Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.8% → 86.3%

Time: 17.7s

Alternatives: 21

Speedup: 3.7×

• 28.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.15 \cdot 10^{-155}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{\cos k}{\left(t\_m \cdot {k}^{2}\right) \cdot {\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(\sqrt[3]{\sin k} \cdot \left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right) \cdot \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.15e-155)
    (*
     2.0
     (* (pow l 2.0) (/ (cos k) (* (* t_m (pow k 2.0)) (pow (sin k) 2.0)))))
    (/
     2.0
     (pow
      (*
       (* (cbrt (sin k)) (* t_m (pow (cbrt l) -2.0)))
       (cbrt (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))))
      3.0)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.15e-155) {
		tmp = 2.0 * (pow(l, 2.0) * (cos(k) / ((t_m * pow(k, 2.0)) * pow(sin(k), 2.0))));
	} else {
		tmp = 2.0 / pow(((cbrt(sin(k)) * (t_m * pow(cbrt(l), -2.0))) * cbrt((tan(k) * (2.0 + pow((k / t_m), 2.0))))), 3.0);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.15e-155) {
		tmp = 2.0 * (Math.pow(l, 2.0) * (Math.cos(k) / ((t_m * Math.pow(k, 2.0)) * Math.pow(Math.sin(k), 2.0))));
	} else {
		tmp = 2.0 / Math.pow(((Math.cbrt(Math.sin(k)) * (t_m * Math.pow(Math.cbrt(l), -2.0))) * Math.cbrt((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0))))), 3.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.15e-155)
		tmp = Float64(2.0 * Float64((l ^ 2.0) * Float64(cos(k) / Float64(Float64(t_m * (k ^ 2.0)) * (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(2.0 + (Float64(k / t_m) ^ 2.0))))) ^ 3.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.15e-155], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.15000000000000004e-155

   1. Initial program 51.1%

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

   3. Simplified 51.1%

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

      1. add-cube-cbrt 51.1%

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

      2. pow3 51.1%

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

      3. associate-/r* 54.6%

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

      4. *-commutative 54.6%

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

      5. cbrt-prod 54.6%

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

      6. associate-/r* 51.1%

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

      7. cbrt-div 51.1%

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

      8. rem-cbrt-cube 59.4%

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

      9. cbrt-prod 66.2%

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

      10. pow2 66.2%

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

   6. Applied egg-rr 66.2%

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

      1. cube-mult 66.2%

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

      2. div-inv 66.2%

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

      3. pow-flip 66.2%

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

      4. metadata-eval 66.2%

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

      5. pow2 66.2%

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

      6. div-inv 66.1%

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

      7. pow-flip 66.2%

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

      8. metadata-eval 66.2%

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

   8. Applied egg-rr 66.2%

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

      1. unpow2 66.2%

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

      2. cube-unmult 66.2%

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

   10. Simplified 66.2%

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

       1. add-cube-cbrt 66.1%

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

       2. pow3 66.1%

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

   12. Applied egg-rr 72.3%

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

   13. Taylor expanded in k around inf 67.7%

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

       1. associate-/l* 67.7%

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

       2. associate-*r* 67.8%

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

   15. Simplified 67.8%

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

   if 2.15000000000000004e-155 < t

   1. Initial program 63.4%

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

   3. Simplified 63.4%

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

      1. add-cube-cbrt 63.3%

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

      2. pow3 63.3%

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

      3. associate-/r* 67.5%

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

      4. *-commutative 67.5%

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

      5. cbrt-prod 67.5%

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

      6. associate-/r* 63.3%

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

      7. cbrt-div 65.3%

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

      8. rem-cbrt-cube 76.0%

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

      9. cbrt-prod 89.0%

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

      10. pow2 89.0%

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

   6. Applied egg-rr 89.0%

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

      1. cube-mult 89.0%

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

      2. div-inv 89.0%

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

      3. pow-flip 89.0%

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

      4. metadata-eval 89.0%

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

      5. pow2 89.0%

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

      6. div-inv 89.1%

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

      7. pow-flip 89.1%

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

      8. metadata-eval 89.1%

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

   8. Applied egg-rr 89.1%

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

      1. unpow2 89.1%

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

      2. cube-unmult 89.1%

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

   10. Simplified 89.1%

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

       1. add-cube-cbrt 89.1%

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

       2. pow3 89.1%

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

   12. Applied egg-rr 95.3%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.15 \cdot 10^{-155}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{\cos k}{\left(t \cdot {k}^{2}\right) \cdot {\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(\sqrt[3]{\sin k} \cdot \left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right) \cdot \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.05 \cdot 10^{-153}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{\cos k}{\left(t\_m \cdot {k}^{2}\right) \cdot {\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k \cdot {\left(\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.05e-153)
    (*
     2.0
     (* (pow l 2.0) (/ (cos k) (* (* t_m (pow k 2.0)) (pow (sin k) 2.0)))))
    (/
     2.0
     (*
      (sin k)
      (pow
       (*
        (* t_m (pow (cbrt l) -2.0))
        (cbrt (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))))
       3.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.05e-153) {
		tmp = 2.0 * (pow(l, 2.0) * (cos(k) / ((t_m * pow(k, 2.0)) * pow(sin(k), 2.0))));
	} else {
		tmp = 2.0 / (sin(k) * pow(((t_m * pow(cbrt(l), -2.0)) * cbrt((tan(k) * (2.0 + pow((k / t_m), 2.0))))), 3.0));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.05e-153) {
		tmp = 2.0 * (Math.pow(l, 2.0) * (Math.cos(k) / ((t_m * Math.pow(k, 2.0)) * Math.pow(Math.sin(k), 2.0))));
	} else {
		tmp = 2.0 / (Math.sin(k) * Math.pow(((t_m * Math.pow(Math.cbrt(l), -2.0)) * Math.cbrt((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0))))), 3.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.05e-153)
		tmp = Float64(2.0 * Float64((l ^ 2.0) * Float64(cos(k) / Float64(Float64(t_m * (k ^ 2.0)) * (sin(k) ^ 2.0)))));
	else
		tmp = Float64(2.0 / Float64(sin(k) * (Float64(Float64(t_m * (cbrt(l) ^ -2.0)) * cbrt(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))))) ^ 3.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.05e-153], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.05000000000000002e-153

   1. Initial program 51.1%

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

   3. Simplified 51.1%

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

      1. add-cube-cbrt 51.1%

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

      2. pow3 51.1%

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

      3. associate-/r* 54.6%

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

      4. *-commutative 54.6%

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

      5. cbrt-prod 54.6%

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

      6. associate-/r* 51.1%

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

      7. cbrt-div 51.1%

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

      8. rem-cbrt-cube 59.4%

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

      9. cbrt-prod 66.2%

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

      10. pow2 66.2%

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

   6. Applied egg-rr 66.2%

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

      1. cube-mult 66.2%

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

      2. div-inv 66.2%

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

      3. pow-flip 66.2%

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

      4. metadata-eval 66.2%

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

      5. pow2 66.2%

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

      6. div-inv 66.1%

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

      7. pow-flip 66.2%

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

      8. metadata-eval 66.2%

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

   8. Applied egg-rr 66.2%

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

      1. unpow2 66.2%

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

      2. cube-unmult 66.2%

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

   10. Simplified 66.2%

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

       1. add-cube-cbrt 66.1%

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

       2. pow3 66.1%

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

   12. Applied egg-rr 72.3%

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

   13. Taylor expanded in k around inf 67.7%

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

       1. associate-/l* 67.7%

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

       2. associate-*r* 67.8%

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

   15. Simplified 67.8%

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

   if 1.05000000000000002e-153 < t

   1. Initial program 63.4%

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

   3. Simplified 63.4%

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

      1. add-cube-cbrt 63.3%

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

      2. pow3 63.3%

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

      3. associate-/r* 67.5%

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

      4. *-commutative 67.5%

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

      5. cbrt-prod 67.5%

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

      6. associate-/r* 63.3%

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

      7. cbrt-div 65.3%

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

      8. rem-cbrt-cube 76.0%

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

      9. cbrt-prod 89.0%

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

      10. pow2 89.0%

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

   6. Applied egg-rr 89.0%

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

      1. cube-mult 89.0%

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

      2. div-inv 89.0%

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

      3. pow-flip 89.0%

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

      4. metadata-eval 89.0%

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

      5. pow2 89.0%

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

      6. div-inv 89.1%

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

      7. pow-flip 89.1%

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

      8. metadata-eval 89.1%

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

   8. Applied egg-rr 89.1%

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

      1. unpow2 89.1%

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

      2. cube-unmult 89.1%

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

   10. Simplified 89.1%

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

       1. add-cube-cbrt 89.1%

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

       2. pow3 89.1%

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

   12. Applied egg-rr 95.3%

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

       1. associate-*l* 95.2%

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

       2. unpow-prod-down 90.9%

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

       3. unpow3 90.9%

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

       4. add-cube-cbrt 91.0%

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

   14. Applied egg-rr 91.0%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.05 \cdot 10^{-153}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{\cos k}{\left(t \cdot {k}^{2}\right) \cdot {\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 0.0045:\\ \;\;\;\;\frac{2}{{\left(\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \left(\sqrt[3]{2 \cdot k} \cdot \sqrt[3]{k}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{\cos k}{\left(t\_m \cdot {k}^{2}\right) \cdot {\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 0.0045)
    (/
     2.0
     (pow (* (* t_m (pow (cbrt l) -2.0)) (* (cbrt (* 2.0 k)) (cbrt k))) 3.0))
    (*
     2.0
     (* (pow l 2.0) (/ (cos k) (* (* t_m (pow k 2.0)) (pow (sin k) 2.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 0.0045) {
		tmp = 2.0 / pow(((t_m * pow(cbrt(l), -2.0)) * (cbrt((2.0 * k)) * cbrt(k))), 3.0);
	} else {
		tmp = 2.0 * (pow(l, 2.0) * (cos(k) / ((t_m * pow(k, 2.0)) * pow(sin(k), 2.0))));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 0.0045) {
		tmp = 2.0 / Math.pow(((t_m * Math.pow(Math.cbrt(l), -2.0)) * (Math.cbrt((2.0 * k)) * Math.cbrt(k))), 3.0);
	} else {
		tmp = 2.0 * (Math.pow(l, 2.0) * (Math.cos(k) / ((t_m * Math.pow(k, 2.0)) * Math.pow(Math.sin(k), 2.0))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 0.0045)
		tmp = Float64(2.0 / (Float64(Float64(t_m * (cbrt(l) ^ -2.0)) * Float64(cbrt(Float64(2.0 * k)) * cbrt(k))) ^ 3.0));
	else
		tmp = Float64(2.0 * Float64((l ^ 2.0) * Float64(cos(k) / Float64(Float64(t_m * (k ^ 2.0)) * (sin(k) ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 0.0045], N[(2.0 / N[Power[N[(N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(2.0 * k), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[k, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 0.00449999999999999966

   1. Initial program 60.8%

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

   3. Simplified 59.0%

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

   5. Taylor expanded in k around 0 58.9%

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

      1. add-cube-cbrt 58.9%

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

      2. pow3 58.9%

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

      3. cbrt-prod 58.9%

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

      4. associate-/l/ 55.9%

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

      5. unpow2 55.9%

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

      6. cbrt-div 56.5%

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

      7. unpow3 56.5%

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

      8. add-cbrt-cube 62.8%

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

      9. unpow2 62.8%

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

      10. cbrt-prod 67.7%

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

      11. unpow2 67.7%

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

      12. div-inv 67.6%

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

      13. pow-flip 67.7%

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

      14. metadata-eval 67.7%

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

   7. Applied egg-rr 67.7%

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

      1. pow2 67.7%

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

      2. associate-*r* 67.7%

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

      3. cbrt-prod 78.8%

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

   9. Applied egg-rr 78.8%

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

   if 0.00449999999999999966 < k

   1. Initial program 43.6%

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

   3. Simplified 43.6%

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

      1. add-cube-cbrt 43.5%

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

      2. pow3 43.5%

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

      3. associate-/r* 46.5%

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

      4. *-commutative 46.5%

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

      5. cbrt-prod 46.5%

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

      6. associate-/r* 43.5%

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

      7. cbrt-div 43.5%

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

      8. rem-cbrt-cube 55.4%

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

      9. cbrt-prod 60.8%

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

      10. pow2 60.8%

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

   6. Applied egg-rr 60.8%

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

      1. cube-mult 60.8%

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

      2. div-inv 60.8%

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

      3. pow-flip 60.8%

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

      4. metadata-eval 60.8%

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

      5. pow2 60.8%

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

      6. div-inv 60.8%

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

      7. pow-flip 60.8%

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

      8. metadata-eval 60.8%

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

   8. Applied egg-rr 60.8%

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

      1. unpow2 60.8%

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

      2. cube-unmult 60.8%

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

   10. Simplified 60.8%

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

       1. add-cube-cbrt 60.7%

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

       2. pow3 60.7%

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

   12. Applied egg-rr 65.6%

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

   13. Taylor expanded in k around inf 72.3%

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

       1. associate-/l* 72.4%

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

       2. associate-*r* 72.4%

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

   15. Simplified 72.4%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.0045:\\ \;\;\;\;\frac{2}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \left(\sqrt[3]{2 \cdot k} \cdot \sqrt[3]{k}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{\cos k}{\left(t \cdot {k}^{2}\right) \cdot {\sin k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 0.0065:\\ \;\;\;\;\frac{2}{{\left(\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \left(\sqrt[3]{2 \cdot k} \cdot \sqrt[3]{k}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos k \cdot \frac{{\ell}^{2}}{\left(t\_m \cdot {k}^{2}\right) \cdot {\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 0.0065)
    (/
     2.0
     (pow (* (* t_m (pow (cbrt l) -2.0)) (* (cbrt (* 2.0 k)) (cbrt k))) 3.0))
    (*
     2.0
     (* (cos k) (/ (pow l 2.0) (* (* t_m (pow k 2.0)) (pow (sin k) 2.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 0.0065) {
		tmp = 2.0 / pow(((t_m * pow(cbrt(l), -2.0)) * (cbrt((2.0 * k)) * cbrt(k))), 3.0);
	} else {
		tmp = 2.0 * (cos(k) * (pow(l, 2.0) / ((t_m * pow(k, 2.0)) * pow(sin(k), 2.0))));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 0.0065) {
		tmp = 2.0 / Math.pow(((t_m * Math.pow(Math.cbrt(l), -2.0)) * (Math.cbrt((2.0 * k)) * Math.cbrt(k))), 3.0);
	} else {
		tmp = 2.0 * (Math.cos(k) * (Math.pow(l, 2.0) / ((t_m * Math.pow(k, 2.0)) * Math.pow(Math.sin(k), 2.0))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 0.0065)
		tmp = Float64(2.0 / (Float64(Float64(t_m * (cbrt(l) ^ -2.0)) * Float64(cbrt(Float64(2.0 * k)) * cbrt(k))) ^ 3.0));
	else
		tmp = Float64(2.0 * Float64(cos(k) * Float64((l ^ 2.0) / Float64(Float64(t_m * (k ^ 2.0)) * (sin(k) ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 0.0065], N[(2.0 / N[Power[N[(N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(2.0 * k), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[k, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Cos[k], $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 0.0064999999999999997

   1. Initial program 60.8%

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

   3. Simplified 59.0%

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

   5. Taylor expanded in k around 0 58.9%

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

      1. add-cube-cbrt 58.9%

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

      2. pow3 58.9%

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

      3. cbrt-prod 58.9%

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

      4. associate-/l/ 55.9%

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

      5. unpow2 55.9%

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

      6. cbrt-div 56.5%

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

      7. unpow3 56.5%

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

      8. add-cbrt-cube 62.8%

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

      9. unpow2 62.8%

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

      10. cbrt-prod 67.7%

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

      11. unpow2 67.7%

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

      12. div-inv 67.6%

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

      13. pow-flip 67.7%

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

      14. metadata-eval 67.7%

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

   7. Applied egg-rr 67.7%

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

      1. pow2 67.7%

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

      2. associate-*r* 67.7%

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

      3. cbrt-prod 78.8%

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

   9. Applied egg-rr 78.8%

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

   if 0.0064999999999999997 < k

   1. Initial program 43.6%

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

   3. Simplified 43.6%

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

      1. add-cube-cbrt 43.5%

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

      2. pow3 43.5%

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

      3. associate-/r* 46.5%

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

      4. *-commutative 46.5%

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

      5. cbrt-prod 46.5%

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

      6. associate-/r* 43.5%

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

      7. cbrt-div 43.5%

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

      8. rem-cbrt-cube 55.4%

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

      9. cbrt-prod 60.8%

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

      10. pow2 60.8%

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

   6. Applied egg-rr 60.8%

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

   7. Taylor expanded in k around inf 72.3%

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

      1. *-commutative 72.3%

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

      2. *-commutative 72.3%

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

      3. *-commutative 72.3%

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

      4. associate-*r* 72.3%

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

      5. associate-/l* 72.3%

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

      6. *-commutative 72.3%

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

   9. Simplified 72.3%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.0065:\\ \;\;\;\;\frac{2}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \left(\sqrt[3]{2 \cdot k} \cdot \sqrt[3]{k}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos k \cdot \frac{{\ell}^{2}}{\left(t \cdot {k}^{2}\right) \cdot {\sin k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 0.0053:\\ \;\;\;\;\frac{2}{{\left(\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \left(\sqrt[3]{2 \cdot k} \cdot \sqrt[3]{k}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{\sin k}^{2} \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 0.0053)
    (/
     2.0
     (pow (* (* t_m (pow (cbrt l) -2.0)) (* (cbrt (* 2.0 k)) (cbrt k))) 3.0))
    (/
     (* 2.0 (* (pow l 2.0) (cos k)))
     (* (pow (sin k) 2.0) (* t_m (* k k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 0.0053) {
		tmp = 2.0 / pow(((t_m * pow(cbrt(l), -2.0)) * (cbrt((2.0 * k)) * cbrt(k))), 3.0);
	} else {
		tmp = (2.0 * (pow(l, 2.0) * cos(k))) / (pow(sin(k), 2.0) * (t_m * (k * k)));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 0.0053) {
		tmp = 2.0 / Math.pow(((t_m * Math.pow(Math.cbrt(l), -2.0)) * (Math.cbrt((2.0 * k)) * Math.cbrt(k))), 3.0);
	} else {
		tmp = (2.0 * (Math.pow(l, 2.0) * Math.cos(k))) / (Math.pow(Math.sin(k), 2.0) * (t_m * (k * k)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 0.0053)
		tmp = Float64(2.0 / (Float64(Float64(t_m * (cbrt(l) ^ -2.0)) * Float64(cbrt(Float64(2.0 * k)) * cbrt(k))) ^ 3.0));
	else
		tmp = Float64(Float64(2.0 * Float64((l ^ 2.0) * cos(k))) / Float64((sin(k) ^ 2.0) * Float64(t_m * Float64(k * k))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 0.0053], N[(2.0 / N[Power[N[(N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(2.0 * k), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[k, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 0.00530000000000000002

   1. Initial program 60.8%

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

   3. Simplified 59.0%

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

   5. Taylor expanded in k around 0 58.9%

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

      1. add-cube-cbrt 58.9%

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

      2. pow3 58.9%

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

      3. cbrt-prod 58.9%

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

      4. associate-/l/ 55.9%

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

      5. unpow2 55.9%

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

      6. cbrt-div 56.5%

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

      7. unpow3 56.5%

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

      8. add-cbrt-cube 62.8%

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

      9. unpow2 62.8%

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

      10. cbrt-prod 67.7%

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

      11. unpow2 67.7%

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

      12. div-inv 67.6%

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

      13. pow-flip 67.7%

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

      14. metadata-eval 67.7%

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

   7. Applied egg-rr 67.7%

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

      1. pow2 67.7%

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

      2. associate-*r* 67.7%

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

      3. cbrt-prod 78.8%

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

   9. Applied egg-rr 78.8%

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

   if 0.00530000000000000002 < k

   1. Initial program 43.6%

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

   3. Simplified 43.6%

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

   5. Taylor expanded in t around 0 72.3%

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

      1. associate-*r/ 72.3%

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

      2. associate-*r* 72.3%

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

   7. Simplified 72.3%

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

      1. unpow2 72.3%

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

   9. Applied egg-rr 72.3%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.0053:\\ \;\;\;\;\frac{2}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \left(\sqrt[3]{2 \cdot k} \cdot \sqrt[3]{k}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\sin k}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 360:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left(\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{k}\right)}^{3}}\\ \mathbf{elif}\;k \leq 6.4 \cdot 10^{+63}:\\ \;\;\;\;\frac{2}{\frac{\frac{{t\_m}^{3}}{\ell}}{\ell} \cdot \left(2 \cdot \frac{t\_2}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot {\ell}^{2}}{\left(t\_m \cdot {k}^{2}\right) \cdot t\_2}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (sin k) 2.0)))
   (*
    t_s
    (if (<= k 360.0)
      (/ 2.0 (* (* 2.0 k) (pow (* (* t_m (pow (cbrt l) -2.0)) (cbrt k)) 3.0)))
      (if (<= k 6.4e+63)
        (/ 2.0 (* (/ (/ (pow t_m 3.0) l) l) (* 2.0 (/ t_2 (cos k)))))
        (/ (* 2.0 (pow l 2.0)) (* (* t_m (pow k 2.0)) t_2)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow(sin(k), 2.0);
	double tmp;
	if (k <= 360.0) {
		tmp = 2.0 / ((2.0 * k) * pow(((t_m * pow(cbrt(l), -2.0)) * cbrt(k)), 3.0));
	} else if (k <= 6.4e+63) {
		tmp = 2.0 / (((pow(t_m, 3.0) / l) / l) * (2.0 * (t_2 / cos(k))));
	} else {
		tmp = (2.0 * pow(l, 2.0)) / ((t_m * pow(k, 2.0)) * t_2);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.pow(Math.sin(k), 2.0);
	double tmp;
	if (k <= 360.0) {
		tmp = 2.0 / ((2.0 * k) * Math.pow(((t_m * Math.pow(Math.cbrt(l), -2.0)) * Math.cbrt(k)), 3.0));
	} else if (k <= 6.4e+63) {
		tmp = 2.0 / (((Math.pow(t_m, 3.0) / l) / l) * (2.0 * (t_2 / Math.cos(k))));
	} else {
		tmp = (2.0 * Math.pow(l, 2.0)) / ((t_m * Math.pow(k, 2.0)) * t_2);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = sin(k) ^ 2.0
	tmp = 0.0
	if (k <= 360.0)
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * (Float64(Float64(t_m * (cbrt(l) ^ -2.0)) * cbrt(k)) ^ 3.0)));
	elseif (k <= 6.4e+63)
		tmp = Float64(2.0 / Float64(Float64(Float64((t_m ^ 3.0) / l) / l) * Float64(2.0 * Float64(t_2 / cos(k)))));
	else
		tmp = Float64(Float64(2.0 * (l ^ 2.0)) / Float64(Float64(t_m * (k ^ 2.0)) * t_2));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 360.0], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[Power[N[(N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[Power[k, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 6.4e+63], N[(2.0 / N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * N[(2.0 * N[(t$95$2 / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 360

   1. Initial program 60.7%

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

   3. Simplified 60.7%

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

      1. add-cube-cbrt 60.6%

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

      2. pow3 60.6%

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

      3. associate-/r* 64.7%

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

      4. *-commutative 64.7%

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

      5. cbrt-prod 64.7%

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

      6. associate-/r* 60.7%

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

      7. cbrt-div 61.7%

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

      8. rem-cbrt-cube 70.5%

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

      9. cbrt-prod 81.2%

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

      10. pow2 81.2%

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

   6. Applied egg-rr 81.2%

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

      1. cube-mult 81.2%

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

      2. div-inv 81.2%

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

      3. pow-flip 81.2%

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

      4. metadata-eval 81.2%

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

      5. pow2 81.2%

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

      6. div-inv 81.2%

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

      7. pow-flip 81.2%

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

      8. metadata-eval 81.2%

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

   8. Applied egg-rr 81.2%

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

      1. unpow2 81.2%

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

      2. cube-unmult 81.2%

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

   10. Simplified 81.2%

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

   11. Taylor expanded in k around 0 74.6%

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

   12. Taylor expanded in k around 0 75.2%

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

   if 360 < k < 6.40000000000000022e63

   1. Initial program 25.2%

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

   3. Simplified 37.8%

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

   5. Taylor expanded in t around inf 63.9%

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

   if 6.40000000000000022e63 < k

   1. Initial program 49.1%

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

   3. Simplified 49.1%

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

   5. Taylor expanded in t around 0 71.4%

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

      1. associate-*r/ 71.4%

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

      2. associate-*r* 71.4%

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

   7. Simplified 71.4%

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

   8. Taylor expanded in k around 0 69.5%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 360:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{k}\right)}^{3}}\\ \mathbf{elif}\;k \leq 6.4 \cdot 10^{+63}:\\ \;\;\;\;\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(2 \cdot \frac{{\sin k}^{2}}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot {\ell}^{2}}{\left(t \cdot {k}^{2}\right) \cdot {\sin k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 0.0055:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left(\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{k}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{\sin k}^{2} \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 0.0055)
    (/ 2.0 (* (* 2.0 k) (pow (* (* t_m (pow (cbrt l) -2.0)) (cbrt k)) 3.0)))
    (/
     (* 2.0 (* (pow l 2.0) (cos k)))
     (* (pow (sin k) 2.0) (* t_m (* k k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 0.0055) {
		tmp = 2.0 / ((2.0 * k) * pow(((t_m * pow(cbrt(l), -2.0)) * cbrt(k)), 3.0));
	} else {
		tmp = (2.0 * (pow(l, 2.0) * cos(k))) / (pow(sin(k), 2.0) * (t_m * (k * k)));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 0.0055) {
		tmp = 2.0 / ((2.0 * k) * Math.pow(((t_m * Math.pow(Math.cbrt(l), -2.0)) * Math.cbrt(k)), 3.0));
	} else {
		tmp = (2.0 * (Math.pow(l, 2.0) * Math.cos(k))) / (Math.pow(Math.sin(k), 2.0) * (t_m * (k * k)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 0.0055)
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * (Float64(Float64(t_m * (cbrt(l) ^ -2.0)) * cbrt(k)) ^ 3.0)));
	else
		tmp = Float64(Float64(2.0 * Float64((l ^ 2.0) * cos(k))) / Float64((sin(k) ^ 2.0) * Float64(t_m * Float64(k * k))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 0.0055], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[Power[N[(N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[Power[k, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 0.0054999999999999997

   1. Initial program 60.8%

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

   3. Simplified 60.8%

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

      1. add-cube-cbrt 60.7%

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

      2. pow3 60.8%

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

      3. associate-/r* 64.8%

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

      4. *-commutative 64.8%

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

      5. cbrt-prod 64.8%

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

      6. associate-/r* 60.8%

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

      7. cbrt-div 61.9%

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

      8. rem-cbrt-cube 70.2%

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

      9. cbrt-prod 81.0%

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

      10. pow2 81.0%

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

   6. Applied egg-rr 81.0%

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

      1. cube-mult 81.0%

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

      2. div-inv 81.0%

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

      3. pow-flip 81.0%

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

      4. metadata-eval 81.0%

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

      5. pow2 81.0%

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

      6. div-inv 81.0%

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

      7. pow-flip 81.0%

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

      8. metadata-eval 81.0%

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

   8. Applied egg-rr 81.0%

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

      1. unpow2 81.0%

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

      2. cube-unmult 81.0%

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

   10. Simplified 81.0%

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

   11. Taylor expanded in k around 0 74.8%

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

   12. Taylor expanded in k around 0 75.4%

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

   if 0.0054999999999999997 < k

   1. Initial program 43.6%

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

   3. Simplified 43.6%

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

   5. Taylor expanded in t around 0 72.3%

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

      1. associate-*r/ 72.3%

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

      2. associate-*r* 72.3%

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

   7. Simplified 72.3%

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

      1. unpow2 72.3%

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

   9. Applied egg-rr 72.3%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.0055:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{k}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{+287}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left(t\_m \cdot \sqrt[3]{\frac{k}{{\ell}^{2}}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{2 \cdot {k}^{2}}}{{\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 1e+287)
    (/ 2.0 (* (* 2.0 k) (pow (* t_m (cbrt (/ k (pow l 2.0)))) 3.0)))
    (/ (/ 2.0 (* 2.0 (pow k 2.0))) (pow (* t_m (pow (cbrt l) -2.0)) 3.0)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 1e+287) {
		tmp = 2.0 / ((2.0 * k) * pow((t_m * cbrt((k / pow(l, 2.0)))), 3.0));
	} else {
		tmp = (2.0 / (2.0 * pow(k, 2.0))) / pow((t_m * pow(cbrt(l), -2.0)), 3.0);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 1e+287) {
		tmp = 2.0 / ((2.0 * k) * Math.pow((t_m * Math.cbrt((k / Math.pow(l, 2.0)))), 3.0));
	} else {
		tmp = (2.0 / (2.0 * Math.pow(k, 2.0))) / Math.pow((t_m * Math.pow(Math.cbrt(l), -2.0)), 3.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 1e+287)
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * (Float64(t_m * cbrt(Float64(k / (l ^ 2.0)))) ^ 3.0)));
	else
		tmp = Float64(Float64(2.0 / Float64(2.0 * (k ^ 2.0))) / (Float64(t_m * (cbrt(l) ^ -2.0)) ^ 3.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 1e+287], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[Power[N[(t$95$m * N[Power[N[(k / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1.0000000000000001e287

   1. Initial program 61.7%

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

   3. Simplified 61.7%

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

      1. add-cube-cbrt 61.6%

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

      2. pow3 61.6%

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

      3. associate-/r* 65.5%

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

      4. *-commutative 65.5%

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

      5. cbrt-prod 65.5%

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

      6. associate-/r* 61.6%

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

      7. cbrt-div 62.8%

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

      8. rem-cbrt-cube 73.9%

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

      9. cbrt-prod 78.4%

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

      10. pow2 78.4%

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

   6. Applied egg-rr 78.4%

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

      1. cube-mult 78.4%

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

      2. div-inv 78.4%

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

      3. pow-flip 78.4%

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

      4. metadata-eval 78.4%

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

      5. pow2 78.4%

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

      6. div-inv 78.4%

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

      7. pow-flip 78.4%

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

      8. metadata-eval 78.4%

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

   8. Applied egg-rr 78.4%

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

      1. unpow2 78.4%

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

      2. cube-unmult 78.4%

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

   10. Simplified 78.4%

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

   11. Taylor expanded in k around 0 67.0%

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

   12. Taylor expanded in k around 0 71.7%

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

   if 1.0000000000000001e287 < (*.f64 l l)

   1. Initial program 42.8%

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

   3. Simplified 46.2%

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

   5. Taylor expanded in k around 0 55.6%

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

      1. add-cube-cbrt 55.6%

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

      2. pow3 55.6%

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

      3. cbrt-prod 55.6%

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

      4. associate-/l/ 52.1%

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

      5. unpow2 52.1%

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

      6. cbrt-div 52.1%

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

      7. unpow3 52.1%

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

      8. add-cbrt-cube 54.3%

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

      9. unpow2 54.3%

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

      10. cbrt-prod 63.0%

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

      11. unpow2 63.0%

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

      12. div-inv 63.0%

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

      13. pow-flip 63.0%

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

      14. metadata-eval 63.0%

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

   7. Applied egg-rr 63.0%

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

      1. *-un-lft-identity 63.0%

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

      2. *-commutative 63.0%

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

      3. unpow-prod-down 62.8%

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

      4. pow1/3 62.5%

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

      5. pow2 62.5%

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

      6. pow-pow 62.8%

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

      7. metadata-eval 62.8%

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

      8. pow1 62.8%

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

      9. pow2 62.8%

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

   9. Applied egg-rr 62.8%

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

       1. *-lft-identity 62.8%

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

       2. associate-/r* 62.8%

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

   11. Simplified 62.8%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{+287}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left(t \cdot \sqrt[3]{\frac{k}{{\ell}^{2}}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{2 \cdot {k}^{2}}}{{\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{+287}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left(t\_m \cdot \sqrt[3]{\frac{k}{{\ell}^{2}}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3} \cdot \left(2 \cdot {k}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 1e+287)
    (/ 2.0 (* (* 2.0 k) (pow (* t_m (cbrt (/ k (pow l 2.0)))) 3.0)))
    (/ 2.0 (* (pow (* t_m (pow (cbrt l) -2.0)) 3.0) (* 2.0 (pow k 2.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 1e+287) {
		tmp = 2.0 / ((2.0 * k) * pow((t_m * cbrt((k / pow(l, 2.0)))), 3.0));
	} else {
		tmp = 2.0 / (pow((t_m * pow(cbrt(l), -2.0)), 3.0) * (2.0 * pow(k, 2.0)));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 1e+287) {
		tmp = 2.0 / ((2.0 * k) * Math.pow((t_m * Math.cbrt((k / Math.pow(l, 2.0)))), 3.0));
	} else {
		tmp = 2.0 / (Math.pow((t_m * Math.pow(Math.cbrt(l), -2.0)), 3.0) * (2.0 * Math.pow(k, 2.0)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 1e+287)
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * (Float64(t_m * cbrt(Float64(k / (l ^ 2.0)))) ^ 3.0)));
	else
		tmp = Float64(2.0 / Float64((Float64(t_m * (cbrt(l) ^ -2.0)) ^ 3.0) * Float64(2.0 * (k ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 1e+287], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[Power[N[(t$95$m * N[Power[N[(k / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1.0000000000000001e287

   1. Initial program 61.7%

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

   3. Simplified 61.7%

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

      1. add-cube-cbrt 61.6%

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

      2. pow3 61.6%

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

      3. associate-/r* 65.5%

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

      4. *-commutative 65.5%

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

      5. cbrt-prod 65.5%

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

      6. associate-/r* 61.6%

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

      7. cbrt-div 62.8%

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

      8. rem-cbrt-cube 73.9%

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

      9. cbrt-prod 78.4%

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

      10. pow2 78.4%

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

   6. Applied egg-rr 78.4%

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

      1. cube-mult 78.4%

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

      2. div-inv 78.4%

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

      3. pow-flip 78.4%

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

      4. metadata-eval 78.4%

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

      5. pow2 78.4%

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

      6. div-inv 78.4%

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

      7. pow-flip 78.4%

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

      8. metadata-eval 78.4%

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

   8. Applied egg-rr 78.4%

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

      1. unpow2 78.4%

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

      2. cube-unmult 78.4%

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

   10. Simplified 78.4%

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

   11. Taylor expanded in k around 0 67.0%

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

   12. Taylor expanded in k around 0 71.7%

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

   if 1.0000000000000001e287 < (*.f64 l l)

   1. Initial program 42.8%

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

   3. Simplified 46.2%

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

   5. Taylor expanded in k around 0 55.6%

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

      1. add-cube-cbrt 55.6%

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

      2. pow3 55.6%

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

      3. cbrt-prod 55.6%

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

      4. associate-/l/ 52.1%

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

      5. unpow2 52.1%

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

      6. cbrt-div 52.1%

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

      7. unpow3 52.1%

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

      8. add-cbrt-cube 54.3%

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

      9. unpow2 54.3%

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

      10. cbrt-prod 63.0%

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

      11. unpow2 63.0%

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

      12. div-inv 63.0%

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

      13. pow-flip 63.0%

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

      14. metadata-eval 63.0%

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

   7. Applied egg-rr 63.0%

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

      1. cube-prod 62.8%

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

      2. rem-cube-cbrt 62.8%

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

   9. Simplified 62.8%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{+287}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left(t \cdot \sqrt[3]{\frac{k}{{\ell}^{2}}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3} \cdot \left(2 \cdot {k}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 0.007:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left(\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{k}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{t\_m \cdot {k}^{4}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 0.007)
    (/ 2.0 (* (* 2.0 k) (pow (* (* t_m (pow (cbrt l) -2.0)) (cbrt k)) 3.0)))
    (/ (* 2.0 (* (pow l 2.0) (cos k))) (* t_m (pow k 4.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 0.007) {
		tmp = 2.0 / ((2.0 * k) * pow(((t_m * pow(cbrt(l), -2.0)) * cbrt(k)), 3.0));
	} else {
		tmp = (2.0 * (pow(l, 2.0) * cos(k))) / (t_m * pow(k, 4.0));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 0.007) {
		tmp = 2.0 / ((2.0 * k) * Math.pow(((t_m * Math.pow(Math.cbrt(l), -2.0)) * Math.cbrt(k)), 3.0));
	} else {
		tmp = (2.0 * (Math.pow(l, 2.0) * Math.cos(k))) / (t_m * Math.pow(k, 4.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 0.007)
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * (Float64(Float64(t_m * (cbrt(l) ^ -2.0)) * cbrt(k)) ^ 3.0)));
	else
		tmp = Float64(Float64(2.0 * Float64((l ^ 2.0) * cos(k))) / Float64(t_m * (k ^ 4.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 0.007], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[Power[N[(N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[Power[k, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 0.00700000000000000015

   1. Initial program 60.8%

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

   3. Simplified 60.8%

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

      1. add-cube-cbrt 60.7%

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

      2. pow3 60.8%

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

      3. associate-/r* 64.8%

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

      4. *-commutative 64.8%

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

      5. cbrt-prod 64.8%

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

      6. associate-/r* 60.8%

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

      7. cbrt-div 61.9%

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

      8. rem-cbrt-cube 70.2%

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

      9. cbrt-prod 81.0%

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

      10. pow2 81.0%

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

   6. Applied egg-rr 81.0%

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

      1. cube-mult 81.0%

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

      2. div-inv 81.0%

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

      3. pow-flip 81.0%

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

      4. metadata-eval 81.0%

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

      5. pow2 81.0%

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

      6. div-inv 81.0%

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

      7. pow-flip 81.0%

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

      8. metadata-eval 81.0%

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

   8. Applied egg-rr 81.0%

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

      1. unpow2 81.0%

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

      2. cube-unmult 81.0%

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

   10. Simplified 81.0%

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

   11. Taylor expanded in k around 0 74.8%

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

   12. Taylor expanded in k around 0 75.4%

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

   if 0.00700000000000000015 < k

   1. Initial program 43.6%

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

   3. Simplified 43.6%

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

   5. Taylor expanded in t around 0 72.3%

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

      1. associate-*r/ 72.3%

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

      2. associate-*r* 72.3%

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

   7. Simplified 72.3%

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

   8. Taylor expanded in k around 0 62.3%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.007:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{k}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{t \cdot {k}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6.4 \cdot 10^{+47}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot {\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot {\ell}^{2}}{\left(t\_m \cdot {k}^{2}\right) \cdot {\sin k}^{2}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 6.4e+47)
    (/ 2.0 (* (* 2.0 k) (* (sin k) (pow (* t_m (pow (cbrt l) -2.0)) 3.0))))
    (/ (* 2.0 (pow l 2.0)) (* (* t_m (pow k 2.0)) (pow (sin k) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 6.4e+47) {
		tmp = 2.0 / ((2.0 * k) * (sin(k) * pow((t_m * pow(cbrt(l), -2.0)), 3.0)));
	} else {
		tmp = (2.0 * pow(l, 2.0)) / ((t_m * pow(k, 2.0)) * pow(sin(k), 2.0));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 6.4e+47) {
		tmp = 2.0 / ((2.0 * k) * (Math.sin(k) * Math.pow((t_m * Math.pow(Math.cbrt(l), -2.0)), 3.0)));
	} else {
		tmp = (2.0 * Math.pow(l, 2.0)) / ((t_m * Math.pow(k, 2.0)) * Math.pow(Math.sin(k), 2.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 6.4e+47)
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * Float64(sin(k) * (Float64(t_m * (cbrt(l) ^ -2.0)) ^ 3.0))));
	else
		tmp = Float64(Float64(2.0 * (l ^ 2.0)) / Float64(Float64(t_m * (k ^ 2.0)) * (sin(k) ^ 2.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 6.4e+47], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 6.4e47

   1. Initial program 58.4%

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

   3. Simplified 58.4%

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

      1. add-cube-cbrt 58.3%

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

      2. pow3 58.3%

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

      3. associate-/r* 63.1%

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

      4. *-commutative 63.1%

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

      5. cbrt-prod 63.1%

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

      6. associate-/r* 58.3%

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

      7. cbrt-div 59.4%

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

      8. rem-cbrt-cube 68.1%

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

      9. cbrt-prod 79.7%

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

      10. pow2 79.7%

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

   6. Applied egg-rr 79.7%

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

      1. cube-mult 79.7%

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

      2. div-inv 79.7%

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

      3. pow-flip 79.7%

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

      4. metadata-eval 79.7%

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

      5. pow2 79.7%

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

      6. div-inv 79.7%

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

      7. pow-flip 79.7%

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

      8. metadata-eval 79.7%

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

   8. Applied egg-rr 79.7%

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

      1. unpow2 79.7%

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

      2. cube-unmult 79.7%

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

   10. Simplified 79.7%

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

   11. Taylor expanded in k around 0 73.1%

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

       1. *-commutative 73.1%

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

       2. unpow-prod-down 68.4%

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

       3. pow3 68.4%

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

       4. add-cube-cbrt 68.4%

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

   13. Applied egg-rr 68.4%

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

   if 6.4e47 < k

   1. Initial program 48.3%

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

   3. Simplified 48.3%

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

   5. Taylor expanded in t around 0 72.8%

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

      1. associate-*r/ 72.8%

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

      2. associate-*r* 72.8%

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

   7. Simplified 72.8%

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

   8. Taylor expanded in k around 0 68.8%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.4 \cdot 10^{+47}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot {\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot {\ell}^{2}}{\left(t \cdot {k}^{2}\right) \cdot {\sin k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 65.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{+287}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left(t\_m \cdot \sqrt[3]{\frac{k}{{\ell}^{2}}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 1e+287)
    (/ 2.0 (* (* 2.0 k) (pow (* t_m (cbrt (/ k (pow l 2.0)))) 3.0)))
    (/ 2.0 (* (pow (/ t_m (pow (cbrt l) 2.0)) 3.0) (* 2.0 (* k k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 1e+287) {
		tmp = 2.0 / ((2.0 * k) * pow((t_m * cbrt((k / pow(l, 2.0)))), 3.0));
	} else {
		tmp = 2.0 / (pow((t_m / pow(cbrt(l), 2.0)), 3.0) * (2.0 * (k * k)));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 1e+287) {
		tmp = 2.0 / ((2.0 * k) * Math.pow((t_m * Math.cbrt((k / Math.pow(l, 2.0)))), 3.0));
	} else {
		tmp = 2.0 / (Math.pow((t_m / Math.pow(Math.cbrt(l), 2.0)), 3.0) * (2.0 * (k * k)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 1e+287)
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * (Float64(t_m * cbrt(Float64(k / (l ^ 2.0)))) ^ 3.0)));
	else
		tmp = Float64(2.0 / Float64((Float64(t_m / (cbrt(l) ^ 2.0)) ^ 3.0) * Float64(2.0 * Float64(k * k))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 1e+287], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[Power[N[(t$95$m * N[Power[N[(k / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1.0000000000000001e287

   1. Initial program 61.7%

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

   3. Simplified 61.7%

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

      1. add-cube-cbrt 61.6%

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

      2. pow3 61.6%

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

      3. associate-/r* 65.5%

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

      4. *-commutative 65.5%

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

      5. cbrt-prod 65.5%

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

      6. associate-/r* 61.6%

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

      7. cbrt-div 62.8%

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

      8. rem-cbrt-cube 73.9%

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

      9. cbrt-prod 78.4%

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

      10. pow2 78.4%

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

   6. Applied egg-rr 78.4%

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

      1. cube-mult 78.4%

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

      2. div-inv 78.4%

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

      3. pow-flip 78.4%

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

      4. metadata-eval 78.4%

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

      5. pow2 78.4%

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

      6. div-inv 78.4%

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

      7. pow-flip 78.4%

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

      8. metadata-eval 78.4%

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

   8. Applied egg-rr 78.4%

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

      1. unpow2 78.4%

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

      2. cube-unmult 78.4%

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

   10. Simplified 78.4%

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

   11. Taylor expanded in k around 0 67.0%

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

   12. Taylor expanded in k around 0 71.7%

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

   if 1.0000000000000001e287 < (*.f64 l l)

   1. Initial program 42.8%

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

   3. Simplified 46.2%

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

   5. Taylor expanded in k around 0 55.6%

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

      1. unpow2 58.3%

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

   7. Applied egg-rr 55.6%

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

      1. add-cube-cbrt 55.6%

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

      2. pow3 55.6%

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

      3. associate-/r* 52.1%

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

      4. cbrt-div 52.1%

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

      5. rem-cbrt-cube 54.3%

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

      6. cbrt-prod 62.8%

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

      7. pow2 62.8%

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

   9. Applied egg-rr 62.8%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{+287}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left(t \cdot \sqrt[3]{\frac{k}{{\ell}^{2}}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 63.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.2 \cdot 10^{-22}:\\ \;\;\;\;\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{t\_m \cdot {k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left(t\_m \cdot \sqrt[3]{\frac{k}{{\ell}^{2}}}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 6.2e-22)
    (/ (* 2.0 (* (pow l 2.0) (cos k))) (* t_m (pow k 4.0)))
    (/ 2.0 (* (* 2.0 k) (pow (* t_m (cbrt (/ k (pow l 2.0)))) 3.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 6.2e-22) {
		tmp = (2.0 * (pow(l, 2.0) * cos(k))) / (t_m * pow(k, 4.0));
	} else {
		tmp = 2.0 / ((2.0 * k) * pow((t_m * cbrt((k / pow(l, 2.0)))), 3.0));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 6.2e-22) {
		tmp = (2.0 * (Math.pow(l, 2.0) * Math.cos(k))) / (t_m * Math.pow(k, 4.0));
	} else {
		tmp = 2.0 / ((2.0 * k) * Math.pow((t_m * Math.cbrt((k / Math.pow(l, 2.0)))), 3.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 6.2e-22)
		tmp = Float64(Float64(2.0 * Float64((l ^ 2.0) * cos(k))) / Float64(t_m * (k ^ 4.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * (Float64(t_m * cbrt(Float64(k / (l ^ 2.0)))) ^ 3.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 6.2e-22], N[(N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[Power[N[(t$95$m * N[Power[N[(k / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 6.20000000000000025e-22

   1. Initial program 52.1%

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

   3. Simplified 52.1%

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

   5. Taylor expanded in t around 0 69.4%

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

      1. associate-*r/ 69.4%

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

      2. associate-*r* 69.4%

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

   7. Simplified 69.4%

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

   8. Taylor expanded in k around 0 63.1%

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

   if 6.20000000000000025e-22 < t

   1. Initial program 66.8%

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

   3. Simplified 66.8%

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

      1. add-cube-cbrt 66.7%

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

      2. pow3 66.7%

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

      3. associate-/r* 70.0%

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

      4. *-commutative 70.0%

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

      5. cbrt-prod 70.0%

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

      6. associate-/r* 66.7%

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

      7. cbrt-div 69.7%

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

      8. rem-cbrt-cube 77.0%

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

      9. cbrt-prod 92.4%

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

      10. pow2 92.4%

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

   6. Applied egg-rr 92.4%

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

      1. cube-mult 92.4%

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

      2. div-inv 92.4%

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

      3. pow-flip 92.4%

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

      4. metadata-eval 92.4%

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

      5. pow2 92.4%

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

      6. div-inv 92.4%

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

      7. pow-flip 92.4%

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

      8. metadata-eval 92.4%

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

   8. Applied egg-rr 92.4%

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

      1. unpow2 92.4%

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

      2. cube-unmult 92.4%

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

   10. Simplified 92.4%

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

   11. Taylor expanded in k around 0 80.7%

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

   12. Taylor expanded in k around 0 71.6%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.2 \cdot 10^{-22}:\\ \;\;\;\;\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{t \cdot {k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left(t \cdot \sqrt[3]{\frac{k}{{\ell}^{2}}}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 61.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.6 \cdot 10^{-25}:\\ \;\;\;\;\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{t\_m \cdot {k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \frac{k \cdot {t\_m}^{3}}{{\ell}^{2}}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 7.6e-25)
    (/ (* 2.0 (* (pow l 2.0) (cos k))) (* t_m (pow k 4.0)))
    (/ 2.0 (* (* 2.0 k) (/ (* k (pow t_m 3.0)) (pow l 2.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 7.6e-25) {
		tmp = (2.0 * (pow(l, 2.0) * cos(k))) / (t_m * pow(k, 4.0));
	} else {
		tmp = 2.0 / ((2.0 * k) * ((k * pow(t_m, 3.0)) / pow(l, 2.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 7.6d-25) then
        tmp = (2.0d0 * ((l ** 2.0d0) * cos(k))) / (t_m * (k ** 4.0d0))
    else
        tmp = 2.0d0 / ((2.0d0 * k) * ((k * (t_m ** 3.0d0)) / (l ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 7.6e-25) {
		tmp = (2.0 * (Math.pow(l, 2.0) * Math.cos(k))) / (t_m * Math.pow(k, 4.0));
	} else {
		tmp = 2.0 / ((2.0 * k) * ((k * Math.pow(t_m, 3.0)) / Math.pow(l, 2.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 7.6e-25:
		tmp = (2.0 * (math.pow(l, 2.0) * math.cos(k))) / (t_m * math.pow(k, 4.0))
	else:
		tmp = 2.0 / ((2.0 * k) * ((k * math.pow(t_m, 3.0)) / math.pow(l, 2.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 7.6e-25)
		tmp = Float64(Float64(2.0 * Float64((l ^ 2.0) * cos(k))) / Float64(t_m * (k ^ 4.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * Float64(Float64(k * (t_m ^ 3.0)) / (l ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 7.6e-25)
		tmp = (2.0 * ((l ^ 2.0) * cos(k))) / (t_m * (k ^ 4.0));
	else
		tmp = 2.0 / ((2.0 * k) * ((k * (t_m ^ 3.0)) / (l ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 7.6e-25], N[(N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[(N[(k * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 7.5999999999999996e-25

   1. Initial program 52.1%

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

   3. Simplified 52.1%

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

   5. Taylor expanded in t around 0 69.4%

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

      1. associate-*r/ 69.4%

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

      2. associate-*r* 69.4%

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

   7. Simplified 69.4%

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

   8. Taylor expanded in k around 0 63.1%

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

   if 7.5999999999999996e-25 < t

   1. Initial program 66.8%

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

   3. Simplified 66.8%

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

      1. add-cube-cbrt 66.7%

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

      2. pow3 66.7%

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

      3. associate-/r* 70.0%

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

      4. *-commutative 70.0%

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

      5. cbrt-prod 70.0%

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

      6. associate-/r* 66.7%

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

      7. cbrt-div 69.7%

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

      8. rem-cbrt-cube 77.0%

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

      9. cbrt-prod 92.4%

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

      10. pow2 92.4%

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

   6. Applied egg-rr 92.4%

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

      1. cube-mult 92.4%

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

      2. div-inv 92.4%

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

      3. pow-flip 92.4%

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

      4. metadata-eval 92.4%

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

      5. pow2 92.4%

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

      6. div-inv 92.4%

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

      7. pow-flip 92.4%

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

      8. metadata-eval 92.4%

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

   8. Applied egg-rr 92.4%

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

      1. unpow2 92.4%

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

      2. cube-unmult 92.4%

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

   10. Simplified 92.4%

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

   11. Taylor expanded in k around 0 80.7%

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

   12. Taylor expanded in k around 0 68.4%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.6 \cdot 10^{-25}:\\ \;\;\;\;\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{t \cdot {k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \frac{k \cdot {t}^{3}}{{\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 60.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.2 \cdot 10^{-25}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{{k}^{3} \cdot \left(t\_m \cdot \sin k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \frac{k \cdot {t\_m}^{3}}{{\ell}^{2}}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4.2e-25)
    (* 2.0 (/ (pow l 2.0) (* (pow k 3.0) (* t_m (sin k)))))
    (/ 2.0 (* (* 2.0 k) (/ (* k (pow t_m 3.0)) (pow l 2.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.2e-25) {
		tmp = 2.0 * (pow(l, 2.0) / (pow(k, 3.0) * (t_m * sin(k))));
	} else {
		tmp = 2.0 / ((2.0 * k) * ((k * pow(t_m, 3.0)) / pow(l, 2.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 4.2d-25) then
        tmp = 2.0d0 * ((l ** 2.0d0) / ((k ** 3.0d0) * (t_m * sin(k))))
    else
        tmp = 2.0d0 / ((2.0d0 * k) * ((k * (t_m ** 3.0d0)) / (l ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.2e-25) {
		tmp = 2.0 * (Math.pow(l, 2.0) / (Math.pow(k, 3.0) * (t_m * Math.sin(k))));
	} else {
		tmp = 2.0 / ((2.0 * k) * ((k * Math.pow(t_m, 3.0)) / Math.pow(l, 2.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 4.2e-25:
		tmp = 2.0 * (math.pow(l, 2.0) / (math.pow(k, 3.0) * (t_m * math.sin(k))))
	else:
		tmp = 2.0 / ((2.0 * k) * ((k * math.pow(t_m, 3.0)) / math.pow(l, 2.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 4.2e-25)
		tmp = Float64(2.0 * Float64((l ^ 2.0) / Float64((k ^ 3.0) * Float64(t_m * sin(k)))));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * Float64(Float64(k * (t_m ^ 3.0)) / (l ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 4.2e-25)
		tmp = 2.0 * ((l ^ 2.0) / ((k ^ 3.0) * (t_m * sin(k))));
	else
		tmp = 2.0 / ((2.0 * k) * ((k * (t_m ^ 3.0)) / (l ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 4.2e-25], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / N[(N[Power[k, 3.0], $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[(N[(k * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.20000000000000005e-25

   1. Initial program 52.1%

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

   3. Simplified 51.6%

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

   5. Taylor expanded in k around 0 49.6%

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

   6. Taylor expanded in k around inf 60.9%

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

   if 4.20000000000000005e-25 < t

   1. Initial program 66.8%

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

   3. Simplified 66.8%

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

      1. add-cube-cbrt 66.7%

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

      2. pow3 66.7%

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

      3. associate-/r* 70.0%

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

      4. *-commutative 70.0%

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

      5. cbrt-prod 70.0%

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

      6. associate-/r* 66.7%

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

      7. cbrt-div 69.7%

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

      8. rem-cbrt-cube 77.0%

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

      9. cbrt-prod 92.4%

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

      10. pow2 92.4%

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

   6. Applied egg-rr 92.4%

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

      1. cube-mult 92.4%

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

      2. div-inv 92.4%

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

      3. pow-flip 92.4%

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

      4. metadata-eval 92.4%

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

      5. pow2 92.4%

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

      6. div-inv 92.4%

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

      7. pow-flip 92.4%

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

      8. metadata-eval 92.4%

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

   8. Applied egg-rr 92.4%

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

      1. unpow2 92.4%

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

      2. cube-unmult 92.4%

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

   10. Simplified 92.4%

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

   11. Taylor expanded in k around 0 80.7%

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

   12. Taylor expanded in k around 0 68.4%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{-25}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{{k}^{3} \cdot \left(t \cdot \sin k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \frac{k \cdot {t}^{3}}{{\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 60.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.7 \cdot 10^{-25}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{t\_m \cdot {k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \frac{k \cdot {t\_m}^{3}}{{\ell}^{2}}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.7e-25)
    (* 2.0 (/ (pow l 2.0) (* t_m (pow k 4.0))))
    (/ 2.0 (* (* 2.0 k) (/ (* k (pow t_m 3.0)) (pow l 2.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.7e-25) {
		tmp = 2.0 * (pow(l, 2.0) / (t_m * pow(k, 4.0)));
	} else {
		tmp = 2.0 / ((2.0 * k) * ((k * pow(t_m, 3.0)) / pow(l, 2.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.7d-25) then
        tmp = 2.0d0 * ((l ** 2.0d0) / (t_m * (k ** 4.0d0)))
    else
        tmp = 2.0d0 / ((2.0d0 * k) * ((k * (t_m ** 3.0d0)) / (l ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.7e-25) {
		tmp = 2.0 * (Math.pow(l, 2.0) / (t_m * Math.pow(k, 4.0)));
	} else {
		tmp = 2.0 / ((2.0 * k) * ((k * Math.pow(t_m, 3.0)) / Math.pow(l, 2.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.7e-25:
		tmp = 2.0 * (math.pow(l, 2.0) / (t_m * math.pow(k, 4.0)))
	else:
		tmp = 2.0 / ((2.0 * k) * ((k * math.pow(t_m, 3.0)) / math.pow(l, 2.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.7e-25)
		tmp = Float64(2.0 * Float64((l ^ 2.0) / Float64(t_m * (k ^ 4.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * Float64(Float64(k * (t_m ^ 3.0)) / (l ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.7e-25)
		tmp = 2.0 * ((l ^ 2.0) / (t_m * (k ^ 4.0)));
	else
		tmp = 2.0 / ((2.0 * k) * ((k * (t_m ^ 3.0)) / (l ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.7e-25], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[(N[(k * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.70000000000000001e-25

   1. Initial program 52.1%

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

   3. Simplified 52.1%

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

   5. Taylor expanded in t around 0 69.4%

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

      1. associate-*r/ 69.4%

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

      2. associate-*r* 69.4%

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

   7. Simplified 69.4%

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

   8. Taylor expanded in k around 0 60.8%

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

   if 1.70000000000000001e-25 < t

   1. Initial program 66.8%

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

   3. Simplified 66.8%

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

      1. add-cube-cbrt 66.7%

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

      2. pow3 66.7%

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

      3. associate-/r* 70.0%

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

      4. *-commutative 70.0%

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

      5. cbrt-prod 70.0%

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

      6. associate-/r* 66.7%

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

      7. cbrt-div 69.7%

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

      8. rem-cbrt-cube 77.0%

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

      9. cbrt-prod 92.4%

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

      10. pow2 92.4%

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

   6. Applied egg-rr 92.4%

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

      1. cube-mult 92.4%

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

      2. div-inv 92.4%

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

      3. pow-flip 92.4%

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

      4. metadata-eval 92.4%

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

      5. pow2 92.4%

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

      6. div-inv 92.4%

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

      7. pow-flip 92.4%

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

      8. metadata-eval 92.4%

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

   8. Applied egg-rr 92.4%

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

      1. unpow2 92.4%

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

      2. cube-unmult 92.4%

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

   10. Simplified 92.4%

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

   11. Taylor expanded in k around 0 80.7%

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

   12. Taylor expanded in k around 0 68.4%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.7 \cdot 10^{-25}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{t \cdot {k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \frac{k \cdot {t}^{3}}{{\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 59.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.8 \cdot 10^{-21}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{t\_m \cdot {k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(k \cdot \frac{{t\_m}^{3}}{{\ell}^{2}}\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.8e-21)
    (* 2.0 (/ (pow l 2.0) (* t_m (pow k 4.0))))
    (/ 2.0 (* (* 2.0 k) (* k (/ (pow t_m 3.0) (pow l 2.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.8e-21) {
		tmp = 2.0 * (pow(l, 2.0) / (t_m * pow(k, 4.0)));
	} else {
		tmp = 2.0 / ((2.0 * k) * (k * (pow(t_m, 3.0) / pow(l, 2.0))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2.8d-21) then
        tmp = 2.0d0 * ((l ** 2.0d0) / (t_m * (k ** 4.0d0)))
    else
        tmp = 2.0d0 / ((2.0d0 * k) * (k * ((t_m ** 3.0d0) / (l ** 2.0d0))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.8e-21) {
		tmp = 2.0 * (Math.pow(l, 2.0) / (t_m * Math.pow(k, 4.0)));
	} else {
		tmp = 2.0 / ((2.0 * k) * (k * (Math.pow(t_m, 3.0) / Math.pow(l, 2.0))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2.8e-21:
		tmp = 2.0 * (math.pow(l, 2.0) / (t_m * math.pow(k, 4.0)))
	else:
		tmp = 2.0 / ((2.0 * k) * (k * (math.pow(t_m, 3.0) / math.pow(l, 2.0))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.8e-21)
		tmp = Float64(2.0 * Float64((l ^ 2.0) / Float64(t_m * (k ^ 4.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * Float64(k * Float64((t_m ^ 3.0) / (l ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2.8e-21)
		tmp = 2.0 * ((l ^ 2.0) / (t_m * (k ^ 4.0)));
	else
		tmp = 2.0 / ((2.0 * k) * (k * ((t_m ^ 3.0) / (l ^ 2.0))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.8e-21], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[(k * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.80000000000000004e-21

   1. Initial program 52.1%

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

   3. Simplified 52.1%

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

   5. Taylor expanded in t around 0 69.4%

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

      1. associate-*r/ 69.4%

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

      2. associate-*r* 69.4%

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

   7. Simplified 69.4%

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

   8. Taylor expanded in k around 0 60.8%

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

   if 2.80000000000000004e-21 < t

   1. Initial program 66.8%

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

   3. Simplified 66.8%

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

      1. add-cube-cbrt 66.7%

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

      2. pow3 66.7%

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

      3. associate-/r* 70.0%

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

      4. *-commutative 70.0%

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

      5. cbrt-prod 70.0%

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

      6. associate-/r* 66.7%

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

      7. cbrt-div 69.7%

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

      8. rem-cbrt-cube 77.0%

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

      9. cbrt-prod 92.4%

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

      10. pow2 92.4%

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

   6. Applied egg-rr 92.4%

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

      1. cube-mult 92.4%

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

      2. div-inv 92.4%

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

      3. pow-flip 92.4%

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

      4. metadata-eval 92.4%

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

      5. pow2 92.4%

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

      6. div-inv 92.4%

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

      7. pow-flip 92.4%

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

      8. metadata-eval 92.4%

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

   8. Applied egg-rr 92.4%

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

      1. unpow2 92.4%

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

      2. cube-unmult 92.4%

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

   10. Simplified 92.4%

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

   11. Taylor expanded in k around 0 80.7%

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

   12. Taylor expanded in k around 0 68.4%

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

       1. associate-/l* 65.4%

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

   14. Simplified 65.4%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.8 \cdot 10^{-21}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{t \cdot {k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(k \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 58.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.25 \cdot 10^{-34}:\\ \;\;\;\;\frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{t\_m \cdot \frac{{t\_m}^{2}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{t\_m \cdot {k}^{4}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.25e-34)
    (/ 2.0 (* (* 2.0 (* k k)) (/ (* t_m (/ (pow t_m 2.0) l)) l)))
    (* 2.0 (/ (pow l 2.0) (* t_m (pow k 4.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.25e-34) {
		tmp = 2.0 / ((2.0 * (k * k)) * ((t_m * (pow(t_m, 2.0) / l)) / l));
	} else {
		tmp = 2.0 * (pow(l, 2.0) / (t_m * pow(k, 4.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.25d-34) then
        tmp = 2.0d0 / ((2.0d0 * (k * k)) * ((t_m * ((t_m ** 2.0d0) / l)) / l))
    else
        tmp = 2.0d0 * ((l ** 2.0d0) / (t_m * (k ** 4.0d0)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.25e-34) {
		tmp = 2.0 / ((2.0 * (k * k)) * ((t_m * (Math.pow(t_m, 2.0) / l)) / l));
	} else {
		tmp = 2.0 * (Math.pow(l, 2.0) / (t_m * Math.pow(k, 4.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 1.25e-34:
		tmp = 2.0 / ((2.0 * (k * k)) * ((t_m * (math.pow(t_m, 2.0) / l)) / l))
	else:
		tmp = 2.0 * (math.pow(l, 2.0) / (t_m * math.pow(k, 4.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.25e-34)
		tmp = Float64(2.0 / Float64(Float64(2.0 * Float64(k * k)) * Float64(Float64(t_m * Float64((t_m ^ 2.0) / l)) / l)));
	else
		tmp = Float64(2.0 * Float64((l ^ 2.0) / Float64(t_m * (k ^ 4.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 1.25e-34)
		tmp = 2.0 / ((2.0 * (k * k)) * ((t_m * ((t_m ^ 2.0) / l)) / l));
	else
		tmp = 2.0 * ((l ^ 2.0) / (t_m * (k ^ 4.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1.25e-34], N[(2.0 / N[(N[(2.0 * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.2500000000000001e-34

   1. Initial program 60.8%

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

   3. Simplified 58.8%

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

   5. Taylor expanded in k around 0 59.2%

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

      1. unpow2 59.1%

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

   7. Applied egg-rr 59.2%

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

      1. associate-/r* 56.2%

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

      2. unpow3 56.2%

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

      3. times-frac 60.4%

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

      4. pow2 60.4%

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

   9. Applied egg-rr 60.4%

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

       1. associate-*r/ 60.4%

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

   11. Applied egg-rr 60.4%

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

   if 1.2500000000000001e-34 < k

   1. Initial program 45.7%

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

   3. Simplified 45.7%

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

   5. Taylor expanded in t around 0 72.1%

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

      1. associate-*r/ 72.1%

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

      2. associate-*r* 72.1%

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

   7. Simplified 72.1%

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

   8. Taylor expanded in k around 0 59.4%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.25 \cdot 10^{-34}:\\ \;\;\;\;\frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{t \cdot \frac{{t}^{2}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{t \cdot {k}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 58.1% accurate, 3.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{t\_m \cdot \frac{{t\_m}^{2}}{\ell}}{\ell}} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ 2.0 (* (* 2.0 (* k k)) (/ (* t_m (/ (pow t_m 2.0) l)) l)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / ((2.0 * (k * k)) * ((t_m * (pow(t_m, 2.0) / l)) / l)));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 / ((2.0d0 * (k * k)) * ((t_m * ((t_m ** 2.0d0) / l)) / l)))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / ((2.0 * (k * k)) * ((t_m * (Math.pow(t_m, 2.0) / l)) / l)));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 / ((2.0 * (k * k)) * ((t_m * (math.pow(t_m, 2.0) / l)) / l)))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 / Float64(Float64(2.0 * Float64(k * k)) * Float64(Float64(t_m * Float64((t_m ^ 2.0) / l)) / l))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 / ((2.0 * (k * k)) * ((t_m * ((t_m ^ 2.0) / l)) / l)));
end

Wolfram

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

Derivation

1. Initial program 56.2%

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

3. Simplified 55.6%

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

5. Taylor expanded in k around 0 56.9%

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

   1. unpow2 63.1%

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

7. Applied egg-rr 56.9%

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

   1. associate-/r* 53.9%

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

   2. unpow3 53.9%

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

   3. times-frac 58.1%

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

   4. pow2 58.1%

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

9. Applied egg-rr 58.1%

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

    1. associate-*r/ 58.1%

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

11. Applied egg-rr 58.1%

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

12. Final simplification 58.1%

    \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{t \cdot \frac{{t}^{2}}{\ell}}{\ell}} \]
13. Add Preprocessing

Alternative 20: 58.4% accurate, 3.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{{t\_m}^{2}}{\ell} \cdot \frac{t\_m}{\ell}\right)} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ 2.0 (* (* 2.0 (* k k)) (* (/ (pow t_m 2.0) l) (/ t_m l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / ((2.0 * (k * k)) * ((pow(t_m, 2.0) / l) * (t_m / l))));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 / ((2.0d0 * (k * k)) * (((t_m ** 2.0d0) / l) * (t_m / l))))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / ((2.0 * (k * k)) * ((Math.pow(t_m, 2.0) / l) * (t_m / l))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 / ((2.0 * (k * k)) * ((math.pow(t_m, 2.0) / l) * (t_m / l))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 / Float64(Float64(2.0 * Float64(k * k)) * Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l)))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 / ((2.0 * (k * k)) * (((t_m ^ 2.0) / l) * (t_m / l))));
end

Wolfram

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

Derivation

1. Initial program 56.2%

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

3. Simplified 55.6%

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

5. Taylor expanded in k around 0 56.9%

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

   1. unpow2 63.1%

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

7. Applied egg-rr 56.9%

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

   1. associate-/r* 53.9%

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

   2. unpow3 53.9%

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

   3. times-frac 58.1%

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

   4. pow2 58.1%

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

9. Applied egg-rr 58.1%

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

10. Final simplification 58.1%

    \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \]
11. Add Preprocessing

Alternative 21: 55.4% accurate, 3.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{\frac{\frac{{t\_m}^{3}}{\ell}}{\ell} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \end{array} \]

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 (/ 2.0 (* (/ (/ (pow t_m 3.0) l) l) (* 2.0 (* k k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / (((pow(t_m, 3.0) / l) / l) * (2.0 * (k * k))));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 / ((((t_m ** 3.0d0) / l) / l) * (2.0d0 * (k * k))))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / (((Math.pow(t_m, 3.0) / l) / l) * (2.0 * (k * k))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 / (((math.pow(t_m, 3.0) / l) / l) * (2.0 * (k * k))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 / Float64(Float64(Float64((t_m ^ 3.0) / l) / l) * Float64(2.0 * Float64(k * k)))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 / ((((t_m ^ 3.0) / l) / l) * (2.0 * (k * k))));
end

Wolfram

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

Derivation

1. Initial program 56.2%

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

3. Simplified 55.6%

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

5. Taylor expanded in k around 0 56.9%

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

   1. unpow2 63.1%

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

7. Applied egg-rr 56.9%

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

Reproduce

herbie shell --seed 2024146 
(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))))