, 2.0) / (t_m * Math.pow(k, 4.0)));
	} else {
		tmp = 2.0 / ((2.0 * (k * k)) * (((t_m * t_m) / 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 <= 3.4e-125:
		tmp = 2.0 * (math.pow(l, 2.0) / (t_m * math.pow(k, 4.0)))
	else:
		tmp = 2.0 / ((2.0 * (k * k)) * (((t_m * t_m) / 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 <= 3.4e-125)
		tmp = Float64(2.0 * Float64((l ^ 2.0) / Float64(t_m * (k ^ 4.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * Float64(k * k)) * Float64(Float64(Float64(t_m * t_m) / 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 <= 3.4e-125)
		tmp = 2.0 * ((l ^ 2.0) / (t_m * (k ^ 4.0)));
	else
		tmp = 2.0 / ((2.0 * (k * k)) * (((t_m * t_m) / 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, 3.4e-125], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.39999999999999975e-125

   1. Initial program 52.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 52.6%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 66.9%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. associate-*r/ 66.9%

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. times-frac 66.4%

         \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2} \cdot \cos k}{t \cdot {\sin k}^{2}}} \]

      3. times-frac 68.2%

         \[\leadsto \frac{2}{{k}^{2}} \cdot \color{blue}{\left(\frac{{\ell}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)} \]

   7. Simplified 68.2%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \left(\frac{{\ell}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)} \]

   8. Taylor expanded in k around 0 58.3%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

   if 3.39999999999999975e-125 < t

   1. Initial program 57.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 57.1%

      \[\leadsto \color{blue}{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 51.4%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(2 \cdot {k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. unpow2 44.2%

         \[\leadsto \frac{2}{\color{blue}{k \cdot k}} \cdot \left(\frac{{\ell}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right) \]

   7. Applied egg-rr 51.4%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]
   8. Step-by-step derivation

      1. associate-/r* 45.4%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

      2. unpow3 45.4%

         \[\leadsto \frac{2}{\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

      3. times-frac 53.6%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

      4. pow2 53.6%

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{2}}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

   9. Applied egg-rr 53.6%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 53.6%

          \[\leadsto \frac{2}{\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

   11. Applied egg-rr 53.6%

       \[\leadsto \frac{2}{\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.4 \cdot 10^{-125}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{t \cdot {k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 57.8% accurate, 24.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{t\_m \cdot t\_m}{\ell} \cdot \frac{t\_m}{\ell}\right)} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ 2.0 (* (* 2.0 (* k k)) (* (/ (* t_m t_m) 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) {
	return t_s * (2.0 / ((2.0 * (k * k)) * (((t_m * t_m) / l) * (t_m / l))));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 / ((2.0d0 * (k * k)) * (((t_m * t_m) / l) * (t_m / l))))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / ((2.0 * (k * k)) * (((t_m * t_m) / l) * (t_m / l))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 / ((2.0 * (k * k)) * (((t_m * t_m) / l) * (t_m / l))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 / Float64(Float64(2.0 * Float64(k * k)) * Float64(Float64(Float64(t_m * t_m) / l) * Float64(t_m / l)))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 / ((2.0 * (k * k)) * (((t_m * t_m) / l) * (t_m / l))));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(2.0 / N[(N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.4%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

3. Simplified 55.1%

   \[\leadsto \color{blue}{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in k around 0 54.9%

   \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(2 \cdot {k}^{2}\right)}} \]
6. Step-by-step derivation

   1. unpow2 59.4%

      \[\leadsto \frac{2}{\color{blue}{k \cdot k}} \cdot \left(\frac{{\ell}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right) \]

7. Applied egg-rr 54.9%

   \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]
8. Step-by-step derivation

   1. associate-/r* 51.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

   2. unpow3 51.3%

      \[\leadsto \frac{2}{\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

   3. times-frac 57.7%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

   4. pow2 57.7%

      \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{2}}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

9. Applied egg-rr 57.7%

   \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]
10. Step-by-step derivation

    1. unpow2 57.7%

       \[\leadsto \frac{2}{\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

11. Applied egg-rr 57.7%

    \[\leadsto \frac{2}{\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

12. Final simplification 57.7%

    \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024186 
(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))))