Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.9% → 88.6%

Time: 15.5s

Alternatives: 18

Speedup: 8.7×

• 26.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.9% 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: 88.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.8 \cdot 10^{-60}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\frac{{\sin k}^{2}}{\ell}}{\cos k \cdot \ell} \cdot \mathsf{fma}\left(2, \frac{\frac{{t\_m}^{3}}{k}}{k}, t\_m\right)\right) \cdot k\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(k, \frac{1}{\frac{t\_m}{k} \cdot t\_m}, 2\right) \cdot \left(\left(\left(\tan k \cdot t\_m\right) \cdot \frac{\sin k \cdot t\_m}{\ell}\right) \cdot \frac{t\_m}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4.8e-60)
    (/
     2.0
     (*
      (*
       (*
        (/ (/ (pow (sin k) 2.0) l) (* (cos k) l))
        (fma 2.0 (/ (/ (pow t_m 3.0) k) k) t_m))
       k)
      k))
    (/
     2.0
     (*
      (fma k (/ 1.0 (* (/ t_m k) t_m)) 2.0)
      (* (* (* (tan k) t_m) (/ (* (sin k) t_m) l)) (/ t_m l)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.8e-60) {
		tmp = 2.0 / (((((pow(sin(k), 2.0) / l) / (cos(k) * l)) * fma(2.0, ((pow(t_m, 3.0) / k) / k), t_m)) * k) * k);
	} else {
		tmp = 2.0 / (fma(k, (1.0 / ((t_m / k) * t_m)), 2.0) * (((tan(k) * t_m) * ((sin(k) * t_m) / l)) * (t_m / l)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 4.8e-60)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64((sin(k) ^ 2.0) / l) / Float64(cos(k) * l)) * fma(2.0, Float64(Float64((t_m ^ 3.0) / k) / k), t_m)) * k) * k));
	else
		tmp = Float64(2.0 / Float64(fma(k, Float64(1.0 / Float64(Float64(t_m / k) * t_m)), 2.0) * Float64(Float64(Float64(tan(k) * t_m) * Float64(Float64(sin(k) * t_m) / l)) * Float64(t_m / l))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 4.8e-60], N[(2.0 / N[(N[(N[(N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / k), $MachinePrecision] / k), $MachinePrecision] + t$95$m), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * N[(1.0 / N[(N[(t$95$m / k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(N[(N[Tan[k], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.80000000000000019e-60

   1. Initial program 50.5%

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

   3. Taylor expanded in k around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 75.8%

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

   if 4.80000000000000019e-60 < t

   1. Initial program 67.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow3 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 77.2

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

   4. Applied rewrites 77.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 84.5

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 84.5

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

   6. Applied rewrites 84.5%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. frac-times N/A

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

      11. div-inv N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 84.5

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

   8. Applied rewrites 84.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 89.6

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

   10. Applied rewrites 89.6%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.8 \cdot 10^{-60}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\frac{{\sin k}^{2}}{\ell}}{\cos k \cdot \ell} \cdot \mathsf{fma}\left(2, \frac{\frac{{t}^{3}}{k}}{k}, t\right)\right) \cdot k\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(k, \frac{1}{\frac{t}{k} \cdot t}, 2\right) \cdot \left(\left(\left(\tan k \cdot t\right) \cdot \frac{\sin k \cdot t}{\ell}\right) \cdot \frac{t}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 65.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left({\left(\frac{k}{t\_m}\right)}^{2} + 1\right) + 1\right) \cdot \left(\left(\frac{{t\_m}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \leq 0:\\ \;\;\;\;\frac{2}{\left(\left(\left(k \cdot t\_m\right) \cdot t\_m\right) \cdot \left(k \cdot t\_m\right)\right) \cdot \frac{2}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t\_m}{\ell} \cdot t\_m\right) \cdot \frac{t\_m}{\ell}\right) \cdot \left(\left(k \cdot k\right) \cdot 2\right)}\\ \end{array} \end{array} \]

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 (<=
       (/
        2.0
        (*
         (+ (+ (pow (/ k t_m) 2.0) 1.0) 1.0)
         (* (* (/ (pow t_m 3.0) (* l l)) (sin k)) (tan k))))
       0.0)
    (/ 2.0 (* (* (* (* k t_m) t_m) (* k t_m)) (/ 2.0 (* l l))))
    (/ 2.0 (* (* (* (/ t_m l) t_m) (/ t_m l)) (* (* k 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 ((2.0 / (((pow((k / t_m), 2.0) + 1.0) + 1.0) * (((pow(t_m, 3.0) / (l * l)) * sin(k)) * tan(k)))) <= 0.0) {
		tmp = 2.0 / ((((k * t_m) * t_m) * (k * t_m)) * (2.0 / (l * l)));
	} else {
		tmp = 2.0 / ((((t_m / l) * t_m) * (t_m / l)) * ((k * k) * 2.0));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((2.0d0 / (((((k / t_m) ** 2.0d0) + 1.0d0) + 1.0d0) * ((((t_m ** 3.0d0) / (l * l)) * sin(k)) * tan(k)))) <= 0.0d0) then
        tmp = 2.0d0 / ((((k * t_m) * t_m) * (k * t_m)) * (2.0d0 / (l * l)))
    else
        tmp = 2.0d0 / ((((t_m / l) * t_m) * (t_m / l)) * ((k * k) * 2.0d0))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((2.0 / (((Math.pow((k / t_m), 2.0) + 1.0) + 1.0) * (((Math.pow(t_m, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)))) <= 0.0) {
		tmp = 2.0 / ((((k * t_m) * t_m) * (k * t_m)) * (2.0 / (l * l)));
	} else {
		tmp = 2.0 / ((((t_m / l) * t_m) * (t_m / l)) * ((k * k) * 2.0));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if (2.0 / (((math.pow((k / t_m), 2.0) + 1.0) + 1.0) * (((math.pow(t_m, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)))) <= 0.0:
		tmp = 2.0 / ((((k * t_m) * t_m) * (k * t_m)) * (2.0 / (l * l)))
	else:
		tmp = 2.0 / ((((t_m / l) * t_m) * (t_m / l)) * ((k * 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(2.0 / Float64(Float64(Float64((Float64(k / t_m) ^ 2.0) + 1.0) + 1.0) * Float64(Float64(Float64((t_m ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)))) <= 0.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * t_m) * t_m) * Float64(k * t_m)) * Float64(2.0 / Float64(l * l))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m / l) * t_m) * Float64(t_m / l)) * Float64(Float64(k * k) * 2.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

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

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[N[(2.0 / N[(N[(N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(2.0 / N[(N[(N[(N[(k * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < -0.0

   1. Initial program 77.3%

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

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 67.6

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

   5. Applied rewrites 67.6%

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

   7. Applied rewrites 70.4%

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

   9. Applied rewrites 81.7%

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

   11. Applied rewrites 81.8%

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

   if -0.0 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 26.5%

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

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 40.8

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

   5. Applied rewrites 40.8%

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

   7. Applied rewrites 38.6%

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

   9. Applied rewrites 50.8%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \leq 0:\\ \;\;\;\;\frac{2}{\left(\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot \frac{2}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left(\left(k \cdot k\right) \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 65.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left({\left(\frac{k}{t\_m}\right)}^{2} + 1\right) + 1\right) \cdot \left(\left(\frac{{t\_m}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \leq 5 \cdot 10^{+18}:\\ \;\;\;\;\frac{2}{\left(\left(\left(k \cdot t\_m\right) \cdot t\_m\right) \cdot \left(k \cdot t\_m\right)\right) \cdot \frac{2}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \left(\left(\frac{t\_m}{\ell} \cdot \frac{t\_m}{\ell}\right) \cdot t\_m\right)}\\ \end{array} \end{array} \]

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 (<=
       (/
        2.0
        (*
         (+ (+ (pow (/ k t_m) 2.0) 1.0) 1.0)
         (* (* (/ (pow t_m 3.0) (* l l)) (sin k)) (tan k))))
       5e+18)
    (/ 2.0 (* (* (* (* k t_m) t_m) (* k t_m)) (/ 2.0 (* l l))))
    (/ 2.0 (* (* (* k k) 2.0) (* (* (/ t_m l) (/ t_m l)) t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((2.0 / (((pow((k / t_m), 2.0) + 1.0) + 1.0) * (((pow(t_m, 3.0) / (l * l)) * sin(k)) * tan(k)))) <= 5e+18) {
		tmp = 2.0 / ((((k * t_m) * t_m) * (k * t_m)) * (2.0 / (l * l)));
	} else {
		tmp = 2.0 / (((k * k) * 2.0) * (((t_m / l) * (t_m / l)) * t_m));
	}
	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 ((2.0d0 / (((((k / t_m) ** 2.0d0) + 1.0d0) + 1.0d0) * ((((t_m ** 3.0d0) / (l * l)) * sin(k)) * tan(k)))) <= 5d+18) then
        tmp = 2.0d0 / ((((k * t_m) * t_m) * (k * t_m)) * (2.0d0 / (l * l)))
    else
        tmp = 2.0d0 / (((k * k) * 2.0d0) * (((t_m / l) * (t_m / l)) * t_m))
    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 ((2.0 / (((Math.pow((k / t_m), 2.0) + 1.0) + 1.0) * (((Math.pow(t_m, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)))) <= 5e+18) {
		tmp = 2.0 / ((((k * t_m) * t_m) * (k * t_m)) * (2.0 / (l * l)));
	} else {
		tmp = 2.0 / (((k * k) * 2.0) * (((t_m / l) * (t_m / l)) * t_m));
	}
	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 (2.0 / (((math.pow((k / t_m), 2.0) + 1.0) + 1.0) * (((math.pow(t_m, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)))) <= 5e+18:
		tmp = 2.0 / ((((k * t_m) * t_m) * (k * t_m)) * (2.0 / (l * l)))
	else:
		tmp = 2.0 / (((k * k) * 2.0) * (((t_m / l) * (t_m / l)) * t_m))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64((Float64(k / t_m) ^ 2.0) + 1.0) + 1.0) * Float64(Float64(Float64((t_m ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)))) <= 5e+18)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * t_m) * t_m) * Float64(k * t_m)) * Float64(2.0 / Float64(l * l))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * 2.0) * Float64(Float64(Float64(t_m / l) * Float64(t_m / l)) * t_m)));
	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 ((2.0 / (((((k / t_m) ^ 2.0) + 1.0) + 1.0) * ((((t_m ^ 3.0) / (l * l)) * sin(k)) * tan(k)))) <= 5e+18)
		tmp = 2.0 / ((((k * t_m) * t_m) * (k * t_m)) * (2.0 / (l * l)));
	else
		tmp = 2.0 / (((k * k) * 2.0) * (((t_m / l) * (t_m / l)) * t_m));
	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[N[(2.0 / N[(N[(N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+18], N[(2.0 / N[(N[(N[(N[(k * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 5e18

   1. Initial program 77.8%

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

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 67.0

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

   5. Applied rewrites 67.0%

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

   7. Applied rewrites 69.7%

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

   9. Applied rewrites 80.8%

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

   11. Applied rewrites 80.8%

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

   if 5e18 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 24.4%

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

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 40.9

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

   5. Applied rewrites 40.9%

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

   7. Applied rewrites 50.3%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \leq 5 \cdot 10^{+18}:\\ \;\;\;\;\frac{2}{\left(\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot \frac{2}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8 \cdot 10^{-27}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left({t\_m}^{3}, 2, \left(k \cdot t\_m\right) \cdot k\right) \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(k, \frac{1}{\frac{t\_m}{k} \cdot t\_m}, 2\right) \cdot \left(\left(\left(\tan k \cdot t\_m\right) \cdot \frac{\sin k \cdot t\_m}{\ell}\right) \cdot \frac{t\_m}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 8e-27)
    (/
     2.0
     (*
      (fma (pow t_m 3.0) 2.0 (* (* k t_m) k))
      (/ (/ (pow (sin k) 2.0) l) (* (cos k) l))))
    (/
     2.0
     (*
      (fma k (/ 1.0 (* (/ t_m k) t_m)) 2.0)
      (* (* (* (tan k) t_m) (/ (* (sin k) t_m) l)) (/ t_m l)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 8e-27) {
		tmp = 2.0 / (fma(pow(t_m, 3.0), 2.0, ((k * t_m) * k)) * ((pow(sin(k), 2.0) / l) / (cos(k) * l)));
	} else {
		tmp = 2.0 / (fma(k, (1.0 / ((t_m / k) * t_m)), 2.0) * (((tan(k) * t_m) * ((sin(k) * t_m) / l)) * (t_m / l)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 8e-27)
		tmp = Float64(2.0 / Float64(fma((t_m ^ 3.0), 2.0, Float64(Float64(k * t_m) * k)) * Float64(Float64((sin(k) ^ 2.0) / l) / Float64(cos(k) * l))));
	else
		tmp = Float64(2.0 / Float64(fma(k, Float64(1.0 / Float64(Float64(t_m / k) * t_m)), 2.0) * Float64(Float64(Float64(tan(k) * t_m) * Float64(Float64(sin(k) * t_m) / l)) * Float64(t_m / l))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 8e-27], N[(2.0 / N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] * 2.0 + N[(N[(k * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * N[(1.0 / N[(N[(t$95$m / k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(N[(N[Tan[k], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 8.0000000000000003e-27

   1. Initial program 52.0%

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

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 54.9

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

   5. Applied rewrites 54.9%

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

   7. Applied rewrites 56.3%

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

   8. Taylor expanded in t around 0

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

      1. distribute-lft-in N/A

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

   10. Applied rewrites 77.6%

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

   if 8.0000000000000003e-27 < t

   1. Initial program 66.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow3 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 75.9

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

   4. Applied rewrites 75.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 84.7

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 84.7

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

   6. Applied rewrites 84.7%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. frac-times N/A

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

      11. div-inv N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 84.7

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

   8. Applied rewrites 84.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 90.8

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

   10. Applied rewrites 90.8%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8 \cdot 10^{-27}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left({t}^{3}, 2, \left(k \cdot t\right) \cdot k\right) \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(k, \frac{1}{\frac{t}{k} \cdot t}, 2\right) \cdot \left(\left(\left(\tan k \cdot t\right) \cdot \frac{\sin k \cdot t}{\ell}\right) \cdot \frac{t}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.8 \cdot 10^{-64}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\cos k} \cdot \frac{\frac{k \cdot k}{\ell} \cdot t\_m}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(k, \frac{1}{\frac{t\_m}{k} \cdot t\_m}, 2\right) \cdot \left(\left(\left(\tan k \cdot t\_m\right) \cdot \frac{\sin k \cdot t\_m}{\ell}\right) \cdot \frac{t\_m}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.8e-64)
    (/ 2.0 (* (/ (pow (sin k) 2.0) (cos k)) (/ (* (/ (* k k) l) t_m) l)))
    (/
     2.0
     (*
      (fma k (/ 1.0 (* (/ t_m k) t_m)) 2.0)
      (* (* (* (tan k) t_m) (/ (* (sin k) t_m) l)) (/ t_m l)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.8e-64) {
		tmp = 2.0 / ((pow(sin(k), 2.0) / cos(k)) * ((((k * k) / l) * t_m) / l));
	} else {
		tmp = 2.0 / (fma(k, (1.0 / ((t_m / k) * t_m)), 2.0) * (((tan(k) * t_m) * ((sin(k) * t_m) / l)) * (t_m / l)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.8e-64)
		tmp = Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) / cos(k)) * Float64(Float64(Float64(Float64(k * k) / l) * t_m) / l)));
	else
		tmp = Float64(2.0 / Float64(fma(k, Float64(1.0 / Float64(Float64(t_m / k) * t_m)), 2.0) * Float64(Float64(Float64(tan(k) * t_m) * Float64(Float64(sin(k) * t_m) / l)) * Float64(t_m / l))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.8e-64], N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * N[(1.0 / N[(N[(t$95$m / k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(N[(N[Tan[k], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.80000000000000004e-64

   1. Initial program 50.5%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

   5. Applied rewrites 74.6%

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

   if 2.80000000000000004e-64 < t

   1. Initial program 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow3 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 76.5

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

   4. Applied rewrites 76.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 83.5

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 83.5

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

   6. Applied rewrites 83.5%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. frac-times N/A

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

      11. div-inv N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 83.6

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

   8. Applied rewrites 83.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 88.5

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

   10. Applied rewrites 88.5%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.8 \cdot 10^{-64}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\cos k} \cdot \frac{\frac{k \cdot k}{\ell} \cdot t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(k, \frac{1}{\frac{t}{k} \cdot t}, 2\right) \cdot \left(\left(\left(\tan k \cdot t\right) \cdot \frac{\sin k \cdot t}{\ell}\right) \cdot \frac{t}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.8 \cdot 10^{-64}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{k \cdot k}{\ell}}{\ell} \cdot t\_m\right) \cdot \frac{{\sin k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(k, \frac{1}{\frac{t\_m}{k} \cdot t\_m}, 2\right) \cdot \left(\left(\left(\tan k \cdot t\_m\right) \cdot \frac{\sin k \cdot t\_m}{\ell}\right) \cdot \frac{t\_m}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.8e-64)
    (/ 2.0 (* (* (/ (/ (* k k) l) l) t_m) (/ (pow (sin k) 2.0) (cos k))))
    (/
     2.0
     (*
      (fma k (/ 1.0 (* (/ t_m k) t_m)) 2.0)
      (* (* (* (tan k) t_m) (/ (* (sin k) t_m) l)) (/ t_m l)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.8e-64) {
		tmp = 2.0 / (((((k * k) / l) / l) * t_m) * (pow(sin(k), 2.0) / cos(k)));
	} else {
		tmp = 2.0 / (fma(k, (1.0 / ((t_m / k) * t_m)), 2.0) * (((tan(k) * t_m) * ((sin(k) * t_m) / l)) * (t_m / l)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.8e-64)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k * k) / l) / l) * t_m) * Float64((sin(k) ^ 2.0) / cos(k))));
	else
		tmp = Float64(2.0 / Float64(fma(k, Float64(1.0 / Float64(Float64(t_m / k) * t_m)), 2.0) * Float64(Float64(Float64(tan(k) * t_m) * Float64(Float64(sin(k) * t_m) / l)) * Float64(t_m / l))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.8e-64], N[(2.0 / N[(N[(N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * N[(1.0 / N[(N[(t$95$m / k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(N[(N[Tan[k], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.80000000000000004e-64

   1. Initial program 50.5%

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

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 54.1

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

   5. Applied rewrites 54.1%

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

   7. Applied rewrites 55.6%

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

   8. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. times-frac N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

   10. Applied rewrites 73.6%

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

   if 2.80000000000000004e-64 < t

   1. Initial program 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow3 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 76.5

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

   4. Applied rewrites 76.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 83.5

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 83.5

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

   6. Applied rewrites 83.5%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. frac-times N/A

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

      11. div-inv N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 83.6

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

   8. Applied rewrites 83.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 88.5

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

   10. Applied rewrites 88.5%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.8 \cdot 10^{-64}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{k \cdot k}{\ell}}{\ell} \cdot t\right) \cdot \frac{{\sin k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(k, \frac{1}{\frac{t}{k} \cdot t}, 2\right) \cdot \left(\left(\left(\tan k \cdot t\right) \cdot \frac{\sin k \cdot t}{\ell}\right) \cdot \frac{t}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.1 \cdot 10^{-144}:\\ \;\;\;\;\frac{\cos k}{\left(\left({\sin k}^{2} \cdot t\_m\right) \cdot k\right) \cdot k} \cdot \left(\left(\ell \cdot \ell\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(k, \frac{1}{\frac{t\_m}{k} \cdot t\_m}, 2\right) \cdot \left(\left(\left(\tan k \cdot t\_m\right) \cdot \frac{\sin k \cdot t\_m}{\ell}\right) \cdot \frac{t\_m}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.1e-144)
    (* (/ (cos k) (* (* (* (pow (sin k) 2.0) t_m) k) k)) (* (* l l) 2.0))
    (/
     2.0
     (*
      (fma k (/ 1.0 (* (/ t_m k) t_m)) 2.0)
      (* (* (* (tan k) t_m) (/ (* (sin k) t_m) l)) (/ t_m l)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.1e-144) {
		tmp = (cos(k) / (((pow(sin(k), 2.0) * t_m) * k) * k)) * ((l * l) * 2.0);
	} else {
		tmp = 2.0 / (fma(k, (1.0 / ((t_m / k) * t_m)), 2.0) * (((tan(k) * t_m) * ((sin(k) * t_m) / l)) * (t_m / l)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 3.1e-144)
		tmp = Float64(Float64(cos(k) / Float64(Float64(Float64((sin(k) ^ 2.0) * t_m) * k) * k)) * Float64(Float64(l * l) * 2.0));
	else
		tmp = Float64(2.0 / Float64(fma(k, Float64(1.0 / Float64(Float64(t_m / k) * t_m)), 2.0) * Float64(Float64(Float64(tan(k) * t_m) * Float64(Float64(sin(k) * t_m) / l)) * Float64(t_m / l))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 3.1e-144], N[(N[(N[Cos[k], $MachinePrecision] / N[(N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * N[(1.0 / N[(N[(t$95$m / k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(N[(N[Tan[k], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.1000000000000001e-144

   1. Initial program 51.3%

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

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 53.8

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

   5. Applied rewrites 53.8%

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

   6. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 71.2

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

   8. Applied rewrites 71.2%

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

   if 3.1000000000000001e-144 < t

   1. Initial program 63.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow3 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 75.7

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

   4. Applied rewrites 75.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 81.7

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 81.7

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

   6. Applied rewrites 81.7%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. frac-times N/A

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

      11. div-inv N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 81.7

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

   8. Applied rewrites 81.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 85.8

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

   10. Applied rewrites 85.8%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.1 \cdot 10^{-144}:\\ \;\;\;\;\frac{\cos k}{\left(\left({\sin k}^{2} \cdot t\right) \cdot k\right) \cdot k} \cdot \left(\left(\ell \cdot \ell\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(k, \frac{1}{\frac{t}{k} \cdot t}, 2\right) \cdot \left(\left(\left(\tan k \cdot t\right) \cdot \frac{\sin k \cdot t}{\ell}\right) \cdot \frac{t}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.7 \cdot 10^{-115}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{2}{\ell}, \frac{{t\_m}^{3}}{\ell}, \left(\mathsf{fma}\left(0.3333333333333333, t\_m \cdot t\_m, 1\right) \cdot t\_m\right) \cdot \frac{\frac{k \cdot k}{\ell}}{\ell}\right) \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(k, \frac{1}{\frac{t\_m}{k} \cdot t\_m}, 2\right) \cdot \left(\left(\left(\tan k \cdot t\_m\right) \cdot \frac{\sin k \cdot t\_m}{\ell}\right) \cdot \frac{t\_m}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.7e-115)
    (/
     2.0
     (*
      (fma
       (/ 2.0 l)
       (/ (pow t_m 3.0) l)
       (*
        (* (fma 0.3333333333333333 (* t_m t_m) 1.0) t_m)
        (/ (/ (* k k) l) l)))
      (* k k)))
    (/
     2.0
     (*
      (fma k (/ 1.0 (* (/ t_m k) t_m)) 2.0)
      (* (* (* (tan k) t_m) (/ (* (sin k) t_m) l)) (/ t_m l)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.7e-115) {
		tmp = 2.0 / (fma((2.0 / l), (pow(t_m, 3.0) / l), ((fma(0.3333333333333333, (t_m * t_m), 1.0) * t_m) * (((k * k) / l) / l))) * (k * k));
	} else {
		tmp = 2.0 / (fma(k, (1.0 / ((t_m / k) * t_m)), 2.0) * (((tan(k) * t_m) * ((sin(k) * t_m) / l)) * (t_m / l)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.7e-115)
		tmp = Float64(2.0 / Float64(fma(Float64(2.0 / l), Float64((t_m ^ 3.0) / l), Float64(Float64(fma(0.3333333333333333, Float64(t_m * t_m), 1.0) * t_m) * Float64(Float64(Float64(k * k) / l) / l))) * Float64(k * k)));
	else
		tmp = Float64(2.0 / Float64(fma(k, Float64(1.0 / Float64(Float64(t_m / k) * t_m)), 2.0) * Float64(Float64(Float64(tan(k) * t_m) * Float64(Float64(sin(k) * t_m) / l)) * Float64(t_m / l))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.7e-115], N[(2.0 / N[(N[(N[(2.0 / l), $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] + N[(N[(N[(0.3333333333333333 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * N[(1.0 / N[(N[(t$95$m / k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(N[(N[Tan[k], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.6999999999999999e-115

   1. Initial program 50.7%

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

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 54.3

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

   5. Applied rewrites 54.3%

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

   7. Applied rewrites 56.1%

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

   8. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 66.9%

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

   if 1.6999999999999999e-115 < t

   1. Initial program 65.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow3 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 74.6

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

   4. Applied rewrites 74.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 80.8

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 80.8

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

   6. Applied rewrites 80.8%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. frac-times N/A

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

      11. div-inv N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 80.9

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

   8. Applied rewrites 80.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 85.2

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

   10. Applied rewrites 85.2%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.7 \cdot 10^{-115}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{2}{\ell}, \frac{{t}^{3}}{\ell}, \left(\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot t\right) \cdot \frac{\frac{k \cdot k}{\ell}}{\ell}\right) \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(k, \frac{1}{\frac{t}{k} \cdot t}, 2\right) \cdot \left(\left(\left(\tan k \cdot t\right) \cdot \frac{\sin k \cdot t}{\ell}\right) \cdot \frac{t}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.7 \cdot 10^{-115}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{2}{\ell}, \frac{{t\_m}^{3}}{\ell}, \left(\mathsf{fma}\left(0.3333333333333333, t\_m \cdot t\_m, 1\right) \cdot t\_m\right) \cdot \frac{\frac{k \cdot k}{\ell}}{\ell}\right) \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\tan k \cdot t\_m\right) \cdot \left(\frac{\sin k \cdot t\_m}{\ell} \cdot \frac{t\_m}{\ell}\right)\right) \cdot \mathsf{fma}\left(k, \frac{1}{\frac{t\_m}{k} \cdot t\_m}, 2\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.7e-115)
    (/
     2.0
     (*
      (fma
       (/ 2.0 l)
       (/ (pow t_m 3.0) l)
       (*
        (* (fma 0.3333333333333333 (* t_m t_m) 1.0) t_m)
        (/ (/ (* k k) l) l)))
      (* k k)))
    (/
     2.0
     (*
      (* (* (tan k) t_m) (* (/ (* (sin k) t_m) l) (/ t_m l)))
      (fma k (/ 1.0 (* (/ t_m k) t_m)) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.7e-115) {
		tmp = 2.0 / (fma((2.0 / l), (pow(t_m, 3.0) / l), ((fma(0.3333333333333333, (t_m * t_m), 1.0) * t_m) * (((k * k) / l) / l))) * (k * k));
	} else {
		tmp = 2.0 / (((tan(k) * t_m) * (((sin(k) * t_m) / l) * (t_m / l))) * fma(k, (1.0 / ((t_m / k) * t_m)), 2.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.7e-115)
		tmp = Float64(2.0 / Float64(fma(Float64(2.0 / l), Float64((t_m ^ 3.0) / l), Float64(Float64(fma(0.3333333333333333, Float64(t_m * t_m), 1.0) * t_m) * Float64(Float64(Float64(k * k) / l) / l))) * Float64(k * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(tan(k) * t_m) * Float64(Float64(Float64(sin(k) * t_m) / l) * Float64(t_m / l))) * fma(k, Float64(1.0 / Float64(Float64(t_m / k) * t_m)), 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[t$95$m, 1.7e-115], N[(2.0 / N[(N[(N[(2.0 / l), $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] + N[(N[(N[(0.3333333333333333 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Tan[k], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * N[(1.0 / N[(N[(t$95$m / k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.6999999999999999e-115

   1. Initial program 50.7%

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

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 54.3

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

   5. Applied rewrites 54.3%

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

   7. Applied rewrites 56.1%

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

   8. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 66.9%

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

   if 1.6999999999999999e-115 < t

   1. Initial program 65.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow3 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 74.6

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

   4. Applied rewrites 74.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 80.8

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 80.8

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

   6. Applied rewrites 80.8%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. frac-times N/A

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

      11. div-inv N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 80.9

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

   8. Applied rewrites 80.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 79.9

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

   10. Applied rewrites 79.9%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.7 \cdot 10^{-115}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{2}{\ell}, \frac{{t}^{3}}{\ell}, \left(\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot t\right) \cdot \frac{\frac{k \cdot k}{\ell}}{\ell}\right) \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\tan k \cdot t\right) \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \mathsf{fma}\left(k, \frac{1}{\frac{t}{k} \cdot t}, 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 79.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.7 \cdot 10^{-115}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{2}{\ell}, \frac{{t\_m}^{3}}{\ell}, \left(\mathsf{fma}\left(0.3333333333333333, t\_m \cdot t\_m, 1\right) \cdot t\_m\right) \cdot \frac{\frac{k \cdot k}{\ell}}{\ell}\right) \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{\sin k \cdot t\_m}{\ell} \cdot \tan k\right) \cdot \frac{t\_m}{\ell}\right) \cdot t\_m\right) \cdot \mathsf{fma}\left(k, \frac{1}{\frac{t\_m}{k} \cdot t\_m}, 2\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.7e-115)
    (/
     2.0
     (*
      (fma
       (/ 2.0 l)
       (/ (pow t_m 3.0) l)
       (*
        (* (fma 0.3333333333333333 (* t_m t_m) 1.0) t_m)
        (/ (/ (* k k) l) l)))
      (* k k)))
    (/
     2.0
     (*
      (* (* (* (/ (* (sin k) t_m) l) (tan k)) (/ t_m l)) t_m)
      (fma k (/ 1.0 (* (/ t_m k) t_m)) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.7e-115) {
		tmp = 2.0 / (fma((2.0 / l), (pow(t_m, 3.0) / l), ((fma(0.3333333333333333, (t_m * t_m), 1.0) * t_m) * (((k * k) / l) / l))) * (k * k));
	} else {
		tmp = 2.0 / ((((((sin(k) * t_m) / l) * tan(k)) * (t_m / l)) * t_m) * fma(k, (1.0 / ((t_m / k) * t_m)), 2.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.7e-115)
		tmp = Float64(2.0 / Float64(fma(Float64(2.0 / l), Float64((t_m ^ 3.0) / l), Float64(Float64(fma(0.3333333333333333, Float64(t_m * t_m), 1.0) * t_m) * Float64(Float64(Float64(k * k) / l) / l))) * Float64(k * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(sin(k) * t_m) / l) * tan(k)) * Float64(t_m / l)) * t_m) * fma(k, Float64(1.0 / Float64(Float64(t_m / k) * t_m)), 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[t$95$m, 1.7e-115], N[(2.0 / N[(N[(N[(2.0 / l), $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] + N[(N[(N[(0.3333333333333333 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(k * N[(1.0 / N[(N[(t$95$m / k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.6999999999999999e-115

   1. Initial program 50.7%

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

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 54.3

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

   5. Applied rewrites 54.3%

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

   7. Applied rewrites 56.1%

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

   8. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 66.9%

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

   if 1.6999999999999999e-115 < t

   1. Initial program 65.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow3 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 74.6

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

   4. Applied rewrites 74.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 80.8

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 80.8

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

   6. Applied rewrites 80.8%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. frac-times N/A

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

      11. div-inv N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 80.9

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

   8. Applied rewrites 80.9%

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

      1. lift-*.f64 N/A

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

      2. *-rgt-identity 80.9

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

   10. Applied rewrites 80.9%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.7 \cdot 10^{-115}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{2}{\ell}, \frac{{t}^{3}}{\ell}, \left(\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot t\right) \cdot \frac{\frac{k \cdot k}{\ell}}{\ell}\right) \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right) \cdot \frac{t}{\ell}\right) \cdot t\right) \cdot \mathsf{fma}\left(k, \frac{1}{\frac{t}{k} \cdot t}, 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 71.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 1.85 \cdot 10^{-20}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\left(\left(\mathsf{fma}\left(k \cdot k, -0.16666666666666666, 1\right) \cdot \frac{t\_m}{\ell}\right) \cdot k\right) \cdot \tan k\right) \cdot \frac{t\_m}{\ell}\right) \cdot t\_m\right) \cdot \mathsf{fma}\left(k, \frac{1}{\frac{t\_m}{k} \cdot t\_m}, 2\right)}\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{+168}:\\ \;\;\;\;\frac{2}{\left(\left(\left(k \cdot t\_m\right) \cdot \left(k \cdot t\_m\right)\right) \cdot 2\right) \cdot \frac{\frac{t\_m}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\left(\left(\left(\frac{t\_m}{\ell} \cdot \frac{t\_m}{\ell}\right) \cdot t\_m\right) \cdot \sin k\right) \cdot \tan k\right)}\\ \end{array} \end{array} \]

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 1.85e-20)
    (/
     2.0
     (*
      (*
       (*
        (* (* (* (fma (* k k) -0.16666666666666666 1.0) (/ t_m l)) k) (tan k))
        (/ t_m l))
       t_m)
      (fma k (/ 1.0 (* (/ t_m k) t_m)) 2.0)))
    (if (<= l 8e+168)
      (/ 2.0 (* (* (* (* k t_m) (* k t_m)) 2.0) (/ (/ t_m l) l)))
      (/
       2.0
       (* 2.0 (* (* (* (* (/ t_m l) (/ t_m l)) t_m) (sin k)) (tan k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 1.85e-20) {
		tmp = 2.0 / ((((((fma((k * k), -0.16666666666666666, 1.0) * (t_m / l)) * k) * tan(k)) * (t_m / l)) * t_m) * fma(k, (1.0 / ((t_m / k) * t_m)), 2.0));
	} else if (l <= 8e+168) {
		tmp = 2.0 / ((((k * t_m) * (k * t_m)) * 2.0) * ((t_m / l) / l));
	} else {
		tmp = 2.0 / (2.0 * (((((t_m / l) * (t_m / l)) * t_m) * sin(k)) * tan(k)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (l <= 1.85e-20)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(fma(Float64(k * k), -0.16666666666666666, 1.0) * Float64(t_m / l)) * k) * tan(k)) * Float64(t_m / l)) * t_m) * fma(k, Float64(1.0 / Float64(Float64(t_m / k) * t_m)), 2.0)));
	elseif (l <= 8e+168)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * t_m) * Float64(k * t_m)) * 2.0) * Float64(Float64(t_m / l) / l)));
	else
		tmp = Float64(2.0 / Float64(2.0 * Float64(Float64(Float64(Float64(Float64(t_m / l) * Float64(t_m / l)) * t_m) * sin(k)) * tan(k))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[l, 1.85e-20], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(N[(k * k), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(k * N[(1.0 / N[(N[(t$95$m / k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 8e+168], N[(2.0 / N[(N[(N[(N[(k * t$95$m), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(2.0 * N[(N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if l < 1.85e-20

   1. Initial program 59.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow3 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 71.0

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

   4. Applied rewrites 71.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 80.4

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 80.4

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

   6. Applied rewrites 80.4%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. frac-times N/A

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

      11. div-inv N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 79.9

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

   8. Applied rewrites 79.9%

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

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\color{blue}{\left(k \cdot \left(\frac{-1}{6} \cdot \frac{{k}^{2} \cdot t}{\ell} + \frac{t}{\ell}\right)\right)} \cdot \tan k\right)\right)\right) \cdot \mathsf{fma}\left(k \cdot 1, \frac{1}{t \cdot \frac{t}{k}}, 2\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\color{blue}{\left(\left(\frac{-1}{6} \cdot \frac{{k}^{2} \cdot t}{\ell} + \frac{t}{\ell}\right) \cdot k\right)} \cdot \tan k\right)\right)\right) \cdot \mathsf{fma}\left(k \cdot 1, \frac{1}{t \cdot \frac{t}{k}}, 2\right)} \]

       2. +-commutative N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\left(\color{blue}{\left(\frac{t}{\ell} + \frac{-1}{6} \cdot \frac{{k}^{2} \cdot t}{\ell}\right)} \cdot k\right) \cdot \tan k\right)\right)\right) \cdot \mathsf{fma}\left(k \cdot 1, \frac{1}{t \cdot \frac{t}{k}}, 2\right)} \]

       3. *-commutative N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} + \color{blue}{\frac{{k}^{2} \cdot t}{\ell} \cdot \frac{-1}{6}}\right) \cdot k\right) \cdot \tan k\right)\right)\right) \cdot \mathsf{fma}\left(k \cdot 1, \frac{1}{t \cdot \frac{t}{k}}, 2\right)} \]

       4. associate-/l* N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} + \color{blue}{\left({k}^{2} \cdot \frac{t}{\ell}\right)} \cdot \frac{-1}{6}\right) \cdot k\right) \cdot \tan k\right)\right)\right) \cdot \mathsf{fma}\left(k \cdot 1, \frac{1}{t \cdot \frac{t}{k}}, 2\right)} \]

       5. associate-*r* N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} + \color{blue}{{k}^{2} \cdot \left(\frac{t}{\ell} \cdot \frac{-1}{6}\right)}\right) \cdot k\right) \cdot \tan k\right)\right)\right) \cdot \mathsf{fma}\left(k \cdot 1, \frac{1}{t \cdot \frac{t}{k}}, 2\right)} \]

       6. *-commutative N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} + {k}^{2} \cdot \color{blue}{\left(\frac{-1}{6} \cdot \frac{t}{\ell}\right)}\right) \cdot k\right) \cdot \tan k\right)\right)\right) \cdot \mathsf{fma}\left(k \cdot 1, \frac{1}{t \cdot \frac{t}{k}}, 2\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\color{blue}{\left(\left(\frac{t}{\ell} + {k}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{t}{\ell}\right)\right) \cdot k\right)} \cdot \tan k\right)\right)\right) \cdot \mathsf{fma}\left(k \cdot 1, \frac{1}{t \cdot \frac{t}{k}}, 2\right)} \]

       8. associate-*r* N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} + \color{blue}{\left({k}^{2} \cdot \frac{-1}{6}\right) \cdot \frac{t}{\ell}}\right) \cdot k\right) \cdot \tan k\right)\right)\right) \cdot \mathsf{fma}\left(k \cdot 1, \frac{1}{t \cdot \frac{t}{k}}, 2\right)} \]

       9. distribute-rgt1-in N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\left(\color{blue}{\left(\left({k}^{2} \cdot \frac{-1}{6} + 1\right) \cdot \frac{t}{\ell}\right)} \cdot k\right) \cdot \tan k\right)\right)\right) \cdot \mathsf{fma}\left(k \cdot 1, \frac{1}{t \cdot \frac{t}{k}}, 2\right)} \]

       10. lower-*.f64 N/A

           \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\left(\color{blue}{\left(\left({k}^{2} \cdot \frac{-1}{6} + 1\right) \cdot \frac{t}{\ell}\right)} \cdot k\right) \cdot \tan k\right)\right)\right) \cdot \mathsf{fma}\left(k \cdot 1, \frac{1}{t \cdot \frac{t}{k}}, 2\right)} \]

       11. lower-fma.f64 N/A

           \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\left(\left(\color{blue}{\mathsf{fma}\left({k}^{2}, \frac{-1}{6}, 1\right)} \cdot \frac{t}{\ell}\right) \cdot k\right) \cdot \tan k\right)\right)\right) \cdot \mathsf{fma}\left(k \cdot 1, \frac{1}{t \cdot \frac{t}{k}}, 2\right)} \]

       12. unpow2 N/A

           \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\left(\left(\mathsf{fma}\left(\color{blue}{k \cdot k}, \frac{-1}{6}, 1\right) \cdot \frac{t}{\ell}\right) \cdot k\right) \cdot \tan k\right)\right)\right) \cdot \mathsf{fma}\left(k \cdot 1, \frac{1}{t \cdot \frac{t}{k}}, 2\right)} \]

       13. lower-*.f64 N/A

           \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\left(\left(\mathsf{fma}\left(\color{blue}{k \cdot k}, \frac{-1}{6}, 1\right) \cdot \frac{t}{\ell}\right) \cdot k\right) \cdot \tan k\right)\right)\right) \cdot \mathsf{fma}\left(k \cdot 1, \frac{1}{t \cdot \frac{t}{k}}, 2\right)} \]

       14. lower-/.f64 79.3

           \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\left(\left(\mathsf{fma}\left(k \cdot k, -0.16666666666666666, 1\right) \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot k\right) \cdot \tan k\right)\right)\right) \cdot \mathsf{fma}\left(k \cdot 1, \frac{1}{t \cdot \frac{t}{k}}, 2\right)} \]

   11. Applied rewrites 79.3%

       \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\color{blue}{\left(\left(\mathsf{fma}\left(k \cdot k, -0.16666666666666666, 1\right) \cdot \frac{t}{\ell}\right) \cdot k\right)} \cdot \tan k\right)\right)\right) \cdot \mathsf{fma}\left(k \cdot 1, \frac{1}{t \cdot \frac{t}{k}}, 2\right)} \]

   if 1.85e-20 < l < 7.9999999999999995e168

   1. Initial program 42.0%

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

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 49.2

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

   5. Applied rewrites 49.2%

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

   7. Applied rewrites 51.1%

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

   9. Applied rewrites 65.6%

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

   11. Applied rewrites 65.6%

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

   if 7.9999999999999995e168 < l

   1. Initial program 50.0%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 29.6

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

   4. Applied rewrites 29.6%

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

      1. lift-exp.f64 N/A

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

      2. lift--.f64 N/A

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

      3. exp-diff N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. pow-to-exp N/A

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

      7. cube-mult N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. pow-to-exp N/A

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

      12. pow2 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. times-frac N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-/.f64 61.3

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

   6. Applied rewrites 61.3%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   7. Taylor expanded in t around inf

      \[\leadsto \frac{2}{\left(\left(\left(t \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{2}} \]
   8. Step-by-step derivation

   9. Applied rewrites 75.3%

      \[\leadsto \frac{2}{\left(\left(\left(t \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.85 \cdot 10^{-20}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\left(\left(\mathsf{fma}\left(k \cdot k, -0.16666666666666666, 1\right) \cdot \frac{t}{\ell}\right) \cdot k\right) \cdot \tan k\right) \cdot \frac{t}{\ell}\right) \cdot t\right) \cdot \mathsf{fma}\left(k, \frac{1}{\frac{t}{k} \cdot t}, 2\right)}\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{+168}:\\ \;\;\;\;\frac{2}{\left(\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot 2\right) \cdot \frac{\frac{t}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\left(\left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot t\right) \cdot \sin k\right) \cdot \tan k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 73.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.62 \cdot 10^{-115}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{2}{\ell}, \frac{{t\_m}^{3}}{\ell}, \left(\mathsf{fma}\left(0.3333333333333333, t\_m \cdot t\_m, 1\right) \cdot t\_m\right) \cdot \frac{\frac{k \cdot k}{\ell}}{\ell}\right) \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\left(\left(\mathsf{fma}\left(k \cdot k, -0.16666666666666666, 1\right) \cdot \frac{t\_m}{\ell}\right) \cdot k\right) \cdot \tan k\right) \cdot \frac{t\_m}{\ell}\right) \cdot t\_m\right) \cdot \mathsf{fma}\left(k, \frac{1}{\frac{t\_m}{k} \cdot t\_m}, 2\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.62e-115)
    (/
     2.0
     (*
      (fma
       (/ 2.0 l)
       (/ (pow t_m 3.0) l)
       (*
        (* (fma 0.3333333333333333 (* t_m t_m) 1.0) t_m)
        (/ (/ (* k k) l) l)))
      (* k k)))
    (/
     2.0
     (*
      (*
       (*
        (* (* (* (fma (* k k) -0.16666666666666666 1.0) (/ t_m l)) k) (tan k))
        (/ t_m l))
       t_m)
      (fma k (/ 1.0 (* (/ t_m k) t_m)) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.62e-115) {
		tmp = 2.0 / (fma((2.0 / l), (pow(t_m, 3.0) / l), ((fma(0.3333333333333333, (t_m * t_m), 1.0) * t_m) * (((k * k) / l) / l))) * (k * k));
	} else {
		tmp = 2.0 / ((((((fma((k * k), -0.16666666666666666, 1.0) * (t_m / l)) * k) * tan(k)) * (t_m / l)) * t_m) * fma(k, (1.0 / ((t_m / k) * t_m)), 2.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.62e-115)
		tmp = Float64(2.0 / Float64(fma(Float64(2.0 / l), Float64((t_m ^ 3.0) / l), Float64(Float64(fma(0.3333333333333333, Float64(t_m * t_m), 1.0) * t_m) * Float64(Float64(Float64(k * k) / l) / l))) * Float64(k * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(fma(Float64(k * k), -0.16666666666666666, 1.0) * Float64(t_m / l)) * k) * tan(k)) * Float64(t_m / l)) * t_m) * fma(k, Float64(1.0 / Float64(Float64(t_m / k) * t_m)), 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[t$95$m, 1.62e-115], N[(2.0 / N[(N[(N[(2.0 / l), $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] + N[(N[(N[(0.3333333333333333 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(N[(k * k), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(k * N[(1.0 / N[(N[(t$95$m / k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.62e-115

   1. Initial program 50.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{2}{2 \cdot \color{blue}{\left({k}^{2} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right)} \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

      8. associate-/r* N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\color{blue}{\frac{{t}^{3}}{\ell}}}{\ell}} \]

      11. lower-pow.f64 54.3

          \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell}} \]

   5. Applied rewrites 54.3%

      \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 56.1%

      \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\frac{t}{\ell \cdot \ell}}\right)} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot {k}^{2}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot {k}^{2}}} \]

   10. Applied rewrites 66.9%

       \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\frac{2}{\ell}, \frac{{t}^{3}}{\ell}, \left(t \cdot \mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right)\right) \cdot \frac{\frac{k \cdot k}{\ell}}{\ell}\right) \cdot \left(k \cdot k\right)}} \]

   if 1.62e-115 < t

   1. Initial program 65.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{3}} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. unpow3 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)}{\color{blue}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. times-frac N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{t \cdot t}{\ell}} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\frac{t \cdot \sin k}{\ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-*.f64 74.6

          \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{\color{blue}{t \cdot \sin k}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 74.6%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{t \cdot t}{\ell}} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(t \cdot \frac{t}{\ell}\right)} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\color{blue}{\frac{t}{\ell}} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-*.f64 80.8

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\color{blue}{t \cdot \sin k}}{\ell} \cdot \tan k\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. *-commutative N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\color{blue}{\sin k \cdot t}}{\ell} \cdot \tan k\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. lower-*.f64 80.8

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\color{blue}{\sin k \cdot t}}{\ell} \cdot \tan k\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied rewrites 80.8%

      \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right) \cdot \left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right) \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} + 1\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right) \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 + 1\right)\right)}} \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right) \cdot \left(\color{blue}{{\left(\frac{k}{t}\right)}^{2}} + \left(1 + 1\right)\right)} \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right) \cdot \left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + \left(1 + 1\right)\right)} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right) \cdot \left(\color{blue}{\frac{k}{t}} \cdot \frac{k}{t} + \left(1 + 1\right)\right)} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right) \cdot \left(\frac{k}{t} \cdot \color{blue}{\frac{k}{t}} + \left(1 + 1\right)\right)} \]

      9. clear-num N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right) \cdot \left(\frac{k}{t} \cdot \color{blue}{\frac{1}{\frac{t}{k}}} + \left(1 + 1\right)\right)} \]

      10. frac-times N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right) \cdot \left(\color{blue}{\frac{k \cdot 1}{t \cdot \frac{t}{k}}} + \left(1 + 1\right)\right)} \]

      11. div-inv N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right) \cdot \left(\color{blue}{\left(k \cdot 1\right) \cdot \frac{1}{t \cdot \frac{t}{k}}} + \left(1 + 1\right)\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right) \cdot \left(\left(k \cdot 1\right) \cdot \frac{1}{t \cdot \frac{t}{k}} + \color{blue}{2}\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(k \cdot 1, \frac{1}{t \cdot \frac{t}{k}}, 2\right)}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{k \cdot 1}, \frac{1}{t \cdot \frac{t}{k}}, 2\right)} \]

      15. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right) \cdot \mathsf{fma}\left(k \cdot 1, \color{blue}{\frac{1}{t \cdot \frac{t}{k}}}, 2\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right) \cdot \mathsf{fma}\left(k \cdot 1, \frac{1}{\color{blue}{t \cdot \frac{t}{k}}}, 2\right)} \]

      17. lower-/.f64 80.9

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right) \cdot \mathsf{fma}\left(k \cdot 1, \frac{1}{t \cdot \color{blue}{\frac{t}{k}}}, 2\right)} \]

   8. Applied rewrites 80.9%

      \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(k \cdot 1, \frac{1}{t \cdot \frac{t}{k}}, 2\right)}} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\color{blue}{\left(k \cdot \left(\frac{-1}{6} \cdot \frac{{k}^{2} \cdot t}{\ell} + \frac{t}{\ell}\right)\right)} \cdot \tan k\right)\right)\right) \cdot \mathsf{fma}\left(k \cdot 1, \frac{1}{t \cdot \frac{t}{k}}, 2\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\color{blue}{\left(\left(\frac{-1}{6} \cdot \frac{{k}^{2} \cdot t}{\ell} + \frac{t}{\ell}\right) \cdot k\right)} \cdot \tan k\right)\right)\right) \cdot \mathsf{fma}\left(k \cdot 1, \frac{1}{t \cdot \frac{t}{k}}, 2\right)} \]

       2. +-commutative N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\left(\color{blue}{\left(\frac{t}{\ell} + \frac{-1}{6} \cdot \frac{{k}^{2} \cdot t}{\ell}\right)} \cdot k\right) \cdot \tan k\right)\right)\right) \cdot \mathsf{fma}\left(k \cdot 1, \frac{1}{t \cdot \frac{t}{k}}, 2\right)} \]

       3. *-commutative N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} + \color{blue}{\frac{{k}^{2} \cdot t}{\ell} \cdot \frac{-1}{6}}\right) \cdot k\right) \cdot \tan k\right)\right)\right) \cdot \mathsf{fma}\left(k \cdot 1, \frac{1}{t \cdot \frac{t}{k}}, 2\right)} \]

       4. associate-/l* N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} + \color{blue}{\left({k}^{2} \cdot \frac{t}{\ell}\right)} \cdot \frac{-1}{6}\right) \cdot k\right) \cdot \tan k\right)\right)\right) \cdot \mathsf{fma}\left(k \cdot 1, \frac{1}{t \cdot \frac{t}{k}}, 2\right)} \]

       5. associate-*r* N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} + \color{blue}{{k}^{2} \cdot \left(\frac{t}{\ell} \cdot \frac{-1}{6}\right)}\right) \cdot k\right) \cdot \tan k\right)\right)\right) \cdot \mathsf{fma}\left(k \cdot 1, \frac{1}{t \cdot \frac{t}{k}}, 2\right)} \]

       6. *-commutative N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} + {k}^{2} \cdot \color{blue}{\left(\frac{-1}{6} \cdot \frac{t}{\ell}\right)}\right) \cdot k\right) \cdot \tan k\right)\right)\right) \cdot \mathsf{fma}\left(k \cdot 1, \frac{1}{t \cdot \frac{t}{k}}, 2\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\color{blue}{\left(\left(\frac{t}{\ell} + {k}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{t}{\ell}\right)\right) \cdot k\right)} \cdot \tan k\right)\right)\right) \cdot \mathsf{fma}\left(k \cdot 1, \frac{1}{t \cdot \frac{t}{k}}, 2\right)} \]

       8. associate-*r* N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} + \color{blue}{\left({k}^{2} \cdot \frac{-1}{6}\right) \cdot \frac{t}{\ell}}\right) \cdot k\right) \cdot \tan k\right)\right)\right) \cdot \mathsf{fma}\left(k \cdot 1, \frac{1}{t \cdot \frac{t}{k}}, 2\right)} \]

       9. distribute-rgt1-in N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\left(\color{blue}{\left(\left({k}^{2} \cdot \frac{-1}{6} + 1\right) \cdot \frac{t}{\ell}\right)} \cdot k\right) \cdot \tan k\right)\right)\right) \cdot \mathsf{fma}\left(k \cdot 1, \frac{1}{t \cdot \frac{t}{k}}, 2\right)} \]

       10. lower-*.f64 N/A

           \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\left(\color{blue}{\left(\left({k}^{2} \cdot \frac{-1}{6} + 1\right) \cdot \frac{t}{\ell}\right)} \cdot k\right) \cdot \tan k\right)\right)\right) \cdot \mathsf{fma}\left(k \cdot 1, \frac{1}{t \cdot \frac{t}{k}}, 2\right)} \]

       11. lower-fma.f64 N/A

           \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\left(\left(\color{blue}{\mathsf{fma}\left({k}^{2}, \frac{-1}{6}, 1\right)} \cdot \frac{t}{\ell}\right) \cdot k\right) \cdot \tan k\right)\right)\right) \cdot \mathsf{fma}\left(k \cdot 1, \frac{1}{t \cdot \frac{t}{k}}, 2\right)} \]

       12. unpow2 N/A

           \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\left(\left(\mathsf{fma}\left(\color{blue}{k \cdot k}, \frac{-1}{6}, 1\right) \cdot \frac{t}{\ell}\right) \cdot k\right) \cdot \tan k\right)\right)\right) \cdot \mathsf{fma}\left(k \cdot 1, \frac{1}{t \cdot \frac{t}{k}}, 2\right)} \]

       13. lower-*.f64 N/A

           \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\left(\left(\mathsf{fma}\left(\color{blue}{k \cdot k}, \frac{-1}{6}, 1\right) \cdot \frac{t}{\ell}\right) \cdot k\right) \cdot \tan k\right)\right)\right) \cdot \mathsf{fma}\left(k \cdot 1, \frac{1}{t \cdot \frac{t}{k}}, 2\right)} \]

       14. lower-/.f64 77.5

           \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\left(\left(\mathsf{fma}\left(k \cdot k, -0.16666666666666666, 1\right) \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot k\right) \cdot \tan k\right)\right)\right) \cdot \mathsf{fma}\left(k \cdot 1, \frac{1}{t \cdot \frac{t}{k}}, 2\right)} \]

   11. Applied rewrites 77.5%

       \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\color{blue}{\left(\left(\mathsf{fma}\left(k \cdot k, -0.16666666666666666, 1\right) \cdot \frac{t}{\ell}\right) \cdot k\right)} \cdot \tan k\right)\right)\right) \cdot \mathsf{fma}\left(k \cdot 1, \frac{1}{t \cdot \frac{t}{k}}, 2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.62 \cdot 10^{-115}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{2}{\ell}, \frac{{t}^{3}}{\ell}, \left(\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot t\right) \cdot \frac{\frac{k \cdot k}{\ell}}{\ell}\right) \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\left(\left(\mathsf{fma}\left(k \cdot k, -0.16666666666666666, 1\right) \cdot \frac{t}{\ell}\right) \cdot k\right) \cdot \tan k\right) \cdot \frac{t}{\ell}\right) \cdot t\right) \cdot \mathsf{fma}\left(k, \frac{1}{\frac{t}{k} \cdot t}, 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 73.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.62 \cdot 10^{-115}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{2}{\ell}, \frac{{t\_m}^{3}}{\ell}, \left(\mathsf{fma}\left(0.3333333333333333, t\_m \cdot t\_m, 1\right) \cdot t\_m\right) \cdot \frac{\frac{k \cdot k}{\ell}}{\ell}\right) \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\left(\frac{t\_m}{\ell} \cdot k\right) \cdot \tan k\right) \cdot \frac{t\_m}{\ell}\right) \cdot t\_m\right) \cdot \mathsf{fma}\left(k, \frac{1}{\frac{t\_m}{k} \cdot t\_m}, 2\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.62e-115)
    (/
     2.0
     (*
      (fma
       (/ 2.0 l)
       (/ (pow t_m 3.0) l)
       (*
        (* (fma 0.3333333333333333 (* t_m t_m) 1.0) t_m)
        (/ (/ (* k k) l) l)))
      (* k k)))
    (/
     2.0
     (*
      (* (* (* (* (/ t_m l) k) (tan k)) (/ t_m l)) t_m)
      (fma k (/ 1.0 (* (/ t_m k) t_m)) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.62e-115) {
		tmp = 2.0 / (fma((2.0 / l), (pow(t_m, 3.0) / l), ((fma(0.3333333333333333, (t_m * t_m), 1.0) * t_m) * (((k * k) / l) / l))) * (k * k));
	} else {
		tmp = 2.0 / ((((((t_m / l) * k) * tan(k)) * (t_m / l)) * t_m) * fma(k, (1.0 / ((t_m / k) * t_m)), 2.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.62e-115)
		tmp = Float64(2.0 / Float64(fma(Float64(2.0 / l), Float64((t_m ^ 3.0) / l), Float64(Float64(fma(0.3333333333333333, Float64(t_m * t_m), 1.0) * t_m) * Float64(Float64(Float64(k * k) / l) / l))) * Float64(k * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t_m / l) * k) * tan(k)) * Float64(t_m / l)) * t_m) * fma(k, Float64(1.0 / Float64(Float64(t_m / k) * t_m)), 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[t$95$m, 1.62e-115], N[(2.0 / N[(N[(N[(2.0 / l), $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] + N[(N[(N[(0.3333333333333333 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * k), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(k * N[(1.0 / N[(N[(t$95$m / k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.62e-115

   1. Initial program 50.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{2}{2 \cdot \color{blue}{\left({k}^{2} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right)} \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

      8. associate-/r* N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\color{blue}{\frac{{t}^{3}}{\ell}}}{\ell}} \]

      11. lower-pow.f64 54.3

          \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell}} \]

   5. Applied rewrites 54.3%

      \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 56.1%

      \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\frac{t}{\ell \cdot \ell}}\right)} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot {k}^{2}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right) \cdot {k}^{2}}} \]

   10. Applied rewrites 66.9%

       \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\frac{2}{\ell}, \frac{{t}^{3}}{\ell}, \left(t \cdot \mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right)\right) \cdot \frac{\frac{k \cdot k}{\ell}}{\ell}\right) \cdot \left(k \cdot k\right)}} \]

   if 1.62e-115 < t

   1. Initial program 65.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{3}} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. unpow3 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)}{\color{blue}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. times-frac N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{t \cdot t}{\ell}} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\frac{t \cdot \sin k}{\ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-*.f64 74.6

          \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{\color{blue}{t \cdot \sin k}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 74.6%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{t \cdot t}{\ell}} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(t \cdot \frac{t}{\ell}\right)} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\color{blue}{\frac{t}{\ell}} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-*.f64 80.8

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\color{blue}{t \cdot \sin k}}{\ell} \cdot \tan k\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. *-commutative N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\color{blue}{\sin k \cdot t}}{\ell} \cdot \tan k\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. lower-*.f64 80.8

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\color{blue}{\sin k \cdot t}}{\ell} \cdot \tan k\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied rewrites 80.8%

      \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right) \cdot \left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right) \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} + 1\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right) \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 + 1\right)\right)}} \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right) \cdot \left(\color{blue}{{\left(\frac{k}{t}\right)}^{2}} + \left(1 + 1\right)\right)} \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right) \cdot \left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + \left(1 + 1\right)\right)} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right) \cdot \left(\color{blue}{\frac{k}{t}} \cdot \frac{k}{t} + \left(1 + 1\right)\right)} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right) \cdot \left(\frac{k}{t} \cdot \color{blue}{\frac{k}{t}} + \left(1 + 1\right)\right)} \]

      9. clear-num N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right) \cdot \left(\frac{k}{t} \cdot \color{blue}{\frac{1}{\frac{t}{k}}} + \left(1 + 1\right)\right)} \]

      10. frac-times N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right) \cdot \left(\color{blue}{\frac{k \cdot 1}{t \cdot \frac{t}{k}}} + \left(1 + 1\right)\right)} \]

      11. div-inv N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right) \cdot \left(\color{blue}{\left(k \cdot 1\right) \cdot \frac{1}{t \cdot \frac{t}{k}}} + \left(1 + 1\right)\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right) \cdot \left(\left(k \cdot 1\right) \cdot \frac{1}{t \cdot \frac{t}{k}} + \color{blue}{2}\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(k \cdot 1, \frac{1}{t \cdot \frac{t}{k}}, 2\right)}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{k \cdot 1}, \frac{1}{t \cdot \frac{t}{k}}, 2\right)} \]

      15. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right) \cdot \mathsf{fma}\left(k \cdot 1, \color{blue}{\frac{1}{t \cdot \frac{t}{k}}}, 2\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right) \cdot \mathsf{fma}\left(k \cdot 1, \frac{1}{\color{blue}{t \cdot \frac{t}{k}}}, 2\right)} \]

      17. lower-/.f64 80.9

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right) \cdot \mathsf{fma}\left(k \cdot 1, \frac{1}{t \cdot \color{blue}{\frac{t}{k}}}, 2\right)} \]

   8. Applied rewrites 80.9%

      \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(k \cdot 1, \frac{1}{t \cdot \frac{t}{k}}, 2\right)}} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\color{blue}{\frac{k \cdot t}{\ell}} \cdot \tan k\right)\right)\right) \cdot \mathsf{fma}\left(k \cdot 1, \frac{1}{t \cdot \frac{t}{k}}, 2\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\color{blue}{t \cdot k}}{\ell} \cdot \tan k\right)\right)\right) \cdot \mathsf{fma}\left(k \cdot 1, \frac{1}{t \cdot \frac{t}{k}}, 2\right)} \]

       2. associate-*l/ N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\color{blue}{\left(\frac{t}{\ell} \cdot k\right)} \cdot \tan k\right)\right)\right) \cdot \mathsf{fma}\left(k \cdot 1, \frac{1}{t \cdot \frac{t}{k}}, 2\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\color{blue}{\left(\frac{t}{\ell} \cdot k\right)} \cdot \tan k\right)\right)\right) \cdot \mathsf{fma}\left(k \cdot 1, \frac{1}{t \cdot \frac{t}{k}}, 2\right)} \]

       4. lower-/.f64 78.6

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\left(\color{blue}{\frac{t}{\ell}} \cdot k\right) \cdot \tan k\right)\right)\right) \cdot \mathsf{fma}\left(k \cdot 1, \frac{1}{t \cdot \frac{t}{k}}, 2\right)} \]

   11. Applied rewrites 78.6%

       \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\color{blue}{\left(\frac{t}{\ell} \cdot k\right)} \cdot \tan k\right)\right)\right) \cdot \mathsf{fma}\left(k \cdot 1, \frac{1}{t \cdot \frac{t}{k}}, 2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.62 \cdot 10^{-115}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{2}{\ell}, \frac{{t}^{3}}{\ell}, \left(\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot t\right) \cdot \frac{\frac{k \cdot k}{\ell}}{\ell}\right) \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\left(\frac{t}{\ell} \cdot k\right) \cdot \tan k\right) \cdot \frac{t}{\ell}\right) \cdot t\right) \cdot \mathsf{fma}\left(k, \frac{1}{\frac{t}{k} \cdot t}, 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 69.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m}{\ell} \cdot k\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 2.55 \cdot 10^{-182}:\\ \;\;\;\;\frac{2}{\left(\left(\left(t\_2 \cdot \tan k\right) \cdot \frac{t\_m}{\ell}\right) \cdot t\_m\right) \cdot \mathsf{fma}\left(k, \frac{1}{\frac{t\_m}{k} \cdot t\_m}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(t\_2 \cdot \left(k \cdot 2\right)\right) \cdot t\_m\right) \cdot \frac{t\_m}{\ell}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (/ t_m l) k)))
   (*
    t_s
    (if (<= l 2.55e-182)
      (/
       2.0
       (*
        (* (* (* t_2 (tan k)) (/ t_m l)) t_m)
        (fma k (/ 1.0 (* (/ t_m k) t_m)) 2.0)))
      (/ 2.0 (* (* (* t_2 (* k 2.0)) t_m) (/ t_m l)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = (t_m / l) * k;
	double tmp;
	if (l <= 2.55e-182) {
		tmp = 2.0 / ((((t_2 * tan(k)) * (t_m / l)) * t_m) * fma(k, (1.0 / ((t_m / k) * t_m)), 2.0));
	} else {
		tmp = 2.0 / (((t_2 * (k * 2.0)) * t_m) * (t_m / l));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(Float64(t_m / l) * k)
	tmp = 0.0
	if (l <= 2.55e-182)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_2 * tan(k)) * Float64(t_m / l)) * t_m) * fma(k, Float64(1.0 / Float64(Float64(t_m / k) * t_m)), 2.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_2 * Float64(k * 2.0)) * t_m) * Float64(t_m / l)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[(t$95$m / l), $MachinePrecision] * k), $MachinePrecision]}, N[(t$95$s * If[LessEqual[l, 2.55e-182], N[(2.0 / N[(N[(N[(N[(t$95$2 * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(k * N[(1.0 / N[(N[(t$95$m / k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$2 * N[(k * 2.0), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.55000000000000009e-182

   1. Initial program 56.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{3}} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. unpow3 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)}{\color{blue}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. times-frac N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{t \cdot t}{\ell}} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\frac{t \cdot \sin k}{\ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-*.f64 70.5

          \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{\color{blue}{t \cdot \sin k}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 70.5%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{t \cdot t}{\ell}} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(t \cdot \frac{t}{\ell}\right)} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\color{blue}{\frac{t}{\ell}} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-*.f64 80.4

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\color{blue}{t \cdot \sin k}}{\ell} \cdot \tan k\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. *-commutative N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\color{blue}{\sin k \cdot t}}{\ell} \cdot \tan k\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. lower-*.f64 80.4

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\color{blue}{\sin k \cdot t}}{\ell} \cdot \tan k\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied rewrites 80.4%

      \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right) \cdot \left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right) \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} + 1\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right) \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 + 1\right)\right)}} \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right) \cdot \left(\color{blue}{{\left(\frac{k}{t}\right)}^{2}} + \left(1 + 1\right)\right)} \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right) \cdot \left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + \left(1 + 1\right)\right)} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right) \cdot \left(\color{blue}{\frac{k}{t}} \cdot \frac{k}{t} + \left(1 + 1\right)\right)} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right) \cdot \left(\frac{k}{t} \cdot \color{blue}{\frac{k}{t}} + \left(1 + 1\right)\right)} \]

      9. clear-num N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right) \cdot \left(\frac{k}{t} \cdot \color{blue}{\frac{1}{\frac{t}{k}}} + \left(1 + 1\right)\right)} \]

      10. frac-times N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right) \cdot \left(\color{blue}{\frac{k \cdot 1}{t \cdot \frac{t}{k}}} + \left(1 + 1\right)\right)} \]

      11. div-inv N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right) \cdot \left(\color{blue}{\left(k \cdot 1\right) \cdot \frac{1}{t \cdot \frac{t}{k}}} + \left(1 + 1\right)\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right) \cdot \left(\left(k \cdot 1\right) \cdot \frac{1}{t \cdot \frac{t}{k}} + \color{blue}{2}\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(k \cdot 1, \frac{1}{t \cdot \frac{t}{k}}, 2\right)}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{k \cdot 1}, \frac{1}{t \cdot \frac{t}{k}}, 2\right)} \]

      15. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right) \cdot \mathsf{fma}\left(k \cdot 1, \color{blue}{\frac{1}{t \cdot \frac{t}{k}}}, 2\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right) \cdot \mathsf{fma}\left(k \cdot 1, \frac{1}{\color{blue}{t \cdot \frac{t}{k}}}, 2\right)} \]

      17. lower-/.f64 80.4

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right) \cdot \mathsf{fma}\left(k \cdot 1, \frac{1}{t \cdot \color{blue}{\frac{t}{k}}}, 2\right)} \]

   8. Applied rewrites 80.4%

      \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(k \cdot 1, \frac{1}{t \cdot \frac{t}{k}}, 2\right)}} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\color{blue}{\frac{k \cdot t}{\ell}} \cdot \tan k\right)\right)\right) \cdot \mathsf{fma}\left(k \cdot 1, \frac{1}{t \cdot \frac{t}{k}}, 2\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\color{blue}{t \cdot k}}{\ell} \cdot \tan k\right)\right)\right) \cdot \mathsf{fma}\left(k \cdot 1, \frac{1}{t \cdot \frac{t}{k}}, 2\right)} \]

       2. associate-*l/ N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\color{blue}{\left(\frac{t}{\ell} \cdot k\right)} \cdot \tan k\right)\right)\right) \cdot \mathsf{fma}\left(k \cdot 1, \frac{1}{t \cdot \frac{t}{k}}, 2\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\color{blue}{\left(\frac{t}{\ell} \cdot k\right)} \cdot \tan k\right)\right)\right) \cdot \mathsf{fma}\left(k \cdot 1, \frac{1}{t \cdot \frac{t}{k}}, 2\right)} \]

       4. lower-/.f64 75.7

          \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\left(\color{blue}{\frac{t}{\ell}} \cdot k\right) \cdot \tan k\right)\right)\right) \cdot \mathsf{fma}\left(k \cdot 1, \frac{1}{t \cdot \frac{t}{k}}, 2\right)} \]

   11. Applied rewrites 75.7%

       \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\color{blue}{\left(\frac{t}{\ell} \cdot k\right)} \cdot \tan k\right)\right)\right) \cdot \mathsf{fma}\left(k \cdot 1, \frac{1}{t \cdot \frac{t}{k}}, 2\right)} \]

   if 2.55000000000000009e-182 < l

   1. Initial program 53.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{2}{2 \cdot \color{blue}{\left({k}^{2} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right)} \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

      8. associate-/r* N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\color{blue}{\frac{{t}^{3}}{\ell}}}{\ell}} \]

      11. lower-pow.f64 57.7

          \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell}} \]

   5. Applied rewrites 57.7%

      \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 57.4%

      \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\frac{t}{\ell \cdot \ell}}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 61.4%

      \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot t\right)}\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 70.7%

       \[\leadsto \frac{2}{\left(\left(\left(2 \cdot k\right) \cdot \left(k \cdot \frac{t}{\ell}\right)\right) \cdot t\right) \cdot \color{blue}{\frac{t}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.55 \cdot 10^{-182}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\left(\frac{t}{\ell} \cdot k\right) \cdot \tan k\right) \cdot \frac{t}{\ell}\right) \cdot t\right) \cdot \mathsf{fma}\left(k, \frac{1}{\frac{t}{k} \cdot t}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t}{\ell} \cdot k\right) \cdot \left(k \cdot 2\right)\right) \cdot t\right) \cdot \frac{t}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 69.8% accurate, 7.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{\left(\left(\left(\frac{t\_m}{\ell} \cdot k\right) \cdot \left(k \cdot 2\right)\right) \cdot \frac{t\_m}{\ell}\right) \cdot t\_m} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ 2.0 (* (* (* (* (/ t_m l) k) (* k 2.0)) (/ t_m l)) t_m))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / (((((t_m / l) * k) * (k * 2.0)) * (t_m / l)) * t_m));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 / (((((t_m / l) * k) * (k * 2.0d0)) * (t_m / l)) * t_m))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / (((((t_m / l) * k) * (k * 2.0)) * (t_m / l)) * t_m));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 / (((((t_m / l) * k) * (k * 2.0)) * (t_m / l)) * t_m))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t_m / l) * k) * Float64(k * 2.0)) * Float64(t_m / l)) * t_m)))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 / (((((t_m / l) * k) * (k * 2.0)) * (t_m / l)) * t_m));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(2.0 / N[(N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * k), $MachinePrecision] * N[(k * 2.0), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.5%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
4. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \frac{2}{2 \cdot \color{blue}{\left({k}^{2} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

   2. associate-*r* N/A

      \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right)} \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

   5. unpow2 N/A

      \[\leadsto \frac{2}{\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{2}{\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

   7. unpow2 N/A

      \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

   8. associate-/r* N/A

      \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

   9. lower-/.f64 N/A

      \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

   10. lower-/.f64 N/A

       \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\color{blue}{\frac{{t}^{3}}{\ell}}}{\ell}} \]

   11. lower-pow.f64 56.1

       \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell}} \]

5. Applied rewrites 56.1%

   \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]
6. Step-by-step derivation

7. Applied rewrites 56.8%

   \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\frac{t}{\ell \cdot \ell}}\right)} \]
8. Step-by-step derivation

9. Applied rewrites 61.9%

   \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot t\right)}\right)} \]
10. Step-by-step derivation

11. Applied rewrites 74.7%

    \[\leadsto \frac{2}{\left(\left(\left(2 \cdot k\right) \cdot \left(k \cdot \frac{t}{\ell}\right)\right) \cdot \frac{t}{\ell}\right) \cdot \color{blue}{t}} \]

12. Final simplification 74.7%

    \[\leadsto \frac{2}{\left(\left(\left(\frac{t}{\ell} \cdot k\right) \cdot \left(k \cdot 2\right)\right) \cdot \frac{t}{\ell}\right) \cdot t} \]
13. Add Preprocessing

Alternative 16: 69.4% accurate, 7.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{\left(\left(\left(\frac{t\_m}{\ell} \cdot k\right) \cdot \left(k \cdot 2\right)\right) \cdot t\_m\right) \cdot \frac{t\_m}{\ell}} \end{array} \]

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 (* (* (* (* (/ t_m l) k) (* k 2.0)) t_m) (/ t_m l)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / (((((t_m / l) * k) * (k * 2.0)) * t_m) * (t_m / l)));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 / (((((t_m / l) * k) * (k * 2.0d0)) * t_m) * (t_m / l)))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / (((((t_m / l) * k) * (k * 2.0)) * t_m) * (t_m / l)));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 / (((((t_m / l) * k) * (k * 2.0)) * t_m) * (t_m / l)))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t_m / l) * k) * Float64(k * 2.0)) * t_m) * 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 / (((((t_m / l) * k) * (k * 2.0)) * t_m) * (t_m / l)));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(2.0 / N[(N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * k), $MachinePrecision] * N[(k * 2.0), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.5%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
4. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \frac{2}{2 \cdot \color{blue}{\left({k}^{2} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

   2. associate-*r* N/A

      \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right)} \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

   5. unpow2 N/A

      \[\leadsto \frac{2}{\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{2}{\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

   7. unpow2 N/A

      \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

   8. associate-/r* N/A

      \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

   9. lower-/.f64 N/A

      \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

   10. lower-/.f64 N/A

       \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\color{blue}{\frac{{t}^{3}}{\ell}}}{\ell}} \]

   11. lower-pow.f64 56.1

       \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell}} \]

5. Applied rewrites 56.1%

   \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]
6. Step-by-step derivation

7. Applied rewrites 56.8%

   \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\frac{t}{\ell \cdot \ell}}\right)} \]
8. Step-by-step derivation

9. Applied rewrites 61.9%

   \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot t\right)}\right)} \]
10. Step-by-step derivation

11. Applied rewrites 72.9%

    \[\leadsto \frac{2}{\left(\left(\left(2 \cdot k\right) \cdot \left(k \cdot \frac{t}{\ell}\right)\right) \cdot t\right) \cdot \color{blue}{\frac{t}{\ell}}} \]

12. Final simplification 72.9%

    \[\leadsto \frac{2}{\left(\left(\left(\frac{t}{\ell} \cdot k\right) \cdot \left(k \cdot 2\right)\right) \cdot t\right) \cdot \frac{t}{\ell}} \]
13. Add Preprocessing

Alternative 17: 61.1% accurate, 8.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{\left(\left(\left(k \cdot t\_m\right) \cdot t\_m\right) \cdot \left(k \cdot t\_m\right)\right) \cdot \frac{2}{\ell \cdot \ell}} \end{array} \]

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 (* (* (* (* k t_m) t_m) (* k 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 / ((((k * t_m) * t_m) * (k * 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 / ((((k * t_m) * t_m) * (k * 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 / ((((k * t_m) * t_m) * (k * 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 / ((((k * t_m) * t_m) * (k * 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(Float64(Float64(k * t_m) * t_m) * Float64(k * t_m)) * Float64(2.0 / Float64(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 / ((((k * t_m) * t_m) * (k * 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[(N[(N[(k * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.5%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
4. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \frac{2}{2 \cdot \color{blue}{\left({k}^{2} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

   2. associate-*r* N/A

      \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right)} \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

   5. unpow2 N/A

      \[\leadsto \frac{2}{\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{2}{\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

   7. unpow2 N/A

      \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

   8. associate-/r* N/A

      \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

   9. lower-/.f64 N/A

      \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

   10. lower-/.f64 N/A

       \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\color{blue}{\frac{{t}^{3}}{\ell}}}{\ell}} \]

   11. lower-pow.f64 56.1

       \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell}} \]

5. Applied rewrites 56.1%

   \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]
6. Step-by-step derivation

7. Applied rewrites 56.8%

   \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\frac{t}{\ell \cdot \ell}}\right)} \]
8. Step-by-step derivation

9. Applied rewrites 64.7%

   \[\leadsto \frac{2}{\frac{2}{\ell \cdot \ell} \cdot \color{blue}{\frac{{\left(t \cdot k\right)}^{2}}{{t}^{-1}}}} \]
10. Step-by-step derivation

11. Applied rewrites 64.7%

    \[\leadsto \frac{2}{\frac{2}{\ell \cdot \ell} \cdot \left(\left(t \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot t\right)}\right)} \]

12. Final simplification 64.7%

    \[\leadsto \frac{2}{\left(\left(\left(k \cdot t\right) \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot \frac{2}{\ell \cdot \ell}} \]
13. Add Preprocessing

Alternative 18: 53.2% accurate, 8.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{\left(\frac{t\_m}{\ell \cdot \ell} \cdot \left(t\_m \cdot t\_m\right)\right) \cdot \left(\left(k \cdot k\right) \cdot 2\right)} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ 2.0 (* (* (/ t_m (* l l)) (* t_m t_m)) (* (* k 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) {
	return t_s * (2.0 / (((t_m / (l * l)) * (t_m * t_m)) * ((k * k) * 2.0)));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 / (((t_m / (l * l)) * (t_m * t_m)) * ((k * k) * 2.0d0)))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / (((t_m / (l * l)) * (t_m * t_m)) * ((k * k) * 2.0)));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 / (((t_m / (l * l)) * (t_m * t_m)) * ((k * k) * 2.0)))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 / Float64(Float64(Float64(t_m / Float64(l * l)) * Float64(t_m * t_m)) * Float64(Float64(k * k) * 2.0))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 / (((t_m / (l * l)) * (t_m * t_m)) * ((k * k) * 2.0)));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(2.0 / N[(N[(N[(t$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.5%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
4. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \frac{2}{2 \cdot \color{blue}{\left({k}^{2} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

   2. associate-*r* N/A

      \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right)} \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

   5. unpow2 N/A

      \[\leadsto \frac{2}{\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{2}{\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

   7. unpow2 N/A

      \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

   8. associate-/r* N/A

      \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

   9. lower-/.f64 N/A

      \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

   10. lower-/.f64 N/A

       \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\color{blue}{\frac{{t}^{3}}{\ell}}}{\ell}} \]

   11. lower-pow.f64 56.1

       \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell}} \]

5. Applied rewrites 56.1%

   \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]
6. Step-by-step derivation

7. Applied rewrites 56.8%

   \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\frac{t}{\ell \cdot \ell}}\right)} \]

8. Final simplification 56.8%

   \[\leadsto \frac{2}{\left(\frac{t}{\ell \cdot \ell} \cdot \left(t \cdot t\right)\right) \cdot \left(\left(k \cdot k\right) \cdot 2\right)} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024273 
(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))))