Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.5% → 83.5%

Time: 10.1s

Alternatives: 19

Speedup: 7.8×

• 27.3% 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: 55.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 83.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9 \cdot 10^{-11}:\\ \;\;\;\;\frac{2}{\left(\left(\mathsf{fma}\left(2, \frac{\frac{{t\_m}^{3}}{k}}{k}, t\_m\right) \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k \cdot \ell}\right) \cdot k\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\left(\left(\tan k \cdot t\_m\right) \cdot \frac{\frac{\sin k}{\ell} \cdot t\_m}{\ell}\right) \cdot \left({\left(\frac{k}{t\_m}\right)}^{2} + 2\right)}}{t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 9e-11)
    (/
     2.0
     (*
      (*
       (*
        (fma 2.0 (/ (/ (pow t_m 3.0) k) k) t_m)
        (/ (/ (pow (sin k) 2.0) l) (* (cos k) l)))
       k)
      k))
    (/
     (/
      2.0
      (*
       (* (* (tan k) t_m) (/ (* (/ (sin k) l) t_m) l))
       (+ (pow (/ k t_m) 2.0) 2.0)))
     t_m))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 9e-11) {
		tmp = 2.0 / (((fma(2.0, ((pow(t_m, 3.0) / k) / k), t_m) * ((pow(sin(k), 2.0) / l) / (cos(k) * l))) * k) * k);
	} else {
		tmp = (2.0 / (((tan(k) * t_m) * (((sin(k) / l) * t_m) / l)) * (pow((k / t_m), 2.0) + 2.0))) / t_m;
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 9e-11)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(2.0, Float64(Float64((t_m ^ 3.0) / k) / k), t_m) * Float64(Float64((sin(k) ^ 2.0) / l) / Float64(cos(k) * l))) * k) * k));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(Float64(tan(k) * t_m) * Float64(Float64(Float64(sin(k) / l) * t_m) / l)) * Float64((Float64(k / t_m) ^ 2.0) + 2.0))) / t_m);
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 9e-11], N[(2.0 / N[(N[(N[(N[(2.0 * N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / k), $MachinePrecision] / k), $MachinePrecision] + t$95$m), $MachinePrecision] * N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(N[(N[Tan[k], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 8.9999999999999999e-11

   1. Initial program 54.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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 76.0%

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

   if 8.9999999999999999e-11 < 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 76.7

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

   4. Applied rewrites 76.7%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 85.9

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

      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 85.9

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

   6. Applied rewrites 85.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 85.8

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

   8. Applied rewrites 82.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   10. Applied rewrites 88.2%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9 \cdot 10^{-11}:\\ \;\;\;\;\frac{2}{\left(\left(\mathsf{fma}\left(2, \frac{\frac{{t}^{3}}{k}}{k}, t\right) \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k \cdot \ell}\right) \cdot k\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\left(\left(\tan k \cdot t\right) \cdot \frac{\frac{\sin k}{\ell} \cdot t}{\ell}\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.3 \cdot 10^{-121}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\left(\cos k \cdot \ell\right) \cdot \ell} \cdot \left(\left(k \cdot t\_m\right) \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\left(\left(\tan k \cdot t\_m\right) \cdot \frac{\frac{\sin k}{\ell} \cdot t\_m}{\ell}\right) \cdot \left({\left(\frac{k}{t\_m}\right)}^{2} + 2\right)}}{t\_m}\\ \end{array} \end{array} \]

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 3.3e-121)
    (/ 2.0 (* (/ (pow (sin k) 2.0) (* (* (cos k) l) l)) (* (* k t_m) k)))
    (/
     (/
      2.0
      (*
       (* (* (tan k) t_m) (/ (* (/ (sin k) l) t_m) l))
       (+ (pow (/ k t_m) 2.0) 2.0)))
     t_m))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.3e-121) {
		tmp = 2.0 / ((pow(sin(k), 2.0) / ((cos(k) * l) * l)) * ((k * t_m) * k));
	} else {
		tmp = (2.0 / (((tan(k) * t_m) * (((sin(k) / l) * t_m) / l)) * (pow((k / t_m), 2.0) + 2.0))) / 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 (t_m <= 3.3d-121) then
        tmp = 2.0d0 / (((sin(k) ** 2.0d0) / ((cos(k) * l) * l)) * ((k * t_m) * k))
    else
        tmp = (2.0d0 / (((tan(k) * t_m) * (((sin(k) / l) * t_m) / l)) * (((k / t_m) ** 2.0d0) + 2.0d0))) / 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 (t_m <= 3.3e-121) {
		tmp = 2.0 / ((Math.pow(Math.sin(k), 2.0) / ((Math.cos(k) * l) * l)) * ((k * t_m) * k));
	} else {
		tmp = (2.0 / (((Math.tan(k) * t_m) * (((Math.sin(k) / l) * t_m) / l)) * (Math.pow((k / t_m), 2.0) + 2.0))) / 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 t_m <= 3.3e-121:
		tmp = 2.0 / ((math.pow(math.sin(k), 2.0) / ((math.cos(k) * l) * l)) * ((k * t_m) * k))
	else:
		tmp = (2.0 / (((math.tan(k) * t_m) * (((math.sin(k) / l) * t_m) / l)) * (math.pow((k / t_m), 2.0) + 2.0))) / t_m
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 3.3e-121)
		tmp = Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) / Float64(Float64(cos(k) * l) * l)) * Float64(Float64(k * t_m) * k)));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(Float64(tan(k) * t_m) * Float64(Float64(Float64(sin(k) / l) * t_m) / l)) * Float64((Float64(k / t_m) ^ 2.0) + 2.0))) / 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 (t_m <= 3.3e-121)
		tmp = 2.0 / (((sin(k) ^ 2.0) / ((cos(k) * l) * l)) * ((k * t_m) * k));
	else
		tmp = (2.0 / (((tan(k) * t_m) * (((sin(k) / l) * t_m) / l)) * (((k / t_m) ^ 2.0) + 2.0))) / 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[t$95$m, 3.3e-121], N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(N[(k * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(N[(N[Tan[k], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.3000000000000001e-121

   1. Initial program 54.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 56.6

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

   5. Applied rewrites 56.6%

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

   6. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-cos.f64 70.8

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

   8. Applied rewrites 70.8%

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

   if 3.3000000000000001e-121 < t

   1. Initial program 64.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 73.6

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

   4. Applied rewrites 73.6%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 78.7

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

      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 78.7

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

   6. Applied rewrites 78.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 79.2

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

   8. Applied rewrites 77.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   10. Applied rewrites 81.9%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.3 \cdot 10^{-121}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\left(\cos k \cdot \ell\right) \cdot \ell} \cdot \left(\left(k \cdot t\right) \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\left(\left(\tan k \cdot t\right) \cdot \frac{\frac{\sin k}{\ell} \cdot t}{\ell}\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.5 \cdot 10^{+16}:\\ \;\;\;\;\frac{\cos k}{k} \cdot \frac{\left(\ell \cdot \ell\right) \cdot 2}{\left({\sin k}^{2} \cdot t\_m\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right) \cdot \frac{\left(\left(\frac{\sin k \cdot t\_m}{\ell} \cdot t\_m\right) \cdot \tan k\right) \cdot t\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 5.5e+16)
    (* (/ (cos k) k) (/ (* (* l l) 2.0) (* (* (pow (sin k) 2.0) t_m) k)))
    (/
     2.0
     (*
      (fma (/ k t_m) (/ k t_m) 2.0)
      (/ (* (* (* (/ (* (sin k) t_m) l) t_m) (tan k)) 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 <= 5.5e+16) {
		tmp = (cos(k) / k) * (((l * l) * 2.0) / ((pow(sin(k), 2.0) * t_m) * k));
	} else {
		tmp = 2.0 / (fma((k / t_m), (k / t_m), 2.0) * ((((((sin(k) * t_m) / l) * t_m) * tan(k)) * 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 <= 5.5e+16)
		tmp = Float64(Float64(cos(k) / k) * Float64(Float64(Float64(l * l) * 2.0) / Float64(Float64((sin(k) ^ 2.0) * t_m) * k)));
	else
		tmp = Float64(2.0 / Float64(fma(Float64(k / t_m), Float64(k / t_m), 2.0) * Float64(Float64(Float64(Float64(Float64(Float64(sin(k) * t_m) / l) * t_m) * tan(k)) * 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, 5.5e+16], N[(N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision] * N[(N[(N[(l * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5.5e16

   1. Initial program 54.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 56.0

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

   5. Applied rewrites 56.0%

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

   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 69.9

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

   8. Applied rewrites 69.9%

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

   10. Applied rewrites 73.4%

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

   if 5.5e16 < t

   1. Initial program 70.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 78.0

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

   4. Applied rewrites 78.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 78.0

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

   6. Applied rewrites 78.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. associate-*l/ N/A

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

   8. Applied rewrites 88.3%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.5 \cdot 10^{+16}:\\ \;\;\;\;\frac{\cos k}{k} \cdot \frac{\left(\ell \cdot \ell\right) \cdot 2}{\left({\sin k}^{2} \cdot t\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \frac{\left(\left(\frac{\sin k \cdot t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot t}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.4 \cdot 10^{-235}:\\ \;\;\;\;\frac{\cos k}{\left(\left({\sin k}^{2} \cdot t\_m\right) \cdot k\right) \cdot k} \cdot \left(\left(\ell \cdot \ell\right) \cdot 2\right)\\ \mathbf{elif}\;t\_m \leq 325:\\ \;\;\;\;\left(\left(\ell \cdot 2\right) \cdot \frac{\cos k}{{\left(\sin k \cdot k\right)}^{2} \cdot t\_m}\right) \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right) \cdot \frac{\left(\left(\frac{\sin k \cdot t\_m}{\ell} \cdot t\_m\right) \cdot \tan k\right) \cdot t\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.4e-235)
    (* (/ (cos k) (* (* (* (pow (sin k) 2.0) t_m) k) k)) (* (* l l) 2.0))
    (if (<= t_m 325.0)
      (* (* (* l 2.0) (/ (cos k) (* (pow (* (sin k) k) 2.0) t_m))) l)
      (/
       2.0
       (*
        (fma (/ k t_m) (/ k t_m) 2.0)
        (/ (* (* (* (/ (* (sin k) t_m) l) t_m) (tan k)) 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 <= 1.4e-235) {
		tmp = (cos(k) / (((pow(sin(k), 2.0) * t_m) * k) * k)) * ((l * l) * 2.0);
	} else if (t_m <= 325.0) {
		tmp = ((l * 2.0) * (cos(k) / (pow((sin(k) * k), 2.0) * t_m))) * l;
	} else {
		tmp = 2.0 / (fma((k / t_m), (k / t_m), 2.0) * ((((((sin(k) * t_m) / l) * t_m) * tan(k)) * 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 <= 1.4e-235)
		tmp = Float64(Float64(cos(k) / Float64(Float64(Float64((sin(k) ^ 2.0) * t_m) * k) * k)) * Float64(Float64(l * l) * 2.0));
	elseif (t_m <= 325.0)
		tmp = Float64(Float64(Float64(l * 2.0) * Float64(cos(k) / Float64((Float64(sin(k) * k) ^ 2.0) * t_m))) * l);
	else
		tmp = Float64(2.0 / Float64(fma(Float64(k / t_m), Float64(k / t_m), 2.0) * Float64(Float64(Float64(Float64(Float64(Float64(sin(k) * t_m) / l) * t_m) * tan(k)) * 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, 1.4e-235], 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], If[LessEqual[t$95$m, 325.0], N[(N[(N[(l * 2.0), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(N[Power[N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(2.0 / N[(N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.39999999999999998e-235

   1. Initial program 54.2%

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

   3. 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. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 56.3

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

   5. Applied rewrites 56.3%

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

   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 66.9

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

   8. Applied rewrites 66.9%

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

   if 1.39999999999999998e-235 < t < 325

   1. Initial program 54.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 55.1

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

   5. Applied rewrites 55.1%

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

   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 79.6

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

   8. Applied rewrites 79.6%

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

   10. Applied rewrites 82.3%

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

   if 325 < t

   1. Initial program 69.7%

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

      1. 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 77.1

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

   4. Applied rewrites 77.1%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 77.1

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

   6. Applied rewrites 77.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. associate-*l/ N/A

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

   8. Applied rewrites 87.0%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.4 \cdot 10^{-235}:\\ \;\;\;\;\frac{\cos k}{\left(\left({\sin k}^{2} \cdot t\right) \cdot k\right) \cdot k} \cdot \left(\left(\ell \cdot \ell\right) \cdot 2\right)\\ \mathbf{elif}\;t \leq 325:\\ \;\;\;\;\left(\left(\ell \cdot 2\right) \cdot \frac{\cos k}{{\left(\sin k \cdot k\right)}^{2} \cdot t}\right) \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \frac{\left(\left(\frac{\sin k \cdot t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot t}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 325:\\ \;\;\;\;\left(\left(\ell \cdot 2\right) \cdot \frac{\cos k}{{\left(\sin k \cdot k\right)}^{2} \cdot t\_m}\right) \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right) \cdot \frac{\left(\left(\frac{\sin k \cdot t\_m}{\ell} \cdot t\_m\right) \cdot \tan k\right) \cdot t\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 325.0)
    (* (* (* l 2.0) (/ (cos k) (* (pow (* (sin k) k) 2.0) t_m))) l)
    (/
     2.0
     (*
      (fma (/ k t_m) (/ k t_m) 2.0)
      (/ (* (* (* (/ (* (sin k) t_m) l) t_m) (tan k)) 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 <= 325.0) {
		tmp = ((l * 2.0) * (cos(k) / (pow((sin(k) * k), 2.0) * t_m))) * l;
	} else {
		tmp = 2.0 / (fma((k / t_m), (k / t_m), 2.0) * ((((((sin(k) * t_m) / l) * t_m) * tan(k)) * 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 <= 325.0)
		tmp = Float64(Float64(Float64(l * 2.0) * Float64(cos(k) / Float64((Float64(sin(k) * k) ^ 2.0) * t_m))) * l);
	else
		tmp = Float64(2.0 / Float64(fma(Float64(k / t_m), Float64(k / t_m), 2.0) * Float64(Float64(Float64(Float64(Float64(Float64(sin(k) * t_m) / l) * t_m) * tan(k)) * 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, 325.0], N[(N[(N[(l * 2.0), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(N[Power[N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(2.0 / N[(N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 325

   1. Initial program 54.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 56.0

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

   5. Applied rewrites 56.0%

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

   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 70.0

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

   8. Applied rewrites 70.0%

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

   10. Applied rewrites 70.0%

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

   if 325 < t

   1. Initial program 69.7%

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

      1. 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 77.1

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

   4. Applied rewrites 77.1%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 77.1

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

   6. Applied rewrites 77.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. associate-*l/ N/A

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

   8. Applied rewrites 87.0%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 325:\\ \;\;\;\;\left(\left(\ell \cdot 2\right) \cdot \frac{\cos k}{{\left(\sin k \cdot k\right)}^{2} \cdot t}\right) \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \frac{\left(\left(\frac{\sin k \cdot t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot t}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\sin k \cdot t\_m}{\ell}\\ t_3 := \mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8.5 \cdot 10^{-167}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, {t\_m}^{3}, t\_m\right)}{\ell}, \frac{k \cdot k}{\ell}, \frac{{t\_m}^{3}}{\ell} \cdot \frac{2}{\ell}\right) \cdot \left(k \cdot k\right)}\\ \mathbf{elif}\;t\_m \leq 1.38 \cdot 10^{+51}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t\_m \cdot t\_m}{\ell} \cdot t\_2\right) \cdot \tan k\right) \cdot t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_3 \cdot \frac{\left(\left(t\_2 \cdot t\_m\right) \cdot \tan k\right) \cdot t\_m}{\ell}}\\ \end{array} \end{array} \end{array} \]

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 (/ (* (sin k) t_m) l)) (t_3 (fma (/ k t_m) (/ k t_m) 2.0)))
   (*
    t_s
    (if (<= t_m 8.5e-167)
      (/
       2.0
       (*
        (fma
         (/ (fma 0.3333333333333333 (pow t_m 3.0) t_m) l)
         (/ (* k k) l)
         (* (/ (pow t_m 3.0) l) (/ 2.0 l)))
        (* k k)))
      (if (<= t_m 1.38e+51)
        (/ 2.0 (* (* (* (/ (* t_m t_m) l) t_2) (tan k)) t_3))
        (/ 2.0 (* t_3 (/ (* (* (* t_2 t_m) (tan k)) 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 = (sin(k) * t_m) / l;
	double t_3 = fma((k / t_m), (k / t_m), 2.0);
	double tmp;
	if (t_m <= 8.5e-167) {
		tmp = 2.0 / (fma((fma(0.3333333333333333, pow(t_m, 3.0), t_m) / l), ((k * k) / l), ((pow(t_m, 3.0) / l) * (2.0 / l))) * (k * k));
	} else if (t_m <= 1.38e+51) {
		tmp = 2.0 / (((((t_m * t_m) / l) * t_2) * tan(k)) * t_3);
	} else {
		tmp = 2.0 / (t_3 * ((((t_2 * t_m) * tan(k)) * 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(sin(k) * t_m) / l)
	t_3 = fma(Float64(k / t_m), Float64(k / t_m), 2.0)
	tmp = 0.0
	if (t_m <= 8.5e-167)
		tmp = Float64(2.0 / Float64(fma(Float64(fma(0.3333333333333333, (t_m ^ 3.0), t_m) / l), Float64(Float64(k * k) / l), Float64(Float64((t_m ^ 3.0) / l) * Float64(2.0 / l))) * Float64(k * k)));
	elseif (t_m <= 1.38e+51)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t_m * t_m) / l) * t_2) * tan(k)) * t_3));
	else
		tmp = Float64(2.0 / Float64(t_3 * Float64(Float64(Float64(Float64(t_2 * t_m) * tan(k)) * 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[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]}, Block[{t$95$3 = N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 8.5e-167], N[(2.0 / N[(N[(N[(N[(0.3333333333333333 * N[Power[t$95$m, 3.0], $MachinePrecision] + t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] + N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(2.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.38e+51], N[(2.0 / N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision] * t$95$2), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$3 * N[(N[(N[(N[(t$95$2 * t$95$m), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 8.4999999999999994e-167

   1. Initial program 53.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 55.6

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

   5. Applied rewrites 55.6%

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

   7. Applied rewrites 55.6%

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

   9. Applied rewrites 53.2%

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

   10. Taylor expanded in k around 0

       \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}} \]
   11. 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

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

   12. Applied rewrites 66.8%

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

   if 8.4999999999999994e-167 < t < 1.38000000000000006e51

   1. Initial program 64.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 73.1

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

   4. Applied rewrites 73.1%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 73.1

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

   6. Applied rewrites 73.1%

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

   if 1.38000000000000006e51 < t

   1. Initial program 66.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 75.3

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

   4. Applied rewrites 75.3%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 75.3

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

   6. Applied rewrites 75.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. associate-*l/ N/A

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

   8. Applied rewrites 87.5%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.5 \cdot 10^{-167}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, {t}^{3}, t\right)}{\ell}, \frac{k \cdot k}{\ell}, \frac{{t}^{3}}{\ell} \cdot \frac{2}{\ell}\right) \cdot \left(k \cdot k\right)}\\ \mathbf{elif}\;t \leq 1.38 \cdot 10^{+51}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{\sin k \cdot t}{\ell}\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \frac{\left(\left(\frac{\sin k \cdot t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot t}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\sin k \cdot t\_m}{\ell}\\ t_3 := \mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.5 \cdot 10^{-163}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, {t\_m}^{3}, t\_m\right) \cdot k, \frac{\frac{k}{\ell}}{\ell}, \frac{{t\_m}^{3}}{\ell} \cdot \frac{2}{\ell}\right) \cdot \left(k \cdot k\right)}\\ \mathbf{elif}\;t\_m \leq 1.38 \cdot 10^{+51}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t\_m \cdot t\_m}{\ell} \cdot t\_2\right) \cdot \tan k\right) \cdot t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_3 \cdot \frac{\left(\left(t\_2 \cdot t\_m\right) \cdot \tan k\right) \cdot t\_m}{\ell}}\\ \end{array} \end{array} \end{array} \]

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 (/ (* (sin k) t_m) l)) (t_3 (fma (/ k t_m) (/ k t_m) 2.0)))
   (*
    t_s
    (if (<= t_m 1.5e-163)
      (/
       2.0
       (*
        (fma
         (* (fma 0.3333333333333333 (pow t_m 3.0) t_m) k)
         (/ (/ k l) l)
         (* (/ (pow t_m 3.0) l) (/ 2.0 l)))
        (* k k)))
      (if (<= t_m 1.38e+51)
        (/ 2.0 (* (* (* (/ (* t_m t_m) l) t_2) (tan k)) t_3))
        (/ 2.0 (* t_3 (/ (* (* (* t_2 t_m) (tan k)) 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 = (sin(k) * t_m) / l;
	double t_3 = fma((k / t_m), (k / t_m), 2.0);
	double tmp;
	if (t_m <= 1.5e-163) {
		tmp = 2.0 / (fma((fma(0.3333333333333333, pow(t_m, 3.0), t_m) * k), ((k / l) / l), ((pow(t_m, 3.0) / l) * (2.0 / l))) * (k * k));
	} else if (t_m <= 1.38e+51) {
		tmp = 2.0 / (((((t_m * t_m) / l) * t_2) * tan(k)) * t_3);
	} else {
		tmp = 2.0 / (t_3 * ((((t_2 * t_m) * tan(k)) * 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(sin(k) * t_m) / l)
	t_3 = fma(Float64(k / t_m), Float64(k / t_m), 2.0)
	tmp = 0.0
	if (t_m <= 1.5e-163)
		tmp = Float64(2.0 / Float64(fma(Float64(fma(0.3333333333333333, (t_m ^ 3.0), t_m) * k), Float64(Float64(k / l) / l), Float64(Float64((t_m ^ 3.0) / l) * Float64(2.0 / l))) * Float64(k * k)));
	elseif (t_m <= 1.38e+51)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t_m * t_m) / l) * t_2) * tan(k)) * t_3));
	else
		tmp = Float64(2.0 / Float64(t_3 * Float64(Float64(Float64(Float64(t_2 * t_m) * tan(k)) * 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[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]}, Block[{t$95$3 = N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.5e-163], N[(2.0 / N[(N[(N[(N[(0.3333333333333333 * N[Power[t$95$m, 3.0], $MachinePrecision] + t$95$m), $MachinePrecision] * k), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] / l), $MachinePrecision] + N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(2.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.38e+51], N[(2.0 / N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision] * t$95$2), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$3 * N[(N[(N[(N[(t$95$2 * t$95$m), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.5000000000000001e-163

   1. Initial program 53.3%

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

   3. Taylor expanded in k around 0

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

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

   5. Applied rewrites 65.1%

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

   if 1.5000000000000001e-163 < t < 1.38000000000000006e51

   1. Initial program 63.8%

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

      1. 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 72.5

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

   4. Applied rewrites 72.5%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 72.5

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

   6. Applied rewrites 72.5%

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

   if 1.38000000000000006e51 < t

   1. Initial program 66.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 75.3

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

   4. Applied rewrites 75.3%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 75.3

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

   6. Applied rewrites 75.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. associate-*l/ N/A

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

   8. Applied rewrites 87.5%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.5 \cdot 10^{-163}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, {t}^{3}, t\right) \cdot k, \frac{\frac{k}{\ell}}{\ell}, \frac{{t}^{3}}{\ell} \cdot \frac{2}{\ell}\right) \cdot \left(k \cdot k\right)}\\ \mathbf{elif}\;t \leq 1.38 \cdot 10^{+51}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{\sin k \cdot t}{\ell}\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \frac{\left(\left(\frac{\sin k \cdot t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot t}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\sin k \cdot t\_m}{\ell}\\ t_3 := \mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.35 \cdot 10^{-164}:\\ \;\;\;\;\frac{\frac{2}{t\_m}}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t\_m \cdot t\_m, 1\right)}{\ell}, \frac{k \cdot k}{\ell}, \left(\frac{\frac{t\_m}{\ell}}{\ell} \cdot t\_m\right) \cdot 2\right) \cdot \left(k \cdot k\right)}\\ \mathbf{elif}\;t\_m \leq 8.2 \cdot 10^{+55}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t\_m \cdot t\_m}{\ell} \cdot t\_2\right) \cdot \tan k\right) \cdot t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_3 \cdot \frac{\left(\left(t\_2 \cdot t\_m\right) \cdot \tan k\right) \cdot t\_m}{\ell}}\\ \end{array} \end{array} \end{array} \]

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 (/ (* (sin k) t_m) l)) (t_3 (fma (/ k t_m) (/ k t_m) 2.0)))
   (*
    t_s
    (if (<= t_m 3.35e-164)
      (/
       (/ 2.0 t_m)
       (*
        (fma
         (/ (fma 0.3333333333333333 (* t_m t_m) 1.0) l)
         (/ (* k k) l)
         (* (* (/ (/ t_m l) l) t_m) 2.0))
        (* k k)))
      (if (<= t_m 8.2e+55)
        (/ 2.0 (* (* (* (/ (* t_m t_m) l) t_2) (tan k)) t_3))
        (/ 2.0 (* t_3 (/ (* (* (* t_2 t_m) (tan k)) 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 = (sin(k) * t_m) / l;
	double t_3 = fma((k / t_m), (k / t_m), 2.0);
	double tmp;
	if (t_m <= 3.35e-164) {
		tmp = (2.0 / t_m) / (fma((fma(0.3333333333333333, (t_m * t_m), 1.0) / l), ((k * k) / l), ((((t_m / l) / l) * t_m) * 2.0)) * (k * k));
	} else if (t_m <= 8.2e+55) {
		tmp = 2.0 / (((((t_m * t_m) / l) * t_2) * tan(k)) * t_3);
	} else {
		tmp = 2.0 / (t_3 * ((((t_2 * t_m) * tan(k)) * 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(sin(k) * t_m) / l)
	t_3 = fma(Float64(k / t_m), Float64(k / t_m), 2.0)
	tmp = 0.0
	if (t_m <= 3.35e-164)
		tmp = Float64(Float64(2.0 / t_m) / Float64(fma(Float64(fma(0.3333333333333333, Float64(t_m * t_m), 1.0) / l), Float64(Float64(k * k) / l), Float64(Float64(Float64(Float64(t_m / l) / l) * t_m) * 2.0)) * Float64(k * k)));
	elseif (t_m <= 8.2e+55)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t_m * t_m) / l) * t_2) * tan(k)) * t_3));
	else
		tmp = Float64(2.0 / Float64(t_3 * Float64(Float64(Float64(Float64(t_2 * t_m) * tan(k)) * 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[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]}, Block[{t$95$3 = N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 3.35e-164], N[(N[(2.0 / t$95$m), $MachinePrecision] / N[(N[(N[(N[(0.3333333333333333 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / l), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] + N[(N[(N[(N[(t$95$m / l), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 8.2e+55], N[(2.0 / N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision] * t$95$2), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$3 * N[(N[(N[(N[(t$95$2 * t$95$m), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 3.35e-164

   1. Initial program 53.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 64.3

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

   4. Applied rewrites 64.3%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 73.8

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

      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 73.8

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

   6. Applied rewrites 73.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 74.5

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

   8. Applied rewrites 74.6%

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

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   11. Applied rewrites 64.6%

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

   if 3.35e-164 < t < 8.19999999999999962e55

   1. Initial program 61.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 71.7

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

   4. Applied rewrites 71.7%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 71.6

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

   6. Applied rewrites 71.6%

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

   if 8.19999999999999962e55 < t

   1. Initial program 69.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 76.3

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

   4. Applied rewrites 76.3%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 76.3

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

   6. Applied rewrites 76.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. associate-*l/ N/A

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

   8. Applied rewrites 89.0%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.35 \cdot 10^{-164}:\\ \;\;\;\;\frac{\frac{2}{t}}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right)}{\ell}, \frac{k \cdot k}{\ell}, \left(\frac{\frac{t}{\ell}}{\ell} \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}\\ \mathbf{elif}\;t \leq 8.2 \cdot 10^{+55}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{\sin k \cdot t}{\ell}\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \frac{\left(\left(\frac{\sin k \cdot t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot t}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.6 \cdot 10^{-197}:\\ \;\;\;\;\frac{\frac{2}{t\_m}}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t\_m \cdot t\_m, 1\right)}{\ell}, \frac{k \cdot k}{\ell}, \left(\frac{\frac{t\_m}{\ell}}{\ell} \cdot t\_m\right) \cdot 2\right) \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\left(\tan k \cdot t\_m\right) \cdot \frac{\sin k}{\ell}\right) \cdot \frac{t\_m}{\ell}\right) \cdot t\_m\right) \cdot \mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right)}\\ \end{array} \end{array} \]

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 4.6e-197)
    (/
     (/ 2.0 t_m)
     (*
      (fma
       (/ (fma 0.3333333333333333 (* t_m t_m) 1.0) l)
       (/ (* k k) l)
       (* (* (/ (/ t_m l) l) t_m) 2.0))
      (* k k)))
    (/
     2.0
     (*
      (* (* (* (* (tan k) t_m) (/ (sin k) l)) (/ t_m l)) t_m)
      (fma (/ k 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 <= 4.6e-197) {
		tmp = (2.0 / t_m) / (fma((fma(0.3333333333333333, (t_m * t_m), 1.0) / l), ((k * k) / l), ((((t_m / l) / l) * t_m) * 2.0)) * (k * k));
	} else {
		tmp = 2.0 / (((((tan(k) * t_m) * (sin(k) / l)) * (t_m / l)) * t_m) * fma((k / 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 <= 4.6e-197)
		tmp = Float64(Float64(2.0 / t_m) / Float64(fma(Float64(fma(0.3333333333333333, Float64(t_m * t_m), 1.0) / l), Float64(Float64(k * k) / l), Float64(Float64(Float64(Float64(t_m / l) / l) * t_m) * 2.0)) * Float64(k * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(tan(k) * t_m) * Float64(sin(k) / l)) * Float64(t_m / l)) * t_m) * fma(Float64(k / t_m), Float64(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, 4.6e-197], N[(N[(2.0 / t$95$m), $MachinePrecision] / N[(N[(N[(N[(0.3333333333333333 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / l), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] + N[(N[(N[(N[(t$95$m / l), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(N[Tan[k], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.6000000000000001e-197

   1. Initial program 52.8%

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

      1. 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 64.2

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

   4. Applied rewrites 64.2%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 74.1

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

      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 74.1

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

   6. Applied rewrites 74.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 74.8

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

   8. Applied rewrites 74.8%

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

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   11. Applied rewrites 64.4%

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

   if 4.6000000000000001e-197 < t

   1. Initial program 65.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 73.5

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

   4. Applied rewrites 73.5%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 73.5

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

   6. Applied rewrites 73.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 78.0

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

   8. Applied rewrites 76.0%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.6 \cdot 10^{-197}:\\ \;\;\;\;\frac{\frac{2}{t}}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right)}{\ell}, \frac{k \cdot k}{\ell}, \left(\frac{\frac{t}{\ell}}{\ell} \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\left(\tan k \cdot t\right) \cdot \frac{\sin k}{\ell}\right) \cdot \frac{t}{\ell}\right) \cdot t\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 79.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.5 \cdot 10^{-160}:\\ \;\;\;\;\frac{\frac{2}{t\_m}}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t\_m \cdot t\_m, 1\right)}{\ell}, \frac{k \cdot k}{\ell}, \left(\frac{\frac{t\_m}{\ell}}{\ell} \cdot t\_m\right) \cdot 2\right) \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right) \cdot \left(\left(\left(\frac{\sin k \cdot t\_m}{\ell} \cdot \tan k\right) \cdot \frac{t\_m}{\ell}\right) \cdot t\_m\right)}\\ \end{array} \end{array} \]

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 (<= t_m 3.5e-160)
    (/
     (/ 2.0 t_m)
     (*
      (fma
       (/ (fma 0.3333333333333333 (* t_m t_m) 1.0) l)
       (/ (* k k) l)
       (* (* (/ (/ t_m l) l) t_m) 2.0))
      (* k k)))
    (/
     2.0
     (*
      (fma k (/ k (* t_m t_m)) 2.0)
      (* (* (* (/ (* (sin k) t_m) l) (tan k)) (/ 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 (t_m <= 3.5e-160) {
		tmp = (2.0 / t_m) / (fma((fma(0.3333333333333333, (t_m * t_m), 1.0) / l), ((k * k) / l), ((((t_m / l) / l) * t_m) * 2.0)) * (k * k));
	} else {
		tmp = 2.0 / (fma(k, (k / (t_m * t_m)), 2.0) * (((((sin(k) * t_m) / l) * tan(k)) * (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 (t_m <= 3.5e-160)
		tmp = Float64(Float64(2.0 / t_m) / Float64(fma(Float64(fma(0.3333333333333333, Float64(t_m * t_m), 1.0) / l), Float64(Float64(k * k) / l), Float64(Float64(Float64(Float64(t_m / l) / l) * t_m) * 2.0)) * Float64(k * k)));
	else
		tmp = Float64(2.0 / Float64(fma(k, Float64(k / Float64(t_m * t_m)), 2.0) * Float64(Float64(Float64(Float64(Float64(sin(k) * t_m) / l) * tan(k)) * Float64(t_m / l)) * t_m)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 3.5e-160], N[(N[(2.0 / t$95$m), $MachinePrecision] / N[(N[(N[(N[(0.3333333333333333 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / l), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] + N[(N[(N[(N[(t$95$m / l), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * 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]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.5000000000000003e-160

   1. Initial program 53.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 64.3

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

   4. Applied rewrites 64.3%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 73.8

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

      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 73.8

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

   6. Applied rewrites 73.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 74.5

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

   8. Applied rewrites 74.6%

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

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   11. Applied rewrites 64.6%

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

   if 3.5000000000000003e-160 < t

   1. Initial program 65.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 73.9

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

   4. Applied rewrites 73.9%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 78.8

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

      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 78.8

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

   6. Applied rewrites 78.8%

      \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. pow2 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. frac-times N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 77.8

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

   8. Applied rewrites 77.8%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.5 \cdot 10^{-160}:\\ \;\;\;\;\frac{\frac{2}{t}}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right)}{\ell}, \frac{k \cdot k}{\ell}, \left(\frac{\frac{t}{\ell}}{\ell} \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right) \cdot \left(\left(\left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right) \cdot \frac{t}{\ell}\right) \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 76.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{k}{\ell} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 3 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{2}{t\_m}}{\left(t\_2 \cdot t\_2\right) \cdot \left({\left(\frac{k}{t\_m}\right)}^{2} + 2\right)}\\ \mathbf{elif}\;\ell \leq 4 \cdot 10^{+110}:\\ \;\;\;\;\frac{2}{\frac{{\left(k \cdot t\_m\right)}^{2} \cdot 2}{\frac{\ell \cdot \ell}{t\_m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\left(\left(\frac{\sin k \cdot t\_m}{\ell} \cdot \tan k\right) \cdot \frac{t\_m}{\ell}\right) \cdot t\_m\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (/ k l) t_m)))
   (*
    t_s
    (if (<= l 3e-18)
      (/ (/ 2.0 t_m) (* (* t_2 t_2) (+ (pow (/ k t_m) 2.0) 2.0)))
      (if (<= l 4e+110)
        (/ 2.0 (/ (* (pow (* k t_m) 2.0) 2.0) (/ (* l l) t_m)))
        (/
         2.0
         (* 2.0 (* (* (* (/ (* (sin k) t_m) l) (tan k)) (/ 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 t_2 = (k / l) * t_m;
	double tmp;
	if (l <= 3e-18) {
		tmp = (2.0 / t_m) / ((t_2 * t_2) * (pow((k / t_m), 2.0) + 2.0));
	} else if (l <= 4e+110) {
		tmp = 2.0 / ((pow((k * t_m), 2.0) * 2.0) / ((l * l) / t_m));
	} else {
		tmp = 2.0 / (2.0 * (((((sin(k) * t_m) / l) * tan(k)) * (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) :: t_2
    real(8) :: tmp
    t_2 = (k / l) * t_m
    if (l <= 3d-18) then
        tmp = (2.0d0 / t_m) / ((t_2 * t_2) * (((k / t_m) ** 2.0d0) + 2.0d0))
    else if (l <= 4d+110) then
        tmp = 2.0d0 / ((((k * t_m) ** 2.0d0) * 2.0d0) / ((l * l) / t_m))
    else
        tmp = 2.0d0 / (2.0d0 * (((((sin(k) * t_m) / l) * tan(k)) * (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 t_2 = (k / l) * t_m;
	double tmp;
	if (l <= 3e-18) {
		tmp = (2.0 / t_m) / ((t_2 * t_2) * (Math.pow((k / t_m), 2.0) + 2.0));
	} else if (l <= 4e+110) {
		tmp = 2.0 / ((Math.pow((k * t_m), 2.0) * 2.0) / ((l * l) / t_m));
	} else {
		tmp = 2.0 / (2.0 * (((((Math.sin(k) * t_m) / l) * Math.tan(k)) * (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):
	t_2 = (k / l) * t_m
	tmp = 0
	if l <= 3e-18:
		tmp = (2.0 / t_m) / ((t_2 * t_2) * (math.pow((k / t_m), 2.0) + 2.0))
	elif l <= 4e+110:
		tmp = 2.0 / ((math.pow((k * t_m), 2.0) * 2.0) / ((l * l) / t_m))
	else:
		tmp = 2.0 / (2.0 * (((((math.sin(k) * t_m) / l) * math.tan(k)) * (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)
	t_2 = Float64(Float64(k / l) * t_m)
	tmp = 0.0
	if (l <= 3e-18)
		tmp = Float64(Float64(2.0 / t_m) / Float64(Float64(t_2 * t_2) * Float64((Float64(k / t_m) ^ 2.0) + 2.0)));
	elseif (l <= 4e+110)
		tmp = Float64(2.0 / Float64(Float64((Float64(k * t_m) ^ 2.0) * 2.0) / Float64(Float64(l * l) / t_m)));
	else
		tmp = Float64(2.0 / Float64(2.0 * Float64(Float64(Float64(Float64(Float64(sin(k) * t_m) / l) * tan(k)) * 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)
	t_2 = (k / l) * t_m;
	tmp = 0.0;
	if (l <= 3e-18)
		tmp = (2.0 / t_m) / ((t_2 * t_2) * (((k / t_m) ^ 2.0) + 2.0));
	elseif (l <= 4e+110)
		tmp = 2.0 / ((((k * t_m) ^ 2.0) * 2.0) / ((l * l) / t_m));
	else
		tmp = 2.0 / (2.0 * (((((sin(k) * t_m) / l) * tan(k)) * (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_] := Block[{t$95$2 = N[(N[(k / l), $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[l, 3e-18], N[(N[(2.0 / t$95$m), $MachinePrecision] / N[(N[(t$95$2 * t$95$2), $MachinePrecision] * N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 4e+110], N[(2.0 / N[(N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(l * l), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(2.0 * 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]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if l < 2.99999999999999983e-18

   1. Initial program 61.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 72.7

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

   4. Applied rewrites 72.7%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 81.9

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

      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 81.9

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

   6. Applied rewrites 81.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 82.7

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

   8. Applied rewrites 81.7%

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

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. associate-/l* N/A

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

       3. unpow2 N/A

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

       4. unpow2 N/A

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

       5. unpow2 N/A

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

       6. times-frac N/A

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

       7. unswap-sqr N/A

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

       8. lower-*.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-/.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. lower-/.f64 79.7

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

   11. Applied rewrites 79.7%

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

   if 2.99999999999999983e-18 < l < 4.0000000000000001e110

   1. Initial program 64.3%

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

   3. 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. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 68.7

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

   5. Applied rewrites 68.7%

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

   7. Applied rewrites 68.7%

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

   9. Applied rewrites 68.7%

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

   11. Applied rewrites 72.7%

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

   if 4.0000000000000001e110 < l

   1. Initial program 34.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 48.3

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

   4. Applied rewrites 48.3%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 53.1

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

      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 53.1

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

   6. Applied rewrites 53.1%

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

   7. Taylor expanded in t around inf

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

   9. Applied rewrites 65.5%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{2}{t}}{\left(\left(\frac{k}{\ell} \cdot t\right) \cdot \left(\frac{k}{\ell} \cdot t\right)\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)}\\ \mathbf{elif}\;\ell \leq 4 \cdot 10^{+110}:\\ \;\;\;\;\frac{2}{\frac{{\left(k \cdot t\right)}^{2} \cdot 2}{\frac{\ell \cdot \ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\left(\left(\frac{\sin k \cdot t}{\ell} \cdot \tan k\right) \cdot \frac{t}{\ell}\right) \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 76.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{k}{\ell} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{2}{t\_m}}{\left(t\_2 \cdot t\_2\right) \cdot \left({\left(\frac{k}{t\_m}\right)}^{2} + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left({\left(k \cdot t\_m\right)}^{2} \cdot 2\right) \cdot \frac{t\_m}{\ell}}{\ell}}\\ \end{array} \end{array} \end{array} \]

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 (* (/ k l) t_m)))
   (*
    t_s
    (if (<= (* l l) 2e-67)
      (/ (/ 2.0 t_m) (* (* t_2 t_2) (+ (pow (/ k t_m) 2.0) 2.0)))
      (/ 2.0 (/ (* (* (pow (* k t_m) 2.0) 2.0) (/ t_m l)) l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = (k / l) * t_m;
	double tmp;
	if ((l * l) <= 2e-67) {
		tmp = (2.0 / t_m) / ((t_2 * t_2) * (pow((k / t_m), 2.0) + 2.0));
	} else {
		tmp = 2.0 / (((pow((k * t_m), 2.0) * 2.0) * (t_m / l)) / l);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (k / l) * t_m
    if ((l * l) <= 2d-67) then
        tmp = (2.0d0 / t_m) / ((t_2 * t_2) * (((k / t_m) ** 2.0d0) + 2.0d0))
    else
        tmp = 2.0d0 / (((((k * t_m) ** 2.0d0) * 2.0d0) * (t_m / l)) / l)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = (k / l) * t_m;
	double tmp;
	if ((l * l) <= 2e-67) {
		tmp = (2.0 / t_m) / ((t_2 * t_2) * (Math.pow((k / t_m), 2.0) + 2.0));
	} else {
		tmp = 2.0 / (((Math.pow((k * t_m), 2.0) * 2.0) * (t_m / l)) / l);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = (k / l) * t_m
	tmp = 0
	if (l * l) <= 2e-67:
		tmp = (2.0 / t_m) / ((t_2 * t_2) * (math.pow((k / t_m), 2.0) + 2.0))
	else:
		tmp = 2.0 / (((math.pow((k * t_m), 2.0) * 2.0) * (t_m / l)) / l)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(Float64(k / l) * t_m)
	tmp = 0.0
	if (Float64(l * l) <= 2e-67)
		tmp = Float64(Float64(2.0 / t_m) / Float64(Float64(t_2 * t_2) * Float64((Float64(k / t_m) ^ 2.0) + 2.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k * t_m) ^ 2.0) * 2.0) * Float64(t_m / l)) / l));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = (k / l) * t_m;
	tmp = 0.0;
	if ((l * l) <= 2e-67)
		tmp = (2.0 / t_m) / ((t_2 * t_2) * (((k / t_m) ^ 2.0) + 2.0));
	else
		tmp = 2.0 / (((((k * t_m) ^ 2.0) * 2.0) * (t_m / l)) / l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[(k / l), $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 2e-67], N[(N[(2.0 / t$95$m), $MachinePrecision] / N[(N[(t$95$2 * t$95$2), $MachinePrecision] * N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1.99999999999999989e-67

   1. Initial program 65.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 78.4

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

   4. Applied rewrites 78.4%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 88.3

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

      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 88.3

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

   6. Applied rewrites 88.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 89.1

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

   8. Applied rewrites 89.1%

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

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. associate-/l* N/A

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

       3. unpow2 N/A

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

       4. unpow2 N/A

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

       5. unpow2 N/A

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

       6. times-frac N/A

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

       7. unswap-sqr N/A

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

       8. lower-*.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-/.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. lower-/.f64 91.8

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

   11. Applied rewrites 91.8%

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

   if 1.99999999999999989e-67 < (*.f64 l l)

   1. Initial program 50.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 54.4

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

   5. Applied rewrites 54.4%

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

   7. Applied rewrites 55.1%

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

   9. Applied rewrites 53.2%

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

   11. Applied rewrites 63.3%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{2}{t}}{\left(\left(\frac{k}{\ell} \cdot t\right) \cdot \left(\frac{k}{\ell} \cdot t\right)\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left({\left(k \cdot t\right)}^{2} \cdot 2\right) \cdot \frac{t}{\ell}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 70.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.2 \cdot 10^{-26}:\\ \;\;\;\;\frac{\frac{2}{t\_m}}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t\_m \cdot t\_m, 1\right)}{\ell}, \frac{k \cdot k}{\ell}, \left(\frac{\frac{t\_m}{\ell}}{\ell} \cdot t\_m\right) \cdot 2\right) \cdot \left(k \cdot k\right)}\\ \mathbf{elif}\;t\_m \leq 3.2 \cdot 10^{+150}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\ell}{t\_m}} \cdot \frac{k \cdot 2}{\frac{\ell}{t\_m \cdot t\_m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left({\left(k \cdot t\_m\right)}^{2} \cdot 2\right) \cdot t\_m}{\ell \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 5.2e-26)
    (/
     (/ 2.0 t_m)
     (*
      (fma
       (/ (fma 0.3333333333333333 (* t_m t_m) 1.0) l)
       (/ (* k k) l)
       (* (* (/ (/ t_m l) l) t_m) 2.0))
      (* k k)))
    (if (<= t_m 3.2e+150)
      (/ 2.0 (* (/ k (/ l t_m)) (/ (* k 2.0) (/ l (* t_m t_m)))))
      (/ 2.0 (/ (* (* (pow (* k t_m) 2.0) 2.0) t_m) (* l l)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 5.2e-26) {
		tmp = (2.0 / t_m) / (fma((fma(0.3333333333333333, (t_m * t_m), 1.0) / l), ((k * k) / l), ((((t_m / l) / l) * t_m) * 2.0)) * (k * k));
	} else if (t_m <= 3.2e+150) {
		tmp = 2.0 / ((k / (l / t_m)) * ((k * 2.0) / (l / (t_m * t_m))));
	} else {
		tmp = 2.0 / (((pow((k * t_m), 2.0) * 2.0) * t_m) / (l * l));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 5.2e-26)
		tmp = Float64(Float64(2.0 / t_m) / Float64(fma(Float64(fma(0.3333333333333333, Float64(t_m * t_m), 1.0) / l), Float64(Float64(k * k) / l), Float64(Float64(Float64(Float64(t_m / l) / l) * t_m) * 2.0)) * Float64(k * k)));
	elseif (t_m <= 3.2e+150)
		tmp = Float64(2.0 / Float64(Float64(k / Float64(l / t_m)) * Float64(Float64(k * 2.0) / Float64(l / Float64(t_m * t_m)))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k * t_m) ^ 2.0) * 2.0) * t_m) / Float64(l * l)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 5.2e-26], N[(N[(2.0 / t$95$m), $MachinePrecision] / N[(N[(N[(N[(0.3333333333333333 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / l), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] + N[(N[(N[(N[(t$95$m / l), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.2e+150], N[(2.0 / N[(N[(k / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(k * 2.0), $MachinePrecision] / N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] * t$95$m), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 5.2000000000000002e-26

   1. Initial program 54.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 65.2

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

   4. Applied rewrites 65.2%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 73.4

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

      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 73.4

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

   6. Applied rewrites 73.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 74.0

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

   8. Applied rewrites 74.0%

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

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   11. Applied rewrites 65.0%

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

   if 5.2000000000000002e-26 < t < 3.20000000000000016e150

   1. Initial program 62.6%

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

   3. 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. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 54.9

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

   5. Applied rewrites 54.9%

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

   7. Applied rewrites 72.2%

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

   if 3.20000000000000016e150 < t

   1. Initial program 69.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 60.2

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

   5. Applied rewrites 60.2%

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

   7. Applied rewrites 60.2%

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

   9. Applied rewrites 59.6%

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

   11. Applied rewrites 87.4%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.2 \cdot 10^{-26}:\\ \;\;\;\;\frac{\frac{2}{t}}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right)}{\ell}, \frac{k \cdot k}{\ell}, \left(\frac{\frac{t}{\ell}}{\ell} \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{+150}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\ell}{t}} \cdot \frac{k \cdot 2}{\frac{\ell}{t \cdot t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left({\left(k \cdot t\right)}^{2} \cdot 2\right) \cdot t}{\ell \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 70.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 4.4 \cdot 10^{-60}:\\ \;\;\;\;\frac{2}{\frac{\left({\left(k \cdot t\_m\right)}^{2} \cdot 2\right) \cdot \frac{t\_m}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{t\_m}}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t\_m \cdot t\_m, 1\right)}{\ell}, \frac{k \cdot k}{\ell}, \left(\frac{\frac{t\_m}{\ell}}{\ell} \cdot t\_m\right) \cdot 2\right) \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

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 (<= k 4.4e-60)
    (/ 2.0 (/ (* (* (pow (* k t_m) 2.0) 2.0) (/ t_m l)) l))
    (/
     (/ 2.0 t_m)
     (*
      (fma
       (/ (fma 0.3333333333333333 (* t_m t_m) 1.0) l)
       (/ (* k k) l)
       (* (* (/ (/ t_m l) l) t_m) 2.0))
      (* k k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 4.4e-60) {
		tmp = 2.0 / (((pow((k * t_m), 2.0) * 2.0) * (t_m / l)) / l);
	} else {
		tmp = (2.0 / t_m) / (fma((fma(0.3333333333333333, (t_m * t_m), 1.0) / l), ((k * k) / l), ((((t_m / l) / l) * t_m) * 2.0)) * (k * k));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 4.4e-60)
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k * t_m) ^ 2.0) * 2.0) * Float64(t_m / l)) / l));
	else
		tmp = Float64(Float64(2.0 / t_m) / Float64(fma(Float64(fma(0.3333333333333333, Float64(t_m * t_m), 1.0) / l), Float64(Float64(k * k) / l), Float64(Float64(Float64(Float64(t_m / l) / l) * t_m) * 2.0)) * Float64(k * k)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 4.4e-60], N[(2.0 / N[(N[(N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / t$95$m), $MachinePrecision] / N[(N[(N[(N[(0.3333333333333333 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / l), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] + N[(N[(N[(N[(t$95$m / l), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 4.3999999999999998e-60

   1. Initial program 60.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 58.0

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

   5. Applied rewrites 58.0%

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

   7. Applied rewrites 58.6%

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

   9. Applied rewrites 55.0%

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

   11. Applied rewrites 73.7%

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

   if 4.3999999999999998e-60 < k

   1. Initial program 52.5%

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

      1. 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 61.1

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

   4. Applied rewrites 61.1%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 69.6

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

      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 69.6

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

   6. Applied rewrites 69.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 70.2

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

   8. Applied rewrites 70.2%

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

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   11. Applied rewrites 67.4%

       \[\leadsto \frac{\frac{2}{t}}{\color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right)}{\ell}, \frac{k \cdot k}{\ell}, \left(t \cdot \frac{\frac{t}{\ell}}{\ell}\right) \cdot 2\right) \cdot \left(k \cdot k\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.4 \cdot 10^{-60}:\\ \;\;\;\;\frac{2}{\frac{\left({\left(k \cdot t\right)}^{2} \cdot 2\right) \cdot \frac{t}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{t}}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right)}{\ell}, \frac{k \cdot k}{\ell}, \left(\frac{\frac{t}{\ell}}{\ell} \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 67.9% accurate, 4.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 4.4 \cdot 10^{-60}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\ell}{t\_m}} \cdot \frac{k \cdot 2}{\frac{\ell}{t\_m \cdot t\_m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{t\_m}}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t\_m \cdot t\_m, 1\right)}{\ell}, \frac{k \cdot k}{\ell}, \left(\frac{\frac{t\_m}{\ell}}{\ell} \cdot t\_m\right) \cdot 2\right) \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

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 (<= k 4.4e-60)
    (/ 2.0 (* (/ k (/ l t_m)) (/ (* k 2.0) (/ l (* t_m t_m)))))
    (/
     (/ 2.0 t_m)
     (*
      (fma
       (/ (fma 0.3333333333333333 (* t_m t_m) 1.0) l)
       (/ (* k k) l)
       (* (* (/ (/ t_m l) l) t_m) 2.0))
      (* k k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 4.4e-60) {
		tmp = 2.0 / ((k / (l / t_m)) * ((k * 2.0) / (l / (t_m * t_m))));
	} else {
		tmp = (2.0 / t_m) / (fma((fma(0.3333333333333333, (t_m * t_m), 1.0) / l), ((k * k) / l), ((((t_m / l) / l) * t_m) * 2.0)) * (k * k));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 4.4e-60)
		tmp = Float64(2.0 / Float64(Float64(k / Float64(l / t_m)) * Float64(Float64(k * 2.0) / Float64(l / Float64(t_m * t_m)))));
	else
		tmp = Float64(Float64(2.0 / t_m) / Float64(fma(Float64(fma(0.3333333333333333, Float64(t_m * t_m), 1.0) / l), Float64(Float64(k * k) / l), Float64(Float64(Float64(Float64(t_m / l) / l) * t_m) * 2.0)) * Float64(k * k)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 4.4e-60], N[(2.0 / N[(N[(k / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(k * 2.0), $MachinePrecision] / N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / t$95$m), $MachinePrecision] / N[(N[(N[(N[(0.3333333333333333 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / l), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] + N[(N[(N[(N[(t$95$m / l), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 4.3999999999999998e-60

   1. Initial program 60.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 58.0

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

   5. Applied rewrites 58.0%

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

   7. Applied rewrites 70.2%

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

   if 4.3999999999999998e-60 < k

   1. Initial program 52.5%

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

      1. 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 61.1

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

   4. Applied rewrites 61.1%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 69.6

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

      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 69.6

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

   6. Applied rewrites 69.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 70.2

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

   8. Applied rewrites 70.2%

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

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   11. Applied rewrites 67.4%

       \[\leadsto \frac{\frac{2}{t}}{\color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right)}{\ell}, \frac{k \cdot k}{\ell}, \left(t \cdot \frac{\frac{t}{\ell}}{\ell}\right) \cdot 2\right) \cdot \left(k \cdot k\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.4 \cdot 10^{-60}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\ell}{t}} \cdot \frac{k \cdot 2}{\frac{\ell}{t \cdot t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{t}}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right)}{\ell}, \frac{k \cdot k}{\ell}, \left(\frac{\frac{t}{\ell}}{\ell} \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 66.6% accurate, 6.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.5 \cdot 10^{-79}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\ell}{t\_m}} \cdot \frac{k \cdot 2}{\frac{\ell}{t\_m \cdot t\_m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{t\_m}}{\left(\left(\left(\frac{\frac{t\_m}{\ell}}{\ell} \cdot t\_m\right) \cdot 2\right) \cdot k\right) \cdot k}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 2.5e-79)
    (/ 2.0 (* (/ k (/ l t_m)) (/ (* k 2.0) (/ l (* t_m t_m)))))
    (/ (/ 2.0 t_m) (* (* (* (* (/ (/ t_m l) l) t_m) 2.0) k) k)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.5e-79) {
		tmp = 2.0 / ((k / (l / t_m)) * ((k * 2.0) / (l / (t_m * t_m))));
	} else {
		tmp = (2.0 / t_m) / ((((((t_m / l) / l) * t_m) * 2.0) * k) * k);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.5d-79) then
        tmp = 2.0d0 / ((k / (l / t_m)) * ((k * 2.0d0) / (l / (t_m * t_m))))
    else
        tmp = (2.0d0 / t_m) / ((((((t_m / l) / l) * t_m) * 2.0d0) * k) * k)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.5e-79) {
		tmp = 2.0 / ((k / (l / t_m)) * ((k * 2.0) / (l / (t_m * t_m))));
	} else {
		tmp = (2.0 / t_m) / ((((((t_m / l) / l) * t_m) * 2.0) * k) * k);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 2.5e-79:
		tmp = 2.0 / ((k / (l / t_m)) * ((k * 2.0) / (l / (t_m * t_m))))
	else:
		tmp = (2.0 / t_m) / ((((((t_m / l) / l) * t_m) * 2.0) * k) * k)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 2.5e-79)
		tmp = Float64(2.0 / Float64(Float64(k / Float64(l / t_m)) * Float64(Float64(k * 2.0) / Float64(l / Float64(t_m * t_m)))));
	else
		tmp = Float64(Float64(2.0 / t_m) / Float64(Float64(Float64(Float64(Float64(Float64(t_m / l) / l) * t_m) * 2.0) * k) * k));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 2.5e-79)
		tmp = 2.0 / ((k / (l / t_m)) * ((k * 2.0) / (l / (t_m * t_m))));
	else
		tmp = (2.0 / t_m) / ((((((t_m / l) / l) * t_m) * 2.0) * k) * k);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 2.5e-79], N[(2.0 / N[(N[(k / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(k * 2.0), $MachinePrecision] / N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / t$95$m), $MachinePrecision] / N[(N[(N[(N[(N[(N[(t$95$m / l), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.5e-79

   1. Initial program 60.6%

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

   3. 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. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 58.1

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

   5. Applied rewrites 58.1%

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

   7. Applied rewrites 70.4%

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

   if 2.5e-79 < k

   1. Initial program 52.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 60.9

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

   4. Applied rewrites 60.9%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 70.2

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

      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 70.2

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

   6. Applied rewrites 70.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 70.8

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

   8. Applied rewrites 69.8%

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

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. associate-*l/ N/A

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

       3. associate-*l* N/A

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

       4. unpow2 N/A

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

       5. associate-*r* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. associate-/l* N/A

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

       12. lower-*.f64 N/A

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

       13. unpow2 N/A

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

       14. associate-/r* N/A

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

       15. lower-/.f64 N/A

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

       16. lower-/.f64 63.0

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

   11. Applied rewrites 63.0%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.5 \cdot 10^{-79}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\ell}{t}} \cdot \frac{k \cdot 2}{\frac{\ell}{t \cdot t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{t}}{\left(\left(\left(\frac{\frac{t}{\ell}}{\ell} \cdot t\right) \cdot 2\right) \cdot k\right) \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 67.3% accurate, 7.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{\frac{2}{t\_m}}{\left(\left(\left(\frac{\frac{t\_m}{\ell}}{\ell} \cdot t\_m\right) \cdot 2\right) \cdot k\right) \cdot k} \end{array} \]

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) (* (* (* (* (/ (/ t_m l) l) t_m) 2.0) k) k))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((2.0 / t_m) / ((((((t_m / l) / l) * t_m) * 2.0) * k) * k));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((2.0d0 / t_m) / ((((((t_m / l) / l) * t_m) * 2.0d0) * k) * k))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * ((2.0 / t_m) / ((((((t_m / l) / l) * t_m) * 2.0) * k) * k));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * ((2.0 / t_m) / ((((((t_m / l) / l) * t_m) * 2.0) * k) * k))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(2.0 / t_m) / Float64(Float64(Float64(Float64(Float64(Float64(t_m / l) / l) * t_m) * 2.0) * k) * k)))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * ((2.0 / t_m) / ((((((t_m / l) / l) * t_m) * 2.0) * k) * k));
end

Wolfram

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

Derivation

1. Initial program 57.6%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
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 67.8

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

4. Applied rewrites 67.8%

   \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
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 75.6

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

   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 75.6

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

6. Applied rewrites 75.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 76.2

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

8. Applied rewrites 75.5%

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

9. Taylor expanded in k around 0

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

    1. *-commutative N/A

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

    2. associate-*l/ N/A

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

    3. associate-*l* N/A

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

    4. unpow2 N/A

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

    5. associate-*r* N/A

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

    6. lower-*.f64 N/A

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

    7. lower-*.f64 N/A

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

    8. *-commutative N/A

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

    9. lower-*.f64 N/A

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

    10. unpow2 N/A

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

    11. associate-/l* N/A

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

    12. lower-*.f64 N/A

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

    13. unpow2 N/A

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

    14. associate-/r* N/A

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

    15. lower-/.f64 N/A

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

    16. lower-/.f64 66.2

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

11. Applied rewrites 66.2%

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

12. Final simplification 66.2%

    \[\leadsto \frac{\frac{2}{t}}{\left(\left(\left(\frac{\frac{t}{\ell}}{\ell} \cdot t\right) \cdot 2\right) \cdot k\right) \cdot k} \]
13. Add Preprocessing

Alternative 18: 62.2% accurate, 7.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{\left(\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{\frac{t\_m}{\ell} \cdot t\_m}{\ell}\right) \cdot t\_m} \end{array} \]

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 (* (* (* (* 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) {
	return t_s * (2.0 / ((((k * k) * 2.0) * (((t_m / l) * 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 / ((((k * k) * 2.0d0) * (((t_m / l) * 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 / ((((k * k) * 2.0) * (((t_m / l) * 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 / ((((k * k) * 2.0) * (((t_m / l) * 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(k * k) * 2.0) * Float64(Float64(Float64(t_m / l) * 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 / ((((k * k) * 2.0) * (((t_m / l) * 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[(k * k), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(N[(t$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.6%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
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. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. associate-/r* N/A

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

   12. lower-/.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-pow.f64 56.7

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

5. Applied rewrites 56.7%

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

7. Applied rewrites 32.0%

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

9. Applied rewrites 62.5%

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

10. Final simplification 62.5%

    \[\leadsto \frac{2}{\left(\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{\frac{t}{\ell} \cdot t}{\ell}\right) \cdot t} \]
11. Add Preprocessing

Alternative 19: 53.9% 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 57.6%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
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. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

      \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot k\right)} \cdot 2\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

   10. unpow2 N/A

       \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

   11. associate-/r* N/A

       \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

   12. lower-/.f64 N/A

       \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

   13. lower-/.f64 N/A

       \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{\color{blue}{\frac{{t}^{3}}{\ell}}}{\ell}} \]

   14. lower-pow.f64 56.7

       \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell}} \]

5. Applied rewrites 56.7%

   \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]
6. Step-by-step derivation

7. Applied rewrites 57.5%

   \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \frac{t \cdot \frac{t \cdot t}{\ell}}{\ell}} \]
8. Step-by-step derivation

9. Applied rewrites 55.1%

   \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot 2\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\frac{t}{\ell \cdot \ell}}\right)} \]

10. Final simplification 55.1%

    \[\leadsto \frac{2}{\left(\frac{t}{\ell \cdot \ell} \cdot \left(t \cdot t\right)\right) \cdot \left(\left(k \cdot k\right) \cdot 2\right)} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024304 
(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))))