Toniolo and Linder, Equation (10+)

Percentage Accurate: 53.7% → 89.2%

Time: 20.6s

Alternatives: 22

Speedup: 3.8×

• 26.9% 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: 53.7% 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: 89.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 + {\left(\frac{k}{t\_m}\right)}^{2}\\ t_3 := \sqrt[3]{\sin k} \cdot \frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.4 \cdot 10^{-55}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \left(\left(\ell \cdot {k}^{-2}\right) \cdot \frac{\cos k}{t\_m \cdot {\sin k}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t\_3 \cdot \left(\sqrt[3]{\tan k} \cdot \sqrt[3]{t\_2}\right)\right)}^{2} \cdot \left(t\_3 \cdot \sqrt[3]{\tan k \cdot t\_2}\right)}\\ \end{array} \end{array} \end{array} \]

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 (+ 2.0 (pow (/ k t_m) 2.0)))
        (t_3 (* (cbrt (sin k)) (/ t_m (pow (cbrt l) 2.0)))))
   (*
    t_s
    (if (<= t_m 1.4e-55)
      (*
       l
       (* 2.0 (* (* l (pow k -2.0)) (/ (cos k) (* t_m (pow (sin k) 2.0))))))
      (/
       2.0
       (*
        (pow (* t_3 (* (cbrt (tan k)) (cbrt t_2))) 2.0)
        (* t_3 (cbrt (* (tan k) 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 t_3 = cbrt(sin(k)) * (t_m / pow(cbrt(l), 2.0));
	double tmp;
	if (t_m <= 1.4e-55) {
		tmp = l * (2.0 * ((l * pow(k, -2.0)) * (cos(k) / (t_m * pow(sin(k), 2.0)))));
	} else {
		tmp = 2.0 / (pow((t_3 * (cbrt(tan(k)) * cbrt(t_2))), 2.0) * (t_3 * cbrt((tan(k) * 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 = 2.0 + Math.pow((k / t_m), 2.0);
	double t_3 = Math.cbrt(Math.sin(k)) * (t_m / Math.pow(Math.cbrt(l), 2.0));
	double tmp;
	if (t_m <= 1.4e-55) {
		tmp = l * (2.0 * ((l * Math.pow(k, -2.0)) * (Math.cos(k) / (t_m * Math.pow(Math.sin(k), 2.0)))));
	} else {
		tmp = 2.0 / (Math.pow((t_3 * (Math.cbrt(Math.tan(k)) * Math.cbrt(t_2))), 2.0) * (t_3 * Math.cbrt((Math.tan(k) * 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))
	t_3 = Float64(cbrt(sin(k)) * Float64(t_m / (cbrt(l) ^ 2.0)))
	tmp = 0.0
	if (t_m <= 1.4e-55)
		tmp = Float64(l * Float64(2.0 * Float64(Float64(l * (k ^ -2.0)) * Float64(cos(k) / Float64(t_m * (sin(k) ^ 2.0))))));
	else
		tmp = Float64(2.0 / Float64((Float64(t_3 * Float64(cbrt(tan(k)) * cbrt(t_2))) ^ 2.0) * Float64(t_3 * cbrt(Float64(tan(k) * 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[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.4e-55], N[(l * N[(2.0 * N[(N[(l * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(t$95$3 * N[(N[Power[N[Tan[k], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[t$95$2, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$3 * N[Power[N[(N[Tan[k], $MachinePrecision] * t$95$2), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1.39999999999999992e-55

   1. Initial program 51.2%

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

   3. Simplified 49.3%

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

      1. associate-*r* 57.1%

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

      2. associate-*l/ 55.8%

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

   6. Applied egg-rr 55.8%

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

   7. Taylor expanded in k around inf 71.0%

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

      1. times-frac 71.6%

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

   9. Simplified 71.6%

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

       1. *-un-lft-identity 71.6%

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

       2. div-inv 71.6%

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

       3. pow-flip 71.8%

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

       4. metadata-eval 71.8%

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

   11. Applied egg-rr 71.8%

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

       1. *-lft-identity 71.8%

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

   13. Simplified 71.8%

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

   if 1.39999999999999992e-55 < t

   1. Initial program 64.7%

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

      1. unpow3 64.7%

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

      2. times-frac 74.7%

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

      3. pow2 74.7%

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

   4. Applied egg-rr 74.7%

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

   6. Applied egg-rr 96.0%

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

      1. cbrt-prod 96.1%

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

      2. unpow2 96.1%

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

      3. hypot-undefine 96.1%

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

      4. metadata-eval 96.1%

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

      5. unpow2 96.1%

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

      6. hypot-undefine 96.1%

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

      7. metadata-eval 96.1%

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

      8. unpow2 96.1%

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

      9. add-sqr-sqrt 96.1%

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

   8. Applied egg-rr 96.1%

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

      1. associate-+r+ 96.1%

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

      2. metadata-eval 96.1%

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

   10. Simplified 96.1%

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

       1. *-un-lft-identity 96.1%

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

       2. unpow2 96.1%

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

       3. hypot-undefine 96.1%

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

       4. metadata-eval 96.1%

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

       5. unpow2 96.1%

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

       6. hypot-undefine 96.1%

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

       7. metadata-eval 96.1%

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

       8. unpow2 96.1%

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

       9. add-sqr-sqrt 96.1%

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

   12. Applied egg-rr 96.1%

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

       1. *-lft-identity 96.1%

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

       2. associate-+r+ 96.1%

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

       3. metadata-eval 96.1%

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

   14. Simplified 96.1%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.4 \cdot 10^{-55}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \left(\left(\ell \cdot {k}^{-2}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right) \cdot \left(\sqrt[3]{\tan k} \cdot \sqrt[3]{2 + {\left(\frac{k}{t}\right)}^{2}}\right)\right)}^{2} \cdot \left(\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right) \cdot \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6 \cdot 10^{-55}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \left(\left(\ell \cdot {k}^{-2}\right) \cdot \frac{\cos k}{t\_m \cdot {\sin k}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(\sqrt[3]{\sin k} \cdot \frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right) \cdot \sqrt[3]{\tan k \cdot \left(1 + {\left(\mathsf{hypot}\left(1, \frac{k}{t\_m}\right)\right)}^{2}\right)}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 6e-55)
    (* l (* 2.0 (* (* l (pow k -2.0)) (/ (cos k) (* t_m (pow (sin k) 2.0))))))
    (/
     2.0
     (pow
      (*
       (* (cbrt (sin k)) (/ t_m (pow (cbrt l) 2.0)))
       (cbrt (* (tan k) (+ 1.0 (pow (hypot 1.0 (/ k t_m)) 2.0)))))
      3.0)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 6e-55) {
		tmp = l * (2.0 * ((l * pow(k, -2.0)) * (cos(k) / (t_m * pow(sin(k), 2.0)))));
	} else {
		tmp = 2.0 / pow(((cbrt(sin(k)) * (t_m / pow(cbrt(l), 2.0))) * cbrt((tan(k) * (1.0 + pow(hypot(1.0, (k / t_m)), 2.0))))), 3.0);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 6e-55) {
		tmp = l * (2.0 * ((l * Math.pow(k, -2.0)) * (Math.cos(k) / (t_m * Math.pow(Math.sin(k), 2.0)))));
	} else {
		tmp = 2.0 / Math.pow(((Math.cbrt(Math.sin(k)) * (t_m / Math.pow(Math.cbrt(l), 2.0))) * Math.cbrt((Math.tan(k) * (1.0 + Math.pow(Math.hypot(1.0, (k / t_m)), 2.0))))), 3.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 6e-55)
		tmp = Float64(l * Float64(2.0 * Float64(Float64(l * (k ^ -2.0)) * Float64(cos(k) / Float64(t_m * (sin(k) ^ 2.0))))));
	else
		tmp = Float64(2.0 / (Float64(Float64(cbrt(sin(k)) * Float64(t_m / (cbrt(l) ^ 2.0))) * cbrt(Float64(tan(k) * Float64(1.0 + (hypot(1.0, Float64(k / t_m)) ^ 2.0))))) ^ 3.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 6e-55], N[(l * N[(2.0 * N[(N[(l * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[Power[N[Sqrt[1.0 ^ 2 + N[(k / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 6.00000000000000033e-55

   1. Initial program 51.2%

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

   3. Simplified 49.3%

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

      1. associate-*r* 57.1%

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

      2. associate-*l/ 55.8%

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

   6. Applied egg-rr 55.8%

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

   7. Taylor expanded in k around inf 71.0%

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

      1. times-frac 71.6%

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

   9. Simplified 71.6%

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

       1. *-un-lft-identity 71.6%

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

       2. div-inv 71.6%

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

       3. pow-flip 71.8%

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

       4. metadata-eval 71.8%

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

   11. Applied egg-rr 71.8%

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

       1. *-lft-identity 71.8%

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

   13. Simplified 71.8%

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

   if 6.00000000000000033e-55 < t

   1. Initial program 64.7%

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

      1. unpow3 64.7%

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

      2. times-frac 74.7%

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

      3. pow2 74.7%

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

   4. Applied egg-rr 74.7%

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

      1. add-cube-cbrt 74.4%

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

      2. pow3 74.4%

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

   6. Applied egg-rr 96.0%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6 \cdot 10^{-55}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \left(\left(\ell \cdot {k}^{-2}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right) \cdot \sqrt[3]{\tan k \cdot \left(1 + {\left(\mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}^{2}\right)}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt[3]{\sin k}\\ t_3 := 1 + \left({\left(\frac{k}{t\_m}\right)}^{2} + 1\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.95 \cdot 10^{-60}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \left(\left(\ell \cdot {k}^{-2}\right) \cdot \frac{\cos k}{t\_m \cdot {\sin k}^{2}}\right)\right)\\ \mathbf{elif}\;t\_m \leq 1.32 \cdot 10^{+200}:\\ \;\;\;\;\frac{2}{t\_3 \cdot \left(\frac{{t\_m}^{1.5}}{\ell} \cdot \left(\tan k \cdot \frac{\sin k \cdot {t\_m}^{1.5}}{\ell}\right)\right)}\\ \mathbf{elif}\;t\_m \leq 8 \cdot 10^{+271}:\\ \;\;\;\;\frac{2}{t\_3 \cdot {\left(\sqrt[3]{\tan k} \cdot \left(t\_2 \cdot \left(t\_m \cdot \sqrt[3]{{\ell}^{-2}}\right)\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_3 \cdot \left(\tan k \cdot {\left(t\_2 \cdot \frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (cbrt (sin k))) (t_3 (+ 1.0 (+ (pow (/ k t_m) 2.0) 1.0))))
   (*
    t_s
    (if (<= t_m 1.95e-60)
      (*
       l
       (* 2.0 (* (* l (pow k -2.0)) (/ (cos k) (* t_m (pow (sin k) 2.0))))))
      (if (<= t_m 1.32e+200)
        (/
         2.0
         (*
          t_3
          (* (/ (pow t_m 1.5) l) (* (tan k) (/ (* (sin k) (pow t_m 1.5)) l)))))
        (if (<= t_m 8e+271)
          (/
           2.0
           (*
            t_3
            (pow (* (cbrt (tan k)) (* t_2 (* t_m (cbrt (pow l -2.0))))) 3.0)))
          (/
           2.0
           (*
            t_3
            (* (tan k) (pow (* t_2 (/ t_m (pow (cbrt l) 2.0))) 3.0))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = cbrt(sin(k));
	double t_3 = 1.0 + (pow((k / t_m), 2.0) + 1.0);
	double tmp;
	if (t_m <= 1.95e-60) {
		tmp = l * (2.0 * ((l * pow(k, -2.0)) * (cos(k) / (t_m * pow(sin(k), 2.0)))));
	} else if (t_m <= 1.32e+200) {
		tmp = 2.0 / (t_3 * ((pow(t_m, 1.5) / l) * (tan(k) * ((sin(k) * pow(t_m, 1.5)) / l))));
	} else if (t_m <= 8e+271) {
		tmp = 2.0 / (t_3 * pow((cbrt(tan(k)) * (t_2 * (t_m * cbrt(pow(l, -2.0))))), 3.0));
	} else {
		tmp = 2.0 / (t_3 * (tan(k) * pow((t_2 * (t_m / pow(cbrt(l), 2.0))), 3.0)));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.cbrt(Math.sin(k));
	double t_3 = 1.0 + (Math.pow((k / t_m), 2.0) + 1.0);
	double tmp;
	if (t_m <= 1.95e-60) {
		tmp = l * (2.0 * ((l * Math.pow(k, -2.0)) * (Math.cos(k) / (t_m * Math.pow(Math.sin(k), 2.0)))));
	} else if (t_m <= 1.32e+200) {
		tmp = 2.0 / (t_3 * ((Math.pow(t_m, 1.5) / l) * (Math.tan(k) * ((Math.sin(k) * Math.pow(t_m, 1.5)) / l))));
	} else if (t_m <= 8e+271) {
		tmp = 2.0 / (t_3 * Math.pow((Math.cbrt(Math.tan(k)) * (t_2 * (t_m * Math.cbrt(Math.pow(l, -2.0))))), 3.0));
	} else {
		tmp = 2.0 / (t_3 * (Math.tan(k) * Math.pow((t_2 * (t_m / Math.pow(Math.cbrt(l), 2.0))), 3.0)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = cbrt(sin(k))
	t_3 = Float64(1.0 + Float64((Float64(k / t_m) ^ 2.0) + 1.0))
	tmp = 0.0
	if (t_m <= 1.95e-60)
		tmp = Float64(l * Float64(2.0 * Float64(Float64(l * (k ^ -2.0)) * Float64(cos(k) / Float64(t_m * (sin(k) ^ 2.0))))));
	elseif (t_m <= 1.32e+200)
		tmp = Float64(2.0 / Float64(t_3 * Float64(Float64((t_m ^ 1.5) / l) * Float64(tan(k) * Float64(Float64(sin(k) * (t_m ^ 1.5)) / l)))));
	elseif (t_m <= 8e+271)
		tmp = Float64(2.0 / Float64(t_3 * (Float64(cbrt(tan(k)) * Float64(t_2 * Float64(t_m * cbrt((l ^ -2.0))))) ^ 3.0)));
	else
		tmp = Float64(2.0 / Float64(t_3 * Float64(tan(k) * (Float64(t_2 * Float64(t_m / (cbrt(l) ^ 2.0))) ^ 3.0))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$3 = N[(1.0 + N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.95e-60], N[(l * N[(2.0 * N[(N[(l * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.32e+200], N[(2.0 / N[(t$95$3 * N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 8e+271], N[(2.0 / N[(t$95$3 * N[Power[N[(N[Power[N[Tan[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(t$95$2 * N[(t$95$m * N[Power[N[Power[l, -2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$3 * N[(N[Tan[k], $MachinePrecision] * N[Power[N[(t$95$2 * N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if t < 1.9500000000000001e-60

   1. Initial program 50.7%

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

   3. Simplified 48.7%

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

      1. associate-*r* 56.6%

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

      2. associate-*l/ 55.3%

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

   6. Applied egg-rr 55.3%

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

   7. Taylor expanded in k around inf 70.7%

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

      1. times-frac 71.2%

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

   9. Simplified 71.2%

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

       1. *-un-lft-identity 71.2%

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

       2. div-inv 71.3%

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

       3. pow-flip 71.5%

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

       4. metadata-eval 71.5%

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

   11. Applied egg-rr 71.5%

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

       1. *-lft-identity 71.5%

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

   13. Simplified 71.5%

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

   if 1.9500000000000001e-60 < t < 1.3199999999999999e200

   1. Initial program 64.3%

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

      1. add-sqr-sqrt 64.3%

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

      2. associate-*l* 64.3%

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

      3. sqrt-div 64.3%

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

      4. sqrt-pow1 64.3%

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

      5. metadata-eval 64.3%

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

      6. sqrt-prod 32.3%

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

      7. add-sqr-sqrt 49.5%

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

      8. sqrt-div 49.5%

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

      9. sqrt-pow1 52.5%

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

      10. metadata-eval 52.5%

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

      11. sqrt-prod 40.1%

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

      12. add-sqr-sqrt 85.2%

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

   4. Applied egg-rr 85.2%

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

      1. associate-*l* 91.4%

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

      2. associate-*l/ 91.4%

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

   6. Applied egg-rr 91.4%

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

   if 1.3199999999999999e200 < t < 7.99999999999999962e271

   1. Initial program 69.5%

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

      1. add-cube-cbrt 69.5%

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

      2. pow3 69.5%

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

      3. *-commutative 69.5%

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

      4. cbrt-prod 69.5%

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

      5. div-inv 69.5%

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

      6. cbrt-prod 69.5%

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

      7. rem-cbrt-cube 76.0%

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

      8. pow2 76.0%

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

      9. pow-flip 76.0%

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

      10. metadata-eval 76.0%

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

   4. Applied egg-rr 76.0%

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

      1. add-cube-cbrt 76.0%

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

      2. pow3 76.0%

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

      3. cbrt-prod 76.0%

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

      4. unpow3 76.0%

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

      5. add-cbrt-cube 87.3%

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

   6. Applied egg-rr 87.3%

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

   if 7.99999999999999962e271 < t

   1. Initial program 67.9%

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

      1. unpow3 67.9%

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

      2. times-frac 68.2%

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

      3. pow2 68.2%

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

   4. Applied egg-rr 68.2%

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

      1. add-cube-cbrt 68.2%

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

      2. pow3 68.2%

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

      3. *-commutative 68.2%

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

      4. cbrt-prod 68.2%

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

      5. frac-times 67.9%

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

      6. unpow2 67.9%

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

      7. unpow3 67.9%

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

      8. cbrt-div 67.9%

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

      9. unpow3 67.9%

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

      10. add-cbrt-cube 68.7%

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

      11. cbrt-prod 83.8%

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

      12. pow2 83.8%

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

   6. Applied egg-rr 83.8%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.95 \cdot 10^{-60}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \left(\left(\ell \cdot {k}^{-2}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\right)\\ \mathbf{elif}\;t \leq 1.32 \cdot 10^{+200}:\\ \;\;\;\;\frac{2}{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \left(\tan k \cdot \frac{\sin k \cdot {t}^{1.5}}{\ell}\right)\right)}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{+271}:\\ \;\;\;\;\frac{2}{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot {\left(\sqrt[3]{\tan k} \cdot \left(\sqrt[3]{\sin k} \cdot \left(t \cdot \sqrt[3]{{\ell}^{-2}}\right)\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\tan k \cdot {\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.5 \cdot 10^{-55}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \left(\left(\ell \cdot {k}^{-2}\right) \cdot \frac{\cos k}{t\_m \cdot {\sin k}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\tan k} \cdot \left(\sqrt[3]{\sin k} \cdot \left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)\right)}^{3} \cdot \left(1 + \left({\left(\frac{k}{t\_m}\right)}^{2} + 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4.5e-55)
    (* l (* 2.0 (* (* l (pow k -2.0)) (/ (cos k) (* t_m (pow (sin k) 2.0))))))
    (/
     2.0
     (*
      (pow
       (* (cbrt (tan k)) (* (cbrt (sin k)) (* t_m (pow (cbrt l) -2.0))))
       3.0)
      (+ 1.0 (+ (pow (/ k t_m) 2.0) 1.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.5e-55) {
		tmp = l * (2.0 * ((l * pow(k, -2.0)) * (cos(k) / (t_m * pow(sin(k), 2.0)))));
	} else {
		tmp = 2.0 / (pow((cbrt(tan(k)) * (cbrt(sin(k)) * (t_m * pow(cbrt(l), -2.0)))), 3.0) * (1.0 + (pow((k / t_m), 2.0) + 1.0)));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.5e-55) {
		tmp = l * (2.0 * ((l * Math.pow(k, -2.0)) * (Math.cos(k) / (t_m * Math.pow(Math.sin(k), 2.0)))));
	} else {
		tmp = 2.0 / (Math.pow((Math.cbrt(Math.tan(k)) * (Math.cbrt(Math.sin(k)) * (t_m * Math.pow(Math.cbrt(l), -2.0)))), 3.0) * (1.0 + (Math.pow((k / t_m), 2.0) + 1.0)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 4.5e-55)
		tmp = Float64(l * Float64(2.0 * Float64(Float64(l * (k ^ -2.0)) * Float64(cos(k) / Float64(t_m * (sin(k) ^ 2.0))))));
	else
		tmp = Float64(2.0 / Float64((Float64(cbrt(tan(k)) * Float64(cbrt(sin(k)) * Float64(t_m * (cbrt(l) ^ -2.0)))) ^ 3.0) * Float64(1.0 + Float64((Float64(k / t_m) ^ 2.0) + 1.0))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 4.5e-55], N[(l * N[(2.0 * N[(N[(l * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[Power[N[Tan[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(1.0 + N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.4999999999999997e-55

   1. Initial program 51.2%

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

   3. Simplified 49.3%

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

      1. associate-*r* 57.1%

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

      2. associate-*l/ 55.8%

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

   6. Applied egg-rr 55.8%

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

   7. Taylor expanded in k around inf 71.0%

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

      1. times-frac 71.6%

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

   9. Simplified 71.6%

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

       1. *-un-lft-identity 71.6%

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

       2. div-inv 71.6%

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

       3. pow-flip 71.8%

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

       4. metadata-eval 71.8%

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

   11. Applied egg-rr 71.8%

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

       1. *-lft-identity 71.8%

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

   13. Simplified 71.8%

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

   if 4.4999999999999997e-55 < t

   1. Initial program 64.7%

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

      1. unpow3 64.7%

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

      2. times-frac 74.7%

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

      3. pow2 74.7%

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

   4. Applied egg-rr 74.7%

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

      1. add-cube-cbrt 74.6%

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

   6. Applied egg-rr 83.7%

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

      1. unpow2 83.7%

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

      2. unpow3 83.8%

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

      3. cube-prod 78.9%

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

      4. rem-cube-cbrt 79.0%

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

   8. Simplified 79.0%

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

      1. add-cube-cbrt 78.9%

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

      2. pow3 78.9%

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

      3. cbrt-prod 79.0%

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

      4. cbrt-prod 78.9%

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

      5. unpow3 78.9%

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

      6. add-cbrt-cube 94.8%

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

      7. div-inv 94.8%

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

      8. pow-flip 94.9%

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

      9. metadata-eval 94.9%

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

   10. Applied egg-rr 94.9%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.5 \cdot 10^{-55}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \left(\left(\ell \cdot {k}^{-2}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\tan k} \cdot \left(\sqrt[3]{\sin k} \cdot \left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)\right)}^{3} \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.5 \cdot 10^{-54}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \left(\left(\ell \cdot {k}^{-2}\right) \cdot \frac{\cos k}{t\_m \cdot {\sin k}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(1 + \left({\left(\frac{k}{t\_m}\right)}^{2} + 1\right)\right) \cdot \left(\tan k \cdot {\left(\sqrt[3]{\sin k} \cdot \frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}\\ \end{array} \end{array} \]

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 6.5e-54)
    (* l (* 2.0 (* (* l (pow k -2.0)) (/ (cos k) (* t_m (pow (sin k) 2.0))))))
    (/
     2.0
     (*
      (+ 1.0 (+ (pow (/ k t_m) 2.0) 1.0))
      (* (tan k) (pow (* (cbrt (sin k)) (/ t_m (pow (cbrt l) 2.0))) 3.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 6.5e-54) {
		tmp = l * (2.0 * ((l * pow(k, -2.0)) * (cos(k) / (t_m * pow(sin(k), 2.0)))));
	} else {
		tmp = 2.0 / ((1.0 + (pow((k / t_m), 2.0) + 1.0)) * (tan(k) * pow((cbrt(sin(k)) * (t_m / pow(cbrt(l), 2.0))), 3.0)));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 6.5e-54) {
		tmp = l * (2.0 * ((l * Math.pow(k, -2.0)) * (Math.cos(k) / (t_m * Math.pow(Math.sin(k), 2.0)))));
	} else {
		tmp = 2.0 / ((1.0 + (Math.pow((k / t_m), 2.0) + 1.0)) * (Math.tan(k) * Math.pow((Math.cbrt(Math.sin(k)) * (t_m / Math.pow(Math.cbrt(l), 2.0))), 3.0)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 6.5e-54)
		tmp = Float64(l * Float64(2.0 * Float64(Float64(l * (k ^ -2.0)) * Float64(cos(k) / Float64(t_m * (sin(k) ^ 2.0))))));
	else
		tmp = Float64(2.0 / Float64(Float64(1.0 + Float64((Float64(k / t_m) ^ 2.0) + 1.0)) * Float64(tan(k) * (Float64(cbrt(sin(k)) * Float64(t_m / (cbrt(l) ^ 2.0))) ^ 3.0))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 6.5e-54], N[(l * N[(2.0 * N[(N[(l * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(1.0 + N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Power[N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 6.49999999999999991e-54

   1. Initial program 51.2%

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

   3. Simplified 49.3%

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

      1. associate-*r* 57.1%

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

      2. associate-*l/ 55.8%

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

   6. Applied egg-rr 55.8%

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

   7. Taylor expanded in k around inf 71.0%

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

      1. times-frac 71.6%

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

   9. Simplified 71.6%

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

       1. *-un-lft-identity 71.6%

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

       2. div-inv 71.6%

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

       3. pow-flip 71.8%

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

       4. metadata-eval 71.8%

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

   11. Applied egg-rr 71.8%

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

       1. *-lft-identity 71.8%

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

   13. Simplified 71.8%

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

   if 6.49999999999999991e-54 < t

   1. Initial program 64.7%

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

      1. unpow3 64.7%

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

      2. times-frac 74.7%

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

      3. pow2 74.7%

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

   4. Applied egg-rr 74.7%

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

      1. add-cube-cbrt 74.6%

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

      2. pow3 74.5%

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

      3. *-commutative 74.5%

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

      4. cbrt-prod 74.5%

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

      5. frac-times 64.6%

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

      6. unpow2 64.6%

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

      7. unpow3 64.6%

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

      8. cbrt-div 64.6%

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

      9. unpow3 64.6%

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

      10. add-cbrt-cube 70.8%

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

      11. cbrt-prod 83.8%

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

      12. pow2 83.8%

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

   6. Applied egg-rr 83.8%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.5 \cdot 10^{-54}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \left(\left(\ell \cdot {k}^{-2}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\tan k \cdot {\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 1 + \left({\left(\frac{k}{t\_m}\right)}^{2} + 1\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.05 \cdot 10^{-60}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \left(\left(\ell \cdot {k}^{-2}\right) \cdot \frac{\cos k}{t\_m \cdot {\sin k}^{2}}\right)\right)\\ \mathbf{elif}\;t\_m \leq 8 \cdot 10^{+199}:\\ \;\;\;\;\frac{2}{t\_2 \cdot \left(\frac{{t\_m}^{1.5}}{\ell} \cdot \left(\tan k \cdot \frac{\sin k \cdot {t\_m}^{1.5}}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_2 \cdot \left(\tan k \cdot {\left(\sqrt[3]{\sin k} \cdot \left(t\_m \cdot {\ell}^{-0.6666666666666666}\right)\right)}^{3}\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (+ 1.0 (+ (pow (/ k t_m) 2.0) 1.0))))
   (*
    t_s
    (if (<= t_m 2.05e-60)
      (*
       l
       (* 2.0 (* (* l (pow k -2.0)) (/ (cos k) (* t_m (pow (sin k) 2.0))))))
      (if (<= t_m 8e+199)
        (/
         2.0
         (*
          t_2
          (* (/ (pow t_m 1.5) l) (* (tan k) (/ (* (sin k) (pow t_m 1.5)) l)))))
        (/
         2.0
         (*
          t_2
          (*
           (tan k)
           (pow
            (* (cbrt (sin k)) (* 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 = 1.0 + (pow((k / t_m), 2.0) + 1.0);
	double tmp;
	if (t_m <= 2.05e-60) {
		tmp = l * (2.0 * ((l * pow(k, -2.0)) * (cos(k) / (t_m * pow(sin(k), 2.0)))));
	} else if (t_m <= 8e+199) {
		tmp = 2.0 / (t_2 * ((pow(t_m, 1.5) / l) * (tan(k) * ((sin(k) * pow(t_m, 1.5)) / l))));
	} else {
		tmp = 2.0 / (t_2 * (tan(k) * pow((cbrt(sin(k)) * (t_m * pow(l, -0.6666666666666666))), 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 = 1.0 + (Math.pow((k / t_m), 2.0) + 1.0);
	double tmp;
	if (t_m <= 2.05e-60) {
		tmp = l * (2.0 * ((l * Math.pow(k, -2.0)) * (Math.cos(k) / (t_m * Math.pow(Math.sin(k), 2.0)))));
	} else if (t_m <= 8e+199) {
		tmp = 2.0 / (t_2 * ((Math.pow(t_m, 1.5) / l) * (Math.tan(k) * ((Math.sin(k) * Math.pow(t_m, 1.5)) / l))));
	} else {
		tmp = 2.0 / (t_2 * (Math.tan(k) * Math.pow((Math.cbrt(Math.sin(k)) * (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(1.0 + Float64((Float64(k / t_m) ^ 2.0) + 1.0))
	tmp = 0.0
	if (t_m <= 2.05e-60)
		tmp = Float64(l * Float64(2.0 * Float64(Float64(l * (k ^ -2.0)) * Float64(cos(k) / Float64(t_m * (sin(k) ^ 2.0))))));
	elseif (t_m <= 8e+199)
		tmp = Float64(2.0 / Float64(t_2 * Float64(Float64((t_m ^ 1.5) / l) * Float64(tan(k) * Float64(Float64(sin(k) * (t_m ^ 1.5)) / l)))));
	else
		tmp = Float64(2.0 / Float64(t_2 * Float64(tan(k) * (Float64(cbrt(sin(k)) * Float64(t_m * (l ^ -0.6666666666666666))) ^ 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[(1.0 + N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.05e-60], N[(l * N[(2.0 * N[(N[(l * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 8e+199], N[(2.0 / N[(t$95$2 * N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$2 * N[(N[Tan[k], $MachinePrecision] * N[Power[N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(t$95$m * N[Power[l, -0.6666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 2.05000000000000006e-60

   1. Initial program 50.7%

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

   3. Simplified 48.7%

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

      1. associate-*r* 56.6%

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

      2. associate-*l/ 55.3%

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

   6. Applied egg-rr 55.3%

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

   7. Taylor expanded in k around inf 70.7%

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

      1. times-frac 71.2%

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

   9. Simplified 71.2%

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

       1. *-un-lft-identity 71.2%

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

       2. div-inv 71.3%

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

       3. pow-flip 71.5%

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

       4. metadata-eval 71.5%

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

   11. Applied egg-rr 71.5%

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

       1. *-lft-identity 71.5%

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

   13. Simplified 71.5%

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

   if 2.05000000000000006e-60 < t < 8.00000000000000078e199

   1. Initial program 64.3%

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

      1. add-sqr-sqrt 64.3%

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

      2. associate-*l* 64.3%

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

      3. sqrt-div 64.3%

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

      4. sqrt-pow1 64.3%

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

      5. metadata-eval 64.3%

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

      6. sqrt-prod 32.3%

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

      7. add-sqr-sqrt 49.5%

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

      8. sqrt-div 49.5%

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

      9. sqrt-pow1 52.5%

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

      10. metadata-eval 52.5%

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

      11. sqrt-prod 40.1%

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

      12. add-sqr-sqrt 85.2%

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

   4. Applied egg-rr 85.2%

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

      1. associate-*l* 91.4%

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

      2. associate-*l/ 91.4%

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

   6. Applied egg-rr 91.4%

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

   if 8.00000000000000078e199 < t

   1. Initial program 69.0%

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

      1. add-cube-cbrt 69.0%

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

      2. pow3 69.0%

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

      3. *-commutative 69.0%

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

      4. cbrt-prod 69.0%

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

      5. div-inv 69.0%

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

      6. cbrt-prod 69.0%

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

      7. rem-cbrt-cube 74.0%

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

      8. pow2 74.0%

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

      9. pow-flip 74.0%

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

      10. metadata-eval 74.0%

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

   4. Applied egg-rr 74.0%

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

      1. pow1/3 74.0%

         \[\leadsto \frac{2}{\left({\left(\sqrt[3]{\sin k} \cdot \left(t \cdot \color{blue}{{\left({\ell}^{-2}\right)}^{0.3333333333333333}}\right)\right)}^{3} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. pow-pow 45.8%

         \[\leadsto \frac{2}{\left({\left(\sqrt[3]{\sin k} \cdot \left(t \cdot \color{blue}{{\ell}^{\left(-2 \cdot 0.3333333333333333\right)}}\right)\right)}^{3} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. metadata-eval 45.8%

         \[\leadsto \frac{2}{\left({\left(\sqrt[3]{\sin k} \cdot \left(t \cdot {\ell}^{\color{blue}{-0.6666666666666666}}\right)\right)}^{3} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied egg-rr 45.8%

      \[\leadsto \frac{2}{\left({\left(\sqrt[3]{\sin k} \cdot \left(t \cdot \color{blue}{{\ell}^{-0.6666666666666666}}\right)\right)}^{3} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.05 \cdot 10^{-60}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \left(\left(\ell \cdot {k}^{-2}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\right)\\ \mathbf{elif}\;t \leq 8 \cdot 10^{+199}:\\ \;\;\;\;\frac{2}{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \left(\tan k \cdot \frac{\sin k \cdot {t}^{1.5}}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\tan k \cdot {\left(\sqrt[3]{\sin k} \cdot \left(t \cdot {\ell}^{-0.6666666666666666}\right)\right)}^{3}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\frac{k}{t\_m}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.8 \cdot 10^{-60}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \left(\left(\ell \cdot {k}^{-2}\right) \cdot \frac{\cos k}{t\_m \cdot {\sin k}^{2}}\right)\right)\\ \mathbf{elif}\;t\_m \leq 1.32 \cdot 10^{+200}:\\ \;\;\;\;\frac{2}{\left(1 + \left(t\_2 + 1\right)\right) \cdot \left(\frac{{t\_m}^{1.5}}{\ell} \cdot \left(\tan k \cdot \frac{\sin k \cdot {t\_m}^{1.5}}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell \cdot \frac{2}{2 + t\_2}}{{\left(t\_m \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (/ k t_m) 2.0)))
   (*
    t_s
    (if (<= t_m 2.8e-60)
      (*
       l
       (* 2.0 (* (* l (pow k -2.0)) (/ (cos k) (* t_m (pow (sin k) 2.0))))))
      (if (<= t_m 1.32e+200)
        (/
         2.0
         (*
          (+ 1.0 (+ t_2 1.0))
          (* (/ (pow t_m 1.5) l) (* (tan k) (/ (* (sin k) (pow t_m 1.5)) l)))))
        (*
         l
         (/
          (* l (/ 2.0 (+ 2.0 t_2)))
          (pow (* t_m (pow (cbrt k) 2.0)) 3.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 2.8e-60) {
		tmp = l * (2.0 * ((l * pow(k, -2.0)) * (cos(k) / (t_m * pow(sin(k), 2.0)))));
	} else if (t_m <= 1.32e+200) {
		tmp = 2.0 / ((1.0 + (t_2 + 1.0)) * ((pow(t_m, 1.5) / l) * (tan(k) * ((sin(k) * pow(t_m, 1.5)) / l))));
	} else {
		tmp = l * ((l * (2.0 / (2.0 + t_2))) / pow((t_m * pow(cbrt(k), 2.0)), 3.0));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 2.8e-60) {
		tmp = l * (2.0 * ((l * Math.pow(k, -2.0)) * (Math.cos(k) / (t_m * Math.pow(Math.sin(k), 2.0)))));
	} else if (t_m <= 1.32e+200) {
		tmp = 2.0 / ((1.0 + (t_2 + 1.0)) * ((Math.pow(t_m, 1.5) / l) * (Math.tan(k) * ((Math.sin(k) * Math.pow(t_m, 1.5)) / l))));
	} else {
		tmp = l * ((l * (2.0 / (2.0 + t_2))) / Math.pow((t_m * Math.pow(Math.cbrt(k), 2.0)), 3.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(k / t_m) ^ 2.0
	tmp = 0.0
	if (t_m <= 2.8e-60)
		tmp = Float64(l * Float64(2.0 * Float64(Float64(l * (k ^ -2.0)) * Float64(cos(k) / Float64(t_m * (sin(k) ^ 2.0))))));
	elseif (t_m <= 1.32e+200)
		tmp = Float64(2.0 / Float64(Float64(1.0 + Float64(t_2 + 1.0)) * Float64(Float64((t_m ^ 1.5) / l) * Float64(tan(k) * Float64(Float64(sin(k) * (t_m ^ 1.5)) / l)))));
	else
		tmp = Float64(l * Float64(Float64(l * Float64(2.0 / Float64(2.0 + t_2))) / (Float64(t_m * (cbrt(k) ^ 2.0)) ^ 3.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.8e-60], N[(l * N[(2.0 * N[(N[(l * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.32e+200], N[(2.0 / N[(N[(1.0 + N[(t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(l * N[(2.0 / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[(t$95$m * N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 2.8000000000000002e-60

   1. Initial program 50.7%

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

   3. Simplified 48.7%

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

      1. associate-*r* 56.6%

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

      2. associate-*l/ 55.3%

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

   6. Applied egg-rr 55.3%

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

   7. Taylor expanded in k around inf 70.7%

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

      1. times-frac 71.2%

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

   9. Simplified 71.2%

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

       1. *-un-lft-identity 71.2%

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

       2. div-inv 71.3%

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

       3. pow-flip 71.5%

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

       4. metadata-eval 71.5%

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

   11. Applied egg-rr 71.5%

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

       1. *-lft-identity 71.5%

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

   13. Simplified 71.5%

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

   if 2.8000000000000002e-60 < t < 1.3199999999999999e200

   1. Initial program 64.3%

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

      1. add-sqr-sqrt 64.3%

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

      2. associate-*l* 64.3%

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

      3. sqrt-div 64.3%

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

      4. sqrt-pow1 64.3%

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

      5. metadata-eval 64.3%

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

      6. sqrt-prod 32.3%

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

      7. add-sqr-sqrt 49.5%

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

      8. sqrt-div 49.5%

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

      9. sqrt-pow1 52.5%

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

      10. metadata-eval 52.5%

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

      11. sqrt-prod 40.1%

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

      12. add-sqr-sqrt 85.2%

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

   4. Applied egg-rr 85.2%

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

      1. associate-*l* 91.4%

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

      2. associate-*l/ 91.4%

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

   6. Applied egg-rr 91.4%

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

   if 1.3199999999999999e200 < t

   1. Initial program 69.0%

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

   3. Simplified 54.5%

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

      1. associate-*r* 55.0%

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

      2. associate-*l/ 55.0%

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

   6. Applied egg-rr 55.0%

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

   7. Taylor expanded in k around 0 55.0%

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

      1. add-cube-cbrt 55.0%

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

      2. pow2 55.0%

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

      3. *-commutative 55.0%

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

      4. cbrt-prod 55.0%

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

      5. unpow3 55.0%

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

      6. add-cbrt-cube 55.0%

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

      7. unpow2 55.0%

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

      8. cbrt-prod 55.0%

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

      9. pow2 55.0%

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

      10. *-commutative 55.0%

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

      11. cbrt-prod 55.0%

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

      12. unpow3 55.0%

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

      13. add-cbrt-cube 60.4%

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

      14. unpow2 60.4%

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

      15. cbrt-prod 86.7%

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

      16. pow2 86.7%

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

   9. Applied egg-rr 86.7%

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

       1. unpow2 86.7%

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

       2. unpow3 86.7%

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

   11. Simplified 86.7%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.8 \cdot 10^{-60}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \left(\left(\ell \cdot {k}^{-2}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\right)\\ \mathbf{elif}\;t \leq 1.32 \cdot 10^{+200}:\\ \;\;\;\;\frac{2}{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \left(\tan k \cdot \frac{\sin k \cdot {t}^{1.5}}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell \cdot \frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}}}{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\frac{k}{t\_m}\right)}^{2}\\ t_3 := \frac{{t\_m}^{1.5}}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.3 \cdot 10^{-54}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \left(\left(\ell \cdot {k}^{-2}\right) \cdot \frac{\cos k}{t\_m \cdot {\sin k}^{2}}\right)\right)\\ \mathbf{elif}\;t\_m \leq 6 \cdot 10^{+184}:\\ \;\;\;\;\frac{2}{\left(1 + \left(t\_2 + 1\right)\right) \cdot \left(\tan k \cdot \left(t\_3 \cdot \left(\sin k \cdot t\_3\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell \cdot \frac{2}{2 + t\_2}}{{\left(t\_m \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (/ k t_m) 2.0)) (t_3 (/ (pow t_m 1.5) l)))
   (*
    t_s
    (if (<= t_m 2.3e-54)
      (*
       l
       (* 2.0 (* (* l (pow k -2.0)) (/ (cos k) (* t_m (pow (sin k) 2.0))))))
      (if (<= t_m 6e+184)
        (/ 2.0 (* (+ 1.0 (+ t_2 1.0)) (* (tan k) (* t_3 (* (sin k) t_3)))))
        (*
         l
         (/
          (* l (/ 2.0 (+ 2.0 t_2)))
          (pow (* t_m (pow (cbrt k) 2.0)) 3.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow((k / t_m), 2.0);
	double t_3 = pow(t_m, 1.5) / l;
	double tmp;
	if (t_m <= 2.3e-54) {
		tmp = l * (2.0 * ((l * pow(k, -2.0)) * (cos(k) / (t_m * pow(sin(k), 2.0)))));
	} else if (t_m <= 6e+184) {
		tmp = 2.0 / ((1.0 + (t_2 + 1.0)) * (tan(k) * (t_3 * (sin(k) * t_3))));
	} else {
		tmp = l * ((l * (2.0 / (2.0 + t_2))) / pow((t_m * pow(cbrt(k), 2.0)), 3.0));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.pow((k / t_m), 2.0);
	double t_3 = Math.pow(t_m, 1.5) / l;
	double tmp;
	if (t_m <= 2.3e-54) {
		tmp = l * (2.0 * ((l * Math.pow(k, -2.0)) * (Math.cos(k) / (t_m * Math.pow(Math.sin(k), 2.0)))));
	} else if (t_m <= 6e+184) {
		tmp = 2.0 / ((1.0 + (t_2 + 1.0)) * (Math.tan(k) * (t_3 * (Math.sin(k) * t_3))));
	} else {
		tmp = l * ((l * (2.0 / (2.0 + t_2))) / Math.pow((t_m * Math.pow(Math.cbrt(k), 2.0)), 3.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(k / t_m) ^ 2.0
	t_3 = Float64((t_m ^ 1.5) / l)
	tmp = 0.0
	if (t_m <= 2.3e-54)
		tmp = Float64(l * Float64(2.0 * Float64(Float64(l * (k ^ -2.0)) * Float64(cos(k) / Float64(t_m * (sin(k) ^ 2.0))))));
	elseif (t_m <= 6e+184)
		tmp = Float64(2.0 / Float64(Float64(1.0 + Float64(t_2 + 1.0)) * Float64(tan(k) * Float64(t_3 * Float64(sin(k) * t_3)))));
	else
		tmp = Float64(l * Float64(Float64(l * Float64(2.0 / Float64(2.0 + t_2))) / (Float64(t_m * (cbrt(k) ^ 2.0)) ^ 3.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.3e-54], N[(l * N[(2.0 * N[(N[(l * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6e+184], N[(2.0 / N[(N[(1.0 + N[(t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(t$95$3 * N[(N[Sin[k], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(l * N[(2.0 / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[(t$95$m * N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 2.2999999999999999e-54

   1. Initial program 51.2%

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

   3. Simplified 49.3%

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

      1. associate-*r* 57.1%

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

      2. associate-*l/ 55.8%

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

   6. Applied egg-rr 55.8%

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

   7. Taylor expanded in k around inf 71.0%

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

      1. times-frac 71.6%

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

   9. Simplified 71.6%

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

       1. *-un-lft-identity 71.6%

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

       2. div-inv 71.6%

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

       3. pow-flip 71.8%

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

       4. metadata-eval 71.8%

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

   11. Applied egg-rr 71.8%

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

       1. *-lft-identity 71.8%

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

   13. Simplified 71.8%

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

   if 2.2999999999999999e-54 < t < 5.99999999999999973e184

   1. Initial program 62.8%

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

      1. add-sqr-sqrt 62.8%

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

      2. associate-*l* 62.8%

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

      3. sqrt-div 62.8%

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

      4. sqrt-pow1 62.8%

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

      5. metadata-eval 62.8%

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

      6. sqrt-prod 31.5%

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

      7. add-sqr-sqrt 48.3%

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

      8. sqrt-div 48.4%

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

      9. sqrt-pow1 51.7%

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

      10. metadata-eval 51.7%

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

      11. sqrt-prod 40.2%

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

      12. add-sqr-sqrt 85.6%

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

   4. Applied egg-rr 85.6%

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

   if 5.99999999999999973e184 < t

   1. Initial program 68.9%

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

   3. Simplified 56.0%

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

      1. associate-*r* 56.4%

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

      2. associate-*l/ 56.4%

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

   6. Applied egg-rr 56.4%

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

   7. Taylor expanded in k around 0 56.4%

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

      1. add-cube-cbrt 56.4%

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

      2. pow2 56.4%

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

      3. *-commutative 56.4%

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

      4. cbrt-prod 56.4%

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

      5. unpow3 56.4%

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

      6. add-cbrt-cube 56.4%

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

      7. unpow2 56.4%

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

      8. cbrt-prod 56.4%

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

      9. pow2 56.4%

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

      10. *-commutative 56.4%

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

      11. cbrt-prod 56.4%

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

      12. unpow3 56.4%

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

      13. add-cbrt-cube 61.2%

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

      14. unpow2 61.2%

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

      15. cbrt-prod 88.3%

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

      16. pow2 88.3%

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

   9. Applied egg-rr 88.3%

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

       1. unpow2 88.3%

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

       2. unpow3 88.2%

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

   11. Simplified 88.2%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.3 \cdot 10^{-54}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \left(\left(\ell \cdot {k}^{-2}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\right)\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+184}:\\ \;\;\;\;\frac{2}{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\tan k \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \left(\sin k \cdot \frac{{t}^{1.5}}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell \cdot \frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}}}{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\frac{k}{t\_m}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8 \cdot 10^{-55}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \left(\left(\ell \cdot {k}^{-2}\right) \cdot \frac{\cos k}{t\_m \cdot {\sin k}^{2}}\right)\right)\\ \mathbf{elif}\;t\_m \leq 2.2 \cdot 10^{+149}:\\ \;\;\;\;\frac{2}{\left(1 + \left(t\_2 + 1\right)\right) \cdot \left(\tan k \cdot \left(\frac{{t\_m}^{2}}{\ell} \cdot \left(\sin k \cdot \frac{t\_m}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell \cdot \frac{2}{2 + t\_2}}{{\left(t\_m \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (/ k t_m) 2.0)))
   (*
    t_s
    (if (<= t_m 8e-55)
      (*
       l
       (* 2.0 (* (* l (pow k -2.0)) (/ (cos k) (* t_m (pow (sin k) 2.0))))))
      (if (<= t_m 2.2e+149)
        (/
         2.0
         (*
          (+ 1.0 (+ t_2 1.0))
          (* (tan k) (* (/ (pow t_m 2.0) l) (* (sin k) (/ t_m l))))))
        (*
         l
         (/
          (* l (/ 2.0 (+ 2.0 t_2)))
          (pow (* t_m (pow (cbrt k) 2.0)) 3.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 8e-55) {
		tmp = l * (2.0 * ((l * pow(k, -2.0)) * (cos(k) / (t_m * pow(sin(k), 2.0)))));
	} else if (t_m <= 2.2e+149) {
		tmp = 2.0 / ((1.0 + (t_2 + 1.0)) * (tan(k) * ((pow(t_m, 2.0) / l) * (sin(k) * (t_m / l)))));
	} else {
		tmp = l * ((l * (2.0 / (2.0 + t_2))) / pow((t_m * pow(cbrt(k), 2.0)), 3.0));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 8e-55) {
		tmp = l * (2.0 * ((l * Math.pow(k, -2.0)) * (Math.cos(k) / (t_m * Math.pow(Math.sin(k), 2.0)))));
	} else if (t_m <= 2.2e+149) {
		tmp = 2.0 / ((1.0 + (t_2 + 1.0)) * (Math.tan(k) * ((Math.pow(t_m, 2.0) / l) * (Math.sin(k) * (t_m / l)))));
	} else {
		tmp = l * ((l * (2.0 / (2.0 + t_2))) / Math.pow((t_m * Math.pow(Math.cbrt(k), 2.0)), 3.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(k / t_m) ^ 2.0
	tmp = 0.0
	if (t_m <= 8e-55)
		tmp = Float64(l * Float64(2.0 * Float64(Float64(l * (k ^ -2.0)) * Float64(cos(k) / Float64(t_m * (sin(k) ^ 2.0))))));
	elseif (t_m <= 2.2e+149)
		tmp = Float64(2.0 / Float64(Float64(1.0 + Float64(t_2 + 1.0)) * Float64(tan(k) * Float64(Float64((t_m ^ 2.0) / l) * Float64(sin(k) * Float64(t_m / l))))));
	else
		tmp = Float64(l * Float64(Float64(l * Float64(2.0 / Float64(2.0 + t_2))) / (Float64(t_m * (cbrt(k) ^ 2.0)) ^ 3.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 8e-55], N[(l * N[(2.0 * N[(N[(l * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.2e+149], N[(2.0 / N[(N[(1.0 + N[(t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(l * N[(2.0 / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[(t$95$m * N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 7.99999999999999996e-55

   1. Initial program 51.2%

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

   3. Simplified 49.3%

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

      1. associate-*r* 57.1%

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

      2. associate-*l/ 55.8%

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

   6. Applied egg-rr 55.8%

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

   7. Taylor expanded in k around inf 71.0%

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

      1. times-frac 71.6%

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

   9. Simplified 71.6%

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

       1. *-un-lft-identity 71.6%

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

       2. div-inv 71.6%

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

       3. pow-flip 71.8%

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

       4. metadata-eval 71.8%

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

   11. Applied egg-rr 71.8%

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

       1. *-lft-identity 71.8%

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

   13. Simplified 71.8%

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

   if 7.99999999999999996e-55 < t < 2.2e149

   1. Initial program 67.7%

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

      1. unpow3 67.7%

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

      2. times-frac 82.5%

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

      3. pow2 82.5%

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

   4. Applied egg-rr 82.5%

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

      1. associate-*l* 84.6%

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

   6. Applied egg-rr 84.6%

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

   if 2.2e149 < t

   1. Initial program 61.0%

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

   3. Simplified 51.4%

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

      1. associate-*r* 54.8%

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

      2. associate-*l/ 54.8%

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

   6. Applied egg-rr 54.8%

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

   7. Taylor expanded in k around 0 54.8%

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

      1. add-cube-cbrt 54.8%

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

      2. pow2 54.8%

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

      3. *-commutative 54.8%

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

      4. cbrt-prod 54.8%

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

      5. unpow3 54.8%

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

      6. add-cbrt-cube 54.8%

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

      7. unpow2 54.8%

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

      8. cbrt-prod 54.8%

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

      9. pow2 54.8%

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

      10. *-commutative 54.8%

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

      11. cbrt-prod 54.8%

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

      12. unpow3 54.8%

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

      13. add-cbrt-cube 58.6%

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

      14. unpow2 58.6%

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

      15. cbrt-prod 86.0%

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

      16. pow2 86.0%

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

   9. Applied egg-rr 86.0%

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

       1. unpow2 86.0%

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

       2. unpow3 86.0%

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

   11. Simplified 86.0%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8 \cdot 10^{-55}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \left(\left(\ell \cdot {k}^{-2}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\right)\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{+149}:\\ \;\;\;\;\frac{2}{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\tan k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \left(\sin k \cdot \frac{t}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell \cdot \frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}}}{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \ell \cdot \frac{2}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.7 \cdot 10^{-26}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \left(\left(\ell \cdot {k}^{-2}\right) \cdot \frac{\cos k}{t\_m \cdot {\sin k}^{2}}\right)\right)\\ \mathbf{elif}\;t\_m \leq 3.9 \cdot 10^{+96}:\\ \;\;\;\;\ell \cdot \frac{t\_2}{\sin k \cdot \left(\tan k \cdot {t\_m}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{t\_2}{{\left(t\_m \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* l (/ 2.0 (+ 2.0 (pow (/ k t_m) 2.0))))))
   (*
    t_s
    (if (<= t_m 1.7e-26)
      (*
       l
       (* 2.0 (* (* l (pow k -2.0)) (/ (cos k) (* t_m (pow (sin k) 2.0))))))
      (if (<= t_m 3.9e+96)
        (* l (/ t_2 (* (sin k) (* (tan k) (pow t_m 3.0)))))
        (* l (/ t_2 (pow (* t_m (pow (cbrt k) 2.0)) 3.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = l * (2.0 / (2.0 + pow((k / t_m), 2.0)));
	double tmp;
	if (t_m <= 1.7e-26) {
		tmp = l * (2.0 * ((l * pow(k, -2.0)) * (cos(k) / (t_m * pow(sin(k), 2.0)))));
	} else if (t_m <= 3.9e+96) {
		tmp = l * (t_2 / (sin(k) * (tan(k) * pow(t_m, 3.0))));
	} else {
		tmp = l * (t_2 / pow((t_m * pow(cbrt(k), 2.0)), 3.0));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = l * (2.0 / (2.0 + Math.pow((k / t_m), 2.0)));
	double tmp;
	if (t_m <= 1.7e-26) {
		tmp = l * (2.0 * ((l * Math.pow(k, -2.0)) * (Math.cos(k) / (t_m * Math.pow(Math.sin(k), 2.0)))));
	} else if (t_m <= 3.9e+96) {
		tmp = l * (t_2 / (Math.sin(k) * (Math.tan(k) * Math.pow(t_m, 3.0))));
	} else {
		tmp = l * (t_2 / Math.pow((t_m * Math.pow(Math.cbrt(k), 2.0)), 3.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(l * Float64(2.0 / Float64(2.0 + (Float64(k / t_m) ^ 2.0))))
	tmp = 0.0
	if (t_m <= 1.7e-26)
		tmp = Float64(l * Float64(2.0 * Float64(Float64(l * (k ^ -2.0)) * Float64(cos(k) / Float64(t_m * (sin(k) ^ 2.0))))));
	elseif (t_m <= 3.9e+96)
		tmp = Float64(l * Float64(t_2 / Float64(sin(k) * Float64(tan(k) * (t_m ^ 3.0)))));
	else
		tmp = Float64(l * Float64(t_2 / (Float64(t_m * (cbrt(k) ^ 2.0)) ^ 3.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(l * N[(2.0 / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.7e-26], N[(l * N[(2.0 * N[(N[(l * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.9e+96], N[(l * N[(t$95$2 / N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(t$95$2 / N[Power[N[(t$95$m * N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.70000000000000007e-26

   1. Initial program 51.9%

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

   3. Simplified 50.1%

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

      1. associate-*r* 57.7%

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

      2. associate-*l/ 56.5%

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

   6. Applied egg-rr 56.5%

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

   7. Taylor expanded in k around inf 71.3%

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

      1. times-frac 71.8%

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

   9. Simplified 71.8%

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

       1. *-un-lft-identity 71.8%

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

       2. div-inv 71.8%

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

       3. pow-flip 72.1%

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

       4. metadata-eval 72.1%

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

   11. Applied egg-rr 72.1%

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

       1. *-lft-identity 72.1%

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

   13. Simplified 72.1%

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

   if 1.70000000000000007e-26 < t < 3.9e96

   1. Initial program 77.6%

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

   3. Simplified 77.2%

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

      1. associate-*r* 84.7%

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

      2. associate-*l/ 93.3%

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

   6. Applied egg-rr 93.3%

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

      1. *-commutative 93.3%

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

      2. associate-*r* 93.3%

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

   8. Applied egg-rr 93.3%

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

   if 3.9e96 < t

   1. Initial program 54.7%

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

   3. Simplified 47.3%

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

      1. associate-*r* 50.1%

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

      2. associate-*l/ 50.1%

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

   6. Applied egg-rr 50.1%

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

   7. Taylor expanded in k around 0 50.1%

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

      1. add-cube-cbrt 50.1%

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

      2. pow2 50.1%

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

      3. *-commutative 50.1%

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

      4. cbrt-prod 50.1%

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

      5. unpow3 50.1%

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

      6. add-cbrt-cube 50.1%

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

      7. unpow2 50.1%

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

      8. cbrt-prod 50.1%

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

      9. pow2 50.1%

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

      10. *-commutative 50.1%

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

      11. cbrt-prod 50.1%

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

      12. unpow3 50.1%

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

      13. add-cbrt-cube 57.2%

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

      14. unpow2 57.2%

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

      15. cbrt-prod 78.5%

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

      16. pow2 78.5%

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

   9. Applied egg-rr 78.5%

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

       1. unpow2 78.5%

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

       2. unpow3 78.4%

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

   11. Simplified 78.4%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.7 \cdot 10^{-26}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \left(\left(\ell \cdot {k}^{-2}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\right)\\ \mathbf{elif}\;t \leq 3.9 \cdot 10^{+96}:\\ \;\;\;\;\ell \cdot \frac{\ell \cdot \frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}}}{\sin k \cdot \left(\tan k \cdot {t}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell \cdot \frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}}}{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 79.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{2}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.4 \cdot 10^{-60}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \left(\left(\ell \cdot {k}^{-2}\right) \cdot \frac{\cos k}{t\_m \cdot {\sin k}^{2}}\right)\right)\\ \mathbf{elif}\;t\_m \leq 9.2 \cdot 10^{+77}:\\ \;\;\;\;\ell \cdot \left(\frac{t\_2}{{t\_m}^{3}} \cdot \frac{\ell}{\sin k \cdot \tan k}\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell \cdot t\_2}{{\left(t\_m \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ 2.0 (+ 2.0 (pow (/ k t_m) 2.0)))))
   (*
    t_s
    (if (<= t_m 2.4e-60)
      (*
       l
       (* 2.0 (* (* l (pow k -2.0)) (/ (cos k) (* t_m (pow (sin k) 2.0))))))
      (if (<= t_m 9.2e+77)
        (* l (* (/ t_2 (pow t_m 3.0)) (/ l (* (sin k) (tan k)))))
        (* l (/ (* l t_2) (pow (* t_m (pow (cbrt k) 2.0)) 3.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = 2.0 / (2.0 + pow((k / t_m), 2.0));
	double tmp;
	if (t_m <= 2.4e-60) {
		tmp = l * (2.0 * ((l * pow(k, -2.0)) * (cos(k) / (t_m * pow(sin(k), 2.0)))));
	} else if (t_m <= 9.2e+77) {
		tmp = l * ((t_2 / pow(t_m, 3.0)) * (l / (sin(k) * tan(k))));
	} else {
		tmp = l * ((l * t_2) / pow((t_m * pow(cbrt(k), 2.0)), 3.0));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = 2.0 / (2.0 + Math.pow((k / t_m), 2.0));
	double tmp;
	if (t_m <= 2.4e-60) {
		tmp = l * (2.0 * ((l * Math.pow(k, -2.0)) * (Math.cos(k) / (t_m * Math.pow(Math.sin(k), 2.0)))));
	} else if (t_m <= 9.2e+77) {
		tmp = l * ((t_2 / Math.pow(t_m, 3.0)) * (l / (Math.sin(k) * Math.tan(k))));
	} else {
		tmp = l * ((l * t_2) / Math.pow((t_m * Math.pow(Math.cbrt(k), 2.0)), 3.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(2.0 / Float64(2.0 + (Float64(k / t_m) ^ 2.0)))
	tmp = 0.0
	if (t_m <= 2.4e-60)
		tmp = Float64(l * Float64(2.0 * Float64(Float64(l * (k ^ -2.0)) * Float64(cos(k) / Float64(t_m * (sin(k) ^ 2.0))))));
	elseif (t_m <= 9.2e+77)
		tmp = Float64(l * Float64(Float64(t_2 / (t_m ^ 3.0)) * Float64(l / Float64(sin(k) * tan(k)))));
	else
		tmp = Float64(l * Float64(Float64(l * t_2) / (Float64(t_m * (cbrt(k) ^ 2.0)) ^ 3.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(2.0 / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.4e-60], N[(l * N[(2.0 * N[(N[(l * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 9.2e+77], N[(l * N[(N[(t$95$2 / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] * N[(l / N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(l * t$95$2), $MachinePrecision] / N[Power[N[(t$95$m * N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 2.40000000000000009e-60

   1. Initial program 50.7%

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

   3. Simplified 48.7%

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

      1. associate-*r* 56.6%

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

      2. associate-*l/ 55.3%

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

   6. Applied egg-rr 55.3%

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

   7. Taylor expanded in k around inf 70.7%

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

      1. times-frac 71.2%

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

   9. Simplified 71.2%

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

       1. *-un-lft-identity 71.2%

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

       2. div-inv 71.3%

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

       3. pow-flip 71.5%

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

       4. metadata-eval 71.5%

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

   11. Applied egg-rr 71.5%

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

       1. *-lft-identity 71.5%

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

   13. Simplified 71.5%

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

   if 2.40000000000000009e-60 < t < 9.19999999999999979e77

   1. Initial program 80.1%

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

   3. Simplified 80.1%

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

      1. associate-*r* 84.2%

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

      2. *-commutative 84.2%

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

      3. associate-*l/ 91.6%

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

   6. Applied egg-rr 91.6%

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

      1. times-frac 84.2%

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

   8. Simplified 84.2%

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

   if 9.19999999999999979e77 < t

   1. Initial program 54.7%

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

   3. Simplified 47.5%

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

      1. associate-*r* 52.2%

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

      2. associate-*l/ 52.2%

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

   6. Applied egg-rr 52.2%

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

   7. Taylor expanded in k around 0 50.3%

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

      1. add-cube-cbrt 50.3%

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

      2. pow2 50.3%

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

      3. *-commutative 50.3%

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

      4. cbrt-prod 50.2%

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

      5. unpow3 50.2%

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

      6. add-cbrt-cube 50.2%

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

      7. unpow2 50.2%

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

      8. cbrt-prod 50.3%

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

      9. pow2 50.3%

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

      10. *-commutative 50.3%

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

      11. cbrt-prod 50.2%

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

      12. unpow3 50.2%

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

      13. add-cbrt-cube 57.0%

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

      14. unpow2 57.0%

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

      15. cbrt-prod 77.4%

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

      16. pow2 77.4%

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

   9. Applied egg-rr 77.4%

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

       1. unpow2 77.4%

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

       2. unpow3 77.4%

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

   11. Simplified 77.4%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.4 \cdot 10^{-60}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \left(\left(\ell \cdot {k}^{-2}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\right)\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{+77}:\\ \;\;\;\;\ell \cdot \left(\frac{\frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}}}{{t}^{3}} \cdot \frac{\ell}{\sin k \cdot \tan k}\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell \cdot \frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}}}{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 78.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 840000000000:\\ \;\;\;\;\ell \cdot \left(2 \cdot \left(\left(\ell \cdot {k}^{-2}\right) \cdot \frac{\cos k}{t\_m \cdot {\sin k}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell \cdot \frac{2}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}}{{\left(t\_m \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 840000000000.0)
    (* l (* 2.0 (* (* l (pow k -2.0)) (/ (cos k) (* t_m (pow (sin k) 2.0))))))
    (*
     l
     (/
      (* l (/ 2.0 (+ 2.0 (pow (/ k t_m) 2.0))))
      (pow (* t_m (pow (cbrt k) 2.0)) 3.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 840000000000.0) {
		tmp = l * (2.0 * ((l * pow(k, -2.0)) * (cos(k) / (t_m * pow(sin(k), 2.0)))));
	} else {
		tmp = l * ((l * (2.0 / (2.0 + pow((k / t_m), 2.0)))) / pow((t_m * pow(cbrt(k), 2.0)), 3.0));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 840000000000.0) {
		tmp = l * (2.0 * ((l * Math.pow(k, -2.0)) * (Math.cos(k) / (t_m * Math.pow(Math.sin(k), 2.0)))));
	} else {
		tmp = l * ((l * (2.0 / (2.0 + Math.pow((k / t_m), 2.0)))) / Math.pow((t_m * Math.pow(Math.cbrt(k), 2.0)), 3.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 840000000000.0)
		tmp = Float64(l * Float64(2.0 * Float64(Float64(l * (k ^ -2.0)) * Float64(cos(k) / Float64(t_m * (sin(k) ^ 2.0))))));
	else
		tmp = Float64(l * Float64(Float64(l * Float64(2.0 / Float64(2.0 + (Float64(k / t_m) ^ 2.0)))) / (Float64(t_m * (cbrt(k) ^ 2.0)) ^ 3.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 840000000000.0], N[(l * N[(2.0 * N[(N[(l * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(l * N[(2.0 / 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[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 8.4e11

   1. Initial program 53.4%

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

   3. Simplified 51.7%

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

      1. associate-*r* 59.4%

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

      2. associate-*l/ 58.3%

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

   6. Applied egg-rr 58.3%

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

   7. Taylor expanded in k around inf 71.8%

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

      1. times-frac 72.3%

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

   9. Simplified 72.3%

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

       1. *-un-lft-identity 72.3%

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

       2. div-inv 72.3%

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

       3. pow-flip 72.5%

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

       4. metadata-eval 72.5%

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

   11. Applied egg-rr 72.5%

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

       1. *-lft-identity 72.5%

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

   13. Simplified 72.5%

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

   if 8.4e11 < t

   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 56.0%

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

      1. associate-*r* 59.9%

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

      2. associate-*l/ 63.9%

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

   6. Applied egg-rr 63.9%

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

   7. Taylor expanded in k around 0 53.8%

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

      1. add-cube-cbrt 53.8%

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

      2. pow2 53.8%

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

      3. *-commutative 53.8%

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

      4. cbrt-prod 53.7%

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

      5. unpow3 53.7%

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

      6. add-cbrt-cube 53.7%

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

      7. unpow2 53.7%

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

      8. cbrt-prod 53.7%

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

      9. pow2 53.7%

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

      10. *-commutative 53.7%

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

      11. cbrt-prod 53.7%

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

      12. unpow3 53.7%

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

      13. add-cbrt-cube 58.5%

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

      14. unpow2 58.5%

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

      15. cbrt-prod 73.3%

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

      16. pow2 73.3%

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

   9. Applied egg-rr 73.3%

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

       1. unpow2 73.3%

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

       2. unpow3 73.3%

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

   11. Simplified 73.3%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 840000000000:\\ \;\;\;\;\ell \cdot \left(2 \cdot \left(\left(\ell \cdot {k}^{-2}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell \cdot \frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}}}{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 74.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.9 \cdot 10^{+35}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \left(\left(\ell \cdot {k}^{-2}\right) \cdot \frac{\cos k}{t\_m \cdot {\sin k}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(t\_m \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}} \cdot \left(\ell \cdot \ell\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.9e+35)
    (* l (* 2.0 (* (* l (pow k -2.0)) (/ (cos k) (* t_m (pow (sin k) 2.0))))))
    (* (/ 1.0 (pow (* t_m (pow (cbrt k) 2.0)) 3.0)) (* l l)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.9e+35) {
		tmp = l * (2.0 * ((l * pow(k, -2.0)) * (cos(k) / (t_m * pow(sin(k), 2.0)))));
	} else {
		tmp = (1.0 / pow((t_m * pow(cbrt(k), 2.0)), 3.0)) * (l * l);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.9e+35) {
		tmp = l * (2.0 * ((l * Math.pow(k, -2.0)) * (Math.cos(k) / (t_m * Math.pow(Math.sin(k), 2.0)))));
	} else {
		tmp = (1.0 / Math.pow((t_m * Math.pow(Math.cbrt(k), 2.0)), 3.0)) * (l * l);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.9e+35)
		tmp = Float64(l * Float64(2.0 * Float64(Float64(l * (k ^ -2.0)) * Float64(cos(k) / Float64(t_m * (sin(k) ^ 2.0))))));
	else
		tmp = Float64(Float64(1.0 / (Float64(t_m * (cbrt(k) ^ 2.0)) ^ 3.0)) * Float64(l * l));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.9e+35], N[(l * N[(2.0 * N[(N[(l * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Power[N[(t$95$m * N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.89999999999999995e35

   1. Initial program 53.6%

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

   3. Simplified 51.4%

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

      1. associate-*r* 59.1%

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

      2. associate-*l/ 58.8%

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

   6. Applied egg-rr 58.8%

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

   7. Taylor expanded in k around inf 71.0%

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

      1. times-frac 72.0%

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

   9. Simplified 72.0%

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

       1. *-un-lft-identity 72.0%

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

       2. div-inv 72.0%

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

       3. pow-flip 72.5%

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

       4. metadata-eval 72.5%

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

   11. Applied egg-rr 72.5%

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

       1. *-lft-identity 72.5%

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

   13. Simplified 72.5%

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

   if 2.89999999999999995e35 < t

   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 57.2%

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

   5. Taylor expanded in k around 0 52.6%

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

      1. add-cube-cbrt 54.7%

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

      2. pow2 54.7%

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

      3. *-commutative 54.7%

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

      4. cbrt-prod 54.7%

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

      5. unpow3 54.7%

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

      6. add-cbrt-cube 54.7%

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

      7. unpow2 54.7%

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

      8. cbrt-prod 54.7%

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

      9. pow2 54.7%

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

      10. *-commutative 54.7%

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

      11. cbrt-prod 54.6%

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

      12. unpow3 54.6%

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

      13. add-cbrt-cube 59.9%

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

      14. unpow2 59.9%

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

      15. cbrt-prod 75.9%

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

      16. pow2 75.9%

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

   7. Applied egg-rr 70.7%

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

      1. unpow2 75.9%

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

      2. unpow3 75.9%

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

   9. Simplified 70.7%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.9 \cdot 10^{+35}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \left(\left(\ell \cdot {k}^{-2}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}} \cdot \left(\ell \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 74.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.4 \cdot 10^{+35}:\\ \;\;\;\;2 \cdot \left(\left(\ell \cdot {k}^{-2}\right) \cdot \left(\ell \cdot \frac{\frac{\cos k}{t\_m}}{{\sin k}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(t\_m \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}} \cdot \left(\ell \cdot \ell\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.4e+35)
    (* 2.0 (* (* l (pow k -2.0)) (* l (/ (/ (cos k) t_m) (pow (sin k) 2.0)))))
    (* (/ 1.0 (pow (* t_m (pow (cbrt k) 2.0)) 3.0)) (* l l)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.4e+35) {
		tmp = 2.0 * ((l * pow(k, -2.0)) * (l * ((cos(k) / t_m) / pow(sin(k), 2.0))));
	} else {
		tmp = (1.0 / pow((t_m * pow(cbrt(k), 2.0)), 3.0)) * (l * l);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.4e+35) {
		tmp = 2.0 * ((l * Math.pow(k, -2.0)) * (l * ((Math.cos(k) / t_m) / Math.pow(Math.sin(k), 2.0))));
	} else {
		tmp = (1.0 / Math.pow((t_m * Math.pow(Math.cbrt(k), 2.0)), 3.0)) * (l * l);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 3.4e+35)
		tmp = Float64(2.0 * Float64(Float64(l * (k ^ -2.0)) * Float64(l * Float64(Float64(cos(k) / t_m) / (sin(k) ^ 2.0)))));
	else
		tmp = Float64(Float64(1.0 / (Float64(t_m * (cbrt(k) ^ 2.0)) ^ 3.0)) * Float64(l * l));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 3.4e+35], N[(2.0 * N[(N[(l * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision] * N[(l * N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Power[N[(t$95$m * N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.4000000000000001e35

   1. Initial program 53.6%

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

   3. Simplified 51.4%

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

      1. associate-*r* 59.1%

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

      2. associate-*l/ 58.8%

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

   6. Applied egg-rr 58.8%

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

   7. Taylor expanded in k around inf 71.0%

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

      1. times-frac 72.0%

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

   9. Simplified 72.0%

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

       1. *-un-lft-identity 72.0%

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

       2. *-commutative 72.0%

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

       3. div-inv 72.0%

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

       4. pow-flip 72.5%

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

       5. metadata-eval 72.5%

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

       6. associate-/r* 72.5%

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

   11. Applied egg-rr 72.5%

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

       1. *-lft-identity 72.5%

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

       2. *-commutative 72.5%

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

       3. associate-*r* 72.5%

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

       4. associate-*l* 71.2%

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

       5. *-commutative 71.2%

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

   13. Simplified 71.2%

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

   if 3.4000000000000001e35 < t

   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 57.2%

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

   5. Taylor expanded in k around 0 52.6%

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

      1. add-cube-cbrt 54.7%

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

      2. pow2 54.7%

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

      3. *-commutative 54.7%

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

      4. cbrt-prod 54.7%

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

      5. unpow3 54.7%

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

      6. add-cbrt-cube 54.7%

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

      7. unpow2 54.7%

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

      8. cbrt-prod 54.7%

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

      9. pow2 54.7%

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

      10. *-commutative 54.7%

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

      11. cbrt-prod 54.6%

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

      12. unpow3 54.6%

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

      13. add-cbrt-cube 59.9%

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

      14. unpow2 59.9%

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

      15. cbrt-prod 75.9%

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

      16. pow2 75.9%

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

   7. Applied egg-rr 70.7%

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

      1. unpow2 75.9%

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

      2. unpow3 75.9%

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

   9. Simplified 70.7%

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

Alternative 15: 67.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.7 \cdot 10^{-70}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \left(\frac{\ell}{{k}^{2}} \cdot \frac{\cos k}{t\_m \cdot {k}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(t\_m \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}} \cdot \left(\ell \cdot \ell\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.7e-70)
    (* l (* 2.0 (* (/ l (pow k 2.0)) (/ (cos k) (* t_m (pow k 2.0))))))
    (* (/ 1.0 (pow (* t_m (pow (cbrt k) 2.0)) 3.0)) (* l l)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.7e-70) {
		tmp = l * (2.0 * ((l / pow(k, 2.0)) * (cos(k) / (t_m * pow(k, 2.0)))));
	} else {
		tmp = (1.0 / pow((t_m * pow(cbrt(k), 2.0)), 3.0)) * (l * l);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.7e-70) {
		tmp = l * (2.0 * ((l / Math.pow(k, 2.0)) * (Math.cos(k) / (t_m * Math.pow(k, 2.0)))));
	} else {
		tmp = (1.0 / Math.pow((t_m * Math.pow(Math.cbrt(k), 2.0)), 3.0)) * (l * l);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.7e-70)
		tmp = Float64(l * Float64(2.0 * Float64(Float64(l / (k ^ 2.0)) * Float64(cos(k) / Float64(t_m * (k ^ 2.0))))));
	else
		tmp = Float64(Float64(1.0 / (Float64(t_m * (cbrt(k) ^ 2.0)) ^ 3.0)) * Float64(l * l));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.7e-70], N[(l * N[(2.0 * N[(N[(l / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Power[N[(t$95$m * N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.69999999999999998e-70

   1. Initial program 50.7%

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

   3. Simplified 48.7%

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

      1. associate-*r* 56.6%

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

      2. associate-*l/ 55.3%

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

   6. Applied egg-rr 55.3%

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

   7. Taylor expanded in k around inf 70.7%

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

      1. times-frac 71.2%

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

   9. Simplified 71.2%

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

   10. Taylor expanded in k around 0 61.7%

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

   if 1.69999999999999998e-70 < t

   1. Initial program 65.6%

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

   3. Simplified 61.4%

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

   5. Taylor expanded in k around 0 55.8%

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

      1. add-cube-cbrt 58.7%

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

      2. pow2 58.7%

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

      3. *-commutative 58.7%

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

      4. cbrt-prod 58.7%

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

      5. unpow3 58.7%

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

      6. add-cbrt-cube 58.7%

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

      7. unpow2 58.7%

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

      8. cbrt-prod 58.6%

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

      9. pow2 58.6%

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

      10. *-commutative 58.6%

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

      11. cbrt-prod 58.6%

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

      12. unpow3 58.6%

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

      13. add-cbrt-cube 62.5%

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

      14. unpow2 62.5%

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

      15. cbrt-prod 74.1%

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

      16. pow2 74.1%

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

   7. Applied egg-rr 69.0%

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

      1. unpow2 74.1%

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

      2. unpow3 74.1%

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

   9. Simplified 69.0%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.7 \cdot 10^{-70}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \left(\frac{\ell}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {k}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}} \cdot \left(\ell \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 67.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 10^{-70}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \frac{\frac{\ell}{t\_m}}{{k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(t\_m \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}} \cdot \left(\ell \cdot \ell\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1e-70)
    (* l (* 2.0 (/ (/ l t_m) (pow k 4.0))))
    (* (/ 1.0 (pow (* t_m (pow (cbrt k) 2.0)) 3.0)) (* l l)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1e-70) {
		tmp = l * (2.0 * ((l / t_m) / pow(k, 4.0)));
	} else {
		tmp = (1.0 / pow((t_m * pow(cbrt(k), 2.0)), 3.0)) * (l * l);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1e-70) {
		tmp = l * (2.0 * ((l / t_m) / Math.pow(k, 4.0)));
	} else {
		tmp = (1.0 / Math.pow((t_m * Math.pow(Math.cbrt(k), 2.0)), 3.0)) * (l * l);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1e-70)
		tmp = Float64(l * Float64(2.0 * Float64(Float64(l / t_m) / (k ^ 4.0))));
	else
		tmp = Float64(Float64(1.0 / (Float64(t_m * (cbrt(k) ^ 2.0)) ^ 3.0)) * Float64(l * l));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1e-70], N[(l * N[(2.0 * N[(N[(l / t$95$m), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Power[N[(t$95$m * N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 9.99999999999999996e-71

   1. Initial program 50.7%

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

   3. Simplified 48.7%

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

      1. associate-*r* 56.6%

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

      2. associate-*l/ 55.3%

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

   6. Applied egg-rr 55.3%

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

   7. Taylor expanded in k around 0 52.0%

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

   8. Taylor expanded in k around inf 59.2%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{\ell}{{k}^{4} \cdot t}\right)} \cdot \ell \]
   9. Step-by-step derivation

      1. *-commutative 59.2%

         \[\leadsto \left(2 \cdot \frac{\ell}{\color{blue}{t \cdot {k}^{4}}}\right) \cdot \ell \]

      2. associate-/r* 60.6%

         \[\leadsto \left(2 \cdot \color{blue}{\frac{\frac{\ell}{t}}{{k}^{4}}}\right) \cdot \ell \]

   10. Simplified 60.6%

       \[\leadsto \color{blue}{\left(2 \cdot \frac{\frac{\ell}{t}}{{k}^{4}}\right)} \cdot \ell \]

   if 9.99999999999999996e-71 < t

   1. Initial program 65.6%

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

   3. Simplified 61.4%

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

   5. Taylor expanded in k around 0 55.8%

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

      1. add-cube-cbrt 58.7%

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

      2. pow2 58.7%

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

      3. *-commutative 58.7%

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

      4. cbrt-prod 58.7%

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

      5. unpow3 58.7%

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

      6. add-cbrt-cube 58.7%

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

      7. unpow2 58.7%

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

      8. cbrt-prod 58.6%

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

      9. pow2 58.6%

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

      10. *-commutative 58.6%

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

      11. cbrt-prod 58.6%

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

      12. unpow3 58.6%

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

      13. add-cbrt-cube 62.5%

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

      14. unpow2 62.5%

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

      15. cbrt-prod 74.1%

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

      16. pow2 74.1%

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

   7. Applied egg-rr 69.0%

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

      1. unpow2 74.1%

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

      2. unpow3 74.1%

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

   9. Simplified 69.0%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 10^{-70}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \frac{\frac{\ell}{t}}{{k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}} \cdot \left(\ell \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 63.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.25 \cdot 10^{-70}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \frac{\frac{\ell}{t\_m}}{{k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot {\left(\frac{\sqrt[3]{{k}^{-2}}}{t\_m}\right)}^{3}\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.25e-70)
    (* l (* 2.0 (/ (/ l t_m) (pow k 4.0))))
    (* l (* l (pow (/ (cbrt (pow k -2.0)) t_m) 3.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.25e-70) {
		tmp = l * (2.0 * ((l / t_m) / pow(k, 4.0)));
	} else {
		tmp = l * (l * pow((cbrt(pow(k, -2.0)) / t_m), 3.0));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.25e-70) {
		tmp = l * (2.0 * ((l / t_m) / Math.pow(k, 4.0)));
	} else {
		tmp = l * (l * Math.pow((Math.cbrt(Math.pow(k, -2.0)) / t_m), 3.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.25e-70)
		tmp = Float64(l * Float64(2.0 * Float64(Float64(l / t_m) / (k ^ 4.0))));
	else
		tmp = Float64(l * Float64(l * (Float64(cbrt((k ^ -2.0)) / t_m) ^ 3.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.25e-70], N[(l * N[(2.0 * N[(N[(l / t$95$m), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(l * N[Power[N[(N[Power[N[Power[k, -2.0], $MachinePrecision], 1/3], $MachinePrecision] / t$95$m), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.25e-70

   1. Initial program 50.7%

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

   3. Simplified 48.7%

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

      1. associate-*r* 56.6%

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

      2. associate-*l/ 55.3%

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

   6. Applied egg-rr 55.3%

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

   7. Taylor expanded in k around 0 52.0%

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

   8. Taylor expanded in k around inf 59.2%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{\ell}{{k}^{4} \cdot t}\right)} \cdot \ell \]
   9. Step-by-step derivation

      1. *-commutative 59.2%

         \[\leadsto \left(2 \cdot \frac{\ell}{\color{blue}{t \cdot {k}^{4}}}\right) \cdot \ell \]

      2. associate-/r* 60.6%

         \[\leadsto \left(2 \cdot \color{blue}{\frac{\frac{\ell}{t}}{{k}^{4}}}\right) \cdot \ell \]

   10. Simplified 60.6%

       \[\leadsto \color{blue}{\left(2 \cdot \frac{\frac{\ell}{t}}{{k}^{4}}\right)} \cdot \ell \]

   if 1.25e-70 < t

   1. Initial program 65.6%

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

   3. Simplified 61.4%

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

   5. Taylor expanded in k around 0 55.8%

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

      1. associate-*r* 58.8%

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

      2. associate-/r* 58.8%

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

      3. pow-flip 58.8%

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

      4. metadata-eval 58.8%

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

   7. Applied egg-rr 58.8%

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

      1. add-cube-cbrt 58.7%

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

      2. pow3 58.7%

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

      3. cbrt-div 58.7%

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

      4. unpow3 58.7%

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

      5. add-cbrt-cube 62.5%

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

   9. Applied egg-rr 62.5%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.25 \cdot 10^{-70}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \frac{\frac{\ell}{t}}{{k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot {\left(\frac{\sqrt[3]{{k}^{-2}}}{t}\right)}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 61.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8 \cdot 10^{-73}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \frac{\frac{\ell}{t\_m}}{{k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{{t\_m}^{3} \cdot {k}^{2}}\\ \end{array} \end{array} \]

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 (<= t_m 8e-73)
    (* l (* 2.0 (/ (/ l t_m) (pow k 4.0))))
    (* l (/ l (* (pow t_m 3.0) (pow k 2.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 8e-73) {
		tmp = l * (2.0 * ((l / t_m) / pow(k, 4.0)));
	} else {
		tmp = l * (l / (pow(t_m, 3.0) * pow(k, 2.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 8d-73) then
        tmp = l * (2.0d0 * ((l / t_m) / (k ** 4.0d0)))
    else
        tmp = l * (l / ((t_m ** 3.0d0) * (k ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 8e-73) {
		tmp = l * (2.0 * ((l / t_m) / Math.pow(k, 4.0)));
	} else {
		tmp = l * (l / (Math.pow(t_m, 3.0) * Math.pow(k, 2.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 8e-73:
		tmp = l * (2.0 * ((l / t_m) / math.pow(k, 4.0)))
	else:
		tmp = l * (l / (math.pow(t_m, 3.0) * math.pow(k, 2.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 8e-73)
		tmp = Float64(l * Float64(2.0 * Float64(Float64(l / t_m) / (k ^ 4.0))));
	else
		tmp = Float64(l * Float64(l / Float64((t_m ^ 3.0) * (k ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 8e-73)
		tmp = l * (2.0 * ((l / t_m) / (k ^ 4.0)));
	else
		tmp = l * (l / ((t_m ^ 3.0) * (k ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 8e-73], N[(l * N[(2.0 * N[(N[(l / t$95$m), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(l / N[(N[Power[t$95$m, 3.0], $MachinePrecision] * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 7.99999999999999998e-73

   1. Initial program 50.7%

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

   3. Simplified 48.7%

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

      1. associate-*r* 56.6%

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

      2. associate-*l/ 55.3%

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

   6. Applied egg-rr 55.3%

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

   7. Taylor expanded in k around 0 52.0%

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

   8. Taylor expanded in k around inf 59.2%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{\ell}{{k}^{4} \cdot t}\right)} \cdot \ell \]
   9. Step-by-step derivation

      1. *-commutative 59.2%

         \[\leadsto \left(2 \cdot \frac{\ell}{\color{blue}{t \cdot {k}^{4}}}\right) \cdot \ell \]

      2. associate-/r* 60.6%

         \[\leadsto \left(2 \cdot \color{blue}{\frac{\frac{\ell}{t}}{{k}^{4}}}\right) \cdot \ell \]

   10. Simplified 60.6%

       \[\leadsto \color{blue}{\left(2 \cdot \frac{\frac{\ell}{t}}{{k}^{4}}\right)} \cdot \ell \]

   if 7.99999999999999998e-73 < t

   1. Initial program 65.6%

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

   3. Simplified 61.4%

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

      1. associate-*r* 65.8%

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

      2. associate-*l/ 69.0%

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

   6. Applied egg-rr 69.0%

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

   7. Taylor expanded in k around 0 58.8%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8 \cdot 10^{-73}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \frac{\frac{\ell}{t}}{{k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{{t}^{3} \cdot {k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 61.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.5 \cdot 10^{-70}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \frac{\frac{\ell}{t\_m}}{{k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{{k}^{-2}}{{t\_m}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.5e-70)
    (* l (* 2.0 (/ (/ l t_m) (pow k 4.0))))
    (* l (* l (/ (pow k -2.0) (pow t_m 3.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.5e-70) {
		tmp = l * (2.0 * ((l / t_m) / pow(k, 4.0)));
	} else {
		tmp = l * (l * (pow(k, -2.0) / pow(t_m, 3.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.5d-70) then
        tmp = l * (2.0d0 * ((l / t_m) / (k ** 4.0d0)))
    else
        tmp = l * (l * ((k ** (-2.0d0)) / (t_m ** 3.0d0)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.5e-70) {
		tmp = l * (2.0 * ((l / t_m) / Math.pow(k, 4.0)));
	} else {
		tmp = l * (l * (Math.pow(k, -2.0) / Math.pow(t_m, 3.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.5e-70:
		tmp = l * (2.0 * ((l / t_m) / math.pow(k, 4.0)))
	else:
		tmp = l * (l * (math.pow(k, -2.0) / math.pow(t_m, 3.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.5e-70)
		tmp = Float64(l * Float64(2.0 * Float64(Float64(l / t_m) / (k ^ 4.0))));
	else
		tmp = Float64(l * Float64(l * Float64((k ^ -2.0) / (t_m ^ 3.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.5e-70)
		tmp = l * (2.0 * ((l / t_m) / (k ^ 4.0)));
	else
		tmp = l * (l * ((k ^ -2.0) / (t_m ^ 3.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.5e-70], N[(l * N[(2.0 * N[(N[(l / t$95$m), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(l * N[(N[Power[k, -2.0], $MachinePrecision] / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.5000000000000001e-70

   1. Initial program 50.7%

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

   3. Simplified 48.7%

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

      1. associate-*r* 56.6%

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

      2. associate-*l/ 55.3%

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

   6. Applied egg-rr 55.3%

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

   7. Taylor expanded in k around 0 52.0%

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

   8. Taylor expanded in k around inf 59.2%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{\ell}{{k}^{4} \cdot t}\right)} \cdot \ell \]
   9. Step-by-step derivation

      1. *-commutative 59.2%

         \[\leadsto \left(2 \cdot \frac{\ell}{\color{blue}{t \cdot {k}^{4}}}\right) \cdot \ell \]

      2. associate-/r* 60.6%

         \[\leadsto \left(2 \cdot \color{blue}{\frac{\frac{\ell}{t}}{{k}^{4}}}\right) \cdot \ell \]

   10. Simplified 60.6%

       \[\leadsto \color{blue}{\left(2 \cdot \frac{\frac{\ell}{t}}{{k}^{4}}\right)} \cdot \ell \]

   if 1.5000000000000001e-70 < t

   1. Initial program 65.6%

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

   3. Simplified 61.4%

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

   5. Taylor expanded in k around 0 55.8%

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

      1. associate-*r* 58.8%

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

      2. associate-/r* 58.8%

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

      3. pow-flip 58.8%

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

      4. metadata-eval 58.8%

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

   7. Applied egg-rr 58.8%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.5 \cdot 10^{-70}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \frac{\frac{\ell}{t}}{{k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{{k}^{-2}}{{t}^{3}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 61.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2 \cdot 10^{-72}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \frac{\frac{\ell}{t\_m}}{{k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{\frac{1}{k} \cdot \frac{1}{k}}{{t\_m}^{3}}\right)\\ \end{array} \end{array} \]

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-72)
    (* l (* 2.0 (/ (/ l t_m) (pow k 4.0))))
    (* l (* l (/ (* (/ 1.0 k) (/ 1.0 k)) (pow t_m 3.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2e-72) {
		tmp = l * (2.0 * ((l / t_m) / pow(k, 4.0)));
	} else {
		tmp = l * (l * (((1.0 / k) * (1.0 / k)) / pow(t_m, 3.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2d-72) then
        tmp = l * (2.0d0 * ((l / t_m) / (k ** 4.0d0)))
    else
        tmp = l * (l * (((1.0d0 / k) * (1.0d0 / k)) / (t_m ** 3.0d0)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2e-72) {
		tmp = l * (2.0 * ((l / t_m) / Math.pow(k, 4.0)));
	} else {
		tmp = l * (l * (((1.0 / k) * (1.0 / k)) / Math.pow(t_m, 3.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2e-72:
		tmp = l * (2.0 * ((l / t_m) / math.pow(k, 4.0)))
	else:
		tmp = l * (l * (((1.0 / k) * (1.0 / k)) / math.pow(t_m, 3.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2e-72)
		tmp = Float64(l * Float64(2.0 * Float64(Float64(l / t_m) / (k ^ 4.0))));
	else
		tmp = Float64(l * Float64(l * Float64(Float64(Float64(1.0 / k) * Float64(1.0 / k)) / (t_m ^ 3.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2e-72)
		tmp = l * (2.0 * ((l / t_m) / (k ^ 4.0)));
	else
		tmp = l * (l * (((1.0 / k) * (1.0 / k)) / (t_m ^ 3.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2e-72], N[(l * N[(2.0 * N[(N[(l / t$95$m), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(l * N[(N[(N[(1.0 / k), $MachinePrecision] * N[(1.0 / k), $MachinePrecision]), $MachinePrecision] / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.9999999999999999e-72

   1. Initial program 50.7%

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

   3. Simplified 48.7%

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

      1. associate-*r* 56.6%

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

      2. associate-*l/ 55.3%

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

   6. Applied egg-rr 55.3%

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

   7. Taylor expanded in k around 0 52.0%

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

   8. Taylor expanded in k around inf 59.2%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{\ell}{{k}^{4} \cdot t}\right)} \cdot \ell \]
   9. Step-by-step derivation

      1. *-commutative 59.2%

         \[\leadsto \left(2 \cdot \frac{\ell}{\color{blue}{t \cdot {k}^{4}}}\right) \cdot \ell \]

      2. associate-/r* 60.6%

         \[\leadsto \left(2 \cdot \color{blue}{\frac{\frac{\ell}{t}}{{k}^{4}}}\right) \cdot \ell \]

   10. Simplified 60.6%

       \[\leadsto \color{blue}{\left(2 \cdot \frac{\frac{\ell}{t}}{{k}^{4}}\right)} \cdot \ell \]

   if 1.9999999999999999e-72 < t

   1. Initial program 65.6%

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

   3. Simplified 61.4%

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

   5. Taylor expanded in k around 0 55.8%

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

      1. associate-*r* 58.8%

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

      2. associate-/r* 58.8%

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

      3. pow-flip 58.8%

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

      4. metadata-eval 58.8%

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

   7. Applied egg-rr 58.8%

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

      1. sqr-pow 58.8%

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

      2. metadata-eval 58.8%

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

      3. unpow-1 58.8%

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

      4. metadata-eval 58.8%

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

      5. unpow-1 58.8%

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

   9. Applied egg-rr 58.8%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2 \cdot 10^{-72}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \frac{\frac{\ell}{t}}{{k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{\frac{1}{k} \cdot \frac{1}{k}}{{t}^{3}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 55.4% accurate, 3.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\ell \cdot \left(2 \cdot \frac{\frac{\ell}{t\_m}}{{k}^{4}}\right)\right) \end{array} \]

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 (* l (* 2.0 (/ (/ l t_m) (pow k 4.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (l * (2.0 * ((l / t_m) / pow(k, 4.0))));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (l * (2.0d0 * ((l / t_m) / (k ** 4.0d0))))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (l * (2.0 * ((l / t_m) / Math.pow(k, 4.0))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (l * (2.0 * ((l / t_m) / math.pow(k, 4.0))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(l * Float64(2.0 * Float64(Float64(l / t_m) / (k ^ 4.0)))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (l * (2.0 * ((l / t_m) / (k ^ 4.0))));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(l * N[(2.0 * N[(N[(l / t$95$m), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.4%

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

3. Simplified 52.8%

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

   1. associate-*r* 59.5%

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

   2. associate-*l/ 59.7%

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

6. Applied egg-rr 59.7%

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

7. Taylor expanded in k around 0 54.2%

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

8. Taylor expanded in k around inf 54.6%

   \[\leadsto \color{blue}{\left(2 \cdot \frac{\ell}{{k}^{4} \cdot t}\right)} \cdot \ell \]
9. Step-by-step derivation

   1. *-commutative 54.6%

      \[\leadsto \left(2 \cdot \frac{\ell}{\color{blue}{t \cdot {k}^{4}}}\right) \cdot \ell \]

   2. associate-/r* 55.9%

      \[\leadsto \left(2 \cdot \color{blue}{\frac{\frac{\ell}{t}}{{k}^{4}}}\right) \cdot \ell \]

10. Simplified 55.9%

    \[\leadsto \color{blue}{\left(2 \cdot \frac{\frac{\ell}{t}}{{k}^{4}}\right)} \cdot \ell \]

11. Final simplification 55.9%

    \[\leadsto \ell \cdot \left(2 \cdot \frac{\frac{\ell}{t}}{{k}^{4}}\right) \]
12. Add Preprocessing

Alternative 22: 54.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\ell \cdot \left(2 \cdot \frac{\ell}{t\_m \cdot {k}^{4}}\right)\right) \end{array} \]

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 (* l (* 2.0 (/ l (* t_m (pow k 4.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (l * (2.0 * (l / (t_m * pow(k, 4.0)))));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (l * (2.0d0 * (l / (t_m * (k ** 4.0d0)))))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (l * (2.0 * (l / (t_m * Math.pow(k, 4.0)))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (l * (2.0 * (l / (t_m * math.pow(k, 4.0)))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(l * Float64(2.0 * Float64(l / Float64(t_m * (k ^ 4.0))))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (l * (2.0 * (l / (t_m * (k ^ 4.0)))));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(l * N[(2.0 * N[(l / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.4%

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

3. Simplified 52.8%

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

   1. associate-*r* 59.5%

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

   2. associate-*l/ 59.7%

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

6. Applied egg-rr 59.7%

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

7. Taylor expanded in k around inf 64.0%

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

   1. times-frac 65.2%

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

9. Simplified 65.2%

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

10. Taylor expanded in k around 0 54.6%

    \[\leadsto \left(2 \cdot \color{blue}{\frac{\ell}{{k}^{4} \cdot t}}\right) \cdot \ell \]

11. Final simplification 54.6%

    \[\leadsto \ell \cdot \left(2 \cdot \frac{\ell}{t \cdot {k}^{4}}\right) \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(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))))