Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.1% → 89.1%

Time: 17.5s

Alternatives: 20

Speedup: 12.5×

• 27.6% 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: 55.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \sin k\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.55 \cdot 10^{-98}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell \cdot \left(2 \cdot \cos k\right)}{\left(t\_m \cdot k\right) \cdot \mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right)}}{k}\\ \mathbf{elif}\;t\_m \leq 1.65 \cdot 10^{+163}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot t\_2}{\ell} \cdot \left(\frac{t\_m}{\ell} \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(t\_m \cdot \left(\frac{t\_m}{\ell} \cdot \frac{t\_2}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* t_m (sin k))))
   (*
    t_s
    (if (<= t_m 1.55e-98)
      (/
       (*
        l
        (/ (* l (* 2.0 (cos k))) (* (* t_m k) (fma (cos (+ k k)) -0.5 0.5))))
       k)
      (if (<= t_m 1.65e+163)
        (/
         2.0
         (*
          (/ (* t_m t_2) l)
          (* (/ t_m l) (* (tan k) (fma k (/ k (* t_m t_m)) 2.0)))))
        (/
         2.0
         (*
          (* (tan k) (* t_m (* (/ t_m l) (/ t_2 l))))
          (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 t_2 = t_m * sin(k);
	double tmp;
	if (t_m <= 1.55e-98) {
		tmp = (l * ((l * (2.0 * cos(k))) / ((t_m * k) * fma(cos((k + k)), -0.5, 0.5)))) / k;
	} else if (t_m <= 1.65e+163) {
		tmp = 2.0 / (((t_m * t_2) / l) * ((t_m / l) * (tan(k) * fma(k, (k / (t_m * t_m)), 2.0))));
	} else {
		tmp = 2.0 / ((tan(k) * (t_m * ((t_m / l) * (t_2 / l)))) * 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)
	t_2 = Float64(t_m * sin(k))
	tmp = 0.0
	if (t_m <= 1.55e-98)
		tmp = Float64(Float64(l * Float64(Float64(l * Float64(2.0 * cos(k))) / Float64(Float64(t_m * k) * fma(cos(Float64(k + k)), -0.5, 0.5)))) / k);
	elseif (t_m <= 1.65e+163)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * t_2) / l) * Float64(Float64(t_m / l) * Float64(tan(k) * fma(k, Float64(k / Float64(t_m * t_m)), 2.0)))));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(t_m * Float64(Float64(t_m / l) * Float64(t_2 / l)))) * 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_] := Block[{t$95$2 = N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.55e-98], N[(N[(l * N[(N[(l * N[(2.0 * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$m * k), $MachinePrecision] * N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision], If[LessEqual[t$95$m, 1.65e+163], N[(2.0 / N[(N[(N[(t$95$m * t$95$2), $MachinePrecision] / l), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(t$95$m * N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$2 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 3 regimes

2. if t < 1.55e-98

   1. Initial program 47.9%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 63.1

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

   5. Simplified 63.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. associate-/r* N/A

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

      14. lower-/.f64 N/A

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

   7. Applied egg-rr 63.1%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \cos k\right)}{0.5 - 0.5 \cdot \cos \left(k + k\right)}}{k \cdot t}}{k}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l/ N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. associate-/l* N/A

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

   9. Applied egg-rr 71.8%

      \[\leadsto \frac{\color{blue}{\ell \cdot \frac{\ell \cdot \left(2 \cdot \cos k\right)}{\left(t \cdot k\right) \cdot \mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right)}}}{k} \]

   if 1.55e-98 < t < 1.65e163

   1. Initial program 79.3%

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

   4. Applied egg-rr 76.0%

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

   6. Applied egg-rr 96.0%

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

   if 1.65e163 < t

   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. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 88.8

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

   4. Applied egg-rr 88.8%

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

      1. lift-/.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 96.3

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

   6. Applied egg-rr 96.3%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 96.3

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

   8. Applied egg-rr 96.3%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.55 \cdot 10^{-98}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell \cdot \left(2 \cdot \cos k\right)}{\left(t \cdot k\right) \cdot \mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right)}}{k}\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{+163}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(t \cdot \sin k\right)}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.55 \cdot 10^{-98}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell \cdot \left(2 \cdot \cos k\right)}{\left(t\_m \cdot k\right) \cdot \mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right)}}{k}\\ \mathbf{elif}\;t\_m \leq 5 \cdot 10^{+155}:\\ \;\;\;\;\frac{2}{\left(\frac{t\_m}{\ell} \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right)\right)\right) \cdot \left(\sin k \cdot \left(t\_m \cdot \frac{t\_m}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\tan k \cdot \left(t\_m \cdot \left(\frac{t\_m}{\ell} \cdot \frac{t\_m \cdot \sin k}{\ell}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.55e-98)
    (/
     (* l (/ (* l (* 2.0 (cos k))) (* (* t_m k) (fma (cos (+ k k)) -0.5 0.5))))
     k)
    (if (<= t_m 5e+155)
      (/
       2.0
       (*
        (* (/ t_m l) (* (tan k) (fma k (/ k (* t_m t_m)) 2.0)))
        (* (sin k) (* t_m (/ t_m l)))))
      (/
       2.0
       (* 2.0 (* (tan k) (* t_m (* (/ t_m l) (/ (* t_m (sin k)) 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 <= 1.55e-98) {
		tmp = (l * ((l * (2.0 * cos(k))) / ((t_m * k) * fma(cos((k + k)), -0.5, 0.5)))) / k;
	} else if (t_m <= 5e+155) {
		tmp = 2.0 / (((t_m / l) * (tan(k) * fma(k, (k / (t_m * t_m)), 2.0))) * (sin(k) * (t_m * (t_m / l))));
	} else {
		tmp = 2.0 / (2.0 * (tan(k) * (t_m * ((t_m / l) * ((t_m * sin(k)) / 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 <= 1.55e-98)
		tmp = Float64(Float64(l * Float64(Float64(l * Float64(2.0 * cos(k))) / Float64(Float64(t_m * k) * fma(cos(Float64(k + k)), -0.5, 0.5)))) / k);
	elseif (t_m <= 5e+155)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l) * Float64(tan(k) * fma(k, Float64(k / Float64(t_m * t_m)), 2.0))) * Float64(sin(k) * Float64(t_m * Float64(t_m / l)))));
	else
		tmp = Float64(2.0 / Float64(2.0 * Float64(tan(k) * Float64(t_m * Float64(Float64(t_m / l) * Float64(Float64(t_m * sin(k)) / 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, 1.55e-98], N[(N[(l * N[(N[(l * N[(2.0 * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$m * k), $MachinePrecision] * N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision], If[LessEqual[t$95$m, 5e+155], N[(2.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(t$95$m * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(2.0 * N[(N[Tan[k], $MachinePrecision] * N[(t$95$m * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.55e-98

   1. Initial program 47.9%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 63.1

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

   5. Simplified 63.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. associate-/r* N/A

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

      14. lower-/.f64 N/A

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

   7. Applied egg-rr 63.1%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \cos k\right)}{0.5 - 0.5 \cdot \cos \left(k + k\right)}}{k \cdot t}}{k}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l/ N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. associate-/l* N/A

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

   9. Applied egg-rr 71.8%

      \[\leadsto \frac{\color{blue}{\ell \cdot \frac{\ell \cdot \left(2 \cdot \cos k\right)}{\left(t \cdot k\right) \cdot \mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right)}}}{k} \]

   if 1.55e-98 < t < 4.9999999999999999e155

   1. Initial program 79.3%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 92.3

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

   4. Applied egg-rr 92.3%

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

      1. lift-/.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 92.3

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

   6. Applied egg-rr 92.3%

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

   8. Applied egg-rr 94.3%

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

   if 4.9999999999999999e155 < t

   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. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 88.8

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

   4. Applied egg-rr 88.8%

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

      1. lift-/.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 96.3

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

   6. Applied egg-rr 96.3%

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

   7. Taylor expanded in k around 0

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

   9. Simplified 92.8%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.55 \cdot 10^{-98}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell \cdot \left(2 \cdot \cos k\right)}{\left(t \cdot k\right) \cdot \mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right)}}{k}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+155}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right)\right) \cdot \left(\sin k \cdot \left(t \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\tan k \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.55 \cdot 10^{-98}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell \cdot \left(2 \cdot \cos k\right)}{\left(t\_m \cdot k\right) \cdot \mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right)}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot \left(t\_m \cdot \sin k\right)}{\ell} \cdot \left(\frac{t\_m}{\ell} \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.55e-98)
    (/
     (* l (/ (* l (* 2.0 (cos k))) (* (* t_m k) (fma (cos (+ k k)) -0.5 0.5))))
     k)
    (/
     2.0
     (*
      (/ (* t_m (* t_m (sin k))) l)
      (* (/ t_m l) (* (tan k) (fma k (/ k (* t_m t_m)) 2.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.55e-98) {
		tmp = (l * ((l * (2.0 * cos(k))) / ((t_m * k) * fma(cos((k + k)), -0.5, 0.5)))) / k;
	} else {
		tmp = 2.0 / (((t_m * (t_m * sin(k))) / l) * ((t_m / l) * (tan(k) * fma(k, (k / (t_m * t_m)), 2.0))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.55e-98)
		tmp = Float64(Float64(l * Float64(Float64(l * Float64(2.0 * cos(k))) / Float64(Float64(t_m * k) * fma(cos(Float64(k + k)), -0.5, 0.5)))) / k);
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * Float64(t_m * sin(k))) / l) * Float64(Float64(t_m / l) * Float64(tan(k) * fma(k, Float64(k / Float64(t_m * t_m)), 2.0)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.55e-98], N[(N[(l * N[(N[(l * N[(2.0 * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$m * k), $MachinePrecision] * N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.55e-98

   1. Initial program 47.9%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 63.1

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

   5. Simplified 63.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. associate-/r* N/A

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

      14. lower-/.f64 N/A

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

   7. Applied egg-rr 63.1%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \cos k\right)}{0.5 - 0.5 \cdot \cos \left(k + k\right)}}{k \cdot t}}{k}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l/ N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. associate-/l* N/A

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

   9. Applied egg-rr 71.8%

      \[\leadsto \frac{\color{blue}{\ell \cdot \frac{\ell \cdot \left(2 \cdot \cos k\right)}{\left(t \cdot k\right) \cdot \mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right)}}}{k} \]

   if 1.55e-98 < t

   1. Initial program 71.2%

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

   4. Applied egg-rr 64.9%

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

   6. Applied egg-rr 94.8%

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

Alternative 4: 76.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\ell}{t\_m \cdot k}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.7 \cdot 10^{-33}:\\ \;\;\;\;\ell \cdot \frac{1}{\frac{t\_m \cdot \left(t\_m \cdot k\right)}{t\_2}}\\ \mathbf{elif}\;k \leq 3.9 \cdot 10^{+87}:\\ \;\;\;\;\frac{1}{\left(\sin k \cdot \tan k\right) \cdot \frac{k \cdot \left(t\_m \cdot k\right)}{2 \cdot \left(\ell \cdot \ell\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2 \cdot \frac{\ell \cdot \left(2 \cdot \cos k\right)}{\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right)}}{k}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ l (* t_m k))))
   (*
    t_s
    (if (<= k 2.7e-33)
      (* l (/ 1.0 (/ (* t_m (* t_m k)) t_2)))
      (if (<= k 3.9e+87)
        (/ 1.0 (* (* (sin k) (tan k)) (/ (* k (* t_m k)) (* 2.0 (* l l)))))
        (/
         (* t_2 (/ (* l (* 2.0 (cos k))) (fma (cos (+ k k)) -0.5 0.5)))
         k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = l / (t_m * k);
	double tmp;
	if (k <= 2.7e-33) {
		tmp = l * (1.0 / ((t_m * (t_m * k)) / t_2));
	} else if (k <= 3.9e+87) {
		tmp = 1.0 / ((sin(k) * tan(k)) * ((k * (t_m * k)) / (2.0 * (l * l))));
	} else {
		tmp = (t_2 * ((l * (2.0 * cos(k))) / fma(cos((k + k)), -0.5, 0.5))) / k;
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(l / Float64(t_m * k))
	tmp = 0.0
	if (k <= 2.7e-33)
		tmp = Float64(l * Float64(1.0 / Float64(Float64(t_m * Float64(t_m * k)) / t_2)));
	elseif (k <= 3.9e+87)
		tmp = Float64(1.0 / Float64(Float64(sin(k) * tan(k)) * Float64(Float64(k * Float64(t_m * k)) / Float64(2.0 * Float64(l * l)))));
	else
		tmp = Float64(Float64(t_2 * Float64(Float64(l * Float64(2.0 * cos(k))) / fma(cos(Float64(k + k)), -0.5, 0.5))) / k);
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 2.7e-33], N[(l * N[(1.0 / N[(N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 3.9e+87], N[(1.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(k * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 * N[(N[(l * N[(2.0 * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 2.7000000000000001e-33

   1. Initial program 55.9%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 51.7

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

   5. Simplified 51.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 56.3

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 59.6

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

   7. Applied egg-rr 59.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 64.4

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

   9. Applied egg-rr 64.4%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. associate-/l/ N/A

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

       6. clear-num N/A

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

       7. lower-/.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. associate-*r* N/A

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

       12. lift-*.f64 N/A

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

       13. *-commutative N/A

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

       14. lower-*.f64 N/A

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

       15. lift-*.f64 N/A

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

       16. *-commutative N/A

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

       17. lower-*.f64 N/A

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

       18. lower-/.f64 76.3

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

       19. lift-*.f64 N/A

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

       20. *-commutative N/A

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

       21. lower-*.f64 76.3

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

   11. Applied egg-rr 76.3%

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

   if 2.7000000000000001e-33 < k < 3.9000000000000002e87

   1. Initial program 39.7%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 75.4

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

   5. Simplified 75.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. clear-num N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. times-frac N/A

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

   7. Applied egg-rr 79.1%

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

   if 3.9000000000000002e87 < k

   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 t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 70.6

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

   5. Simplified 70.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. associate-/r* N/A

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

      14. lower-/.f64 N/A

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

   7. Applied egg-rr 84.0%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \cos k\right)}{0.5 - 0.5 \cdot \cos \left(k + k\right)}}{k \cdot t}}{k}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l/ N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. times-frac N/A

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

   9. Applied egg-rr 93.2%

      \[\leadsto \frac{\color{blue}{\frac{\ell}{t \cdot k} \cdot \frac{\ell \cdot \left(2 \cdot \cos k\right)}{\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right)}}}{k} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.7 \cdot 10^{-33}:\\ \;\;\;\;\ell \cdot \frac{1}{\frac{t \cdot \left(t \cdot k\right)}{\frac{\ell}{t \cdot k}}}\\ \mathbf{elif}\;k \leq 3.9 \cdot 10^{+87}:\\ \;\;\;\;\frac{1}{\left(\sin k \cdot \tan k\right) \cdot \frac{k \cdot \left(t \cdot k\right)}{2 \cdot \left(\ell \cdot \ell\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot k} \cdot \frac{\ell \cdot \left(2 \cdot \cos k\right)}{\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right)}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.7 \cdot 10^{-33}:\\ \;\;\;\;\ell \cdot \frac{1}{\frac{t\_m \cdot \left(t\_m \cdot k\right)}{\frac{\ell}{t\_m \cdot k}}}\\ \mathbf{elif}\;k \leq 3.9 \cdot 10^{+87}:\\ \;\;\;\;\frac{1}{\left(\sin k \cdot \tan k\right) \cdot \frac{k \cdot \left(t\_m \cdot k\right)}{2 \cdot \left(\ell \cdot \ell\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell \cdot \left(2 \cdot \cos k\right)}{\left(t\_m \cdot k\right) \cdot \mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right)}}{k}\\ \end{array} \end{array} \]

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 2.7e-33)
    (* l (/ 1.0 (/ (* t_m (* t_m k)) (/ l (* t_m k)))))
    (if (<= k 3.9e+87)
      (/ 1.0 (* (* (sin k) (tan k)) (/ (* k (* t_m k)) (* 2.0 (* l l)))))
      (/
       (*
        l
        (/ (* l (* 2.0 (cos k))) (* (* t_m k) (fma (cos (+ k k)) -0.5 0.5))))
       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 <= 2.7e-33) {
		tmp = l * (1.0 / ((t_m * (t_m * k)) / (l / (t_m * k))));
	} else if (k <= 3.9e+87) {
		tmp = 1.0 / ((sin(k) * tan(k)) * ((k * (t_m * k)) / (2.0 * (l * l))));
	} else {
		tmp = (l * ((l * (2.0 * cos(k))) / ((t_m * k) * fma(cos((k + k)), -0.5, 0.5)))) / 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 <= 2.7e-33)
		tmp = Float64(l * Float64(1.0 / Float64(Float64(t_m * Float64(t_m * k)) / Float64(l / Float64(t_m * k)))));
	elseif (k <= 3.9e+87)
		tmp = Float64(1.0 / Float64(Float64(sin(k) * tan(k)) * Float64(Float64(k * Float64(t_m * k)) / Float64(2.0 * Float64(l * l)))));
	else
		tmp = Float64(Float64(l * Float64(Float64(l * Float64(2.0 * cos(k))) / Float64(Float64(t_m * k) * fma(cos(Float64(k + k)), -0.5, 0.5)))) / 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, 2.7e-33], N[(l * N[(1.0 / N[(N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 3.9e+87], N[(1.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(k * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * N[(N[(l * N[(2.0 * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$m * k), $MachinePrecision] * N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 2.7000000000000001e-33

   1. Initial program 55.9%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 51.7

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

   5. Simplified 51.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 56.3

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 59.6

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

   7. Applied egg-rr 59.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 64.4

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

   9. Applied egg-rr 64.4%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. associate-/l/ N/A

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

       6. clear-num N/A

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

       7. lower-/.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. associate-*r* N/A

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

       12. lift-*.f64 N/A

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

       13. *-commutative N/A

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

       14. lower-*.f64 N/A

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

       15. lift-*.f64 N/A

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

       16. *-commutative N/A

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

       17. lower-*.f64 N/A

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

       18. lower-/.f64 76.3

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

       19. lift-*.f64 N/A

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

       20. *-commutative N/A

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

       21. lower-*.f64 76.3

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

   11. Applied egg-rr 76.3%

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

   if 2.7000000000000001e-33 < k < 3.9000000000000002e87

   1. Initial program 39.7%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 75.4

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

   5. Simplified 75.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. clear-num N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. times-frac N/A

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

   7. Applied egg-rr 79.1%

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

   if 3.9000000000000002e87 < k

   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 t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 70.6

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

   5. Simplified 70.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. associate-/r* N/A

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

      14. lower-/.f64 N/A

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

   7. Applied egg-rr 84.0%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \cos k\right)}{0.5 - 0.5 \cdot \cos \left(k + k\right)}}{k \cdot t}}{k}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l/ N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. associate-/l* N/A

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

   9. Applied egg-rr 93.2%

      \[\leadsto \frac{\color{blue}{\ell \cdot \frac{\ell \cdot \left(2 \cdot \cos k\right)}{\left(t \cdot k\right) \cdot \mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right)}}}{k} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.7 \cdot 10^{-33}:\\ \;\;\;\;\ell \cdot \frac{1}{\frac{t \cdot \left(t \cdot k\right)}{\frac{\ell}{t \cdot k}}}\\ \mathbf{elif}\;k \leq 3.9 \cdot 10^{+87}:\\ \;\;\;\;\frac{1}{\left(\sin k \cdot \tan k\right) \cdot \frac{k \cdot \left(t \cdot k\right)}{2 \cdot \left(\ell \cdot \ell\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell \cdot \left(2 \cdot \cos k\right)}{\left(t \cdot k\right) \cdot \mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right)}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.7 \cdot 10^{-16}:\\ \;\;\;\;\ell \cdot \frac{1}{\frac{t\_m \cdot \left(t\_m \cdot k\right)}{\frac{\ell}{t\_m \cdot k}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot 2\right) \cdot \left(\ell \cdot \frac{\cos k}{\left(k \cdot \left(t\_m \cdot k\right)\right) \cdot \left(0.5 - \cos \left(k + k\right) \cdot 0.5\right)}\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 3.7e-16)
    (* l (/ 1.0 (/ (* t_m (* t_m k)) (/ l (* t_m k)))))
    (*
     (* l 2.0)
     (* l (/ (cos k) (* (* k (* t_m k)) (- 0.5 (* (cos (+ k k)) 0.5)))))))))

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 <= 3.7e-16) {
		tmp = l * (1.0 / ((t_m * (t_m * k)) / (l / (t_m * k))));
	} else {
		tmp = (l * 2.0) * (l * (cos(k) / ((k * (t_m * k)) * (0.5 - (cos((k + k)) * 0.5)))));
	}
	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 <= 3.7d-16) then
        tmp = l * (1.0d0 / ((t_m * (t_m * k)) / (l / (t_m * k))))
    else
        tmp = (l * 2.0d0) * (l * (cos(k) / ((k * (t_m * k)) * (0.5d0 - (cos((k + k)) * 0.5d0)))))
    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 <= 3.7e-16) {
		tmp = l * (1.0 / ((t_m * (t_m * k)) / (l / (t_m * k))));
	} else {
		tmp = (l * 2.0) * (l * (Math.cos(k) / ((k * (t_m * k)) * (0.5 - (Math.cos((k + k)) * 0.5)))));
	}
	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 <= 3.7e-16:
		tmp = l * (1.0 / ((t_m * (t_m * k)) / (l / (t_m * k))))
	else:
		tmp = (l * 2.0) * (l * (math.cos(k) / ((k * (t_m * k)) * (0.5 - (math.cos((k + k)) * 0.5)))))
	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 <= 3.7e-16)
		tmp = Float64(l * Float64(1.0 / Float64(Float64(t_m * Float64(t_m * k)) / Float64(l / Float64(t_m * k)))));
	else
		tmp = Float64(Float64(l * 2.0) * Float64(l * Float64(cos(k) / Float64(Float64(k * Float64(t_m * k)) * Float64(0.5 - Float64(cos(Float64(k + k)) * 0.5))))));
	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 <= 3.7e-16)
		tmp = l * (1.0 / ((t_m * (t_m * k)) / (l / (t_m * k))));
	else
		tmp = (l * 2.0) * (l * (cos(k) / ((k * (t_m * k)) * (0.5 - (cos((k + k)) * 0.5)))));
	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, 3.7e-16], N[(l * N[(1.0 / N[(N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * 2.0), $MachinePrecision] * N[(l * N[(N[Cos[k], $MachinePrecision] / N[(N[(k * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.7e-16

   1. Initial program 56.0%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 52.0

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

   5. Simplified 52.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 56.5

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 59.7

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

   7. Applied egg-rr 59.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 64.4

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

   9. Applied egg-rr 64.4%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. associate-/l/ N/A

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

       6. clear-num N/A

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

       7. lower-/.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. associate-*r* N/A

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

       12. lift-*.f64 N/A

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

       13. *-commutative N/A

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

       14. lower-*.f64 N/A

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

       15. lift-*.f64 N/A

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

       16. *-commutative N/A

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

       17. lower-*.f64 N/A

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

       18. lower-/.f64 76.0

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

       19. lift-*.f64 N/A

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

       20. *-commutative N/A

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

       21. lower-*.f64 76.0

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

   11. Applied egg-rr 76.0%

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

   if 3.7e-16 < k

   1. Initial program 52.3%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 72.1

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

   5. Simplified 72.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 75.6

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

   7. Applied egg-rr 79.0%

      \[\leadsto \color{blue}{\left(2 \cdot \ell\right) \cdot \left(\ell \cdot \frac{\cos k}{\left(k \cdot \left(k \cdot t\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.7 \cdot 10^{-16}:\\ \;\;\;\;\ell \cdot \frac{1}{\frac{t \cdot \left(t \cdot k\right)}{\frac{\ell}{t \cdot k}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot 2\right) \cdot \left(\ell \cdot \frac{\cos k}{\left(k \cdot \left(t \cdot k\right)\right) \cdot \left(0.5 - \cos \left(k + k\right) \cdot 0.5\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.7 \cdot 10^{-16}:\\ \;\;\;\;\ell \cdot \frac{1}{\frac{t\_m \cdot \left(t\_m \cdot k\right)}{\frac{\ell}{t\_m \cdot k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \left(\ell \cdot \left(2 \cdot \cos k\right)\right)}{k \cdot \left(\left(t\_m \cdot k\right) \cdot \mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 3.7e-16)
    (* l (/ 1.0 (/ (* t_m (* t_m k)) (/ l (* t_m k)))))
    (/
     (* l (* l (* 2.0 (cos k))))
     (* k (* (* t_m k) (fma (cos (+ k k)) -0.5 0.5)))))))

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 <= 3.7e-16) {
		tmp = l * (1.0 / ((t_m * (t_m * k)) / (l / (t_m * k))));
	} else {
		tmp = (l * (l * (2.0 * cos(k)))) / (k * ((t_m * k) * fma(cos((k + k)), -0.5, 0.5)));
	}
	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 <= 3.7e-16)
		tmp = Float64(l * Float64(1.0 / Float64(Float64(t_m * Float64(t_m * k)) / Float64(l / Float64(t_m * k)))));
	else
		tmp = Float64(Float64(l * Float64(l * Float64(2.0 * cos(k)))) / Float64(k * Float64(Float64(t_m * k) * fma(cos(Float64(k + k)), -0.5, 0.5))));
	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, 3.7e-16], N[(l * N[(1.0 / N[(N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * N[(l * N[(2.0 * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * N[(N[(t$95$m * k), $MachinePrecision] * N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.7e-16

   1. Initial program 56.0%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 52.0

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

   5. Simplified 52.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 56.5

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 59.7

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

   7. Applied egg-rr 59.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 64.4

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

   9. Applied egg-rr 64.4%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. associate-/l/ N/A

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

       6. clear-num N/A

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

       7. lower-/.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. associate-*r* N/A

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

       12. lift-*.f64 N/A

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

       13. *-commutative N/A

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

       14. lower-*.f64 N/A

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

       15. lift-*.f64 N/A

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

       16. *-commutative N/A

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

       17. lower-*.f64 N/A

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

       18. lower-/.f64 76.0

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

       19. lift-*.f64 N/A

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

       20. *-commutative N/A

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

       21. lower-*.f64 76.0

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

   11. Applied egg-rr 76.0%

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

   if 3.7e-16 < k

   1. Initial program 52.3%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 72.1

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

   5. Simplified 72.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. associate-/r* N/A

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

      14. lower-/.f64 N/A

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

   7. Applied egg-rr 80.6%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \cos k\right)}{0.5 - 0.5 \cdot \cos \left(k + k\right)}}{k \cdot t}}{k}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l/ N/A

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

      11. associate-/l/ N/A

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

      12. lower-/.f64 N/A

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

   9. Applied egg-rr 75.9%

      \[\leadsto \color{blue}{\frac{\ell \cdot \left(\ell \cdot \left(2 \cdot \cos k\right)\right)}{k \cdot \left(\left(t \cdot k\right) \cdot \mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.7 \cdot 10^{-16}:\\ \;\;\;\;\ell \cdot \frac{1}{\frac{t \cdot \left(t \cdot k\right)}{\frac{\ell}{t \cdot k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \left(\ell \cdot \left(2 \cdot \cos k\right)\right)}{k \cdot \left(\left(t \cdot k\right) \cdot \mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.46 \cdot 10^{-204}:\\ \;\;\;\;\frac{\frac{\frac{\left(2 \cdot \cos k\right) \cdot \left(\ell \cdot \ell\right)}{k \cdot k}}{t\_m \cdot k}}{k}\\ \mathbf{elif}\;t\_m \leq 68:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right) \cdot \left(\tan k \cdot \left(t\_m \cdot \left(\frac{t\_m}{\ell} \cdot \left(k \cdot \mathsf{fma}\left(k, k \cdot \mathsf{fma}\left(0.008333333333333333, t\_m \cdot \frac{k \cdot k}{\ell}, \frac{t\_m \cdot -0.16666666666666666}{\ell}\right), \frac{t\_m}{\ell}\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{1}{\frac{t\_m \cdot \left(t\_m \cdot k\right)}{\frac{\ell}{t\_m \cdot k}}}\\ \end{array} \end{array} \]

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 (<= t_m 1.46e-204)
    (/ (/ (/ (* (* 2.0 (cos k)) (* l l)) (* k k)) (* t_m k)) k)
    (if (<= t_m 68.0)
      (/
       2.0
       (*
        (fma (/ k t_m) (/ k t_m) 2.0)
        (*
         (tan k)
         (*
          t_m
          (*
           (/ t_m l)
           (*
            k
            (fma
             k
             (*
              k
              (fma
               0.008333333333333333
               (* t_m (/ (* k k) l))
               (/ (* t_m -0.16666666666666666) l)))
             (/ t_m l))))))))
      (* l (/ 1.0 (/ (* t_m (* t_m k)) (/ l (* t_m k)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.46e-204) {
		tmp = ((((2.0 * cos(k)) * (l * l)) / (k * k)) / (t_m * k)) / k;
	} else if (t_m <= 68.0) {
		tmp = 2.0 / (fma((k / t_m), (k / t_m), 2.0) * (tan(k) * (t_m * ((t_m / l) * (k * fma(k, (k * fma(0.008333333333333333, (t_m * ((k * k) / l)), ((t_m * -0.16666666666666666) / l))), (t_m / l)))))));
	} else {
		tmp = l * (1.0 / ((t_m * (t_m * k)) / (l / (t_m * k))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.46e-204)
		tmp = Float64(Float64(Float64(Float64(Float64(2.0 * cos(k)) * Float64(l * l)) / Float64(k * k)) / Float64(t_m * k)) / k);
	elseif (t_m <= 68.0)
		tmp = Float64(2.0 / Float64(fma(Float64(k / t_m), Float64(k / t_m), 2.0) * Float64(tan(k) * Float64(t_m * Float64(Float64(t_m / l) * Float64(k * fma(k, Float64(k * fma(0.008333333333333333, Float64(t_m * Float64(Float64(k * k) / l)), Float64(Float64(t_m * -0.16666666666666666) / l))), Float64(t_m / l))))))));
	else
		tmp = Float64(l * Float64(1.0 / Float64(Float64(t_m * Float64(t_m * k)) / Float64(l / Float64(t_m * k)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.46e-204], N[(N[(N[(N[(N[(2.0 * N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision], If[LessEqual[t$95$m, 68.0], N[(2.0 / N[(N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(t$95$m * N[(N[(t$95$m / l), $MachinePrecision] * N[(k * N[(k * N[(k * N[(0.008333333333333333 * N[(t$95$m * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$m * -0.16666666666666666), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(1.0 / N[(N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.45999999999999998e-204

   1. Initial program 46.2%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 61.1

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

   5. Simplified 61.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2}}}{\color{blue}{t \cdot \left(k \cdot k\right)}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2}}}{t \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2}}}{\color{blue}{\left(t \cdot k\right) \cdot k}} \]

      13. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2}}}{t \cdot k}}{k}} \]

      14. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2}}}{t \cdot k}}{k}} \]

   7. Applied egg-rr 61.1%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \cos k\right)}{0.5 - 0.5 \cdot \cos \left(k + k\right)}}{k \cdot t}}{k}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{\frac{\frac{\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \cos k\right)}{\color{blue}{{k}^{2}}}}{k \cdot t}}{k} \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\frac{\frac{\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \cos k\right)}{\color{blue}{k \cdot k}}}{k \cdot t}}{k} \]

      2. lower-*.f64 55.0

         \[\leadsto \frac{\frac{\frac{\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \cos k\right)}{\color{blue}{k \cdot k}}}{k \cdot t}}{k} \]

   10. Simplified 55.0%

       \[\leadsto \frac{\frac{\frac{\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \cos k\right)}{\color{blue}{k \cdot k}}}{k \cdot t}}{k} \]

   if 1.45999999999999998e-204 < t < 68

   1. Initial program 69.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\sin k}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. associate-*l/ N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{{t}^{3} \cdot \sin k}{\color{blue}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{3}} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. cube-mult N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \sin k\right)}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. times-frac N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{t}{\ell}} \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \color{blue}{\frac{\left(t \cdot t\right) \cdot \sin k}{\ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. associate-*l* N/A

          \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \frac{\color{blue}{t \cdot \left(t \cdot \sin k\right)}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \frac{\color{blue}{t \cdot \left(t \cdot \sin k\right)}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. lower-*.f64 78.3

          \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \frac{t \cdot \color{blue}{\left(t \cdot \sin k\right)}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied egg-rr 78.3%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{t \cdot \left(t \cdot \sin k\right)}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{t}{\ell}} \cdot \frac{t \cdot \left(t \cdot \sin k\right)}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \frac{t \cdot \left(t \cdot \color{blue}{\sin k}\right)}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \frac{t \cdot \color{blue}{\left(t \cdot \sin k\right)}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \frac{\color{blue}{t \cdot \left(t \cdot \sin k\right)}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \color{blue}{\frac{t \cdot \left(t \cdot \sin k\right)}{\ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot \left(t \cdot \sin k\right)}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{t \cdot \left(t \cdot \sin k\right)}{\ell}} \cdot \frac{t}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{t \cdot \left(t \cdot \sin k\right)}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(t \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \frac{t}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. associate-*l* N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(t \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(t \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(t \cdot \color{blue}{\left(\frac{t \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-/.f64 87.9

          \[\leadsto \frac{2}{\left(\left(t \cdot \left(\color{blue}{\frac{t \cdot \sin k}{\ell}} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied egg-rr 87.9%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(t \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\color{blue}{\left(\frac{k}{t}\right)}}^{2}\right) + 1\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right) + 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} + 1\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 + 1\right)\right)}} \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\color{blue}{{\left(\frac{k}{t}\right)}^{2}} + \left(1 + 1\right)\right)} \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + \left(1 + 1\right)\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\frac{k}{t} \cdot \frac{k}{t} + \color{blue}{2}\right)} \]

      8. lower-fma.f64 87.9

         \[\leadsto \frac{2}{\left(\left(t \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}} \]

   8. Applied egg-rr 87.9%

      \[\leadsto \frac{2}{\left(\left(t \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\left(t \cdot \left(\color{blue}{\left(k \cdot \left({k}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{t}{\ell} + \frac{1}{120} \cdot \frac{{k}^{2} \cdot t}{\ell}\right) + \frac{t}{\ell}\right)\right)} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(t \cdot \left(\color{blue}{\left(k \cdot \left({k}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{t}{\ell} + \frac{1}{120} \cdot \frac{{k}^{2} \cdot t}{\ell}\right) + \frac{t}{\ell}\right)\right)} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)} \]

       2. unpow2 N/A

          \[\leadsto \frac{2}{\left(\left(t \cdot \left(\left(k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \left(\frac{-1}{6} \cdot \frac{t}{\ell} + \frac{1}{120} \cdot \frac{{k}^{2} \cdot t}{\ell}\right) + \frac{t}{\ell}\right)\right) \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)} \]

       3. associate-*l* N/A

          \[\leadsto \frac{2}{\left(\left(t \cdot \left(\left(k \cdot \left(\color{blue}{k \cdot \left(k \cdot \left(\frac{-1}{6} \cdot \frac{t}{\ell} + \frac{1}{120} \cdot \frac{{k}^{2} \cdot t}{\ell}\right)\right)} + \frac{t}{\ell}\right)\right) \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)} \]

       4. lower-fma.f64 N/A

          \[\leadsto \frac{2}{\left(\left(t \cdot \left(\left(k \cdot \color{blue}{\mathsf{fma}\left(k, k \cdot \left(\frac{-1}{6} \cdot \frac{t}{\ell} + \frac{1}{120} \cdot \frac{{k}^{2} \cdot t}{\ell}\right), \frac{t}{\ell}\right)}\right) \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)} \]

   11. Simplified 73.1%

       \[\leadsto \frac{2}{\left(\left(t \cdot \left(\color{blue}{\left(k \cdot \mathsf{fma}\left(k, k \cdot \mathsf{fma}\left(0.008333333333333333, t \cdot \frac{k \cdot k}{\ell}, \frac{t \cdot -0.16666666666666666}{\ell}\right), \frac{t}{\ell}\right)\right)} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)} \]

   if 68 < t

   1. Initial program 66.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. lower-*.f64 53.7

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 53.7%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot k\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      8. lower-/.f64 55.6

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      11. *-commutative N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      12. associate-*l* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      14. lower-*.f64 59.1

          \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   7. Applied egg-rr 59.1%

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \cdot \ell \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot t\right)}} \cdot \ell \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right)} \cdot \left(t \cdot t\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

      7. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot k\right) \cdot k\right)} \cdot \left(t \cdot t\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      10. *-commutative N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

      12. lower-*.f64 65.6

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

   9. Applied egg-rr 65.6%

      \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot \color{blue}{\left(t \cdot t\right)}\right)} \cdot \ell \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

       4. *-commutative N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)}} \cdot \ell \]

       5. associate-/l/ N/A

          \[\leadsto \color{blue}{\frac{\frac{\ell}{k \cdot t}}{k \cdot \left(t \cdot t\right)}} \cdot \ell \]

       6. clear-num N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{k \cdot \left(t \cdot t\right)}{\frac{\ell}{k \cdot t}}}} \cdot \ell \]

       7. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{k \cdot \left(t \cdot t\right)}{\frac{\ell}{k \cdot t}}}} \cdot \ell \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{k \cdot \left(t \cdot t\right)}{\frac{\ell}{k \cdot t}}}} \cdot \ell \]

       9. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{k \cdot \left(t \cdot t\right)}}{\frac{\ell}{k \cdot t}}} \cdot \ell \]

       10. lift-*.f64 N/A

           \[\leadsto \frac{1}{\frac{k \cdot \color{blue}{\left(t \cdot t\right)}}{\frac{\ell}{k \cdot t}}} \cdot \ell \]

       11. associate-*r* N/A

           \[\leadsto \frac{1}{\frac{\color{blue}{\left(k \cdot t\right) \cdot t}}{\frac{\ell}{k \cdot t}}} \cdot \ell \]

       12. lift-*.f64 N/A

           \[\leadsto \frac{1}{\frac{\color{blue}{\left(k \cdot t\right)} \cdot t}{\frac{\ell}{k \cdot t}}} \cdot \ell \]

       13. *-commutative N/A

           \[\leadsto \frac{1}{\frac{\color{blue}{t \cdot \left(k \cdot t\right)}}{\frac{\ell}{k \cdot t}}} \cdot \ell \]

       14. lower-*.f64 N/A

           \[\leadsto \frac{1}{\frac{\color{blue}{t \cdot \left(k \cdot t\right)}}{\frac{\ell}{k \cdot t}}} \cdot \ell \]

       15. lift-*.f64 N/A

           \[\leadsto \frac{1}{\frac{t \cdot \color{blue}{\left(k \cdot t\right)}}{\frac{\ell}{k \cdot t}}} \cdot \ell \]

       16. *-commutative N/A

           \[\leadsto \frac{1}{\frac{t \cdot \color{blue}{\left(t \cdot k\right)}}{\frac{\ell}{k \cdot t}}} \cdot \ell \]

       17. lower-*.f64 N/A

           \[\leadsto \frac{1}{\frac{t \cdot \color{blue}{\left(t \cdot k\right)}}{\frac{\ell}{k \cdot t}}} \cdot \ell \]

       18. lower-/.f64 86.1

           \[\leadsto \frac{1}{\frac{t \cdot \left(t \cdot k\right)}{\color{blue}{\frac{\ell}{k \cdot t}}}} \cdot \ell \]

       19. lift-*.f64 N/A

           \[\leadsto \frac{1}{\frac{t \cdot \left(t \cdot k\right)}{\frac{\ell}{\color{blue}{k \cdot t}}}} \cdot \ell \]

       20. *-commutative N/A

           \[\leadsto \frac{1}{\frac{t \cdot \left(t \cdot k\right)}{\frac{\ell}{\color{blue}{t \cdot k}}}} \cdot \ell \]

       21. lower-*.f64 86.1

           \[\leadsto \frac{1}{\frac{t \cdot \left(t \cdot k\right)}{\frac{\ell}{\color{blue}{t \cdot k}}}} \cdot \ell \]

   11. Applied egg-rr 86.1%

       \[\leadsto \color{blue}{\frac{1}{\frac{t \cdot \left(t \cdot k\right)}{\frac{\ell}{t \cdot k}}}} \cdot \ell \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.46 \cdot 10^{-204}:\\ \;\;\;\;\frac{\frac{\frac{\left(2 \cdot \cos k\right) \cdot \left(\ell \cdot \ell\right)}{k \cdot k}}{t \cdot k}}{k}\\ \mathbf{elif}\;t \leq 68:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \left(\tan k \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot \mathsf{fma}\left(k, k \cdot \mathsf{fma}\left(0.008333333333333333, t \cdot \frac{k \cdot k}{\ell}, \frac{t \cdot -0.16666666666666666}{\ell}\right), \frac{t}{\ell}\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{1}{\frac{t \cdot \left(t \cdot k\right)}{\frac{\ell}{t \cdot k}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.55 \cdot 10^{-204}:\\ \;\;\;\;\frac{\frac{\frac{\left(2 \cdot \cos k\right) \cdot \left(\ell \cdot \ell\right)}{k \cdot k}}{t\_m \cdot k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right) \cdot \left(\tan k \cdot \left(t\_m \cdot \left(\frac{t\_m}{\ell} \cdot \frac{t\_m \cdot k}{\ell}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.55e-204)
    (/ (/ (/ (* (* 2.0 (cos k)) (* l l)) (* k k)) (* t_m k)) k)
    (/
     2.0
     (*
      (fma (/ k t_m) (/ k t_m) 2.0)
      (* (tan k) (* t_m (* (/ t_m l) (/ (* t_m k) 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 <= 1.55e-204) {
		tmp = ((((2.0 * cos(k)) * (l * l)) / (k * k)) / (t_m * k)) / k;
	} else {
		tmp = 2.0 / (fma((k / t_m), (k / t_m), 2.0) * (tan(k) * (t_m * ((t_m / l) * ((t_m * k) / 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 <= 1.55e-204)
		tmp = Float64(Float64(Float64(Float64(Float64(2.0 * cos(k)) * Float64(l * l)) / Float64(k * k)) / Float64(t_m * k)) / k);
	else
		tmp = Float64(2.0 / Float64(fma(Float64(k / t_m), Float64(k / t_m), 2.0) * Float64(tan(k) * Float64(t_m * Float64(Float64(t_m / l) * Float64(Float64(t_m * k) / 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, 1.55e-204], N[(N[(N[(N[(N[(2.0 * N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision], N[(2.0 / N[(N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(t$95$m * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.55e-204

   1. Initial program 46.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot {\ell}^{2}\right) \cdot \cos k}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot {\ell}^{2}\right) \cdot \cos k}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot {\ell}^{2}\right)} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. unpow2 N/A

         \[\leadsto \frac{\left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\cos k}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)}} \]

      12. lower-pow.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{{\sin k}^{2}} \cdot \left({k}^{2} \cdot t\right)} \]

      13. lower-sin.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\color{blue}{\sin k}}^{2} \cdot \left({k}^{2} \cdot t\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}} \]

      16. unpow2 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

      17. lower-*.f64 61.1

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   5. Simplified 61.1%

      \[\leadsto \color{blue}{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot \left(\ell \cdot \ell\right)\right)} \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      3. lift-cos.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\cos k}}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\color{blue}{\sin k}}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{{\sin k}^{2}} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \]

      9. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2}}}{t \cdot \left(k \cdot k\right)}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2}}}{\color{blue}{t \cdot \left(k \cdot k\right)}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2}}}{t \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2}}}{\color{blue}{\left(t \cdot k\right) \cdot k}} \]

      13. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2}}}{t \cdot k}}{k}} \]

      14. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2}}}{t \cdot k}}{k}} \]

   7. Applied egg-rr 61.1%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \cos k\right)}{0.5 - 0.5 \cdot \cos \left(k + k\right)}}{k \cdot t}}{k}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{\frac{\frac{\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \cos k\right)}{\color{blue}{{k}^{2}}}}{k \cdot t}}{k} \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\frac{\frac{\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \cos k\right)}{\color{blue}{k \cdot k}}}{k \cdot t}}{k} \]

      2. lower-*.f64 55.0

         \[\leadsto \frac{\frac{\frac{\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \cos k\right)}{\color{blue}{k \cdot k}}}{k \cdot t}}{k} \]

   10. Simplified 55.0%

       \[\leadsto \frac{\frac{\frac{\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \cos k\right)}{\color{blue}{k \cdot k}}}{k \cdot t}}{k} \]

   if 1.55e-204 < t

   1. Initial program 67.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\sin k}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. associate-*l/ N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{{t}^{3} \cdot \sin k}{\color{blue}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{3}} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. cube-mult N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \sin k\right)}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. times-frac N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{t}{\ell}} \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \color{blue}{\frac{\left(t \cdot t\right) \cdot \sin k}{\ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. associate-*l* N/A

          \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \frac{\color{blue}{t \cdot \left(t \cdot \sin k\right)}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \frac{\color{blue}{t \cdot \left(t \cdot \sin k\right)}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. lower-*.f64 85.2

          \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \frac{t \cdot \color{blue}{\left(t \cdot \sin k\right)}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied egg-rr 85.2%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{t \cdot \left(t \cdot \sin k\right)}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{t}{\ell}} \cdot \frac{t \cdot \left(t \cdot \sin k\right)}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \frac{t \cdot \left(t \cdot \color{blue}{\sin k}\right)}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \frac{t \cdot \color{blue}{\left(t \cdot \sin k\right)}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \frac{\color{blue}{t \cdot \left(t \cdot \sin k\right)}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \color{blue}{\frac{t \cdot \left(t \cdot \sin k\right)}{\ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot \left(t \cdot \sin k\right)}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{t \cdot \left(t \cdot \sin k\right)}{\ell}} \cdot \frac{t}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{t \cdot \left(t \cdot \sin k\right)}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(t \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \frac{t}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. associate-*l* N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(t \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(t \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(t \cdot \color{blue}{\left(\frac{t \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-/.f64 91.1

          \[\leadsto \frac{2}{\left(\left(t \cdot \left(\color{blue}{\frac{t \cdot \sin k}{\ell}} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied egg-rr 91.1%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(t \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\color{blue}{\left(\frac{k}{t}\right)}}^{2}\right) + 1\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right) + 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} + 1\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 + 1\right)\right)}} \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\color{blue}{{\left(\frac{k}{t}\right)}^{2}} + \left(1 + 1\right)\right)} \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + \left(1 + 1\right)\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\frac{k}{t} \cdot \frac{k}{t} + \color{blue}{2}\right)} \]

      8. lower-fma.f64 91.1

         \[\leadsto \frac{2}{\left(\left(t \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}} \]

   8. Applied egg-rr 91.1%

      \[\leadsto \frac{2}{\left(\left(t \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\left(t \cdot \left(\color{blue}{\frac{k \cdot t}{\ell}} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(t \cdot \left(\color{blue}{\frac{k \cdot t}{\ell}} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)} \]

       2. lower-*.f64 82.6

          \[\leadsto \frac{2}{\left(\left(t \cdot \left(\frac{\color{blue}{k \cdot t}}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)} \]

   11. Simplified 82.6%

       \[\leadsto \frac{2}{\left(\left(t \cdot \left(\color{blue}{\frac{k \cdot t}{\ell}} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.55 \cdot 10^{-204}:\\ \;\;\;\;\frac{\frac{\frac{\left(2 \cdot \cos k\right) \cdot \left(\ell \cdot \ell\right)}{k \cdot k}}{t \cdot k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \left(\tan k \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot \frac{t \cdot k}{\ell}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.7 \cdot 10^{-33}:\\ \;\;\;\;\ell \cdot \frac{1}{\frac{t\_m \cdot \left(t\_m \cdot k\right)}{\frac{\ell}{t\_m \cdot k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\left(2 \cdot \cos k\right) \cdot \left(\ell \cdot \ell\right)}{k \cdot k}}{t\_m \cdot k}}{k}\\ \end{array} \end{array} \]

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 2.7e-33)
    (* l (/ 1.0 (/ (* t_m (* t_m k)) (/ l (* t_m k)))))
    (/ (/ (/ (* (* 2.0 (cos k)) (* l l)) (* k k)) (* t_m k)) k))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.7e-33) {
		tmp = l * (1.0 / ((t_m * (t_m * k)) / (l / (t_m * k))));
	} else {
		tmp = ((((2.0 * cos(k)) * (l * l)) / (k * k)) / (t_m * 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 <= 2.7d-33) then
        tmp = l * (1.0d0 / ((t_m * (t_m * k)) / (l / (t_m * k))))
    else
        tmp = ((((2.0d0 * cos(k)) * (l * l)) / (k * k)) / (t_m * 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 <= 2.7e-33) {
		tmp = l * (1.0 / ((t_m * (t_m * k)) / (l / (t_m * k))));
	} else {
		tmp = ((((2.0 * Math.cos(k)) * (l * l)) / (k * k)) / (t_m * 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 <= 2.7e-33:
		tmp = l * (1.0 / ((t_m * (t_m * k)) / (l / (t_m * k))))
	else:
		tmp = ((((2.0 * math.cos(k)) * (l * l)) / (k * k)) / (t_m * k)) / k
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 2.7e-33)
		tmp = Float64(l * Float64(1.0 / Float64(Float64(t_m * Float64(t_m * k)) / Float64(l / Float64(t_m * k)))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(2.0 * cos(k)) * Float64(l * l)) / Float64(k * k)) / Float64(t_m * 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 <= 2.7e-33)
		tmp = l * (1.0 / ((t_m * (t_m * k)) / (l / (t_m * k))));
	else
		tmp = ((((2.0 * cos(k)) * (l * l)) / (k * k)) / (t_m * 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, 2.7e-33], N[(l * N[(1.0 / N[(N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(2.0 * N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.7000000000000001e-33

   1. Initial program 55.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. lower-*.f64 51.7

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 51.7%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot k\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      8. lower-/.f64 56.3

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      11. *-commutative N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      12. associate-*l* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      14. lower-*.f64 59.6

          \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   7. Applied egg-rr 59.6%

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \cdot \ell \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot t\right)}} \cdot \ell \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right)} \cdot \left(t \cdot t\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

      7. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot k\right) \cdot k\right)} \cdot \left(t \cdot t\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      10. *-commutative N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

      12. lower-*.f64 64.4

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

   9. Applied egg-rr 64.4%

      \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot \color{blue}{\left(t \cdot t\right)}\right)} \cdot \ell \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

       4. *-commutative N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)}} \cdot \ell \]

       5. associate-/l/ N/A

          \[\leadsto \color{blue}{\frac{\frac{\ell}{k \cdot t}}{k \cdot \left(t \cdot t\right)}} \cdot \ell \]

       6. clear-num N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{k \cdot \left(t \cdot t\right)}{\frac{\ell}{k \cdot t}}}} \cdot \ell \]

       7. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{k \cdot \left(t \cdot t\right)}{\frac{\ell}{k \cdot t}}}} \cdot \ell \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{k \cdot \left(t \cdot t\right)}{\frac{\ell}{k \cdot t}}}} \cdot \ell \]

       9. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{k \cdot \left(t \cdot t\right)}}{\frac{\ell}{k \cdot t}}} \cdot \ell \]

       10. lift-*.f64 N/A

           \[\leadsto \frac{1}{\frac{k \cdot \color{blue}{\left(t \cdot t\right)}}{\frac{\ell}{k \cdot t}}} \cdot \ell \]

       11. associate-*r* N/A

           \[\leadsto \frac{1}{\frac{\color{blue}{\left(k \cdot t\right) \cdot t}}{\frac{\ell}{k \cdot t}}} \cdot \ell \]

       12. lift-*.f64 N/A

           \[\leadsto \frac{1}{\frac{\color{blue}{\left(k \cdot t\right)} \cdot t}{\frac{\ell}{k \cdot t}}} \cdot \ell \]

       13. *-commutative N/A

           \[\leadsto \frac{1}{\frac{\color{blue}{t \cdot \left(k \cdot t\right)}}{\frac{\ell}{k \cdot t}}} \cdot \ell \]

       14. lower-*.f64 N/A

           \[\leadsto \frac{1}{\frac{\color{blue}{t \cdot \left(k \cdot t\right)}}{\frac{\ell}{k \cdot t}}} \cdot \ell \]

       15. lift-*.f64 N/A

           \[\leadsto \frac{1}{\frac{t \cdot \color{blue}{\left(k \cdot t\right)}}{\frac{\ell}{k \cdot t}}} \cdot \ell \]

       16. *-commutative N/A

           \[\leadsto \frac{1}{\frac{t \cdot \color{blue}{\left(t \cdot k\right)}}{\frac{\ell}{k \cdot t}}} \cdot \ell \]

       17. lower-*.f64 N/A

           \[\leadsto \frac{1}{\frac{t \cdot \color{blue}{\left(t \cdot k\right)}}{\frac{\ell}{k \cdot t}}} \cdot \ell \]

       18. lower-/.f64 76.3

           \[\leadsto \frac{1}{\frac{t \cdot \left(t \cdot k\right)}{\color{blue}{\frac{\ell}{k \cdot t}}}} \cdot \ell \]

       19. lift-*.f64 N/A

           \[\leadsto \frac{1}{\frac{t \cdot \left(t \cdot k\right)}{\frac{\ell}{\color{blue}{k \cdot t}}}} \cdot \ell \]

       20. *-commutative N/A

           \[\leadsto \frac{1}{\frac{t \cdot \left(t \cdot k\right)}{\frac{\ell}{\color{blue}{t \cdot k}}}} \cdot \ell \]

       21. lower-*.f64 76.3

           \[\leadsto \frac{1}{\frac{t \cdot \left(t \cdot k\right)}{\frac{\ell}{\color{blue}{t \cdot k}}}} \cdot \ell \]

   11. Applied egg-rr 76.3%

       \[\leadsto \color{blue}{\frac{1}{\frac{t \cdot \left(t \cdot k\right)}{\frac{\ell}{t \cdot k}}}} \cdot \ell \]

   if 2.7000000000000001e-33 < k

   1. Initial program 52.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot {\ell}^{2}\right) \cdot \cos k}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot {\ell}^{2}\right) \cdot \cos k}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot {\ell}^{2}\right)} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. unpow2 N/A

         \[\leadsto \frac{\left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\cos k}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)}} \]

      12. lower-pow.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{{\sin k}^{2}} \cdot \left({k}^{2} \cdot t\right)} \]

      13. lower-sin.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\color{blue}{\sin k}}^{2} \cdot \left({k}^{2} \cdot t\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}} \]

      16. unpow2 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

      17. lower-*.f64 72.4

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   5. Simplified 72.4%

      \[\leadsto \color{blue}{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot \left(\ell \cdot \ell\right)\right)} \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      3. lift-cos.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\cos k}}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\color{blue}{\sin k}}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{{\sin k}^{2}} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \]

      9. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2}}}{t \cdot \left(k \cdot k\right)}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2}}}{\color{blue}{t \cdot \left(k \cdot k\right)}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2}}}{t \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2}}}{\color{blue}{\left(t \cdot k\right) \cdot k}} \]

      13. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2}}}{t \cdot k}}{k}} \]

      14. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2}}}{t \cdot k}}{k}} \]

   7. Applied egg-rr 76.3%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \cos k\right)}{0.5 - 0.5 \cdot \cos \left(k + k\right)}}{k \cdot t}}{k}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{\frac{\frac{\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \cos k\right)}{\color{blue}{{k}^{2}}}}{k \cdot t}}{k} \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\frac{\frac{\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \cos k\right)}{\color{blue}{k \cdot k}}}{k \cdot t}}{k} \]

      2. lower-*.f64 62.4

         \[\leadsto \frac{\frac{\frac{\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \cos k\right)}{\color{blue}{k \cdot k}}}{k \cdot t}}{k} \]

   10. Simplified 62.4%

       \[\leadsto \frac{\frac{\frac{\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \cos k\right)}{\color{blue}{k \cdot k}}}{k \cdot t}}{k} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.7 \cdot 10^{-33}:\\ \;\;\;\;\ell \cdot \frac{1}{\frac{t \cdot \left(t \cdot k\right)}{\frac{\ell}{t \cdot k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\left(2 \cdot \cos k\right) \cdot \left(\ell \cdot \ell\right)}{k \cdot k}}{t \cdot k}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 69.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.7 \cdot 10^{-33}:\\ \;\;\;\;\ell \cdot \frac{1}{\frac{t\_m \cdot \left(t\_m \cdot k\right)}{\frac{\ell}{t\_m \cdot k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k \cdot \left(2 \cdot \left(\ell \cdot \ell\right)\right)}{\left(k \cdot k\right) \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 2.7e-33)
    (* l (/ 1.0 (/ (* t_m (* t_m k)) (/ l (* t_m k)))))
    (/ (* (cos k) (* 2.0 (* l l))) (* (* k k) (* t_m (* k k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.7e-33) {
		tmp = l * (1.0 / ((t_m * (t_m * k)) / (l / (t_m * k))));
	} else {
		tmp = (cos(k) * (2.0 * (l * l))) / ((k * k) * (t_m * (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 <= 2.7d-33) then
        tmp = l * (1.0d0 / ((t_m * (t_m * k)) / (l / (t_m * k))))
    else
        tmp = (cos(k) * (2.0d0 * (l * l))) / ((k * k) * (t_m * (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 <= 2.7e-33) {
		tmp = l * (1.0 / ((t_m * (t_m * k)) / (l / (t_m * k))));
	} else {
		tmp = (Math.cos(k) * (2.0 * (l * l))) / ((k * k) * (t_m * (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 <= 2.7e-33:
		tmp = l * (1.0 / ((t_m * (t_m * k)) / (l / (t_m * k))))
	else:
		tmp = (math.cos(k) * (2.0 * (l * l))) / ((k * k) * (t_m * (k * k)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 2.7e-33)
		tmp = Float64(l * Float64(1.0 / Float64(Float64(t_m * Float64(t_m * k)) / Float64(l / Float64(t_m * k)))));
	else
		tmp = Float64(Float64(cos(k) * Float64(2.0 * Float64(l * l))) / Float64(Float64(k * k) * Float64(t_m * Float64(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 <= 2.7e-33)
		tmp = l * (1.0 / ((t_m * (t_m * k)) / (l / (t_m * k))));
	else
		tmp = (cos(k) * (2.0 * (l * l))) / ((k * k) * (t_m * (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, 2.7e-33], N[(l * N[(1.0 / N[(N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[k], $MachinePrecision] * N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.7000000000000001e-33

   1. Initial program 55.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. lower-*.f64 51.7

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 51.7%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot k\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      8. lower-/.f64 56.3

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      11. *-commutative N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      12. associate-*l* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      14. lower-*.f64 59.6

          \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   7. Applied egg-rr 59.6%

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \cdot \ell \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot t\right)}} \cdot \ell \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right)} \cdot \left(t \cdot t\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

      7. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot k\right) \cdot k\right)} \cdot \left(t \cdot t\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      10. *-commutative N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

      12. lower-*.f64 64.4

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

   9. Applied egg-rr 64.4%

      \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot \color{blue}{\left(t \cdot t\right)}\right)} \cdot \ell \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

       4. *-commutative N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)}} \cdot \ell \]

       5. associate-/l/ N/A

          \[\leadsto \color{blue}{\frac{\frac{\ell}{k \cdot t}}{k \cdot \left(t \cdot t\right)}} \cdot \ell \]

       6. clear-num N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{k \cdot \left(t \cdot t\right)}{\frac{\ell}{k \cdot t}}}} \cdot \ell \]

       7. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{k \cdot \left(t \cdot t\right)}{\frac{\ell}{k \cdot t}}}} \cdot \ell \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{k \cdot \left(t \cdot t\right)}{\frac{\ell}{k \cdot t}}}} \cdot \ell \]

       9. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{k \cdot \left(t \cdot t\right)}}{\frac{\ell}{k \cdot t}}} \cdot \ell \]

       10. lift-*.f64 N/A

           \[\leadsto \frac{1}{\frac{k \cdot \color{blue}{\left(t \cdot t\right)}}{\frac{\ell}{k \cdot t}}} \cdot \ell \]

       11. associate-*r* N/A

           \[\leadsto \frac{1}{\frac{\color{blue}{\left(k \cdot t\right) \cdot t}}{\frac{\ell}{k \cdot t}}} \cdot \ell \]

       12. lift-*.f64 N/A

           \[\leadsto \frac{1}{\frac{\color{blue}{\left(k \cdot t\right)} \cdot t}{\frac{\ell}{k \cdot t}}} \cdot \ell \]

       13. *-commutative N/A

           \[\leadsto \frac{1}{\frac{\color{blue}{t \cdot \left(k \cdot t\right)}}{\frac{\ell}{k \cdot t}}} \cdot \ell \]

       14. lower-*.f64 N/A

           \[\leadsto \frac{1}{\frac{\color{blue}{t \cdot \left(k \cdot t\right)}}{\frac{\ell}{k \cdot t}}} \cdot \ell \]

       15. lift-*.f64 N/A

           \[\leadsto \frac{1}{\frac{t \cdot \color{blue}{\left(k \cdot t\right)}}{\frac{\ell}{k \cdot t}}} \cdot \ell \]

       16. *-commutative N/A

           \[\leadsto \frac{1}{\frac{t \cdot \color{blue}{\left(t \cdot k\right)}}{\frac{\ell}{k \cdot t}}} \cdot \ell \]

       17. lower-*.f64 N/A

           \[\leadsto \frac{1}{\frac{t \cdot \color{blue}{\left(t \cdot k\right)}}{\frac{\ell}{k \cdot t}}} \cdot \ell \]

       18. lower-/.f64 76.3

           \[\leadsto \frac{1}{\frac{t \cdot \left(t \cdot k\right)}{\color{blue}{\frac{\ell}{k \cdot t}}}} \cdot \ell \]

       19. lift-*.f64 N/A

           \[\leadsto \frac{1}{\frac{t \cdot \left(t \cdot k\right)}{\frac{\ell}{\color{blue}{k \cdot t}}}} \cdot \ell \]

       20. *-commutative N/A

           \[\leadsto \frac{1}{\frac{t \cdot \left(t \cdot k\right)}{\frac{\ell}{\color{blue}{t \cdot k}}}} \cdot \ell \]

       21. lower-*.f64 76.3

           \[\leadsto \frac{1}{\frac{t \cdot \left(t \cdot k\right)}{\frac{\ell}{\color{blue}{t \cdot k}}}} \cdot \ell \]

   11. Applied egg-rr 76.3%

       \[\leadsto \color{blue}{\frac{1}{\frac{t \cdot \left(t \cdot k\right)}{\frac{\ell}{t \cdot k}}}} \cdot \ell \]

   if 2.7000000000000001e-33 < k

   1. Initial program 52.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot {\ell}^{2}\right) \cdot \cos k}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot {\ell}^{2}\right) \cdot \cos k}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot {\ell}^{2}\right)} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. unpow2 N/A

         \[\leadsto \frac{\left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\cos k}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)}} \]

      12. lower-pow.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{{\sin k}^{2}} \cdot \left({k}^{2} \cdot t\right)} \]

      13. lower-sin.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\color{blue}{\sin k}}^{2} \cdot \left({k}^{2} \cdot t\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}} \]

      16. unpow2 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

      17. lower-*.f64 72.4

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   5. Simplified 72.4%

      \[\leadsto \color{blue}{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{{k}^{2}} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{\left(k \cdot k\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      2. lower-*.f64 60.9

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{\left(k \cdot k\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

   8. Simplified 60.9%

      \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{\left(k \cdot k\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.7 \cdot 10^{-33}:\\ \;\;\;\;\ell \cdot \frac{1}{\frac{t \cdot \left(t \cdot k\right)}{\frac{\ell}{t \cdot k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k \cdot \left(2 \cdot \left(\ell \cdot \ell\right)\right)}{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 69.1% accurate, 7.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.7 \cdot 10^{-33}:\\ \;\;\;\;\ell \cdot \frac{1}{\frac{t\_m \cdot \left(t\_m \cdot k\right)}{\frac{\ell}{t\_m \cdot k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t\_m \cdot \left(k \cdot \left(k \cdot k\right)\right)}}{k}\\ \end{array} \end{array} \]

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 2.7e-33)
    (* l (/ 1.0 (/ (* t_m (* t_m k)) (/ l (* t_m k)))))
    (/ (/ (* 2.0 (* l l)) (* t_m (* k (* k 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 <= 2.7e-33) {
		tmp = l * (1.0 / ((t_m * (t_m * k)) / (l / (t_m * k))));
	} else {
		tmp = ((2.0 * (l * l)) / (t_m * (k * (k * 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 <= 2.7d-33) then
        tmp = l * (1.0d0 / ((t_m * (t_m * k)) / (l / (t_m * k))))
    else
        tmp = ((2.0d0 * (l * l)) / (t_m * (k * (k * 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 <= 2.7e-33) {
		tmp = l * (1.0 / ((t_m * (t_m * k)) / (l / (t_m * k))));
	} else {
		tmp = ((2.0 * (l * l)) / (t_m * (k * (k * 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 <= 2.7e-33:
		tmp = l * (1.0 / ((t_m * (t_m * k)) / (l / (t_m * k))))
	else:
		tmp = ((2.0 * (l * l)) / (t_m * (k * (k * 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 <= 2.7e-33)
		tmp = Float64(l * Float64(1.0 / Float64(Float64(t_m * Float64(t_m * k)) / Float64(l / Float64(t_m * k)))));
	else
		tmp = Float64(Float64(Float64(2.0 * Float64(l * l)) / Float64(t_m * Float64(k * Float64(k * 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 <= 2.7e-33)
		tmp = l * (1.0 / ((t_m * (t_m * k)) / (l / (t_m * k))));
	else
		tmp = ((2.0 * (l * l)) / (t_m * (k * (k * 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, 2.7e-33], N[(l * N[(1.0 / N[(N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.7000000000000001e-33

   1. Initial program 55.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. lower-*.f64 51.7

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 51.7%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot k\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      8. lower-/.f64 56.3

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      11. *-commutative N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      12. associate-*l* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      14. lower-*.f64 59.6

          \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   7. Applied egg-rr 59.6%

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \cdot \ell \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot t\right)}} \cdot \ell \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right)} \cdot \left(t \cdot t\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

      7. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot k\right) \cdot k\right)} \cdot \left(t \cdot t\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      10. *-commutative N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

      12. lower-*.f64 64.4

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

   9. Applied egg-rr 64.4%

      \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot \color{blue}{\left(t \cdot t\right)}\right)} \cdot \ell \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

       4. *-commutative N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)}} \cdot \ell \]

       5. associate-/l/ N/A

          \[\leadsto \color{blue}{\frac{\frac{\ell}{k \cdot t}}{k \cdot \left(t \cdot t\right)}} \cdot \ell \]

       6. clear-num N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{k \cdot \left(t \cdot t\right)}{\frac{\ell}{k \cdot t}}}} \cdot \ell \]

       7. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{k \cdot \left(t \cdot t\right)}{\frac{\ell}{k \cdot t}}}} \cdot \ell \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{k \cdot \left(t \cdot t\right)}{\frac{\ell}{k \cdot t}}}} \cdot \ell \]

       9. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{k \cdot \left(t \cdot t\right)}}{\frac{\ell}{k \cdot t}}} \cdot \ell \]

       10. lift-*.f64 N/A

           \[\leadsto \frac{1}{\frac{k \cdot \color{blue}{\left(t \cdot t\right)}}{\frac{\ell}{k \cdot t}}} \cdot \ell \]

       11. associate-*r* N/A

           \[\leadsto \frac{1}{\frac{\color{blue}{\left(k \cdot t\right) \cdot t}}{\frac{\ell}{k \cdot t}}} \cdot \ell \]

       12. lift-*.f64 N/A

           \[\leadsto \frac{1}{\frac{\color{blue}{\left(k \cdot t\right)} \cdot t}{\frac{\ell}{k \cdot t}}} \cdot \ell \]

       13. *-commutative N/A

           \[\leadsto \frac{1}{\frac{\color{blue}{t \cdot \left(k \cdot t\right)}}{\frac{\ell}{k \cdot t}}} \cdot \ell \]

       14. lower-*.f64 N/A

           \[\leadsto \frac{1}{\frac{\color{blue}{t \cdot \left(k \cdot t\right)}}{\frac{\ell}{k \cdot t}}} \cdot \ell \]

       15. lift-*.f64 N/A

           \[\leadsto \frac{1}{\frac{t \cdot \color{blue}{\left(k \cdot t\right)}}{\frac{\ell}{k \cdot t}}} \cdot \ell \]

       16. *-commutative N/A

           \[\leadsto \frac{1}{\frac{t \cdot \color{blue}{\left(t \cdot k\right)}}{\frac{\ell}{k \cdot t}}} \cdot \ell \]

       17. lower-*.f64 N/A

           \[\leadsto \frac{1}{\frac{t \cdot \color{blue}{\left(t \cdot k\right)}}{\frac{\ell}{k \cdot t}}} \cdot \ell \]

       18. lower-/.f64 76.3

           \[\leadsto \frac{1}{\frac{t \cdot \left(t \cdot k\right)}{\color{blue}{\frac{\ell}{k \cdot t}}}} \cdot \ell \]

       19. lift-*.f64 N/A

           \[\leadsto \frac{1}{\frac{t \cdot \left(t \cdot k\right)}{\frac{\ell}{\color{blue}{k \cdot t}}}} \cdot \ell \]

       20. *-commutative N/A

           \[\leadsto \frac{1}{\frac{t \cdot \left(t \cdot k\right)}{\frac{\ell}{\color{blue}{t \cdot k}}}} \cdot \ell \]

       21. lower-*.f64 76.3

           \[\leadsto \frac{1}{\frac{t \cdot \left(t \cdot k\right)}{\frac{\ell}{\color{blue}{t \cdot k}}}} \cdot \ell \]

   11. Applied egg-rr 76.3%

       \[\leadsto \color{blue}{\frac{1}{\frac{t \cdot \left(t \cdot k\right)}{\frac{\ell}{t \cdot k}}}} \cdot \ell \]

   if 2.7000000000000001e-33 < k

   1. Initial program 52.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot {\ell}^{2}\right) \cdot \cos k}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot {\ell}^{2}\right) \cdot \cos k}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot {\ell}^{2}\right)} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. unpow2 N/A

         \[\leadsto \frac{\left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\cos k}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)}} \]

      12. lower-pow.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{{\sin k}^{2}} \cdot \left({k}^{2} \cdot t\right)} \]

      13. lower-sin.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\color{blue}{\sin k}}^{2} \cdot \left({k}^{2} \cdot t\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}} \]

      16. unpow2 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

      17. lower-*.f64 72.4

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   5. Simplified 72.4%

      \[\leadsto \color{blue}{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot \left(\ell \cdot \ell\right)\right)} \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      3. lift-cos.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\cos k}}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\color{blue}{\sin k}}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{{\sin k}^{2}} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \]

      9. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2}}}{t \cdot \left(k \cdot k\right)}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2}}}{\color{blue}{t \cdot \left(k \cdot k\right)}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2}}}{t \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2}}}{\color{blue}{\left(t \cdot k\right) \cdot k}} \]

      13. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2}}}{t \cdot k}}{k}} \]

      14. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2}}}{t \cdot k}}{k}} \]

   7. Applied egg-rr 76.3%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \cos k\right)}{0.5 - 0.5 \cdot \cos \left(k + k\right)}}{k \cdot t}}{k}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{3} \cdot t}}}{k} \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{3} \cdot t}}}{k} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{3} \cdot t}}}{k} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{2 \cdot {\ell}^{2}}}{{k}^{3} \cdot t}}{k} \]

      4. unpow2 N/A

         \[\leadsto \frac{\frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{3} \cdot t}}{k} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{3} \cdot t}}{k} \]

      6. *-commutative N/A

         \[\leadsto \frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t \cdot {k}^{3}}}}{k} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t \cdot {k}^{3}}}}{k} \]

      8. cube-mult N/A

         \[\leadsto \frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \color{blue}{\left(k \cdot \left(k \cdot k\right)\right)}}}{k} \]

      9. unpow2 N/A

         \[\leadsto \frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(k \cdot \color{blue}{{k}^{2}}\right)}}{k} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \color{blue}{\left(k \cdot {k}^{2}\right)}}}{k} \]

      11. unpow2 N/A

          \[\leadsto \frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(k \cdot \color{blue}{\left(k \cdot k\right)}\right)}}{k} \]

      12. lower-*.f64 59.3

          \[\leadsto \frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(k \cdot \color{blue}{\left(k \cdot k\right)}\right)}}{k} \]

   10. Simplified 59.3%

       \[\leadsto \frac{\color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(k \cdot \left(k \cdot k\right)\right)}}}{k} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.7 \cdot 10^{-33}:\\ \;\;\;\;\ell \cdot \frac{1}{\frac{t \cdot \left(t \cdot k\right)}{\frac{\ell}{t \cdot k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(k \cdot \left(k \cdot k\right)\right)}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 69.0% accurate, 8.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8.5 \cdot 10^{-34}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \ell\right)}{t\_m \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}\\ \mathbf{elif}\;t\_m \leq 1.05 \cdot 10^{+102}:\\ \;\;\;\;\frac{\ell}{t\_m \cdot \left(k \cdot \left(t\_m \cdot t\_m\right)\right)} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(k \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 8.5e-34)
    (/ (* 2.0 (* l l)) (* t_m (* (* k k) (* k k))))
    (if (<= t_m 1.05e+102)
      (* (/ l (* t_m (* k (* t_m t_m)))) (/ l k))
      (* l (/ l (* t_m (* k (* t_m (* t_m k))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 8.5e-34) {
		tmp = (2.0 * (l * l)) / (t_m * ((k * k) * (k * k)));
	} else if (t_m <= 1.05e+102) {
		tmp = (l / (t_m * (k * (t_m * t_m)))) * (l / k);
	} else {
		tmp = l * (l / (t_m * (k * (t_m * (t_m * 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 (t_m <= 8.5d-34) then
        tmp = (2.0d0 * (l * l)) / (t_m * ((k * k) * (k * k)))
    else if (t_m <= 1.05d+102) then
        tmp = (l / (t_m * (k * (t_m * t_m)))) * (l / k)
    else
        tmp = l * (l / (t_m * (k * (t_m * (t_m * 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 (t_m <= 8.5e-34) {
		tmp = (2.0 * (l * l)) / (t_m * ((k * k) * (k * k)));
	} else if (t_m <= 1.05e+102) {
		tmp = (l / (t_m * (k * (t_m * t_m)))) * (l / k);
	} else {
		tmp = l * (l / (t_m * (k * (t_m * (t_m * 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 t_m <= 8.5e-34:
		tmp = (2.0 * (l * l)) / (t_m * ((k * k) * (k * k)))
	elif t_m <= 1.05e+102:
		tmp = (l / (t_m * (k * (t_m * t_m)))) * (l / k)
	else:
		tmp = l * (l / (t_m * (k * (t_m * (t_m * k)))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 8.5e-34)
		tmp = Float64(Float64(2.0 * Float64(l * l)) / Float64(t_m * Float64(Float64(k * k) * Float64(k * k))));
	elseif (t_m <= 1.05e+102)
		tmp = Float64(Float64(l / Float64(t_m * Float64(k * Float64(t_m * t_m)))) * Float64(l / k));
	else
		tmp = Float64(l * Float64(l / Float64(t_m * Float64(k * Float64(t_m * Float64(t_m * 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 (t_m <= 8.5e-34)
		tmp = (2.0 * (l * l)) / (t_m * ((k * k) * (k * k)));
	elseif (t_m <= 1.05e+102)
		tmp = (l / (t_m * (k * (t_m * t_m)))) * (l / k);
	else
		tmp = l * (l / (t_m * (k * (t_m * (t_m * 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[t$95$m, 8.5e-34], N[(N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(N[(k * k), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.05e+102], N[(N[(l / N[(t$95$m * N[(k * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision], N[(l * N[(l / N[(t$95$m * N[(k * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 8.5000000000000001e-34

   1. Initial program 49.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot {\ell}^{2}\right) \cdot \cos k}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot {\ell}^{2}\right) \cdot \cos k}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot {\ell}^{2}\right)} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. unpow2 N/A

         \[\leadsto \frac{\left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\cos k}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)}} \]

      12. lower-pow.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{{\sin k}^{2}} \cdot \left({k}^{2} \cdot t\right)} \]

      13. lower-sin.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\color{blue}{\sin k}}^{2} \cdot \left({k}^{2} \cdot t\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}} \]

      16. unpow2 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

      17. lower-*.f64 64.0

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   5. Simplified 64.0%

      \[\leadsto \color{blue}{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot \left(\ell \cdot \ell\right)\right)} \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      3. lift-cos.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\cos k}}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\color{blue}{\sin k}}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{{\sin k}^{2}} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \]

      9. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2}}}{t \cdot \left(k \cdot k\right)}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2}}}{\color{blue}{t \cdot \left(k \cdot k\right)}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2}}}{t \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2}}}{\color{blue}{\left(t \cdot k\right) \cdot k}} \]

      13. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2}}}{t \cdot k}}{k}} \]

      14. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2}}}{t \cdot k}}{k}} \]

   7. Applied egg-rr 63.2%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \cos k\right)}{0.5 - 0.5 \cdot \cos \left(k + k\right)}}{k \cdot t}}{k}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{2 \cdot {\ell}^{2}}}{{k}^{4} \cdot t} \]

      4. unpow2 N/A

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot {k}^{\color{blue}{\left(2 \cdot 2\right)}}} \]

      9. pow-sqr N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \color{blue}{\left({k}^{2} \cdot {k}^{2}\right)}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \color{blue}{\left({k}^{2} \cdot {k}^{2}\right)}} \]

      11. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot {k}^{2}\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot {k}^{2}\right)} \]

      13. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

      14. lower-*.f64 55.6

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   10. Simplified 55.6%

       \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]

   if 8.5000000000000001e-34 < t < 1.05000000000000001e102

   1. Initial program 83.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. lower-*.f64 57.7

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 57.7%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot k\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right) \cdot k}} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot k} \cdot \frac{\ell}{k}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot k} \cdot \frac{\ell}{k}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot k}} \cdot \frac{\ell}{k} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot k} \cdot \frac{\ell}{k} \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot k\right)}} \cdot \frac{\ell}{k} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot k\right)}} \cdot \frac{\ell}{k} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot k\right)}} \cdot \frac{\ell}{k} \]

      11. lower-/.f64 73.2

          \[\leadsto \frac{\ell}{t \cdot \left(\left(t \cdot t\right) \cdot k\right)} \cdot \color{blue}{\frac{\ell}{k}} \]

   7. Applied egg-rr 73.2%

      \[\leadsto \color{blue}{\frac{\ell}{t \cdot \left(\left(t \cdot t\right) \cdot k\right)} \cdot \frac{\ell}{k}} \]

   if 1.05000000000000001e102 < t

   1. Initial program 57.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. lower-*.f64 51.6

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 51.6%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot k\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      8. lower-/.f64 54.5

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      11. *-commutative N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      12. associate-*l* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      14. lower-*.f64 60.2

          \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   7. Applied egg-rr 60.2%

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \cdot \ell \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot t\right)}} \cdot \ell \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right)} \cdot \left(t \cdot t\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

      7. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot k\right) \cdot k\right)} \cdot \left(t \cdot t\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      10. *-commutative N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

      12. lower-*.f64 68.5

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

   9. Applied egg-rr 68.5%

      \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot \color{blue}{\left(t \cdot t\right)}\right)} \cdot \ell \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

       4. *-commutative N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)}} \cdot \ell \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot t\right)}} \cdot \ell \]

       6. associate-*r* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(k \cdot \left(t \cdot t\right)\right) \cdot k\right) \cdot t}} \cdot \ell \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(k \cdot \left(t \cdot t\right)\right) \cdot k\right) \cdot t}} \cdot \ell \]

       8. lower-*.f64 68.5

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(k \cdot \left(t \cdot t\right)\right) \cdot k\right)} \cdot t} \cdot \ell \]

       9. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\color{blue}{\left(k \cdot \left(t \cdot t\right)\right)} \cdot k\right) \cdot t} \cdot \ell \]

       10. lift-*.f64 N/A

           \[\leadsto \frac{\ell}{\left(\left(k \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot k\right) \cdot t} \cdot \ell \]

       11. associate-*r* N/A

           \[\leadsto \frac{\ell}{\left(\color{blue}{\left(\left(k \cdot t\right) \cdot t\right)} \cdot k\right) \cdot t} \cdot \ell \]

       12. lift-*.f64 N/A

           \[\leadsto \frac{\ell}{\left(\left(\color{blue}{\left(k \cdot t\right)} \cdot t\right) \cdot k\right) \cdot t} \cdot \ell \]

       13. *-commutative N/A

           \[\leadsto \frac{\ell}{\left(\color{blue}{\left(t \cdot \left(k \cdot t\right)\right)} \cdot k\right) \cdot t} \cdot \ell \]

       14. lower-*.f64 94.6

           \[\leadsto \frac{\ell}{\left(\color{blue}{\left(t \cdot \left(k \cdot t\right)\right)} \cdot k\right) \cdot t} \cdot \ell \]

       15. lift-*.f64 N/A

           \[\leadsto \frac{\ell}{\left(\left(t \cdot \color{blue}{\left(k \cdot t\right)}\right) \cdot k\right) \cdot t} \cdot \ell \]

       16. *-commutative N/A

           \[\leadsto \frac{\ell}{\left(\left(t \cdot \color{blue}{\left(t \cdot k\right)}\right) \cdot k\right) \cdot t} \cdot \ell \]

       17. lower-*.f64 94.6

           \[\leadsto \frac{\ell}{\left(\left(t \cdot \color{blue}{\left(t \cdot k\right)}\right) \cdot k\right) \cdot t} \cdot \ell \]

   11. Applied egg-rr 94.6%

       \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot \left(t \cdot k\right)\right) \cdot k\right) \cdot t}} \cdot \ell \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.5 \cdot 10^{-34}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{+102}:\\ \;\;\;\;\frac{\ell}{t \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(k \cdot \left(t \cdot \left(t \cdot k\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 68.8% accurate, 8.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 4.8 \cdot 10^{-33}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{t\_m \cdot k}}{t\_m \cdot \left(t\_m \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t\_m \cdot \left(k \cdot \left(k \cdot k\right)\right)}}{k}\\ \end{array} \end{array} \]

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 4.8e-33)
    (/ (* l (/ l (* t_m k))) (* t_m (* t_m k)))
    (/ (/ (* 2.0 (* l l)) (* t_m (* k (* k 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 <= 4.8e-33) {
		tmp = (l * (l / (t_m * k))) / (t_m * (t_m * k));
	} else {
		tmp = ((2.0 * (l * l)) / (t_m * (k * (k * 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 <= 4.8d-33) then
        tmp = (l * (l / (t_m * k))) / (t_m * (t_m * k))
    else
        tmp = ((2.0d0 * (l * l)) / (t_m * (k * (k * 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 <= 4.8e-33) {
		tmp = (l * (l / (t_m * k))) / (t_m * (t_m * k));
	} else {
		tmp = ((2.0 * (l * l)) / (t_m * (k * (k * 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 <= 4.8e-33:
		tmp = (l * (l / (t_m * k))) / (t_m * (t_m * k))
	else:
		tmp = ((2.0 * (l * l)) / (t_m * (k * (k * 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 <= 4.8e-33)
		tmp = Float64(Float64(l * Float64(l / Float64(t_m * k))) / Float64(t_m * Float64(t_m * k)));
	else
		tmp = Float64(Float64(Float64(2.0 * Float64(l * l)) / Float64(t_m * Float64(k * Float64(k * 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 <= 4.8e-33)
		tmp = (l * (l / (t_m * k))) / (t_m * (t_m * k));
	else
		tmp = ((2.0 * (l * l)) / (t_m * (k * (k * 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, 4.8e-33], N[(N[(l * N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 4.8e-33

   1. Initial program 55.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. lower-*.f64 51.7

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 51.7%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot k\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      8. lower-/.f64 56.3

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      11. *-commutative N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      12. associate-*l* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      14. lower-*.f64 59.6

          \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   7. Applied egg-rr 59.6%

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \cdot \ell \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot t\right)}} \cdot \ell \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right)} \cdot \left(t \cdot t\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

      7. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot k\right) \cdot k\right)} \cdot \left(t \cdot t\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      10. *-commutative N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

      12. lower-*.f64 64.4

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

   9. Applied egg-rr 64.4%

      \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

   11. Applied egg-rr 77.1%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{t \cdot k} \cdot \ell}{t \cdot \left(t \cdot k\right)}} \]

   if 4.8e-33 < k

   1. Initial program 52.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot {\ell}^{2}\right) \cdot \cos k}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot {\ell}^{2}\right) \cdot \cos k}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot {\ell}^{2}\right)} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. unpow2 N/A

         \[\leadsto \frac{\left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\cos k}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)}} \]

      12. lower-pow.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{{\sin k}^{2}} \cdot \left({k}^{2} \cdot t\right)} \]

      13. lower-sin.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\color{blue}{\sin k}}^{2} \cdot \left({k}^{2} \cdot t\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}} \]

      16. unpow2 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

      17. lower-*.f64 72.4

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   5. Simplified 72.4%

      \[\leadsto \color{blue}{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot \left(\ell \cdot \ell\right)\right)} \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      3. lift-cos.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\cos k}}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\color{blue}{\sin k}}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{{\sin k}^{2}} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \]

      9. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2}}}{t \cdot \left(k \cdot k\right)}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2}}}{\color{blue}{t \cdot \left(k \cdot k\right)}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2}}}{t \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2}}}{\color{blue}{\left(t \cdot k\right) \cdot k}} \]

      13. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2}}}{t \cdot k}}{k}} \]

      14. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2}}}{t \cdot k}}{k}} \]

   7. Applied egg-rr 76.3%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \cos k\right)}{0.5 - 0.5 \cdot \cos \left(k + k\right)}}{k \cdot t}}{k}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{3} \cdot t}}}{k} \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{3} \cdot t}}}{k} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{3} \cdot t}}}{k} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{2 \cdot {\ell}^{2}}}{{k}^{3} \cdot t}}{k} \]

      4. unpow2 N/A

         \[\leadsto \frac{\frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{3} \cdot t}}{k} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{3} \cdot t}}{k} \]

      6. *-commutative N/A

         \[\leadsto \frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t \cdot {k}^{3}}}}{k} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t \cdot {k}^{3}}}}{k} \]

      8. cube-mult N/A

         \[\leadsto \frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \color{blue}{\left(k \cdot \left(k \cdot k\right)\right)}}}{k} \]

      9. unpow2 N/A

         \[\leadsto \frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(k \cdot \color{blue}{{k}^{2}}\right)}}{k} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \color{blue}{\left(k \cdot {k}^{2}\right)}}}{k} \]

      11. unpow2 N/A

          \[\leadsto \frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(k \cdot \color{blue}{\left(k \cdot k\right)}\right)}}{k} \]

      12. lower-*.f64 59.3

          \[\leadsto \frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(k \cdot \color{blue}{\left(k \cdot k\right)}\right)}}{k} \]

   10. Simplified 59.3%

       \[\leadsto \frac{\color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(k \cdot \left(k \cdot k\right)\right)}}}{k} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.8 \cdot 10^{-33}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{t \cdot k}}{t \cdot \left(t \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(k \cdot \left(k \cdot k\right)\right)}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 68.6% accurate, 9.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 8.6 \cdot 10^{-33}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{t\_m \cdot k}}{t\_m \cdot \left(t\_m \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \ell\right)}{t\_m \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 8.6e-33)
    (/ (* l (/ l (* t_m k))) (* t_m (* t_m k)))
    (/ (* 2.0 (* l l)) (* t_m (* (* k k) (* 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 <= 8.6e-33) {
		tmp = (l * (l / (t_m * k))) / (t_m * (t_m * k));
	} else {
		tmp = (2.0 * (l * l)) / (t_m * ((k * k) * (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 <= 8.6d-33) then
        tmp = (l * (l / (t_m * k))) / (t_m * (t_m * k))
    else
        tmp = (2.0d0 * (l * l)) / (t_m * ((k * k) * (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 <= 8.6e-33) {
		tmp = (l * (l / (t_m * k))) / (t_m * (t_m * k));
	} else {
		tmp = (2.0 * (l * l)) / (t_m * ((k * k) * (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 <= 8.6e-33:
		tmp = (l * (l / (t_m * k))) / (t_m * (t_m * k))
	else:
		tmp = (2.0 * (l * l)) / (t_m * ((k * k) * (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 <= 8.6e-33)
		tmp = Float64(Float64(l * Float64(l / Float64(t_m * k))) / Float64(t_m * Float64(t_m * k)));
	else
		tmp = Float64(Float64(2.0 * Float64(l * l)) / Float64(t_m * Float64(Float64(k * k) * Float64(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 <= 8.6e-33)
		tmp = (l * (l / (t_m * k))) / (t_m * (t_m * k));
	else
		tmp = (2.0 * (l * l)) / (t_m * ((k * k) * (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, 8.6e-33], N[(N[(l * N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(N[(k * k), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 8.60000000000000062e-33

   1. Initial program 55.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. lower-*.f64 51.7

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 51.7%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot k\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      8. lower-/.f64 56.3

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      11. *-commutative N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      12. associate-*l* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      14. lower-*.f64 59.6

          \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   7. Applied egg-rr 59.6%

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \cdot \ell \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot t\right)}} \cdot \ell \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right)} \cdot \left(t \cdot t\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

      7. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot k\right) \cdot k\right)} \cdot \left(t \cdot t\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      10. *-commutative N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

      12. lower-*.f64 64.4

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

   9. Applied egg-rr 64.4%

      \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

   11. Applied egg-rr 77.1%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{t \cdot k} \cdot \ell}{t \cdot \left(t \cdot k\right)}} \]

   if 8.60000000000000062e-33 < k

   1. Initial program 52.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot {\ell}^{2}\right) \cdot \cos k}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot {\ell}^{2}\right) \cdot \cos k}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot {\ell}^{2}\right)} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. unpow2 N/A

         \[\leadsto \frac{\left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\cos k}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)}} \]

      12. lower-pow.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{{\sin k}^{2}} \cdot \left({k}^{2} \cdot t\right)} \]

      13. lower-sin.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\color{blue}{\sin k}}^{2} \cdot \left({k}^{2} \cdot t\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}} \]

      16. unpow2 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

      17. lower-*.f64 72.4

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   5. Simplified 72.4%

      \[\leadsto \color{blue}{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot \left(\ell \cdot \ell\right)\right)} \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      3. lift-cos.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\cos k}}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\color{blue}{\sin k}}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{{\sin k}^{2}} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \]

      9. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2}}}{t \cdot \left(k \cdot k\right)}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2}}}{\color{blue}{t \cdot \left(k \cdot k\right)}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2}}}{t \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2}}}{\color{blue}{\left(t \cdot k\right) \cdot k}} \]

      13. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2}}}{t \cdot k}}{k}} \]

      14. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2}}}{t \cdot k}}{k}} \]

   7. Applied egg-rr 76.3%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \cos k\right)}{0.5 - 0.5 \cdot \cos \left(k + k\right)}}{k \cdot t}}{k}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{2 \cdot {\ell}^{2}}}{{k}^{4} \cdot t} \]

      4. unpow2 N/A

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot {k}^{\color{blue}{\left(2 \cdot 2\right)}}} \]

      9. pow-sqr N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \color{blue}{\left({k}^{2} \cdot {k}^{2}\right)}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \color{blue}{\left({k}^{2} \cdot {k}^{2}\right)}} \]

      11. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot {k}^{2}\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot {k}^{2}\right)} \]

      13. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

      14. lower-*.f64 57.9

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   10. Simplified 57.9%

       \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.6 \cdot 10^{-33}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{t \cdot k}}{t \cdot \left(t \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 66.9% accurate, 9.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.7 \cdot 10^{-33}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t\_m \cdot k\right) \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \ell\right)}{t\_m \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 2.7e-33)
    (* l (/ l (* (* t_m k) (* t_m (* t_m k)))))
    (/ (* 2.0 (* l l)) (* t_m (* (* k k) (* 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 <= 2.7e-33) {
		tmp = l * (l / ((t_m * k) * (t_m * (t_m * k))));
	} else {
		tmp = (2.0 * (l * l)) / (t_m * ((k * k) * (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 <= 2.7d-33) then
        tmp = l * (l / ((t_m * k) * (t_m * (t_m * k))))
    else
        tmp = (2.0d0 * (l * l)) / (t_m * ((k * k) * (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 <= 2.7e-33) {
		tmp = l * (l / ((t_m * k) * (t_m * (t_m * k))));
	} else {
		tmp = (2.0 * (l * l)) / (t_m * ((k * k) * (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 <= 2.7e-33:
		tmp = l * (l / ((t_m * k) * (t_m * (t_m * k))))
	else:
		tmp = (2.0 * (l * l)) / (t_m * ((k * k) * (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 <= 2.7e-33)
		tmp = Float64(l * Float64(l / Float64(Float64(t_m * k) * Float64(t_m * Float64(t_m * k)))));
	else
		tmp = Float64(Float64(2.0 * Float64(l * l)) / Float64(t_m * Float64(Float64(k * k) * Float64(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 <= 2.7e-33)
		tmp = l * (l / ((t_m * k) * (t_m * (t_m * k))));
	else
		tmp = (2.0 * (l * l)) / (t_m * ((k * k) * (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, 2.7e-33], N[(l * N[(l / N[(N[(t$95$m * k), $MachinePrecision] * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(N[(k * k), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.7000000000000001e-33

   1. Initial program 55.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. lower-*.f64 51.7

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 51.7%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot k\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      8. lower-/.f64 56.3

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      11. *-commutative N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      12. associate-*l* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      14. lower-*.f64 59.6

          \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   7. Applied egg-rr 59.6%

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \cdot \ell \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot t\right)}} \cdot \ell \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right)} \cdot \left(t \cdot t\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

      7. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot k\right) \cdot k\right)} \cdot \left(t \cdot t\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      10. *-commutative N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

      12. lower-*.f64 64.4

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

   9. Applied egg-rr 64.4%

      \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot \color{blue}{\left(t \cdot t\right)}\right)} \cdot \ell \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

       4. *-commutative N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)}} \cdot \ell \]

       5. lower-*.f64 64.4

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)}} \cdot \ell \]

       6. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot t\right)} \cdot \ell \]

       7. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

       8. associate-*r* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(k \cdot t\right) \cdot t\right)} \cdot \left(k \cdot t\right)} \cdot \ell \]

       9. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\color{blue}{\left(k \cdot t\right)} \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

       10. *-commutative N/A

           \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot t\right)\right)} \cdot \left(k \cdot t\right)} \cdot \ell \]

       11. lower-*.f64 72.7

           \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot t\right)\right)} \cdot \left(k \cdot t\right)} \cdot \ell \]

       12. lift-*.f64 N/A

           \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(k \cdot t\right)}\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

       13. *-commutative N/A

           \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(t \cdot k\right)}\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

       14. lower-*.f64 72.7

           \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(t \cdot k\right)}\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

       15. lift-*.f64 N/A

           \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \color{blue}{\left(k \cdot t\right)}} \cdot \ell \]

       16. *-commutative N/A

           \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \color{blue}{\left(t \cdot k\right)}} \cdot \ell \]

       17. lower-*.f64 72.7

           \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \color{blue}{\left(t \cdot k\right)}} \cdot \ell \]

   11. Applied egg-rr 72.7%

       \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(t \cdot k\right)}} \cdot \ell \]

   if 2.7000000000000001e-33 < k

   1. Initial program 52.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot {\ell}^{2}\right) \cdot \cos k}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot {\ell}^{2}\right) \cdot \cos k}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot {\ell}^{2}\right)} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. unpow2 N/A

         \[\leadsto \frac{\left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\cos k}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)}} \]

      12. lower-pow.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{{\sin k}^{2}} \cdot \left({k}^{2} \cdot t\right)} \]

      13. lower-sin.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\color{blue}{\sin k}}^{2} \cdot \left({k}^{2} \cdot t\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}} \]

      16. unpow2 N/A

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

      17. lower-*.f64 72.4

          \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   5. Simplified 72.4%

      \[\leadsto \color{blue}{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot \left(\ell \cdot \ell\right)\right)} \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      3. lift-cos.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \color{blue}{\cos k}}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\color{blue}{\sin k}}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{{\sin k}^{2}} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2} \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \]

      9. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2}}}{t \cdot \left(k \cdot k\right)}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2}}}{\color{blue}{t \cdot \left(k \cdot k\right)}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2}}}{t \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2}}}{\color{blue}{\left(t \cdot k\right) \cdot k}} \]

      13. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2}}}{t \cdot k}}{k}} \]

      14. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{\frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{{\sin k}^{2}}}{t \cdot k}}{k}} \]

   7. Applied egg-rr 76.3%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \cos k\right)}{0.5 - 0.5 \cdot \cos \left(k + k\right)}}{k \cdot t}}{k}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{2 \cdot {\ell}^{2}}}{{k}^{4} \cdot t} \]

      4. unpow2 N/A

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot {k}^{\color{blue}{\left(2 \cdot 2\right)}}} \]

      9. pow-sqr N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \color{blue}{\left({k}^{2} \cdot {k}^{2}\right)}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \color{blue}{\left({k}^{2} \cdot {k}^{2}\right)}} \]

      11. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot {k}^{2}\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot {k}^{2}\right)} \]

      13. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

      14. lower-*.f64 57.9

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   10. Simplified 57.9%

       \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.7 \cdot 10^{-33}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t \cdot k\right) \cdot \left(t \cdot \left(t \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 67.7% accurate, 10.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-105}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t\_m \cdot k\right) \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 5e-105)
    (* l (/ l (* (* t_m k) (* t_m (* t_m k)))))
    (* l (/ l (* t_m (* t_m (* t_m (* k k)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 5e-105) {
		tmp = l * (l / ((t_m * k) * (t_m * (t_m * k))));
	} else {
		tmp = l * (l / (t_m * (t_m * (t_m * (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 <= 5d-105) then
        tmp = l * (l / ((t_m * k) * (t_m * (t_m * k))))
    else
        tmp = l * (l / (t_m * (t_m * (t_m * (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 <= 5e-105) {
		tmp = l * (l / ((t_m * k) * (t_m * (t_m * k))));
	} else {
		tmp = l * (l / (t_m * (t_m * (t_m * (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 <= 5e-105:
		tmp = l * (l / ((t_m * k) * (t_m * (t_m * k))))
	else:
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 5e-105)
		tmp = Float64(l * Float64(l / Float64(Float64(t_m * k) * Float64(t_m * Float64(t_m * k)))));
	else
		tmp = Float64(l * Float64(l / Float64(t_m * Float64(t_m * Float64(t_m * Float64(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 <= 5e-105)
		tmp = l * (l / ((t_m * k) * (t_m * (t_m * k))));
	else
		tmp = l * (l / (t_m * (t_m * (t_m * (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, 5e-105], N[(l * N[(l / N[(N[(t$95$m * k), $MachinePrecision] * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(l / N[(t$95$m * N[(t$95$m * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 4.99999999999999963e-105

   1. Initial program 55.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. lower-*.f64 51.4

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 51.4%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot k\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      8. lower-/.f64 55.3

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      11. *-commutative N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      12. associate-*l* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      14. lower-*.f64 58.2

          \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   7. Applied egg-rr 58.2%

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \cdot \ell \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot t\right)}} \cdot \ell \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right)} \cdot \left(t \cdot t\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

      7. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot k\right) \cdot k\right)} \cdot \left(t \cdot t\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      10. *-commutative N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

      12. lower-*.f64 63.5

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

   9. Applied egg-rr 63.5%

      \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot \color{blue}{\left(t \cdot t\right)}\right)} \cdot \ell \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

       4. *-commutative N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)}} \cdot \ell \]

       5. lower-*.f64 63.5

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)}} \cdot \ell \]

       6. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot t\right)} \cdot \ell \]

       7. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

       8. associate-*r* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(k \cdot t\right) \cdot t\right)} \cdot \left(k \cdot t\right)} \cdot \ell \]

       9. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\color{blue}{\left(k \cdot t\right)} \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

       10. *-commutative N/A

           \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot t\right)\right)} \cdot \left(k \cdot t\right)} \cdot \ell \]

       11. lower-*.f64 72.3

           \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot t\right)\right)} \cdot \left(k \cdot t\right)} \cdot \ell \]

       12. lift-*.f64 N/A

           \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(k \cdot t\right)}\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

       13. *-commutative N/A

           \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(t \cdot k\right)}\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

       14. lower-*.f64 72.3

           \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(t \cdot k\right)}\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

       15. lift-*.f64 N/A

           \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \color{blue}{\left(k \cdot t\right)}} \cdot \ell \]

       16. *-commutative N/A

           \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \color{blue}{\left(t \cdot k\right)}} \cdot \ell \]

       17. lower-*.f64 72.3

           \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \color{blue}{\left(t \cdot k\right)}} \cdot \ell \]

   11. Applied egg-rr 72.3%

       \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(t \cdot k\right)}} \cdot \ell \]

   if 4.99999999999999963e-105 < k

   1. Initial program 53.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. lower-*.f64 48.5

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 48.5%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot k\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      8. lower-/.f64 50.3

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      11. *-commutative N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      12. associate-*l* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      14. lower-*.f64 52.9

          \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   7. Applied egg-rr 52.9%

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \cdot \ell \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot t\right)}} \cdot \ell \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right)} \cdot \left(t \cdot t\right)} \cdot \ell \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

      7. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot k\right) \cdot k\right)} \cdot \left(t \cdot t\right)} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      10. *-commutative N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

      12. lower-*.f64 51.7

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

   9. Applied egg-rr 51.7%

      \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot \color{blue}{\left(t \cdot t\right)}\right)} \cdot \ell \]

       3. associate-*r* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \left(t \cdot t\right)}} \cdot \ell \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \color{blue}{\left(t \cdot t\right)}} \cdot \ell \]

       5. associate-*r* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(\left(k \cdot t\right) \cdot k\right) \cdot t\right) \cdot t}} \cdot \ell \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(\left(k \cdot t\right) \cdot k\right) \cdot t\right) \cdot t}} \cdot \ell \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(\left(k \cdot t\right) \cdot k\right) \cdot t\right)} \cdot t} \cdot \ell \]

       8. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(\left(\color{blue}{\left(k \cdot t\right)} \cdot k\right) \cdot t\right) \cdot t} \cdot \ell \]

       9. *-commutative N/A

          \[\leadsto \frac{\ell}{\left(\left(\color{blue}{\left(t \cdot k\right)} \cdot k\right) \cdot t\right) \cdot t} \cdot \ell \]

       10. associate-*l* N/A

           \[\leadsto \frac{\ell}{\left(\color{blue}{\left(t \cdot \left(k \cdot k\right)\right)} \cdot t\right) \cdot t} \cdot \ell \]

       11. lift-*.f64 N/A

           \[\leadsto \frac{\ell}{\left(\left(t \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot t\right) \cdot t} \cdot \ell \]

       12. lower-*.f64 56.7

           \[\leadsto \frac{\ell}{\left(\color{blue}{\left(t \cdot \left(k \cdot k\right)\right)} \cdot t\right) \cdot t} \cdot \ell \]

   11. Applied egg-rr 56.7%

       \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot \left(k \cdot k\right)\right) \cdot t\right) \cdot t}} \cdot \ell \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-105}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t \cdot k\right) \cdot \left(t \cdot \left(t \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 66.7% accurate, 12.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\ell \cdot \frac{\ell}{\left(t\_m \cdot k\right) \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* l (/ l (* (* t_m k) (* t_m (* t_m 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 * (l * (l / ((t_m * k) * (t_m * (t_m * 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 * (l * (l / ((t_m * k) * (t_m * (t_m * 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 * (l * (l / ((t_m * k) * (t_m * (t_m * 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 * (l * (l / ((t_m * k) * (t_m * (t_m * 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(l * Float64(l / Float64(Float64(t_m * k) * Float64(t_m * Float64(t_m * 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 * (l * (l / ((t_m * k) * (t_m * (t_m * 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[(l * N[(l / N[(N[(t$95$m * k), $MachinePrecision] * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.1%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

   2. unpow2 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   6. cube-mult N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

   7. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

   9. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

   11. unpow2 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   12. lower-*.f64 50.5

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

5. Simplified 50.5%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot k\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

   5. associate-/l* N/A

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

   8. lower-/.f64 53.7

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

   11. *-commutative N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

   12. associate-*l* N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   14. lower-*.f64 56.5

       \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

7. Applied egg-rr 56.5%

   \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \cdot \ell \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   4. *-commutative N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot t\right)}} \cdot \ell \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right)} \cdot \left(t \cdot t\right)} \cdot \ell \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

   7. associate-*r* N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot k\right) \cdot k\right)} \cdot \left(t \cdot t\right)} \cdot \ell \]

   8. associate-*l* N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

   10. *-commutative N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

   12. lower-*.f64 59.7

       \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

9. Applied egg-rr 59.7%

   \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot \color{blue}{\left(t \cdot t\right)}\right)} \cdot \ell \]

    2. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

    3. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

    4. *-commutative N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)}} \cdot \ell \]

    5. lower-*.f64 59.7

       \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)}} \cdot \ell \]

    6. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot t\right)} \cdot \ell \]

    7. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(k \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

    8. associate-*r* N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(\left(k \cdot t\right) \cdot t\right)} \cdot \left(k \cdot t\right)} \cdot \ell \]

    9. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(\color{blue}{\left(k \cdot t\right)} \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

    10. *-commutative N/A

        \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot t\right)\right)} \cdot \left(k \cdot t\right)} \cdot \ell \]

    11. lower-*.f64 65.8

        \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot t\right)\right)} \cdot \left(k \cdot t\right)} \cdot \ell \]

    12. lift-*.f64 N/A

        \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(k \cdot t\right)}\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

    13. *-commutative N/A

        \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(t \cdot k\right)}\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

    14. lower-*.f64 65.8

        \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(t \cdot k\right)}\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

    15. lift-*.f64 N/A

        \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \color{blue}{\left(k \cdot t\right)}} \cdot \ell \]

    16. *-commutative N/A

        \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \color{blue}{\left(t \cdot k\right)}} \cdot \ell \]

    17. lower-*.f64 65.8

        \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \color{blue}{\left(t \cdot k\right)}} \cdot \ell \]

11. Applied egg-rr 65.8%

    \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(t \cdot k\right)}} \cdot \ell \]

12. Final simplification 65.8%

    \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot k\right) \cdot \left(t \cdot \left(t \cdot k\right)\right)} \]
13. Add Preprocessing

Alternative 19: 64.7% accurate, 12.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\ell \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(k \cdot \left(t\_m \cdot k\right)\right)\right)}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* l (/ l (* t_m (* t_m (* k (* t_m 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 * (l * (l / (t_m * (t_m * (k * (t_m * 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 * (l * (l / (t_m * (t_m * (k * (t_m * 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 * (l * (l / (t_m * (t_m * (k * (t_m * 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 * (l * (l / (t_m * (t_m * (k * (t_m * 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(l * Float64(l / Float64(t_m * Float64(t_m * Float64(k * Float64(t_m * 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 * (l * (l / (t_m * (t_m * (k * (t_m * 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[(l * N[(l / N[(t$95$m * N[(t$95$m * N[(k * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.1%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

   2. unpow2 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   6. cube-mult N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

   7. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

   9. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

   11. unpow2 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   12. lower-*.f64 50.5

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

5. Simplified 50.5%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot k\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

   5. associate-/l* N/A

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

   8. lower-/.f64 53.7

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

   11. *-commutative N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

   12. associate-*l* N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   14. lower-*.f64 56.5

       \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

7. Applied egg-rr 56.5%

   \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \cdot \ell \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   3. associate-*l* N/A

      \[\leadsto \frac{\ell}{\color{blue}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}} \cdot \ell \]

   4. *-commutative N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right) \cdot t}} \cdot \ell \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right) \cdot t}} \cdot \ell \]

   6. lower-*.f64 61.1

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)} \cdot t} \cdot \ell \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right) \cdot t} \cdot \ell \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)\right) \cdot t} \cdot \ell \]

   9. associate-*r* N/A

      \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}\right) \cdot t} \cdot \ell \]

   10. *-commutative N/A

       \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(k \cdot \left(t \cdot k\right)\right)}\right) \cdot t} \cdot \ell \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(k \cdot \left(t \cdot k\right)\right)}\right) \cdot t} \cdot \ell \]

   12. *-commutative N/A

       \[\leadsto \frac{\ell}{\left(t \cdot \left(k \cdot \color{blue}{\left(k \cdot t\right)}\right)\right) \cdot t} \cdot \ell \]

   13. lower-*.f64 62.6

       \[\leadsto \frac{\ell}{\left(t \cdot \left(k \cdot \color{blue}{\left(k \cdot t\right)}\right)\right) \cdot t} \cdot \ell \]

9. Applied egg-rr 62.6%

   \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot \left(k \cdot t\right)\right)\right) \cdot t}} \cdot \ell \]

10. Final simplification 62.6%

    \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot \left(t \cdot k\right)\right)\right)} \]
11. Add Preprocessing

Alternative 20: 63.2% accurate, 12.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\ell \cdot \frac{\ell}{\left(t\_m \cdot k\right) \cdot \left(k \cdot \left(t\_m \cdot t\_m\right)\right)}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* l (/ l (* (* t_m k) (* k (* t_m 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 * (l * (l / ((t_m * k) * (k * (t_m * 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 * (l * (l / ((t_m * k) * (k * (t_m * 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 * (l * (l / ((t_m * k) * (k * (t_m * 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 * (l * (l / ((t_m * k) * (k * (t_m * 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(l * Float64(l / Float64(Float64(t_m * k) * Float64(k * Float64(t_m * 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 * (l * (l / ((t_m * k) * (k * (t_m * 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[(l * N[(l / N[(N[(t$95$m * k), $MachinePrecision] * N[(k * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.1%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

   2. unpow2 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   6. cube-mult N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

   7. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

   9. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

   11. unpow2 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   12. lower-*.f64 50.5

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

5. Simplified 50.5%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(k \cdot k\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

   5. associate-/l* N/A

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

   8. lower-/.f64 53.7

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

   11. *-commutative N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

   12. associate-*l* N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   14. lower-*.f64 56.5

       \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

7. Applied egg-rr 56.5%

   \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \cdot \ell \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   4. *-commutative N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot t\right)}} \cdot \ell \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right)} \cdot \left(t \cdot t\right)} \cdot \ell \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \left(t \cdot t\right)} \cdot \ell \]

   7. associate-*r* N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot k\right) \cdot k\right)} \cdot \left(t \cdot t\right)} \cdot \ell \]

   8. associate-*l* N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

   10. *-commutative N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

   12. lower-*.f64 59.7

       \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

9. Applied egg-rr 59.7%

   \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

10. Final simplification 59.7%

    \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024208 
(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))))